how long does phd in math take

Guide to Graduate Studies

The PhD Program The Ph.D. program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. Enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in one’s own way. For this reason, a Ph.D. dissertation involving some original research is a fundamental part of the program. The stages in this program may be described as follows:

• Acquiring a broad basic knowledge of mathematics on which to build a future mathematical culture and more detailed knowledge of a field of specialization.
• Choosing a field of specialization within mathematics and obtaining enough knowledge of this specialized field to arrive at the point of current thinking.
• Making a first original contribution to mathematics within this chosen special area.

Students are expected to take the initiative in pacing themselves through the Ph.D. program. In theory, a future research mathematician should be able to go through all three stages with the help of only a good library. In practice, many of the more subtle aspects of mathematics, such as a sense of taste or relative importance and feeling for a particular subject, are primarily communicated by personal contact. In addition, it is not at all trivial to find one’s way through the ever-burgeoning literature of mathematics, and one can go through the stages outlined above with much less lost motion if one has some access to a group of older and more experienced mathematicians who can guide one’s reading, supplement it with seminars and courses, and evaluate one’s first attempts at research. The presence of other graduate students of comparable ability and level of enthusiasm is also very helpful.

University Requirements

The University requires a minimum of two years of academic residence (16 half-courses) for the Ph.D. degree. On the other hand, five years in residence is the maximum usually allowed by the department. Most students complete the Ph.D. in four or five years. Please review the program requirements timeline .

There is no prescribed set of course requirements, but students are required to register and enroll in four courses each term to maintain full-time status with the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Qualifying Exam

The department gives the qualifying examination at the beginning of the fall and spring terms. The qualifying examination covers algebra, algebraic geometry, algebraic topology, complex analysis, differential geometry, and real analysis. Students are required to take the exam at the beginning of the first term. More details about the qualifying exams can be found here .

Students are expected to pass the qualifying exam before the end of their second year. After passing the qualifying exam students are expected to find a Ph.D. dissertation advisor.

Minor Thesis

The minor thesis is complementary to the qualifying exam. In the course of mathematical research, students will inevitably encounter areas in which they have gaps in knowledge. The minor thesis is an exercise in confronting those gaps to learn what is necessary to understand a specific area of math. Students choose a topic outside their area of expertise and, working independently, learns it well and produces a written exposition of the subject.

The topic is selected in consultation with a faculty member, other than the student’s Ph.D. dissertation advisor, chosen by the student. The topic should not be in the area of the student’s Ph.D. dissertation. For example, students working in number theory might do a minor thesis in analysis or geometry. At the end of three weeks time (four if teaching), students submit to the faculty member a written account of the subject and are prepared to answer questions on the topic.

The minor thesis must be completed before the start of the third year in residence.

Language Exam

Mathematics is an international subject in which the principal languages are English, French, German, and Russian. Almost all important work is published in one of these four languages. Accordingly, students are required to demonstrate the ability to read mathematics in French, German, or Russian by passing a two-hour, written language examination. Students are asked to translate one page of mathematics into English with the help of a dictionary. Students may request to substitute the Italian language exam if it is relevant to their area of mathematics. The language requirement should be fulfilled by the end of the second year. For more information on the graduate program requirements, a timeline can be viewed at here .

Non-native English speakers who have received a Bachelor’s degree in mathematics from an institution where classes are taught in a language other than English may request to waive the language requirement.

Upon completion of the language exam and eight upper-level math courses, students can apply for a continuing Master’s Degree.

Teaching Requirement

Most research mathematicians are also university teachers. In preparation for this role, all students are required to participate in the department’s teaching apprenticeship program and to complete two semesters of classroom teaching experience, usually as a teaching fellow. During the teaching apprenticeship, students are paired with a member of the department’s teaching staff. Students attend some of the advisor’s classes and then prepare (with help) and present their own class, which will be videotaped. Apprentices will receive feedback both from the advisor and from members of the class.

Teaching fellows are responsible for teaching calculus to a class of about 25 undergraduates. They meet with their class three hours a week. They have a course assistant (an advanced undergraduate) to grade homework and to take a weekly problem session. Usually, there are several classes following the same syllabus and with common exams. A course head (a member of the department teaching staff) coordinates the various classes following the same syllabus and is available to advise teaching fellows. Other teaching options are available: graduate course assistantships for advanced math courses and tutorials for advanced undergraduate math concentrators.

Final Stages

How students proceed through the second and third stages of the program varies considerably among individuals. While preparing for the qualifying examination or immediately after, students should begin taking more advanced courses to help with choosing a field of specialization. Unless prepared to work independently, students should choose a field that falls within the interests of a member of the faculty who is willing to serve as dissertation advisor. Members of the faculty vary in the way that they go about dissertation supervision; some faculty members expect more initiative and independence than others and some variation in how busy they are with current advisees. Students should consider their own advising needs as well as the faculty member’s field when choosing an advisor. Students must take the initiative to ask a professor if she or he will act as a dissertation advisor. Students having difficulty deciding under whom to work, may want to spend a term reading under the direction of two or more faculty members simultaneously. The sooner students choose an advisor, the sooner they can begin research. Students should have a provisional advisor by the second year.

It is important to keep in mind that there is no technique for teaching students to have ideas. All that faculty can do is to provide an ambiance in which one’s nascent abilities and insights can blossom. Ph.D. dissertations vary enormously in quality, from hard exercises to highly original advances. Many good research mathematicians begin very slowly, and their dissertations and first few papers could be of minor interest. The ideal attitude is: (1) a love of the subject for its own sake, accompanied by inquisitiveness about things which aren’t known; and (2) a somewhat fatalistic attitude concerning “creative ability” and recognition that hard work is, in the end, much more important.

PhD Program

More information and a full list of requirements for the PhD program in Mathematics can be found in the University Bulletin .

During their first year in the program, students typically engage in coursework and seminars which prepare them for the  Qualifying Examinations .  Currently, these two exams test the student’s breadth of knowledge in algebra and real analysis. 

Starting in Autumn 2023, students will choose 2 out of 4 qualifying exam topics: 

• real analysis
• geometry and topology
• applied mathematics

Course Requirements for students starting prior to Autumn 2023

To qualify for candidacy, the student must have successfully completed 27 units of Math graduate courses numbered between 200 and 297.

Within the 27 units, students must satisfactorily complete a course sequence. This can be fulfilled in one of the following ways:

• Math 215A, B, & C: Algebraic Topology, Differential Topology, and Differential Geometry
• Math 216A, B, & C: Introduction to Algebraic Geometry
• Math 230A, B, & C: Theory of Probability
• 3 quarter course sequence in a single subject approved in advance by the Director of Graduate Studies.

Course Requirements for students starting in Autumn 2023 and later

To qualify for candidacy, the student must have successfully completed 27 units of Math graduate courses numbered between 200 and 297. The course sequence requirement is discontinued for students starting in Autumn 2023 and later.

By the end of Spring Quarter of their second year in the program, students must have a dissertation advisor and apply for Candidacy.

During their third year, students will take their Area Examination , which must be completed by the end of Winter Quarter. This exam assesses the student’s breadth of knowledge in their particular area of research. The Area Examination is also used as an opportunity for the student to present their committee with a summary of research conducted to date as well as a detailed plan for the remaining research.

Years 4&5

Typically during the latter part of the fourth or early part of the fifth year of study, students are expected to finish their dissertation research. At this time, students defend their dissertation as they sit for their University Oral Examination. Following the dissertation defense, students take a short time to make final revisions to their actual papers and submit the dissertation to their reading committee for final approval.

Throughout the PhD Program

All students continue through each year of the program serving some form of Assistantship: Course, Teaching or Research, unless they have funding from outside the department.

Our graduate students are very active as both leaders and participants in seminars and colloquia in their chosen areas of interest.

How Long Does It Take to Get a Ph.D. in Mathematics?

Education Requirements for Information Technology

If you love mathematics and have a passion to teach, earning a Ph.D. in mathematics may be the ideal next step. A Ph.D. in mathematics is a gateway to a career in teaching or research at the university level. While someone with a master's degree may be able to get jobs teaching at a community college, a Ph.D. will make candidates more competitive for those jobs, as well. The amount of time it takes to complete a Ph.D. in mathematics depends on a number of variables, including the program, the research project and the individual pace of student work. However, most programs are completed in five to six years in addition to the four years that most students take to complete an undergraduate degree.

The coursework for a Ph.D. program in mathematics focuses on higher-order mathematics and mathematical theory. Many programs also require that students take a language class and demonstrate some proficiency in that language. The number of credit hours required varies by program, but typically, coursework takes two to three years to complete. Some of the required credit hours can usually be applied to dissertation work or time spent as a teaching assistant.

Completing the Dissertation

Completing the dissertation will take up the bulk of the time dedicated to finishing your Ph.D. A dissertation must include original research and is typically a book-length manuscript. Students who have a good idea of the research they would like to conduct before the program begins will usually be able to finish their work faster and graduate sooner. However, many students explore ideas through their coursework and their discussions with professors before deciding on a dissertation topic.

Passing Exams

The final step to completing a Ph.D. program in mathematics is completing qualifying exams. These include both written and oral exams. In some programs, you may have to take several written exams, often at the end of each year of coursework. Oral exams usually involving defending the dissertation and oral exams that test your understanding of the subject matter. Often, other requirements include: a minimum GPA, the presentation of research or work as a teaching assistant.

Job Outlook

The Bureau of Labor Statistics predicts that jobs for mathematicians will grow about as fast as average for other industries by 2020. The agency predicts a growth of 500 jobs, or an increase of about 16 percent. The median pay for mathematicians was $84,760 in 2018. The bureau also notes that mathematicians earning only a master's degree have job opportunities in government agencies and private organizations.

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Maria Magher has been working as a professional writer since 2001. She has worked as an ESL teacher, a freshman composition teacher and an education reporter, writing for regional newspapers and online publications. She has written about parenting for Pampers and other websites. She has a Master's degree in English and creative writing.

Ph.D. Program

Degree requirements.

In outline, to earn the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements.

• Take at least 4 courses, 2 or more of which are graduate courses offered by the Department of Mathematics
• Pass the six-hour written Preliminary Examination covering calculus, real analysis, complex analysis, linear algebra, and abstract algebra; students must pass the prelim before the start of their second year in the program (within three semesters of starting the program)
• Pass a three-hour, oral Qualifying Examination emphasizing, but not exclusively restricted to, the area of specialization. The Qualifying Examination must be attempted within two years of entering the program
• Complete a seminar, giving a talk of at least one-hour duration
• Write a dissertation embodying the results of original research and acceptable to a properly constituted dissertation committee
• Meet the University residence requirement of two years or four semesters

Detailed Regulations

The detailed regulations of the Ph.D. program are the following:

Course Requirements

During the first year of the Ph.D. program, the student must enroll in at least 4 courses. At least 2 of these must be graduate courses offered by the Department of Mathematics. Exceptions can be granted by the Vice-Chair for Graduate Studies.

Preliminary Examination

The Preliminary Examination consists of 6 hours (total) of written work given over a two-day period (3 hours/day). Exam questions are given in calculus, real analysis, complex analysis, linear algebra, and abstract algebra. The Preliminary Examination is offered twice a year during the first week of the fall and spring semesters.

