phd level mathematics

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.

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Department of Mathematics

Requirements for the ph.d. degree.

In order to qualify for the Mathematics Ph.D., all students are required to:

• Complete eight term courses at the graduate level, at least two with Honors grades.
• Pass qualifying examinations on their general mathematical knowledge;
• Submit a dissertation prospectus;
• Participate in the instruction of undergraduates ;
• Be in residence for at least three years;
• Complete a dissertation that clearly advances understanding of the subject it considers.

All students must also complete any other  Graduate School of Arts and Sciences degree requirements  as they appear in the Programs and Policies bulletin.

The normal time for completion of the Ph.D. program is five to six years. Requirement (1) normally includes basic courses in algebra, analysis, and topology.  Students typically complete the eight-course requirement by the end of their third year.  The Honors grades of (1) must be achieved within the first two years.  A sequence of three qualifying examinations (algebra and number theory, real and complex analysis, topology) is offered each term.  All qualifying examinations must be passed by the end of the second year.  There is no limit to the number of times that students can take the exams, and so they are encouraged to take them as soon as possible.

The dissertation prospectus should be submitted during the third year. 

The thesis is expected to be independent work, done under the guidance of an adviser. This adviser should be contacted not long after the student passes the qualifying examinations. A student is admitted to candidacy after completing requirements (1)–(5) and obtaining an adviser.

In addition to all other requirements, students must successfully complete MATH 991a, Ethical Conduct of Research, prior to the end of their first year of study. This requirement must be met prior to registering for a second year of study.

Master’s Degrees :

The M.Phil. and M.S. degrees are conferred only en route to the Ph.D.; there is no separate master’s program in Mathematics.

M.Phil.   Please refer to the Graduate School Degree Requirements

M.S.   A student must complete six term courses with at least one Honors grade, perform adequately on the general qualifying examination, and be in residence at least one year.

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.

Graduate Program

Our graduate program is unique from the other top mathematics institutions in the U.S. in that it emphasizes, from the start, independent research. Each year, we have extremely motivated and talented students among our new Ph.D. candidates who, we are proud to say, will become the next generation of leading researchers in their fields. While we urge independent work and research, there exists a real sense of camaraderie among our graduate students. As a result, the atmosphere created is one of excitement and stimulation as well as of mentoring and support. Furthermore, there exists a strong scholarly relationship between the Math Department and the Institute for Advanced Study, located just a short distance from campus, where students can make contact with members there as well as attend the IAS seminar series.  Our program has minimal requirements and maximal research and educational opportunities. We offer a broad variety of advanced research topics courses as well as more introductory level courses in algebra, analysis, and geometry, which help first-year students strengthen their mathematical background and get involved with faculty through basic course work. In addition to the courses, there are several informal seminars specifically geared toward graduate students: (1) Colloquium Lunch Talk, where experts who have been invited to present at the Department Colloquium give introductory talks, which allows graduate students to understand the afternoon colloquium more easily; (2) Graduate Student Seminar (GSS), which is organized and presented by graduate students for graduate students, creating a vibrant mathematical interaction among them; and, (3) What’s Happening in Fine Hall (WHIFH) seminar where faculty give talks in their own research areas specifically geared towards graduate students. Working or reading seminars in various research fields are also organized by graduate students each semester. First-year students are set on the fast track of research by choosing two advanced topics of research, beyond having a strong knowledge of three more general subjects: algebra, and real and complex analysis, as part of the required General Examination. It is the hope that one, or both, of the advanced topics will lead to the further discovery of a thesis problem. Students are expected to write a thesis in four years but will be provided an additional year to complete their work if deemed necessary. Most of our Ph.D.'s are successfully launched into academic positions at premier mathematical institutions as well as in industry .

Chenyang Xu

Jill leclair.

PhD Qualifying Exams

The requirements for the PhD program in Mathematics have changed for students who enter the program starting in Autumn 2023 and later. 

Requirements for the Qualifying Exams

Students who entered the program prior to autumn 2023.

To qualify for the Ph.D. in Mathematics, students must pass two examinations: one in algebra and one in real analysis. 