Qualifying Examination

To arrange the Qualifying Examination, a student must first settle on an area of concentration, and a prospective Dissertation Advisor (Dissertation Chair), someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective advisor, the student forms an examination committee of 4 members.  All committee members can be faculty in the Mathematics Department and the chair must be in the Mathematics Department. The QE chair and Dissertation Chair cannot be the same person; therefore, t he Math member least likely to serve as the dissertation advisor should be selected as chair of the qualifying exam committee . The syllabus of the examination is to be worked out jointly by the committee and the student, but before final approval, it is to be circulated to all faculty members of the appropriate research sections. The Qualifying Examination must cover material falling in at least 3 subject areas and these must be listed on the application to take the examination. Moreover, the material covered must fall within more than one section of the department. Sample syllabi can be reviewed online or in 910 Evans Hall. The student must attempt the Qualifying Examination within twenty-five months of entering the PhD program. If a student does not pass on the first attempt, then, on the recommendation of the student's examining committee, and subject to the approval of the Graduate Division, the student may repeat the examination once. The examining committee must be the same, and the re-examination must be held within thirty months of the student's entrance into the PhD program. For a student to pass the Qualifying Examination, at least one identified member of the subject area group must be willing to accept the candidate as a dissertation student.

Overview of the PhD Program

For specific information on the Applied Mathematics PhD program, see the navigation links to the right. 

What follows on this page is an overview of all Ph.D. programs at the School; additional information and guidance can be found on the  Graduate Policies  pages. 

General Ph.D. Requirements

• 10 semester-long graduate courses, including at least 8 disciplinary.   At least 5 of the 10 should be graduate-level SEAS "technical" courses (or FAS graduate-level technical courses taught by SEAS faculty), not including seminar/reading/project courses.  Undergraduate-level courses cannot be used.  For details on course requirements, see the school's overall PhD course requirements  and the individual program pages linked therein.
• Program Plan (i.e., the set of courses to be used towards the degree) approval by the  Committee on Higher Degrees  (CHD).
• Minimum full-time academic residency of two years .
• Serve as a Teaching Fellow (TF) in one semester of the second year.
• Oral Qualifying Examination Preparation in the major field is evaluated in an oral examination by a qualifying committee. The examination has the dual purpose of verifying the adequacy of the student's preparation for undertaking research in a chosen field and of assessing the student's ability to synthesize knowledge already acquired. For details on arranging your Qualifying Exam, see the exam policies and the individual program pages linked therein.
• Committee Meetings : PhD students' research committees meet according to the guidelines in each area's "Committee Meetings" listing.  For details see the "G3+ Committee Meetings" section of the Policies of the CHD  and the individual program pages linked therein.
• Final Oral Examination (Defense) This public examination devoted to the field of the dissertation is conducted by the student's research committee. It includes, but is not restricted to, a defense of the dissertation itself.  For details of arranging your final oral exam see the  Ph.D. Timeline  page.
• Dissertation Upon successful completion of the qualifying examination, a committee chaired by the research supervisor is constituted to oversee the dissertation research. The dissertation must, in the judgment of the research committee, meet the standards of significant and original research.

Optional additions to the Ph.D. program

Harvard PhD students may choose to pursue these additional aspects:

• a Secondary Field (which is similar to a "minor" subject area).  SEAS offers PhD Secondary Field programs in  Data Science and in  Computational Science and Engineering .   GSAS  lists  secondary fields offered by other programs.
• a Master of Science (S.M.) degree conferred  en route to the Ph.D in one of several of SEAS's subject areas.  For details see here .
• a Teaching Certificate awarded by the Derek Bok Center for Teaching and Learning .

SEAS PhD students may apply to participate in the  Health Sciences and Technology graduate program  with Harvard Medical School and MIT.  Please check with the HST program for details on eligibility (e.g., only students in their G1 year may apply) and the application process.

In Applied Mathematics

• First-Year Exploration
• Areas of Application
• AM & Economics
• How to Declare
• Who are my Advisors?
• Secondary Field
• Senior Thesis
• Research for Course Credit (AM 91R & AM 99R)
• AB/SM Information
• Peer Concentration Advisors (PCA) Program
• Student Organizations
• How to Apply
• PhD Timeline
• PhD Model Program (Course Guidelines)
• Oral Qualifying Examination
• Committee Meetings
• Committee on Higher Degrees
• Research Interest Comparison
• Collaborations
• Cross-Harvard Engagement
• Clubs & Organizations
• Centers & Initiatives
• Alumni Stories

How Long Does It Take to Get a Ph.D. Degree?

Earning a Ph.D. from a U.S. grad school typically requires nearly six years, federal statistics show.

How Long It Takes to Get a Ph.D. Degree

Caiaimage | Tom Merton | Getty Images

A Ph.D. is most appropriate for someone who is a "lifelong learner."

Students who have excelled within a specific academic discipline and who have a strong interest in that field may choose to pursue a Ph.D. degree. However, Ph.D. degree-holders urge prospective students to think carefully about whether they truly want or need a doctoral degree, since Ph.D. programs last for multiple years.

According to the Survey of Earned Doctorates, a census of recent research doctorate recipients who earned their degree from U.S. institutions, the median amount of time it took individuals who received their doctorates in 2017 to complete their program was 5.8 years. However, there are many types of programs that typically take longer than six years to complete, such as humanities and arts doctorates, where the median time for individuals to earn their degree was 7.1 years, according to the survey.

Some Ph.D. candidates begin doctoral programs after they have already obtained master's degrees, which means the time spent in grad school is a combination of the time spent pursuing a master's and the years invested in a doctorate. In order to receive a Ph.D. degree, a student must produce and successfully defend an original academic dissertation, which must be approved by a dissertation committtee. Writing and defending a dissertation is so difficult that many Ph.D. students drop out of their Ph.D. programs having done most of the work necessary for degree without completing the dissertation component. These Ph.D. program dropouts often use the phrase " all but dissertation " or the abbreviation "ABD" on their resumes.

According to a comprehensive study of Ph.D. completion rates published by The Council of Graduate Schools in 2008, only 56.6% of people who begin Ph.D. programs earn Ph.D. degrees.

Ian Curtis, a founding partner with H&C Education, an educational and admissions consulting firm, who is pursuing a Ph.D. degree in French at Yale University , says there are several steps involved in the process of obtaining a Ph.D. Students typically need to fulfill course requirements and pass comprehensive exams, Curtis warns. "Once these obligations have been completed, how long it takes you to write your dissertation depends on who you are, how you work, what field you're in and what other responsibilities you have in life," he wrote in an email. Though some Ph.D. students can write a dissertation in a single year, that is rare, and the dissertation writing process may last for several years, Curtis says.

Curtis adds that the level of support a Ph.D. student receives from an academic advisor or faculty mentor can be a key factor in determining the length of time it takes to complete a Ph.D. program. "Before you decide to enroll at a specific program, you’ll want to meet your future advisor," Curtis advises. "Also, reach out to his or her current and former students to get a sense of what he or she is like to work with."

Curtis also notes that if there is a gap between the amount of time it takes to complete a Ph.D. and the amount of time a student's funding lasts, this can slow down the Ph.D. completion process. "Keep in mind that if you run out of funding at some point during your doctorate, you will need to find paid work, and this will leave you even less time to focus on writing your dissertation," he says. "If one of the programs you’re looking at has a record of significantly longer – or shorter – times to competition, this is good information to take into consideration."

He adds that prospective Ph.D. students who already have master's degrees in the field they intend to focus their Ph.D. on should investigate whether the courses they took in their master's program would count toward the requirements of a Ph.D. program. "You’ll want to discuss your particular situation with your program to see whether this will be possible, and how many credits you are likely to receive as the result of your master’s work," he says.

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Emmanuel C. Nwaodua, who has a Ph.D. degree in geology, says some Ph.D. programs require candidates to publish a paper in a first-rate, peer-reviewed academic journal. "This could extend your stay by a couple of years," he warns.

Pierre Huguet, the CEO and co-founder of H&C Education, says prospective Ph.D. students should be aware that a Ph.D. is designed to prepare a person for a career as a scholar. "Most of the jobs available to Ph.D. students upon graduation are academic in nature and directly related to their fields of study: professor, researcher, etc.," Huguet wrote in an email. "The truth is that more specialization can mean fewer job opportunities. Before starting a Ph.D., students should be sure that they want to pursue a career in academia, or in research. If not, they should make time during the Ph.D. to show recruiters that they’ve traveled beyond their labs and libraries to gain some professional hands-on experience."

Jack Appleman, a business writing instructor, published author and Ph.D. candidate focusing on organizational communication with the University at Albany—SUNY , says Ph.D. programs require a level of commitment and focus that goes beyond what is necessary for a typical corporate job. A program with flexible course requirements that allow a student to customize his or her curriculum based on academic interests and personal obligations is ideal, he says.

Joan Kee, a professor at the University of Michigan with the university's history of art department, says that the length of time required for a Ph.D. varies widely depending on what subject the Ph.D. focuses on. "Ph.D. program length is very discipline and even field-specific; for example, you can and are expected to finish a Ph.D, in economics in under five years, but that would be impossible in art history (or most of the humanities)," she wrote in an email.

Kee adds that humanities Ph.D. programs often require someone to learn a foreign language, and "fields like anthropology and art history require extensive field research." Kee says funding for a humanities Ph.D. program typically only lasts five years, even though it is uncommon for someone to obtain a Ph.D. degree in a humanities field within that time frame. "Because of this, many if not most Ph.D. students must work to make ends meet, thus further prolonging the time of completion," she says.

Jean Marie Carey, who earned her Ph.D. degree in art history and German from the University of Otago in New Zealand, encourages prospective Ph.D. students to check whether their potential Ph.D. program has published a timeline of how long it takes a Ph.D. student to complete their program. She says it is also prudent to speak with Ph.D. graduates of the school and ask about their experience.

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Kristin Redington Bennett, the founder of the Illumii educational consulting firm in North Carolina, encourages Ph.D. hopefuls to think carefully about whether they want to become a scholar. Bennett, who has a Ph.D. in curriculum and assessment and who previously worked as an assistant professor at Wake Forest University , says a Ph.D. is most appropriate for someone who is a "lifelong learner." She says someone contemplating a Ph.D. should ask themselves the following questions "Are you a very curious person... and are you persistent?"

Bennett urges prospective Ph.D. students to visit the campuses of their target graduate programs since a Ph.D. program takes so much time that it is important to find a school that feels comfortable. She adds that aspiring Ph.D. students who prefer a collaborative learning environment should be wary of graduate programs that have a cut-throat and competitive atmosphere, since such students may not thrive in that type of setting.

Alumni of Ph.D. programs note that the process of obtaining a Ph.D. is arduous, regardless of the type of Ph.D. program. "A Ph.D. is a long commitment of your time, energy and financial resources, so it'll be easier on you if you are passionate about research," says Grace Lee, who has a Ph.D. in neuroscience and is the founder and CEO of Mastery Insights, an education and career coaching company, and the host of the Career Revisionist podcast.

"A Ph.D. isn't about rehashing years of knowledge that is already out there, but rather it is about your ability to generate new knowledge. Your intellectual masterpiece (which is your dissertation) takes a lot of time, intellectual creativity and innovation to put together, so you have to be truly passionate about that," Lee says.

Curtis says a prospective Ph.D. student's enthusiasm for academic work, teaching and research are the key criteria they should use to decide whether to obtain a Ph.D. degree. "While the time it takes to complete a doctorate is an understandable concern for many, my personal belief is that time is not the most important factor to consider," he says. "Good Ph.D. programs provide their students with generous stipends, health care and sometimes even subsidized housing."

Erin Skelly, a graduate admissions counselor at the IvyWise admissions consulting firm, says when a Ph.D. students struggles to complete his or her Ph.D. degree, it may have more to do with the student's academic interests or personal circumstances than his or her program.

"The time to complete a Ph.D. can depend on a number of variables, but the specific discipline or school would only account for a year or two's difference," she wrote in an email. "When a student takes significantly longer to complete a Ph.D. (degree), it's usually related to the student's coursework and research – they need to take additional coursework to complete their comprehensive exams; they change the focus of their program or dissertation, requiring extra coursework or research; or their research doesn't yield the results they hoped for, and they need to generate a new theory and conduct more research."

Skelly warns that the average completion time of a Ph.D. program may be misleading in some cases, if the average is skewed based on one or two outliers. She suggests that instead of focusing on the duration of a particular Ph.D. program, prospective students should investigate the program's attritition and graduation rates.

"It is worthwhile to look at the program requirements and the school's proposed timeline for completion, and meet current students to get their input on how realistic these expectations for completion are," Skelly says. "That can give you an honest idea of how long it will really take to complete the program."

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Doctor of Philosophy (DPhil)

What is a dphil.

A DPhil is Oxford's name for a PhD - a higher research degree which allows you to make an original contribution to mathematics in the form of a thesis. A DPhil takes at least three years to complete, and around two thirds of our postgraduate leavers go on into academia (according to the latest destination data). During your DPhil, you will be supervised by at least one academic, although some students will have more than one supervisor (particularly if they are working across disciplines). Unlike CDT courses (and PhDs in other countries), you will begin to do research straight away and there is no prescribed taught component. However, you are very welcome to attend seminars and you can choose from a wide variety of taught courses and skills training to enhance your broader mathematical knowledge and develop your career. There may also be journal clubs or seminar series specific to your area of study. If you enjoy doing mathematics, and would like to be part of a lively and world-class research institute, then you should take a look at our research groups to see if they align with your own interests. 

All applications should be submitted online through the University's Graduate Application Form . To find out more about how to apply, see the how to apply page, or go to the   University of Oxford's graduate application guide .

For information about scholarships and funding , see the University of Oxford's fees, funding, and scholarship search .

Key Deadlines

Funding deadlines for students applying for EPSRC and Departmental awards

• 5th January 2024 (12:00 GMT)
• 1st March 2024 (12:00 GMT)

Please apply by the 5th January deadline if you would like to be considered for any centrally administered funds. Further information regarding these funds can be found here .

Fees and Funding

Information on University fees and funding can be found here .

Scholarships

Wang scholarship.

The Wang Scholarship is available to DPhil candidates and provides full fees and a stipend for four years.

Charles Coulson Scholarship in Mathematical Physics

This award is available to DPhil candidates in Mathematical Physics and provides full fees and a stipend for four years. 

Heilbronn Doctoral Partnership Scholarship

Oxford-Maryam Mirzakhani Scholarship

Research interests:  group theory, representation theory and algebraic aspects of geometry.

Who's who in Algebra

Find out more about the group

Combinatorics

Research interests: extremal combinatorics, graph theory, and combinatorial number theory.

Who's who in Combinatorics

Functional Analysis

Research interests: operator theory, including unbounded operators, and abstract differential equations.

Who's who in functional analysis

Research interests: algebraic geometry,  geometric representation theory , and differential geometry.

Who's who in Geometry

History of Mathematics

Research interests: history of algebra (19th and 20th century), history of modern algebra, and Soviet mathematics. 

Research interests: analytic topology,  geometric stability theory, and the model theory of p-adic fields and diophantine geometry.

Who's who in Logic

Machine Learning and Data Science

Machine Learning and Data science are being developed using wide ranging mathematical techniques. Our particular research expertise include: applied and computational harmonic analysis, networks, optimisation, random matrix theory, rough paths, topological data analysis, and the application of these methods.

Who's who in machine learning and data science

Mathematical & Computational Finance

Research interests: behavioural finance, financial big data, high dimensional numerical methods, stochastic analysis.

Who's who in Mathematical and Computational Finance

Mathematical Biology

Research interests:  cancer modelling, collective behaviour, gene regulatory networks, multiscale modelling, pattern formation, and sperm dynamics.

Who's who in Mathematical Biology

Mathematical Physics

Research interests: gauge and gravity theories (quantum field theories), string theory, twistor theory, Calabi-Yau manifolds, quantum computation and cryptography.

Who's who in Mathematical Physics

Number Theory

Research interests: analytic number theory, arithmetic geometry, prime number distribution, and Diophantine geometry.

Who's who in Number Theory

Numerical Analysis

Research interests: complexity in optimisation, symmetric cone programming, numerical solutions of PDEs. 

Who's who in Numerical Analysis

Oxford Centre for Industrial and Applied Mathematics

Research interests: energy, industry, geoscience, networks, finance, methodologies.

Who's who in OCIAM

Oxford Centre for Nonlinear Partial Differential Equations

Research interests: geometric analysis, inverse problems, nonlinear hyperbolic systems, specific PDE systems.

Who's who in OxPDE

Stochastic Analysis

Research interests:  rough path theory, Schramm-Loewner evolution, mathematical population genetics, financial mathematics, self-interacting random processes.

Research interests: geometric group theory, 3-manifold topology and knot theory, K-theory, algebraic topology.

Who's who in Topology

Centre for Doctoral Training (CDT)

Department of Mathematics

Graduate program.

Application deadline is December 15th, 2023.

Test requirements:

GRE Subject Test:         GRE Subject Math Test scores are OPTIONAL.

GRE General Test:      GRE General Test scores are OPTIONAL.  

TOEFL or IELTS:      Scores are REQUIRED (the link below contains answers to common questions on these exams including who has to take them).

Standardized Test Questions:       Yale Graduate School of Arts & Sciences

Fee waiver:  if you wish to apply to waive the application fee (105$) please apply for the waiver here:  Application Fees & Fee Waivers | Yale Graduate School of Arts & Sciences . We recommend to do this as early as possible and, at least, several days before the deadline of January 2, 2023. Please note that the department has no control over the waivers. 

Program in Applied Mathematics . Note that there is a separate program in Applied Mathematics. You cannot apply for both programs. Follow  Welcome | Applied Mathematics Program (yale.edu)  for the general information about that program  and  https://applied.math.yale.edu/graduate-program-0  for the information about admissions, requirements, etc.

Welcome to the Yale graduate program in Mathematics.

The transition from mathematics student to working mathematician depends on ability, hard work and independence, but also on community. Yale’s graduate program provides an excellent environment for this, and we are proud of the talented students who come here and the leading faculty with whom they learn the profession.

In their first two years, students focus on building their general knowledge and passing the qualifying exams , but are also encouraged to use the time to think about their areas of interest, work together to explore them, and begin making connections with faculty advisors. There are few formal requirements and this flexibility allows students to develop independence, formulating and following their own goals.

Mathematics, while requiring intense individual focus, also thrives on collaborative work. Students form study groups and seminars together, and also benefit from our excellent cohort of Gibbs Assistant Professors and other Postdoctoral Fellows, who are a source of fresh mathematical perspectives and camaraderie.

Research, and the contribution of new ideas and results to the body of mathematical knowledge, naturally form the main focus of the next few years, and typically students complete their PhD by the end of the 5th (sometimes 6th) year. During this time they also get to know the faculty better, and continue building intellectual and personal connections, horizontally across the discipline and through time to our shared intellectual history and tradition.

Teaching is an important component of our profession, and the department provides support and training to graduate students. Teaching assignments proceed from individual coaching to classroom teaching, with careful mentoring provided by our dedicated team of lecturers.  The Lang Lunch Seminar, in the second year, provides in-depth training to graduate students before they begin to lecture.

Director of Graduate Studies : Van Vu .

Inquiries concerning the graduate program in mathematics should be sent to Van Vu .

Registrar of Graduate Studies: TBA 

Some useful links:

• The mathematics department page in the Graduate School catalog.
• Graduate school homepage for general information.
• Admissions information from the graduate school.
• Mathematics Graduate Program Advising Guidelines

Frequently Asked Questions about the Graduate Admissions Process

1. What are the most important factors in admission decisions? Transcript (do you have a strong enough background to succeed in grad school at Maryland?), Letters of recommendation, GRE Subject score. If you are an international applicant, the TOEFL (or IELTS or PTE) score is important.

2. How many students are accepted? In a recent year, 190 people applied, 30 were offered admission, and 14 accepted our offer.

3. Do you admit students for the Spring semester? No.

4. Do you admit students who are aiming only for a Masters degree? Generally, we admit students only for PhD degree. If you are considering going for a PhD degree, you should apply directly to the PhD program. You should not apply to the Masters program with the intention of getting that degree and then deciding whether to continue. If you are in the PhD program, it is possible to stop at the Masters degree.

5. What score do I need on the GRE Subject test? There is no minimum score, since other factors also play a role, but your chances decrease significantly if your score is much below 700.

6. Do I need the GRE Subject test? Occasionally someone is accepted without the GRE Subject test. However, we accept only about 16% of applicants, so someone with missing information is at a severe disadvantage. (Please note, the GRE General Test is required.)

7. What grade point average do I need? You should have at least a 3.0 grade point average, with mostly A's in your mathematics courses.

8. What score do I need on the TOEFL? For the TOEFL, we want a total score of at least 105, with at least 23 on the speaking portion. Occasionally, we accept someone who misses by a point, but this is rare. We never accept someone with a total score below 100.

We now accept IELTS or PTE in place of the TOEFL. For the IELTS, we want a score of 7.4 or higher. For the PTE, we want a score of 72 or higher.

9. I am a student from a non-English-speaking country who received a Masters degree in the US. The Graduate School has waived the TOEFL requirement. Do I still need the TOEFL?

Yes. Because our graduate students are supported by Teaching Assistantships, it is required that their English skills are higher than the minimum required to attend courses.

10. I have a transcript from my institution that is encrypted. What should I do?

Do not upload an encrypted transcript to the application. It will upload as a black document, and will not be able to be used for evaluation. Please print the document out, scan it, then upload the scanned version. 

11. What if I don't know what area I'm interested in? Am I allowed to change areas when I get to graduate school? You probably have some idea what direction you might be headed, for example, analysis, algebra, logic, geometry, etc. It's best to list one or two areas, since "undecided" could result in your application not being considered as carefully by admission committee members who are assigned applicants in their field to review. You can of course change your area of interest while in graduate school. There are areas of mathematics that you might currently not know anything about, but after taking a course you decide that's what you really want to do. The first two years of grad school are usually spent expanding your horizons and figuring out what area you want to pursue.

12. What if some of my application materials arrive late? Our job is to select the best students, not to enforce arbitrary deadlines. However, when material arrives after a file has been reviewed, sometimes it gets missed. So, a few days late is often no problem, but a few weeks late could hurt your chances.

13. What should be on the Personal Statement and/or Supplementary Application Essay? The personal statement is not like the essay that got you accepted or rejected at the undergraduate school of your dreams. We don't care about literary style, etc. We want to know what interests you mathematically and what interests you about the University of Maryland. For example, which professors might you consider working with? If you have done a research project, say something about it. If there are parts of math that you found exciting, say so. If there is something in your record that is going to raise questions (for example, it's been ten years since you were an undergraduate), answer those questions if appropriate. If this applies to you, you can also discuss your community involvement or service, leadership, or overcoming social, economic or physical barriers. It is acceptable to submit the same text for both the Essay and Statement of Purpose.

14. What courses should I take to be ready for grad school? For almost all areas of math, we want to see real analysis (epsilon-delta proofs, compact sets, Cauchy sequences, etc.) and a theoretical linear algebra course (not just row operations on matrices; there should be linear independence, diagonalization theorems, minimal polynomials). For pure math, a good course in abstract algebra is recommended. For more applied areas, some partial differential equations or complex analysis is good. Generally, the more math, the better. If you are at a school that offers grad courses, then taking a grad course (and doing well) increases your chances. If your school doesn't offer grad courses, then take as many upper-level courses as possible.