Students who entered the program in Autumn 2023 or later

To qualify for the Ph.D. in Mathematics, students must choose and pass examinations in two of the following four areas: 

• real analysis
• geometry and topology
• applied mathematics

The exams each consist of two parts. Students are given three hours for each part.

Topics Covered on the Exams:

• Algebra Syllabus
• Real Analysis Syllabus
• Geometry and Topology Syllabus
• Applied Mathematics Syllabus

Check out some Past and Practice Qualifying Exams to assist your studying.

Because some students have already taken graduate courses as undergraduates, incoming graduate students are allowed to take either or both of the exams in the autumn. If they pass either or both of the exams, they thereby fulfill the requirement in those subjects. However, they are in no way penalized for failing either of the exams.

Students must pass both qualifying exams by the autumn of their second year. Ordinarily first-year students take courses in algebra and real analysis throughout the year to prepare them for the exams. The exams are then taken at the beginning of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Students who started in Autumn 2023 and later

Students must choose and pass two out of the four qualifying exams by the autumn of their second year. Students take courses in algebra, real analysis, geometry and topology, and applied math in the autumn and winter quarters of their first year to prepare them for the exams. The exams are taken during the first week of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Exam Schedule

Unless otherwise noted, the exams will be held each year according to the following schedule:

Autumn Quarter:  The exams are held during the week prior to the first week of the quarter. Spring Quarter:  The exams are held during the first week of the quarter.

The exams are held over two three-hour blocks. The morning block is 9:30am-12:30pm and the afternoon block is 2:00-5:00pm.

For the start date of the current or future years’ quarters please see the  Academic Calendar

Upcoming Exam Dates

Spring 2024.

The exams will be held on the following dates:

Monday, April 1st

Analysis Exam, Room 384H

Wednesday, April 3rd

Algebra Exam, Room 384I

Thursday, April 4th 

Geometry & Topology Exam, Room 384I

Friday, April 5th

Applied Math Exam, Room 384I

The department offers programs covering a broad range of topics leading to the Doctor of Philosophy and the Doctor of Science degrees (the student chooses which to receive; they are functionally equivalent). Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests in both groups. Of the roughly 120 Ph.D. students, about 2/3 are in Pure Mathematics, 1/3 in Applied Mathematics.

The two programs in Pure and Applied Mathematics offer basic and advanced classes in analysis, algebra, geometry, Lie theory, logic, number theory, probability, statistics, topology, astrophysics, combinatorics, fluid dynamics, numerical analysis, mathematics of data, and the theory of computation. In addition, many mathematically-oriented courses are offered by other departments. Students in Applied Mathematics are especially encouraged to take courses in engineering and scientific subjects related to their research.

All students pursue research under the supervision of the faculty , and are encouraged to take advantage of the many seminars and colloquia at MIT and in the Boston area.

Degree Requirements

Degree requirements consist of:

• Oral qualifying exam
• Classroom teaching
• Original thesis and defense

Prospective students are invited to consult the graduate career timeline for more information, and to read about the application procedure .

Graduate Co-Chairs

Graduate Student Issues, math graduate admissions

Jonathan Kelner , Davesh Maulik , and Zhiwei Yun

Mathematics

About the Program

The Department of Mathematics offers both a PhD program in Mathematics and Applied Mathematics.

Students are admitted for specific degree programs: the PhD in Mathematics or PhD in Applied Mathematics. Requirements for the Mathematics and Applied Mathematics PhDs differ only in minor respects, and no distinction is made between the two in day-to-day matters. Graduate students typically take 5-6 years to complete the doctorate.

Continuing students wishing to transfer from one program to another should consult the graduate advisor in 910 Evans Hall. Transfers between the two PhD programs are fairly routine, but must be done prior to taking the qualifying examination. It is a formal policy of the department that an applicant to the PhD program who has previous graduate work in mathematics must present very strong evidence of capability for mathematical research.

Students seeking to transfer to the department's PhD programs from other campus programs, including the Group in Logic and the Methodology of Science, must formally apply and should consult the Vice Chair for Graduate Studies.

Visit Department Website

Admission to the University

Applying for graduate admission.

Thank you for considering UC Berkeley for graduate study! UC Berkeley offers more than 120 graduate programs representing the breadth and depth of interdisciplinary scholarship. The Graduate Division hosts a complete list of graduate academic programs, departments, degrees offered, and application deadlines can be found on the Graduate Division website.