15. How do I support myself during graduate school? All of our accepted students (except a few who are supported by employers) are offered support by the University, usually in the form of a Teaching Assistantship. This covers your tuition and pays a salary of around $23000 per academic year. This is enough to live on, but you don't get rich. Teaching Assistants teach 4 to 6 hours per week, depending on the course. The total time commitment is around 12 to 15 hour per week. Some more advanced grad students are supported as Research Assistants, with the money usually coming from the advisor's research grant.

16. When are accepted students notified? Acceptances are made from mid-February through April 15. Here's what happens: Since we support all accepted students with Teaching Assistantships, we need to end up with a predetermined number of students, with a very small error term. Some acceptances are sent out in mid February, but no one can be required to answer until April 15. Therefore, nothing happens for several weeks. Everyone else starts getting nervous, especially since they see on grad cafe web sites that we've made some acceptances. Finally, around April 1, these accepted students either get rejected by their higher choices and accept our offer, or they have been accepted by a higher choice and are trying to get up the courage to say no to us. (If you end up in this latter category, please be courageous. The graduate director is getting besieged with emails from people who want your place.) When it looks like we'll be below our quota, we send more acceptances. (If it looks like we have too many, we hide from the budget people and hope for rejections.) Around April 10, we have filled about half of our positions, with many offers awaiting responses. By some miracle, over the last five days, the rejections and acceptances come and go, and we hit our projected number.

17. How long does it take to get a PhD? The median time is around 5.5 years.

18. What percentage of the students complete the program? Our recent estimate is that around 75% of entering students will complete their PhD.

19. What do students do after they graduate? (besides celebrate) See the list of recent jobs .

For application FAQs (technical questions), please see:  https://gradschool.umd.edu/admissions/admissions-requirements

Give to the Math Department

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Phd in mathematics: requirements, salary, jobs, & career growth, what is phd in mathematics.

A PhD in Mathematics is a highly specialized degree that entails completing original research in a specific area of mathematics and making a significant contribution to the field.

Students in a PhD program generally take advanced mathematics courses to expand their understanding of the subject and to obtain the abilities needed to undertake research.

They collaborate extensively with faculty mentors and other researchers to create a research project that takes several years to finish.

A PhD in Mathematics study typically results in the discovery of new mathematical theories, proofs, or algorithms that can be applied in a variety of domains such as computer science, physics, finance, and engineering.

A PhD in Mathematics is a tough and rigorous curriculum that needs great mathematical competence, strong analytical skills, and endurance.

Graduates of mathematics PhD programs frequently move on to positions in academia, research, or business, where they can use their mathematical skills to solve challenging issues and make substantial contributions to their field.

How much money do people make with a PhD in Mathematics?

Individuals with a PhD in Mathematics can expect to earn more than those with a bachelor’s or master’s degree in the subject.

As of May 2020, the median annual income for mathematicians and statisticians, which includes individuals with a PhD in Mathematics, was $96,660, according to the US Bureau of Labor Statistics.

However, depending on the field and region, earnings might range from $59,500 to more than $160,000 per year.

I ndividuals with a PhD in Mathematics are frequently employed in academia, where salaries vary according to the type of institution, rank, and years of experience.

For the 2021-2022 academic year, the median income for full-time faculty members in the mathematical sciences was $92,000, according to the American Mathematical Society.

What is expected job growth with PhD in Mathematics?

According to the U.S. Bureau of Labor Statistics, employment of mathematicians and statisticians, which includes people with a PhD in Mathematics, is expected to expand 33 percent from 2020 to 2030, substantially faster than the national average.

The rising use of data and the need for quantitative analysis in a range of areas, including healthcare, finance, and technology, are projected to drive this expansion.

Scientific research and development services, which are expected to grow 48 percent from 2020 to 2030, and management, scientific, and technical consulting services, which are expected to grow 37 percent over the same period, are expected to see the most growth in mathematicians and statisticians’ employment.

What can you do with a PhD in Mathematics?

Individuals holding a PhD in Mathematics have numerous job options. Here are a couple such examples:

1. Academia: Individuals with a PhD in Mathematics can pursue academic careers such as professorships or research positions in universities or research institutions. Graduate students and postdoctoral researchers can also be supervised by them.

2. Data Science: As the amount of data in many businesses grows, there is a great demand for people with extensive mathematical skills who can work as data scientists, data analysts, or statisticians.

3. Financial Services: Individuals with a PhD in Mathematics can work as quantitative analysts, risk managers, or financial analysts in the financial industry.

4. Technology: Individuals with a PhD in Mathematics can work in technology organizations as data scientists or machine learning engineers, thanks to the rising use of artificial intelligence and machine learning.

5. Government Agencies: Individuals with a PhD in Mathematics can work for government organizations such as the National Security Agency (NSA), the Department of Defense, or the National Aeronautics and Space Administration (NASA) in positions such as cryptographers, systems analysts, or aerospace engineers.

6. Consulting: Individuals with a PhD in Mathematics can work for consulting firms, where they can provide analytical and modeling services to clients in a variety of industries.

What are the requirements for a PhD in Mathematics?

The specific requirements for obtaining a PhD in Mathematics can vary depending on the institution and program, but generally, the following are common requirements:

1. Bachelor’s or Master’s Degree: Applicants to most PhD programs in Mathematics must have a Bachelor’s degree from a recognized university. Although it is not usually required, certain schools may accept applicants with a Master’s degree in a related discipline.

2. Academic Transcripts: Applicants are usually expected to present certified transcripts of their undergraduate and graduate education, which demonstrate their academic performance and achievement.

3. Statement of Purpose: Applicants are typically expected to provide a personal statement or statement of purpose detailing their research interests, academic ambitions, and reason for pursuing a PhD in Mathematics.

4. Standardized Test Scores: Applicants to many PhD programs may be required to submit scores from standardized tests such as the Graduate Record Examination (GRE) or other related assessments.

5. Letters of Recommendation: Applicants to PhD programs in Mathematics are frequently required to provide letters of recommendation from academic or professional sources who may speak to the applicant’s academic talents, research potential, and eligibility for a PhD program.

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How long does it take to get a phd in mathematics.

The length of time it takes to earn a PhD in Mathematics can vary depending on a number of factors, such as the program’s structure, the student’s progress, and the research area. However, in general, it takes around 4-6 years to complete a PhD in Mathematics in the United States.

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Do you need a masters in mathematics to get a phd in mathematics.

In most circumstances, a master’s degree in mathematics is not required to pursue a PhD in mathematics.

Many PhD programs in Mathematics accept students who have a bachelor’s degree in Mathematics or a related field and a strong academic record and research potential.

However, some PhD programs may offer a master’s degree as part of the PhD requirements. This is known as a “master’s on the way to a PhD” program.

In this instance, students would normally complete the master’s degree coursework and research requirements before enrolling in the PhD program.

What are the Best PhD in Mathematics Degree programs?

1. massachusetts institute of technology (mit) 2. university of california, berkeley 3. stanford university 4. harvard university 5. princeton university 6. university of chicago 7. california institute of technology (caltech) 8. university of michigan 9. new york university (courant institute) 10. university of texas at austin, leave a comment cancel reply.

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2024 Best Online PhD in Mathematics [Doctorate Guide]

Earning a math PhD can be beneficial for many different professional paths. A doctorate in mathematics will encourage you to explore the boundaries of math, and you’ll conduct research in addition to coursework.

Additionally, some programs ask students to complete a thesis or dissertation to earn the title of Doctor of Mathematics.

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Read on to learn more about what a doctorate degree in mathematics entails and determine if it’s the right path for you.

Online Math PhD Programs

Earning a PhD in Mathematics can be an exciting step for many students. This advanced degree is typically sought out by those who wish to dive deeper into mathematical processes, theory, and research.

Students who earn the title of Doctor of Mathematics are often prepared for a variety of advanced roles in education, STEM fields, and data science and management. In addition to completing core classes, you’ll typically have the opportunity to take courses and complete research that help you complete your dissertation.

A dissertation is often linked to potential career and professional opportunities and studies. Many math PhD programs provide a variety of specialty areas to help students hone in on their area of expertise. Those who graduate with a doctorate degree in mathematics pursue careers in a variety of fields, depending on their area of expertise and research.

Graduates’ advanced skills in understanding numeric principles and mathematical reasoning are applicable to a variety of roles, such as:

• Actuarial analyst or financial strategist for businesses
• Academic leader or postsecondary professor
• Government agency consultant or advisor
• Researcher in STEM fields

A highly developed understanding of mathematics, research, theory, and their applications can be beneficial in a variety of professional settings. Pursuing a math PhD can help you fine-tune your skills and grow your professional qualifications in this lucrative field.

Math Careers & Salaries

Individuals who have earned a PhD in Math often select careers that exercise their skills in reviewing numbers, outcomes, and analyzing information. Many math professionals with a PhD combine their interests with fields such as physics, engineering, or finance.

According to the Bureau of Labor Statistics , here are the median salaries for various careers related to advanced studies in mathematics.

Some mathematics professionals work as analysts, actuaries, or economists, working with businesses or government entities to review figures and review outcome probability. Others pursue work as professors, researchers, or leaders within postsecondary education.

Mathematicians and statisticians are often employed in a variety of fields where an in-depth knowledge of these topics can be important. These fields can include energy production and mining, aerospace and transportation, finance, or pharmaceutical production.

Mathematics Doctorate Curriculum & Courses

While each school offering a doctorate degree in mathematics will have its own unique curriculum, students will typically encounter courses that encourage their growth.

Some courses that you may encounter while studying for your math PhD include:

• Linear Algebra : This course involves the process of plotting mathematical problems via vectors and matrices.
• Ordinary and Partial Differential Equations : In this course, you’ll solve for unknown variables and their derivatives.
• Probability : At the doctorate level, you’ll take this study of likelihood and chance to a more in-depth, theoretical stage.
• Geometry and Topology : This course compares the local and global implications of continuous and discrete moduli, and it explores topics such as Riemannian geometry and homotopy theory.
• Harmonic Analysis : Using the Fourier transform and Fourier series, you’ll explore functions and frequency.
• Estimation Statistics : Students will combine a variety of factors to create and conduct experiments that allow them to analyze incomplete data.
• Graph Theory : This course provides the opportunity to study graphs and the relationship of multiple pieces of data to each other.
• Mathematical Analysis : In this course, you’ll use both real and complex numbers to explore scenarios involving continuous and analytic functions, limits, infinite sequences, and other mathematical objects.
• Algebraic Combinatorics : Students will explore enumeration, permutation, partitions, sets, and cycles to investigate the world of abstract algebra.
• Operator Theory : This course involves a deep dive into linear operators on function spaces, including the study and analysis of various operators.

Your interests, dissertation, and overall goals will typically influence which courses you choose. It can be beneficial to review each program’s curriculum before you decide on a school.

Admissions Requirements

Each college or university will have its own set of admissions requirements for students applying for their PhD in Mathematics. It’s beneficial to review the application process in full before starting.

Some of the items required by many mathematics doctorate programs include:

• GRE or GMAT Scores (only some schools require them)
• Transcripts of previous education
• Master’s degree in a related field
• CV or resume
• Statement of intent

In addition, you will likely be asked to provide academic writing samples as well as letters of recommendation. You may be asked to complete an interview with department staff, too.

Accreditation

When researching schools that offer a PhD in Math online, you may wish to give special attention to schools that hold regional accreditation.