Prospective students must submit an online application to be considered for admission, in addition to any supplemental materials specific to the program for which they are applying. The online application and steps to take to apply can be found on the Graduate Division website .

Admission Requirements

The minimum graduate admission requirements are:

A bachelor’s degree or recognized equivalent from an accredited institution;

A satisfactory scholastic average, usually a minimum grade-point average (GPA) of 3.0 (B) on a 4.0 scale; and

Enough undergraduate training to do graduate work in your chosen field.

For a list of requirements to complete your graduate application, please see the Graduate Division’s Admissions Requirements page . It is also important to check with the program or department of interest, as they may have additional requirements specific to their program of study and degree. Department contact information can be found here .

Where to apply?

Visit the Berkeley Graduate Division application page .

Admission to the Program

Undergraduate students also often take one or more of the following introductory Mathematics graduate courses:

The Math Department admits new graduate students to the fall semester only. The Graduate Division's Online Application will be available in early September at:  http://grad.berkeley.edu/admissions/index.shtml . Please read the information on Graduate Division requirements and information required to complete the application.

Copies of official or unofficial transcripts may be uploaded to your application. Please do  not  mail original transcripts for the review process. 

We require three letters of recommendation, which should be submitted online. Please do  not  mail letters of recommendation for the review process. 

For more information, please review the department's graduate admissions webpage at:  https://math.berkeley.edu/programs/graduate/admissions . We also recommend reviewing our admissions FAQs page at:  https://math.berkeley.edu/programs/graduate/faqs . 

Doctoral Degree Requirements

Prerequisites

The Department of Mathematics offers two PhD degrees, one in Mathematics and one in Applied Mathematics. Applicants for admission to either PhD program are expected to have preparation comparable to the undergraduate major at Berkeley in Mathematics or in Applied Mathematics. These majors consist of two full years of lower division work (covering calculus, linear algebra, differential equations, and multivariable calculus), followed by eight one-semester courses including real analysis, complex analysis, abstract algebra, and linear algebra. These eight courses may include some mathematically based courses in other departments, like physics, engineering, computer science, or economics.

Applicants for admission are considered by the department's Graduate Admissions and M.O.C. Committees. The number of students that can be admitted each year is determined by the Graduate Division and by departmental resources. In making admissions decisions, the committee conducts a comprehensive review of applicants considering broader community impacts, academic performance in mathematics courses, level of mathematical preparation, letters of recommendation, and GRE scores.

Degree Requirements

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

• take at least four courses, two or more of which are graduate courses in mathematics;
• and pass the six-hour written preliminary examination covering primarily undergraduate material. (The exam is given just before the beginning of each semester, and the student must pass it within their first three semesters.)
• 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 offered by the Math department, giving a talk of at least one hour duration.  Research presentations held at Mathematical Sciences Research Institute (MSRI), or Lawrence Berkeley National Lab (LBNL) are also acceptable. A Math Department faculty member must be present at the talk and sign the seminar form confirming.
• 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.

The detailed regulations of the PhD program are as follows:

Course Requirements Students must take and pass at least four 4-unit courses during the first year of the Ph.D. program; at least two courses per semester. At minimum, two of these courses must be graduate courses (200-level) offered by the Department of Mathematics . Two upper division (100-level) undergraduate courses offered by the Department of Mathematics may also be used toward this requirement. Exceptions may also be considered and must be reviewed by the Head Graduate Advisor for approval.

Preliminary Examination The preliminary examination consists of six hours of written work given over a two-day period. Most of the examination covers material, mainly in analysis and algebra, and helps to identify gaps in preparation. The preliminary examination is offered twice a year—during the week before classes start in both the fall and spring semesters. A student may repeat the examination twice. A student who does not pass the preliminary examination within 13 months of the date of entry into the PhD program will not be permitted to remain in the program past the third semester. In exceptional cases, a fourth try may be granted upon appeal to committee omega.