Accredited schools have been carefully evaluated for their academic precedence. As a result, employers often prefer or require candidates to have received their doctorate in mathematics from an accredited institution. Credible regional accrediting organizations are recognized by the Council for Higher Education Accreditation. You can visit CHEA’s website to verify a school’s accreditation status.

Additionally, benefits such as financial aid, certification, and membership to professional organizations may be tied to earning your doctoral degree from an accredited institution.

Financial Aid and Scholarships

If you are interested in needs-based financial aid, you could fill out and submit the FAFSA, or Free Application for Federal Student Aid . This determines your eligibility for federal financial aid. Most state aid programs want to see your FAFSA results as well.

You could also inquire about scholarship opportunities through the school you choose to attend. Many programs offer financial assistance for students based on prior academic performance, chosen specialty, or demographic factors. Your employer may also offer an educational assistance program, such as tuition reimbursement, for workers who are furthering their education.

In addition, some independent parties, corporations, and communities offer scholarship or grant opportunities for doctoral students. The requirements for these scholarships vary and are often as unique as the gifts themselves.

What Can You Do with an Online PhD in Mathematics?

A Doctor of Mathematics degree is designed for those who wish to complete in-depth research and exploration of the various expressions of mathematics in our world.

Those who earn this degree may wish to work in educational, research-driven, financial, economic, or technological fields. They may also choose to work in a variety of STEM-related fields, in areas such as energy, software, corporate, government, and aerospace.

Common career paths for advanced mathematics professionals include mathematician, statistician, actuary, economist, data scientist, and operations research analyst. Those with a PhD may also pursue teaching or research roles in colleges and universities.

How Long Does It Take to Get a PhD in Math Online?

It often takes between 3 to 5 years of full-time study to complete a PhD program. Your overall timeframe, though, could depend on the program you select and your enrollment status.

In many cases, the number of required credit hours combined with the dissertation process will dictate how long it takes to receive your degree. If a doctorate program does not have a dissertation requirement, a full-time student can typically complete the program in 3 years.

You may wish to compare programs’ course catalogs to understand the program structure and course schedule of each school you’re considering.

What’s the Difference Between Applied Mathematics vs. Mathematics PhD Programs?

There are many similarities between the topics studied in a PhD program in applied mathematics versus a PhD program in mathematics. The area of concentration for each program differs a bit in practice.

In each program, students are able to test theories, produce research, and examine principles, though the motivation and results may vary.

Is a Math PhD Degree Worth It?

Yes, a math PhD degree is worth it for many students. Math professionals can use their skills in a variety of areas, including finance, engineering, technology, education, and business.

According to the Bureau of Labor Statistics, overall employment for math occupations is projected to grow by 29% over the next ten years. This is much faster than the average for all occupations. Common math careers include actuaries, data scientists, mathematicians, statisticians, and operations research analysts. A PhD can also help you qualify for positions in research and academia.

Universities Offering Online Doctorate in Mathematics Degree Programs

Methodology: The following school list is in alphabetical order. To be included, a college or university must be regionally accredited and offer degree programs online or in a hybrid format.

Indiana University – Bloomington offers a Doctor of Education in Curriculum and Instruction with a specialization in Mathematics Education. This degree is intended for individuals who have already obtained a master’s in a related field. The curriculum consists of 60 credits, and courses meet online. It covers topics like teaching methods, mathematical thinking, student assessment, and research skills.

Indiana University – Bloomington is accredited by the Higher Learning Commission.

Montana State University offers a Doctor of Philosophy in Mathematics (Education Emphasis). This program is designed for those who wish to pursue careers in K-12 educational settings. Potential courses include Principles of Mathematical Analysis, Models of Assessment in Mathematics, and Applied Educational Research. The completion of at least 60 credits is required to graduate.

Montana State University is accredited by the Northwest Commission on Colleges and Universities.

The University of Colorado – Denver offers a Doctor of Philosophy in Education and Human Development with a concentration in Mathematics Education. The curriculum is designed to teach the skills required to excel in diverse educational environments. The program utilizes a blend of online and on-campus delivery methods. The program’s 75 required credit hours can typically be completed in 4 to 8 years.

The University of Colorado – Denver is accredited by the Higher Learning Commission of the North Central Association of Colleges and Schools.

The University of Wyoming offers a Doctor of Philosophy in Mathematics Education. This program is designed to prepare students for work as scholar-leaders in settings like research institutions and universities. It can be completed entirely online. The curriculum consists of 81 credit hours and covers key concepts like multicultural education, curriculum development, learning and cognition, and assessment.

The University of Wyoming is accredited by the Higher Learning Commission.

Getting Your PhD in Mathematics Online

In our technological and solution-driven world, the horizons and practical applications of mathematics are expanding daily.

A math PhD program could help you grow your professional qualifications in this lucrative field. Much like with a masters in mathematics online  program, earning a PhD in Mathematics online could be a more flexible and convenient format for working professionals. The professional applications of an advanced mathematics degree are diverse. Many graduates use their PhD to advance in research, academia, business, or other areas of industry.

You could start exploring accredited schools today to see which online math PhD programs best align with your interests, schedule, and goals.

Timeline: What to Expect

Getting started at mit.

Orientation advice, instructions, forms, and deadlines are here . In brief, admitted students submit their housing preferences by mid-May and arrive on campus starting in mid-August. Orientation usually takes place during the last week of August.

Course Registration

At the beginning of the first year, each student is assigned a faculty advisor. The advisor approves the student's course registration each term and discusses his/her academic program. Students are expected to register for (at least) three graduate classes in every semester until completion of the oral qualifying exam. In most cases, after completion of the qualifying exam, the thesis advisor takes over as registration advisor.

English Proficiency

Students from non-English-speaking countries should achieve English proficiency as soon as possible, preferably by the end of the first year. Instruction in English as a second language at MIT can be found on the CMSW English Language Studies page .

Academic Evaluation

The overall academic performance of all graduate students is reviewed at the end of each semester, as is their performance in RA or TA positions. Students are expected to maintain at least a B+ average. Ultimately, they must complete 8 one-semester graduate-level subjects with a grade of A or B.

Plan of Study (Applied Math Only)

Students in Applied Mathematics submit a tentative plan of study at the end of the first term. Students submit a more definite plan at the end of the first year.

Choosing a Thesis Advisor

As students progress in their coursework and their interests become more focused, they select an advisor in their intended area of research. They generally make at least a tentative choice by the end of the first year in order to plan for the qualifying examination. Students with advisors outside the department also have an internal advisor.

Qualifying Exam

The qualifying exam is an oral exam given by a committee of faculty members. Students should begin thinking of possible exam topics during their first year. More details on qualifying exams can be found here .

After passing the qualifying examination, students must designate a thesis advisor and register for 18.THG, Thesis, in each subsequent term. Students continue to take regular courses as well as pursuing their research. Grades for thesis research (18.THG) are recorded as J (satisfactory) or U (unsatisfactory).

Students with external advisors submit a summary of their research to their external and internal advisors each semester in order to obtain a grade for 18.THG. The graduate chair reviews all research summaries. See external thesis advisor information .

Teacher Training

Graduate students are strongly encouraged to develop their teaching skills and establish teaching credentials. Toward this end, all students are required to complete at least one semester of classroom teaching. During the semester before their first teaching assignment, prospective teachers are required to participate in teacher training . Foreign students must demonstrate proficiency in English before undertaking the training.

Thesis Proposal

Within one year of passing the qualifying examination or by the beginning of the 6th semester students must submit a thesis proposal of about five pages. See thesis proposal for detailed instructions.

Final Semester

In their first week of their final semester, students must register as degree candidates. This does not constitute a commitment to finish; students can remove their names from the degree list at the end of the term. Students submit their thesis titles at mid-semester and prepare for the thesis defense, which usually takes place in April. Thesis deadlines are here .

Thesis Defense

When the thesis approaches completion a Thesis Examination Committee is formed, consisting of the thesis advisor and two other faculty members in related fields. The student's successful defense of the thesis and submission of the final thesis document complete the doctoral program. Thesis defense instructions are here .

• How Long Does A PhD Take?
• Doing a PhD

Sometimes, just knowing how long a PhD takes can be enough to sway your decision on whether a research degree is for you. So with that in mind, exactly how long does a PhD take?

In the UK, a full-time PhD takes 3 to 4 years to finish whilst a part-time PhD takes twice as long at 6 to 7 years. Alongside these average durations, there are time limits on how long you can be enrolled on to a PhD programme. To discover these limits, the factors which most influence doctoral degree durations and how the UK durations compare to international PhDs, continue reading on.

How Long Does It Take to Get a Full-time PhD?

In the UK, a full-time PhD will typically take you 3 to 4 years. You will usually spend the first three years on the technical aspects of your doctorate. This includes undertaking independent research, designing your research methodology and collecting and analysing data. You will then spend an additional academic year on writing up your PhD thesis and sitting your viva.

How Long Does It Take to Get a Part-time PhD?

In the UK, a part-time PhD will typically take you 6 to 7 years; twice as long as doing a full-time PhD. The reason for this is that as a part-time PhD student, you would dedicate around 20 hours per week to your PhD as opposed to the typical 40 hours full-time students would put into their subject.

How Long Does a Distance Learning PhD Take?

Similarly, distance learning PhD’s take an average of 6 to 7 years to complete. This is because the vast majority of students who undertake a distance learning PhD do so because they can’t relocate closer to the university. Although these commitments will differ, they often mean the student isn’t able to dedicate 40 hours per week to their studies.

Students in STEM disciplines will often take longer to finish a distance learning doctorate degree than those in non-STEM disciplines. This because the progress of a STEM PhD student will be limited by how often they can access a laboratory for experiment work.

How Does Funding Impact a PhD’s Duration?

In reality, the actual time it will take you to complete your PhD degree will depend on your funding situation.

If you’re receiving  funding , it will usually only cover you for 3.5 years if you’re studying full-time or for 7 years at half the stipend if you’re studying part-time. Although this could vary slightly, most PhD funding providers, e.g. Engineering and Physical Sciences Research Council (EPSRC), follow this timescale as indicated on their ‘ length of PhD studentships’ page. Because of this, most students who obtain scholarships try to complete their PhD within the timeframe of their funding so they don’t incur additional fees which they need to cover themselves.

It’s also worth noting that some funded PhD positions have additional conditions attached to them as part of their eligibility requirements. For example, they may require teaching undergraduate students, hosting laboratory sessions or attend presentations and conferences. This will be especially true if you’re on a Graduate Teaching Assistantship (GTA). Although these shouldn’t add considerable time to the length of a PhD programme, they have the potential to do so if they aren’t managed properly.

As self-funded students cover their own annual tuition fees and other associated costs, how long they’ll spend to complete their PhD project will largely depend on their own personal financial situation. Because of this, most self-funded PhD students find it best to complete their PhD study in the shortest time-frame they can manage.

Are There Deadlines?

Yes – unfortunately, all good things must come to an end! Within the UK, the deadline for your PhD is defined as the last date which you must submit your final thesis by. This date is set by your university’s overall regulations and varies depending on the arrangements of your PhD, e.g. whether it’s full or part time. In the vast majority of cases, the adopted deadlines are four years for full-time PhDs and seven years for part-time PhDs from the date you were officially registered onto your programme, as shown below from the University of Leicester’s registration guidance page .

This time-frame may vary from university to university. For example, the University of Sheffield adopts an additional year for part-time PhDs as shown below.

Can I Complete It Faster?

Although it’s possible to complete a full-time PhD in under 3 years, it’s a significant feat that’s rarely heard of. When these feats occur, they’re usually where the doctoral student already has extensive knowledge and experience in their field before undertaking their PhD.