Qualifying Examination To arrange for the qualifying examination, a student must first settle on an area of concentration, and a prospective dissertation supervisor, someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective supervisor, the student forms an examination committee of four members. Committee members must be members of Berkeley's Academic Senate and the Chair must be a faculty member in the Mathematics Department. 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 sections. The qualifying examination must cover material falling in at least three 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 seen on the Qualifying Examination page on the department website. 

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, if asked. The student must obtain an official dissertation supervisor within one semester after passing the qualifying examination or leave the PhD program. For more detailed rules and advice concerning the qualifying examination, consult the graduate advisor in  910 Evans Hall.

Master's Degree Requirements

Eligibility .

At this time, the MA in Mathematics is a simultaneous degree program only offered to students currently enrolled in a doctoral program at UC Berkeley. The doctoral student must be in good standing in their program and have a faculty adviser in the Mathematics Department who is supportive of the addition of the MA in Mathematics and agrees to supervise the MA work. Current doctoral students must apply during the regular admissions cycle for consideration for fall admission. The degree must be completed prior to or in tandem with the PhD degree. Interested students must inquire with the Mathematics Graduate Student Affairs Officer. 

Unit Requirements

Plan I requires at least 20 semester units of upper division and graduate courses and a thesis. At least 8 of these units must be in graduate courses (200 series). These 8 units are normally taken in the Department of Mathematics at Berkeley. In special cases, upon recommendation of the Graduate Adviser and approval of the Dean of the Graduate Division, some of the 8 graduate units may be taken in other departments.

Plan II requires at least 24 semester units of upper division and graduate courses, followed by a comprehensive final examination, the MA examination. At least 12 of these units must be in graduate courses (200 series). These 12 units are normally taken in the Department of Mathematics at Berkeley. In special cases, upon recommendation of the graduate advisor and approval of the dean of the Graduate Division, some of the 12 graduate units may be taken in other departments. All courses fulfilling the above unit requirements must have significant mathematical content. In general, MA students are encouraged to take some courses outside the Department of Mathematics. In many jobs, at least some acquaintance with statistics and computer science is essential; and, for some students, courses in such fields as engineering, biological or physical sciences, or economics are highly desirable.

A breadth requirement consisting of at least one course in each of three fields must be met by all students. Fields include algebra, analysis, geometry, foundations, history of mathematics, numerical analysis, probability and statistics, computer science, and various other fields of applied mathematics. The last category specifically covers courses in a variety of departments, and the graduate adviser may allow more than one such course to count toward the breadth requirement. A depth requirement consisting of a coherent program of three courses all in one of the above fields, at least two of these courses being at the graduate level, must be met. Students interested in a field of applied mathematics are encouraged to take some of these courses outside the department.

• Advancement to Candidacy
• Thesis (Plan I)
• Capstone/Comprehensive Exam (Plan II)
• Capstone/Master's Project (Plan II)

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Terms offered: Fall 2024, Fall 2023, Fall 2022 Metric spaces and general topological spaces. Compactness and connectedness. Characterization of compact metric spaces. Theorems of Tychonoff, Urysohn, Tietze. Complete spaces and the Baire category theorem. Function spaces; Arzela-Ascoli and Stone-Weierstrass theorems. Partitions of unity. Locally compact spaces; one-point compactification. Introduction to measure and integration. Sigma algebras of sets. Measures and outer measures. Lebesgue measure on the line and Rn. Construction of the integral. Dominated convergence theorem. Introduction to Topology and Analysis: Read More [+]

Rules & Requirements

Prerequisites: 104

Hours & Format

Fall and/or spring: 15 weeks - 3 hours of lecture per week

Additional Format: Three hours of Lecture per week for 15 weeks.

Additional Details

Subject/Course Level: Mathematics/Graduate

Grading: Letter grade.

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Prerequisites: 202A and 110

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Prerequisites: 185

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Prerequisites: 202A-202B

Additional Format: Three hours of lecture per week.