Whilst it’s possible to complete a part-time PhD in under 6 years, it largely depends on your commitments outside your studies. For example, if you have a part-time career alongside your PhD, it’s unlikely that you’ll be able to commit the additional hours required to complete your doctorate a year faster.

However, if instead of a steady part-time job you take on occasional work as a freelancer, you’ll be able to set aside many more hours towards your doctoral degree.

Will Having only A Bachelor’s Degree or Being an International Student Limit My Rate of Progression?

Not at all. While there are benefits to having a Master’s degree such as an additional year of learning and greater research experience due to your fourth-year dissertation project, this doesn’t mean not having one would limit you. A PhD is very different to both Bachelor and Master degrees due to being heavily research-based, therefore, both types of students will have just as much to learn on their way to completing their doctorate.

Similarly, whether you’re an international student will bear no influence on the duration of your PhD.

Finding a PhD has never been this easy – search for a PhD by keyword, location or academic area of interest.

How Does This Compare to the Duration of EU and US PhDs?

PhD hosted by universities within the EU, such as those in France, Norway and Spain, have the same programme structure as those within the UK. As a result, there are no noticeable differences in the time to complete a doctorate between UK and EU institutions.

However, this is not the case in the US. Compared to PhDs conducted within the UK or EU, PhDs conducted within the US take considerably longer to obtain. According to a  2017 study conducted by the National Science Foundation, a US government agency which supports research and higher education, the average time to get a PhD within the US is 5.8 years. Besides this, the average completion time can further increase depending on the disciplines. For example, they found doctorates within the humanities and arts to take an average of 7.1 years to achieve.

The primary reason for this difference is the way PhD degrees are structured within the United States. As mentioned previously, PhDs conducted within UK and EU universities are essentially broken into two sections – one covering the analytical aspects and the other covering the writing up aspects. However, within the US, doctorate programmes comprise additional sections. PhD students are first required to undertake 2 to 3 years of courses, which cover a broad range of topics related to their schools’ discipline. This is then followed by coursework and several examinations, which only once passed can the PhD candidate then start working on their research project and dissertation.

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How Long Does It Take to Sober Up After Drinking?

Medically reviewed by Isabel Casimiro, MD, PhD

Enjoying drinks with friends can be a fun way to socialize and unwind, but it's important to be aware of how long it takes to sober up after a night of drinking. Typically, it takes two hours for your body to metabolize one standard drink. However, the effects of alcohol can last long after your last drink, though the time it takes to sober up after drinking is different for everyone.

This article discusses the factors that influence how long it takes to sober up, the impact alcohol has on your body, how long alcohol stays in your system, and what to expect after a night of drinking or when you quit drinking alcohol entirely. 

How Does Alcohol Affect the Body?

Alcohol  (ethanol) is a central nervous system depressant that slows your body's systems and affects your mental and physical functioning. You may feel relaxed or euphoric after your first drink or two, but alcohol's impact on your body and mind extends well beyond these initial effects.

Whether you drink beer, wine, or liquor, alcohol quickly enters your bloodstream and reaches your brain within minutes of your first drink. As alcohol reaches your brain, the relaxed, euphoric feeling sets in and you may have fewer inhibitions. 

While these effects contribute to the sociability associated with drinking, alcohol also slows down communication pathways in your brain. You may feel more sociable and less inhibited but find it harder to think coherently, speak articulately, and make sound decisions. Alcohol also slows how your body communicates with your brain, which slows your reflexes and increases the risk of accidents and injuries.

As your blood alcohol concentration (BAC)—the amount of alcohol in your bloodstream—rises, alcohol affects more than just your brain. It also affects the following organs and body systems:

Your  liver metabolizes (breaks down) alcohol and converts it into acetaldehyde , a toxin and carcinogen. Your body eventually breaks acetaldehyde into acetate, but it can damage the liver before this occurs. Every time you drink, acetaldehyde binds to fat cells and proteins in the liver, triggering an inflammatory response that damages and kills liver cells. Heavy or ongoing alcohol consumption increases the risk of liver diseases, such as cirrhosis (severe scarring of the liver).

Heavy alcohol consumption can raise your heart rate and blood pressure for up to 24 hours after drinking. Over time, alcohol can strain and weaken your heart, increasing the risk of heart disease, stroke, and heart failure.

Digestive System

Your stomach and small intestine absorb alcohol before it moves into your bloodstream. Alcohol consumption increases the production of stomach acids, which can irritate the stomach lining and lead to nausea, vomiting, and heartburn.

Heavy drinking also irritates and inflames the intestines and affects the muscles that control digestion, which is why you might have diarrhea after a night of drinking. Over time, intestinal inflammation increases the risk of gastrointestinal disorders and impairs your body's ability to absorb nutrients from food, leading to nutritional deficiencies.

Immune System

Alcohol suppresses the immune system's ability to fight infection and keep you healthy. A night of drinking five to six drinks suppresses your immune system for up to 24 hours, making you more vulnerable to illnesses like bacterial or viral infections.

Measuring Alcohol Consumption: Blood Alcohol Concentration

BAC is the amount of alcohol in your blood. It shows the percentage of alcohol weight in 100 milliliters of blood. For example, a BAC of 0.10 means that 0.1% of your bloodstream is comprised of alcohol. Blood tests and breath alcohol analyzer devices, such as the Breathalyzer, can measure your BAC, which directly correlates to your level of intoxication.

Here's a breakdown of how different BAC levels can affect you:

• 0.02 BAC : You may feel relaxed and less inhibited, experience mood changes, and lose some judgment.
• 0.05 BAC : Impairment becomes more noticeable; you may experience increased euphoria, impaired judgment and fine motor coordination, and loss of inhibition. 
• 0.08 BAC : Muscle coordination declines, and you may have trouble balancing or walking, have slower reaction times, and experience changes to your vision and hearing.
• 0.10 BAC : Significant intoxication, leading to slurred speech, poor coordination, and slowed thinking. 
• 0.15 BAC : A high level of intoxication, which may cause vomiting, confusion, and significant loss of muscle control and balance. 

Higher BACs can lead to blackouts and loss of consciousness. A BAC of 0.40 or higher causes respiratory failure and can lead to death. 

Factors That Affect How Long It Takes to Sober Up

How long it takes to sober up after drinking depends on your BAC. The higher your BAC, the longer it will take to sober up.

The amount of time it takes to sober up primarily depends on how much alcohol you've consumed. The volume of alcohol consumed and other factors can influence how quickly your BAC rises and how long it takes for alcohol to leave your body, including:

• Alcohol content : The alcohol content in a standard drink varies by the type of alcohol consumed. For example, a shot of liquor contains 40% alcohol per serving (1.5 ounces), a 5-ounce glass of white wine contains 12% alcohol, and regular 12-ounce beer contains 5% alcohol. The higher the alcohol content, the longer it can take to sober up. 
• Amount consumed:  The more alcohol you drink, the longer it will take for it to leave your body.  
• Food in your stomach : Eating before drinking can help slow down the rate of alcohol absorption in your body. 
• Sex:  Women tend to have more body fat and less water per pound of body weight than men. Fat cells do not take up alcohol as quickly as other cells, so men generally metabolize alcohol and sober up more quickly than women.
• Food intake : Eating before drinking can help slow down the rate of alcohol absorption in your body. 
• Body weight : The more you weigh, the higher your body's water content, which helps your body metabolize alcohol more quickly.
• Liver function : The liver plays a central role in metabolizing alcohol. People with healthy liver function process alcohol more efficiently than those with liver damage or disease.

A Note on Gender and Sex Terminology

Verywell Health acknowledges that sex and gender are related concepts, but they are not the same. To reflect our sources accurately, this article uses terms like “female,” “male,” “woman,” and “man” as the sources use them.

How Long Does Alcohol Stay in Your System?

Everyone's body eliminates alcohol at a rate of 0.015% per hour, which means it takes about two hours to metabolize one standard drink no matter your body weight, sex, or the type of alcohol you consume. So, if your BAC is 0.04, it will take about 2.5 hours to sober up. It takes approximately five hours to sober up with a BAC of 0.08. 

Your BAC reaches 0 once there is no remaining alcohol in your bloodstream, but traces of alcohol can stay in your urine, saliva, breath, and hair for varying lengths of time. 

• Urine : Ethyl glucuronide (EtG) is a metabolite of ethanol, produced when the body breaks down alcohol in tissues and organs. A urine test can detect EtG in your urine up to five days after heavy drinking and two days after light drinking.
• Saliva : Alcohol is detectable in saliva 24 to 48 hours after consumption.
• Breath : Alcohol stays in your breath for as long as it remains in your bloodstream, usually up to 12 hours, depending on how much you drank.
• Hair : Traces of alcohol metabolites remain in the hair for up to 90 days after consumption. Hair testing is beneficial for detecting alcohol consumption in people with chronic or heavy drinking habits.

Is There a Faster Way to Sober Up From Alcohol?

You may have heard about quick-fix methods that help you sober up faster than average. Unfortunately, there is no way to speed up the process of eliminating alcohol from your body. The only way to sober up is to allow your liver time to metabolize the alcohol you've consumed.

While coffee or cold showers might make you feel more alert for a short time, they do not impact your BAC level. Eating food before drinking can slow down the absorption of alcohol and buy your body some extra time to process alcohol, but eating after drinking won't speed up its elimination once it is in your system.

What Happens When Alcohol Consumption Stops?

Once you stop drinking, your body continues to metabolize alcohol. You may feel drowsy and fall asleep faster than usual, but alcohol can disrupt your sleep throughout the night, and you may have frequent awakenings, even if you don't remember them.

You might wake up in the morning feeling tired and not well-rested, with gaps in your memory from the previous night (blackouts). The rebound effects of alcohol can also lead to anxiety and mood changes the following day, a side effect termed "hangxiety."

Hangover Signs and Symptoms 

Many people experience a hangover, a set of unpleasant symptoms, after consuming a significant amount of alcohol. These symptoms typically appear several hours after your last drink and can last up to 24 hours. Common hangover symptoms include:

• Increased blood pressure
• Irritability
• Muscle aches
• Sensitivity to light and sound
• Upset stomach

Sobering Up From Alcohol When You're Addicted

Alcohol is a powerful drug that activates your brain's reward system, stimulating pleasant feelings and blunting emotional pain and stress. Like any drug, alcohol use can lead to addiction, or alcohol use disorder (AUD). Signs of alcohol addiction include:

• Drinking more alcohol or for a longer period than you intended to 
• Wanting to cut down on your alcohol consumption or stop drinking entirely but struggling to do so
• Continuing to drink alcohol despite experiencing health problems or relationship issues related to alcohol consumption 
• Needing to drink more alcohol to achieve the same desired effect
• Experiencing withdrawal symptoms when you try to cut back or quit 
• Neglecting responsibilities at work, school, or home due to alcohol use 

Over time, alcohol addiction causes a shift in the brain's motivation to drink. Rather than feeling motivated to drink for the positive feelings alcohol can induce, motivation shifts to a desire to drink to reduce the emotional and physical discomfort alcohol withdrawal brings.

Talk to your healthcare provider if you suspect you have alcohol addiction and want to get sober . When you're addicted to alcohol, sobriety involves more than just waiting for alcohol to leave your system. It can be a challenging journey that causes withdrawal symptoms and requires emotional and sometimes medical support. 

Alcohol Withdrawal Symptoms

When you've been drinking too much for an extended period, you may experience withdrawal symptoms when you quit.  Alcohol withdrawal symptoms  typically begin within eight hours after your last drink and peak within 24 to 72 hours but can continue for weeks.

Common alcohol withdrawal symptoms include: 

• Anxiety, irritability, and mood swings 
• Clouded thinking 
• Depression 
• Fatigue 
• Hand tremors
• Headache 
• Loss of appetite 
• Mood swings
• Nausea and vomiting 
• Shakiness 
• Sleep problems, like insomnia and nightmares
• Sweaty, clammy skin  

Severe alcohol withdrawal ( delirium tremens ) can cause: 

• Hallucinations
• Seizures 
• Severe confusion 

If you have AUD or think you are addicted to alcohol, talk to a healthcare provider. Medical support can help you safely navigate withdrawal and begin your journey to recovery and  long-term sobriety . 

Summary 

Sobering up after drinking alcohol takes time. The average liver in a healthy adult metabolizes one drink in approximately two hours. How much you have to drink determines how much time it takes to get sober, while factors like your sex, body weight, and whether you had something to eat before drinking play a role in how quickly your blood alcohol concentration (BAC) rises.

If you choose to drink, do so mindfully—limit how many drinks you have, avoid getting drunk, and talk to a healthcare provider if you suspect you have an unhealthy relationship with alcohol. 

Read the original article on Verywell Health .

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Strong geomagnetic storm reaches Earth, continues through weekend

NOAA space weather forecasters have observed at least seven coronal mass ejections (CMEs) from the sun, with impacts expected to arrive on Earth as early as midday Friday, May 10, and persist through Sunday, May 12, 2024. 

NOAA’s Space Weather Prediction Center (SWPC) has issued a Geomagnetic Storm Warning for Friday, May 10. Additional solar eruptions could cause geomagnetic storm conditions to persist through the weekend.

• The First of Several CMEs reached Earth on Friday, May 10 at 12:37 pm EDT.  The CME was very strong and SWPC quickly issued a series of geomagnetic storm warnings. SWPC observed G4 conditions at 1:39 pm EDT (G3 at 1:08 pm EDT).
• This storm is ongoing and SWPC will continue to monitor the situation and provide additional warnings as necessary.

This is an unusual and potentially historic event. Clinton Wallace , Director, NOAA’s Space Weather Prediction Center

CMEs are explosions of plasma and magnetic fields from the sun’s corona. They cause geomagnetic storms when they are directed at Earth. Geomagnetic storms can impact infrastructure in near-Earth orbit and on Earth’s surface , potentially disrupting communications, the electric power grid, navigation, radio and satellite operations. SWPC has notified the operators of these systems so they can take protective action.

Geomagnetic storms can also trigger spectacular displays of aurora on Earth . A severe geomagnetic storm includes the potential for aurora to be seen as far south as Alabama and Northern California.

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Stormy Daniels Takes the Stand

The porn star testified for eight hours at donald trump’s hush-money trial. this is how it went..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

It’s 6:41 AM. I’m feeling a little stressed because I’m running late. It’s the fourth week of Donald J. Trump’s criminal trial. It’s a white collar trial. Most of the witnesses we’ve heard from have been, I think, typical white collar witnesses in terms of their professions.

We’ve got a former publisher, a lawyer, accountants. The witness today, a little less typical, Stormy Daniels, porn star in a New York criminal courtroom in front of a jury more accustomed to the types of witnesses they’ve already seen. There’s a lot that could go wrong.

From “The New York Times,” I’m Michael Barbaro. This is “The Daily.”

Today, what happened when Stormy Daniels took the stand for eight hours in the first criminal trial of Donald J. Trump. As before, my colleague Jonah Bromwich was inside the courtroom.

[MUSIC PLAYING]

It’s Friday, May 10th.

So it’s now day 14 of this trial. And I think it’s worth having you briefly, and in broad strokes, catch listeners up on the biggest developments that have occurred since you were last on, which was the day that opening arguments were made by both the defense and the prosecution. So just give us that brief recap.

Sure. It’s all been the prosecution’s case so far. And prosecutors have a saying, which is that the evidence is coming in great. And I think for this prosecution, which is trying to show that Trump falsified business records to cover up a sex scandal, to ease his way into the White House in 2016, the evidence has been coming in pretty well. It’s come in well through David Pecker, former publisher of The National Enquirer, who testified that he entered into a secret plot with Trump and Michael Cohen, his fixer at the time, to suppress negative stories about Trump, the candidate.

It came in pretty well through Keith Davidson, who was a lawyer to Stormy Daniels in 2016 and negotiated the hush money payment. And we’ve seen all these little bits and pieces of evidence that tell the story that prosecutors want to tell. And the case makes sense so far. We can’t tell what the jury is thinking, as we always say.

But we can tell that there’s a narrative that’s coherent and that matches up with the prosecution’s opening statement. Then we come to Tuesday. And that day really marks the first time that the prosecution’s strategy seems a little bit risky because that’s the day that Stormy Daniels gets called to the witness stand.

OK, well, just explain why the prosecution putting Stormy Daniels on the stand would be so risky. And I guess it makes sense to answer that in the context of why the prosecution is calling her as a witness at all.

Well, you can see why it makes sense to have her. The hush money payment was to her. The cover-up of the hush money payment, in some ways, concerns her. And so she’s this character who’s very much at the center of this story. But according to prosecutors, she’s not at the center of the crime. The prosecution is telling a story, and they hope a compelling one. And arguably, that story starts with Stormy Daniels. It starts in 2006, when Stormy Daniels says that she and Trump had sex, which is something that Trump has always denied.

So if prosecutors were to not call Stormy Daniels to the stand, you would have this big hole in the case. It would be like, effect, effect, effect. But where is the cause? Where is the person who set off this chain reaction? But Stormy Daniels is a porn star. She’s there to testify about sex. Sex and pornography are things that the jurors were not asked about during jury selection. And those are subjects that bring up all kinds of different complex reactions in people.

And so, when the prosecutors bring Stormy Daniels to the courtroom, it’s very difficult to know how the jurors will take it, particularly given that she’s about to describe a sexual episode that she says she had with the former president. Will the jurors think that makes sense, as they sit here and try to decide a falsifying business records case, or will they ask themselves, why are we hearing this?

So the reason why this is the first time that the prosecution’s strategy is, for journalists like you, a little bit confusing, is because it’s the first time that the prosecution seems to be taking a genuine risk in what they’re putting before these jurors. Everything else has been kind of cut and dry and a little bit more mechanical. This is just a wild card.

This is like live ammunition, to some extent. Everything else is settled and controlled. And they know what’s going to happen. With Stormy Daniels, that’s not the case.

OK, so walk us through the testimony. When the prosecution brings her to the stand, what actually happens?

It starts, as every witness does, with what’s called direct examination, which is a fancy word for saying prosecutors question Stormy Daniels. And they have her tell her story. First, they have her tell the jury about her education and where she grew up and her professional experience. And because of Stormy Daniels’s biography, that quickly goes into stripping, and then goes into making adult films.

And I thought the prosecutor who questioned her, Susan Hoffinger, had this nice touch in talking about that, because not only did she ask Daniels about acting in adult films. But she asked her about writing and directing them, too, emphasizing the more professional aspects of that work and giving a little more credit to the witness, as if to say, well, you may think this or you may think that. But this is a person with dignity who took what she did seriously. Got it.

What’s your first impression of Daniels as a witness?

It’s very clear that she’s nervous. She’s speaking fast. She’s laughing to herself and making small jokes. But the tension in the room is so serious from the beginning, from the moment she enters, that those jokes aren’t landing. So it just feels, like, really heavy and still and almost oppressive in there. So Daniels talking quickly, seeming nervous, giving more answers than are being asked of her by the prosecution, even before we get to the sexual encounter that she’s about to describe, all of that presents a really discomfiting impression, I would say.

And how does this move towards the encounter that Daniels ultimately has?

It starts at a golf tournament in 2006, in Lake Tahoe, Nevada. Daniels meets Trump there. There are other celebrities there, too. They chatted very briefly. And then she received a dinner invitation from him. She thought it over, she says. And she goes to have dinner with Trump, not at a restaurant, by the way. But she’s invited to join him in the hotel suite.

So she gets to the hotel suite. And his bodyguard is there. And the hotel door is cracked open. And the bodyguard greets her and says she looks nice, this and that. And she goes in. And there’s Donald Trump, just as expected. But what’s not expected, she says, is that he’s not wearing what you would wear to a dinner with a stranger, but instead, she says, silk or satin pajamas. She asked him to change, she says. And he obliges.

He goes, and he puts on a dress shirt and dress pants. And they sit down at the hotel suite’s dining room table. And they have a kind of bizarre dinner. Trump is asking her very personal questions about pornography and safe sex. And she testifies that she teased him about vain and pompous he is. And then at some point, she goes to the bathroom. And she sees that he has got his toiletries in there, his Old Spice, his gold tweezers.

Very specific details.

Yeah, we’re getting a ton of detail in this scene. And the reason we’re getting those is because prosecutors are trying to elicit those details to establish that this is a credible person, that this thing did happen, despite what Donald Trump and his lawyers say. And the reason you can know it happened, prosecutors seem to be saying, is because, look at all these details she can still summon up.

She comes out of the bathroom. And she says that Donald Trump is on the hotel bed. And what stands out to me there is what she describes as a very intense physical reaction. She says that she blacked out. And she quickly clarifies, she doesn’t mean from drugs or alcohol. She means that, she says, that the intensity of this experience was such that, suddenly, she can’t remember every detail. The prosecution asks a question that cuts directly to the sex. Essentially, did you start having sex with him? And Daniels says that she did. And she continues to provide more details than even, I think, the prosecution wanted.

And I think we don’t want to go chapter and verse through this claimed sexual encounter. But I wonder what details stand out and which details feel important, given the prosecution’s strategy here.

All the details stand out because it’s a story about having had sex with a former president. And the more salacious and more private the details feel, the more you’re going to remember them. So we’ll remember that Stormy Daniels said what position they had sex in. We’ll remember that she said he didn’t use a condom. Whether that’s important to the prosecution’s case, now, that’s a much harder question to answer, as we’ve been saying.

But what I can tell you is, as she’s describing having had sex with Donald Trump, and Donald Trump is sitting right there, and Eric Trump, his son, is sitting behind him, seeming to turn a different color as he hears this embarrassment of his father being described to a courtroom full of reporters at this trial, it’s hard to even describe the energy in that room. It was like nothing I had ever experienced. And it was just Daniels’s testimony and, seemingly, the former President’s emotions. And you almost felt like you were trapped in there with both of them as this description was happening.

Well, I think it’s important to try to understand why the prosecution is getting these details, these salacious, carnal, pick your word, graphic details about sex with Donald Trump. What is the value, if other details are clearly making the point that she’s recollecting something?

Well, I think, at this point, we can only speculate. But one thing we can say is, this was uncomfortable. This felt bad. And remember, prosecutor’s story is not about the sex. It’s about trying to hide the sex. So if you’re trying to show a jury why it might be worthwhile to hide a story, it might be worth —

Providing lots of salacious details that a person would want to hide.

— exposing them to how bad that story feels and reminding them that if they had been voters and they had heard that story, and, in fact, they asked Daniels this very question, if you hadn’t accepted hush money, if you hadn’t signed that NDA, is this the story you would have told? And she said, yes. And so where I think they’re going with this, but we can’t really be sure yet, is that they’re going to tell the jurors, hey, that story, you can see why he wanted to cover that up, can’t you?

You mentioned the hush money payments. What testimony does Daniels offer about that? And how does it advance the prosecution’s case of business fraud related to the hush money payments?

So little evidence that it’s almost laughable. She says that she received the hush money. But we actually already heard another witness, her lawyer at the time, Keith Davidson, testify that he had received the hush money payment on her behalf. And she testified about feeling as if she had to sell this story because the election was fast approaching, almost as if her leverage was slipping away because she knew this would be bad for Trump.

That feels important. But just help me understand why it’s important.

Well, what the prosecution has been arguing is that Trump covered up this hush money payment in order to conceal a different crime. And that crime, they say, was to promote his election to the presidency by illegal means.

Right, we’ve talked about this in the past.

So when Daniels ties her side of the payment into the election, it just reminds the jurors maybe, oh, right, this is what they’re arguing.

So how does the prosecution end this very dramatic, and from everything you’re saying, very tense questioning of Stormy Daniels about this encounter?

Well, before they can even end, the defense lawyers go and they consult among themselves. And then, with the jury out of the room, one of them stands up. And he says that the defense is moving for a mistrial.

On what terms?

He says that the testimony offered by Daniels that morning is so prejudicial, so damning to Trump in the eyes of the jury, that the trial can no longer be fair. Like, how could these jurors have heard these details and still be fair when they render their verdict? And he says a memorable expression. He says, you can’t un-ring that bell, meaning they heard it. They can’t un-hear it. It’s over. Throw out this trial. It should be done.

Wow. And what is the response from the judge?

So the judge, Juan Merchan, he hears them out. And he really hears them out. But at the end of their arguments, he says, I do think she went a little too far. He says that. He said, there were things that were better left unsaid.

By Stormy Daniels?

By Stormy Daniels. And he acknowledges that she is a difficult witness. But, he says, the remedy for that is not a mistrial, is not stopping the whole thing right now. The remedy for that is cross-examination. If the defense feels that there are issues with her story, issues with her credibility, they can ask her whatever they want. They can try to win the jury back over. If they think this jury has been poisoned by this witness, well, this is their time to provide the antidote. The antidote is cross-examination. And soon enough, cross-examination starts. And it is exactly as intense and combative as we expected.

We’ll be right back.

So, Jonah, how would you characterize the defense’s overall strategy in this intense cross-examination of Stormy Daniels?

People know the word impeach from presidential impeachments. But it has a meaning in law, too. You impeach a witness, and, specifically, their credibility. And that’s what the defense is going for here. They are going to try to make Stormy Daniels look like a liar, a fraud, an extortionist, a money-grubbing opportunist who wanted to take advantage of Trump and sought to do so by any means necessary.

And what did that impeachment strategy look like in the courtroom?

The defense lawyer who questions Stormy Daniels is a woman named Susan Necheles. She’s defended Trump before. And she’s a bit of a cross-examination specialist. We even saw her during jury selection bring up these past details to confront jurors who had said nasty things about Trump on social media with. And she wants to do the same thing with Daniels. She wants to bring up old interviews and old tweets and things that Daniels has said in the past that don’t match what Daniels is saying from the stand.

What’s a specific example? And do they land?

Some of them land. And some of them don’t. One specific example is that Necheles confronts Daniels with this old tweet, where Daniels says that she’s going to dance down the street if Trump goes to jail. And what she’s trying to show there is that Daniels is out for revenge, that she hates Trump, and that she wants to see him go to jail. And that’s why she’s testifying against him.

And Daniels is very interesting during the cross-examination. It’s almost as if she’s a different person. She kind of squares her shoulders. And she sits up a little straighter. And she leans forward. Daniels is ready to fight. But it doesn’t quite land. The tweet actually says, I’ll dance down the street when he’s selected to go to jail.

And Daniels goes off on this digression about how she knows that people don’t get selected to go to jail. That’s not how it works. But she can’t really unseat this argument, that she’s a political enemy of Donald Trump. So that one kind of sticks, I would say. But there are other moves that Necheles tries to pull that don’t stick.

So unlike the prosecution, which typically used words like adult, adult film, Necheles seems to be taking every chance she can get to say porn, or pornography, or porn star, to make it sound base or dirty. And so when she starts to ask Daniels about actually being in pornography, writing, acting, and directing sex films, she tries to land a punch line, Necheles does. She says, so you have a lot of experience making phony stories about sex appear to be real, right?

As if to say, perhaps this story you have told about entering Trump’s suite in Lake Tahoe and having sex with him was made up.

Just another one of your fictional stories about sex. But Daniels comes back and says, the sex in the films, it’s very much real, just like what happened to me in that room. And so, when you have this kind of combat of a lawyer cross-examining very aggressively and the witness fighting back, you can feel the energy in the room shift as one lands a blow or the other does. But here, Daniels lands one back. And the other issue that I think Susan Necheles runs into is, she tries to draw out disparities from interviews that Daniels gave, particularly to N-TOUCH, very early on once the story was out.

It’s kind of like a tabloid magazine?

But some of the disparities don’t seem to be landing quite like Necheles would want. So she tries to do this complicated thing about where the bodyguard was in the room when Daniels walked into the room, as described in an interview in a magazine. But in that magazine interview, as it turns out, Daniels mentioned that Trump was wearing pajamas. And so, if I’m a juror, I don’t care where the bodyguard is. I’m thinking about, oh, yeah, I remember that Stormy Daniels said now in 2024 that Trump was wearing pajamas.

I’m curious if, as somebody in the room, you felt that the defense was effective in undermining Stormy Daniels’s credibility? Because what I took from the earlier part of our conversation was that Stormy Daniels is in this courtroom on behalf of the prosecution to tell a story that’s uncomfortable and has the kind of details that Donald Trump would be motivated to try to hide. And therefore, this defense strategy is to say, those details about what Trump might want to hide, you can’t trust them. So does this back and forth effectively hurt Stormy Daniels’s credibility, in your estimation?

I don’t think that Stormy Daniels came off as perfectly credible about everything she testified about. There are incidents that were unclear or confusing. There were things she talked about that I found hard to believe, when she, for instance, denied that she had attacked Trump in a tweet or talked about her motivations. But about what prosecutors need, that central story, the story of having had sex with him, we can’t know whether it happened.

But there weren’t that many disparities in these accounts over the years. In terms of things that would make me doubt the story that Daniels was telling, details that don’t add up, those weren’t present. And you don’t have to take my word for that, nor should you. But the judge is in the room. And he says something very, very similar.

What does he say? And why does he say it?

Well, he does it when the defense, again, at the end of the day on Thursday, calls for a mistrial.

With a similar argument as before?

Not only with a similar argument as before, but, like, almost the exact same argument. And I would say that I was astonished to see them do this. But I wasn’t because I’ve covered other trials where Trump is the client. And in those trials, the lawyers, again and again, called for a mistrial.

And what does Judge Marchan say in response to this second effort to seek a mistrial?

Let me say, to this one, he seems a little less patient. He says that after the first mistrial ruling, two days before, he went into his chambers. And he read every decision he had made about the case. He took this moment to reflect on the first decision. And he found that he had, in his own estimation, which is all he has, been fair and not allowed evidence that was prejudicial to Trump into this trial. It could continue. And so he said that again. And then he really almost turned on the defense. And he said that the things that the defense was objecting to were things that the defense had made happen.

He says that in their opening statement, the defense could have taken issue with many elements of the case, about whether there were falsified business records, about any of the other things that prosecutors are saying happened. But instead, he says, they focused their energy on denying that Trump ever had sex with Daniels.

And so that was essentially an invitation to the prosecution to call Stormy Daniels as a witness and have her say from the stand, yes, I had this sexual encounter. The upshot of it is that the judge not only takes the defense to task. But he also just says that he finds Stormy Daniels’s narrative credible. He doesn’t see it as having changed so much from year to year.

Interesting. So in thinking back to our original question here, Jonah, about the idea that putting Stormy Daniels on the stand was risky, I wonder if, by the end of this entire journey, you’re reevaluating that idea because it doesn’t sound like it ended up being super risky. It sounded like it ended up working reasonably well for the prosecution.

Well, let me just assert that it doesn’t really matter what I think. The jury is going to decide this. There’s 12 people. And we can’t know what they’re thinking. But my impression was that, while she was being questioned by the prosecution for the prosecution’s case, Stormy Daniels was a real liability. She was a difficult witness for them.

And the judge said as much. But when the defense cross-examined her, Stormy Daniels became a better witness, in part because their struggles to discredit her may have actually ended up making her story look more credible and stronger. And the reason that matters is because, remember, we said that prosecutors are trying to fill this hole in their case. Well, now, they have. The jury has met Stormy Daniels. They’ve heard her account. They’ve made of it what they will. And now, the sequence of events that prosecutors are trying to line up as they seek prison time for the former President really makes a lot of sense.

It starts with what Stormy Daniels says with sex in a hotel suite in 2006. It picks up years later, as Donald Trump is trying to win an election and, prosecutors say, suppressing negative stories, including Stormy Daniels’s very negative story. And the story that prosecutors are telling ends with Donald Trump orchestrating the falsification of business records to keep that story concealed.

Well, Jonah, thank you very much. We appreciate it.

Of course, thanks for having me.

The prosecution’s next major witness will be Michael Cohen, the former Trump fixer who arranged for the hush money payment to Stormy Daniels. Cohen is expected to take the stand on Monday.

Here’s what else you need to know today. On Thursday, Israeli Prime Minister Benjamin Netanyahu issued a defiant response to warnings from the United States that it would stop supplying weapons to Israel if Israel invades the Southern Gaza City of Rafah. So far, Israel has carried out a limited incursion into the city where a million civilians are sheltering, but has threatened a full invasion. In a statement, Netanyahu said, quote, “if we need to stand alone, we will stand alone.”

Meanwhile, high level ceasefire negotiations between Israel and Hamas have been put on hold in part because of anger over Israel’s incursion into Rafah.

A reminder, tomorrow, we’ll be sharing the latest episode of our colleague’s new show, “The Interview” This week on “The Interview,” Lulu Garcia-Navarro talks with radio host Charlamagne Tha God about his frustrations with how Americans talk about politics.

If me as a Black man, if I criticize Democrats, then I’m supporting MAGA. But if I criticize, you know, Donald Trump and Republicans, then I’m a Democratic shill. Why can’t I just be a person who deals in nuance?

Today’s episode was produced by Olivia Natt and Michael Simon Johnson. It was edited by Lexie Diao, with help from Paige Cowett, contains original music by Will Reid and Marion Lozano, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Michael Barbaro. See you on Monday.

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• May 7, 2024   •   27:43 How Changing Ocean Temperatures Could Upend Life on Earth
• May 6, 2024   •   29:23 R.F.K. Jr.’s Battle to Get on the Ballot
• May 3, 2024   •   25:33 The Protesters and the President
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Hosted by Michael Barbaro

Featuring Jonah E. Bromwich

Produced by Olivia Natt and Michael Simon Johnson

Edited by Lexie Diao

With Paige Cowett

Original music by Will Reid and Marion Lozano

Engineered by Alyssa Moxley

Listen and follow The Daily Apple Podcasts | Spotify | Amazon Music | YouTube

This episode contains descriptions of an alleged sexual liaison.

What happened when Stormy Daniels took the stand for eight hours in the first criminal trial of former President Donald J. Trump?

Jonah Bromwich, one of the lead reporters covering the trial for The Times, was in the room.

On today’s episode

Jonah E. Bromwich , who covers criminal justice in New York for The New York Times.

Background reading

In a second day of cross-examination, Stormy Daniels resisted the implication she had tried to shake down Donald J. Trump by selling her story of a sexual liaison.

Here are six takeaways from Ms. Daniels’s earlier testimony.

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The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Renan Borelli, Maddy Masiello, Isabella Anderson and Nina Lassam.

Jonah E. Bromwich covers criminal justice in New York, with a focus on the Manhattan district attorney’s office and state criminal courts in Manhattan. More about Jonah E. Bromwich

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