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Prerequisites: 206

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Prerequisites: 185 and 202A-202B or their equivalents

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MATH 215A Algebraic Topology 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Fundamental group and covering spaces, simplicial and singular homology theory with applications, cohomology theory, duality theorem. Homotopy theory, fibrations, relations between homotopy and homology, obstruction theory, and topics from spectral sequences, cohomology operations, and characteristic classes. Sequence begins fall. Algebraic Topology: Read More [+]

Prerequisites: 113 and point-set topology (e.g. 202A)

Instructors: 113C, 202A, and 214

Algebraic Topology: Read Less [-]

MATH 215B Algebraic Topology 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Fundamental group and covering spaces, simplicial and singular homology theory with applications, cohomology theory, duality theorem. Homotopy theory, fibrations, relations between homotopy and homology, obstruction theory, and topics from spectral sequences, cohomology operations, and characteristic classes. Sequence begins fall. Algebraic Topology: Read More [+]

Prerequisites: 215A, 214 recommended (can be taken concurrently)

MATH C218A Probability Theory 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 The course is designed as a sequence with Statistics C205B/Mathematics C218B with the following combined syllabus. Measure theory concepts needed for probability. Expection, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations, martingales and martingale convergence theorems. Markov chains. Stationary processes. Brownian motion. Probability Theory: Read More [+]

Also listed as: STAT C205A

Probability Theory: Read Less [-]

MATH C218B Probability Theory 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 The course is designed as a sequence with with Statistics C205A/Mathematics C218A with the following combined syllabus. Measure theory concepts needed for probability. Expection, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations, martingales and martingale convergence theorems. Markov chains. Stationary processes. Brownian motion. Probability Theory: Read More [+]

Also listed as: STAT C205B

MATH 219 Dynamical Systems 4 Units

Terms offered: Fall 2024, Fall 2023, Spring 2022 Diffeomorphisms and flows on manifolds. Ergodic theory. Stable manifolds, generic properties, structural stability. Additional topics selected by the instructor. Dynamical Systems: Read More [+]

Prerequisites: 214

Dynamical Systems: Read Less [-]

MATH 220 Introduction to Probabilistic Methods in Mathematics and the Sciences 4 Units

Terms offered: Spring 2012, Spring 2011, Spring 2010 Brownian motion, Langevin and Fokker-Planck equations, path integrals and Feynman diagrams, time series, an introduction to statistical mechanics, Monte Carlo methods, selected applications. Introduction to Probabilistic Methods in Mathematics and the Sciences: Read More [+]

Prerequisites: Some familiarity with differential equations and their applications

Introduction to Probabilistic Methods in Mathematics and the Sciences: Read Less [-]

MATH 221 Advanced Matrix Computations 4 Units

Terms offered: Fall 2024, Fall 2023, Spring 2022 Direct solution of linear systems, including large sparse systems: error bounds, iteration methods, least square approximation, eigenvalues and eigenvectors of matrices, nonlinear equations, and minimization of functions. Advanced Matrix Computations: Read More [+]

Prerequisites: Consent of instructor

Summer: 8 weeks - 6 hours of lecture per week

Additional Format: Three hours of Lecture per week for 15 weeks. Six hours of Lecture per week for 8 weeks.

Advanced Matrix Computations: Read Less [-]

MATH 222A Partial Differential Equations 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Laplace's equation, heat equation, wave equation, nonlinear first-order equations, conservation laws, Hamilton-Jacobi equations, Fourier transform, Sobolev spaces. Partial Differential Equations: Read More [+]

Prerequisites: 105 or 202A

Partial Differential Equations: Read Less [-]

MATH 222B Partial Differential Equations 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Second-order elliptic equations, parabolic and hyperbolic equations, calculus of variations methods, additional topics selected by instructor. Partial Differential Equations: Read More [+]

MATH C223A Advanced Topics in Probability and Stochastic Process 3 Units

Terms offered: Fall 2024, Fall 2020, Fall 2016, Fall 2014 The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probability offered according to students demand and faculty availability. Advanced Topics in Probability and Stochastic Process: Read More [+]

Prerequisites: Statistics C205A-C205B or consent of instructor

Repeat rules: Course may be repeated for credit with instructor consent.

Also listed as: STAT C206A

Advanced Topics in Probability and Stochastic Process: Read Less [-]

MATH C223B Advanced Topics in Probability and Stochastic Processes 3 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probability offered according to students demand and faculty availability. Advanced Topics in Probability and Stochastic Processes: Read More [+]

Also listed as: STAT C206B

Advanced Topics in Probability and Stochastic Processes: Read Less [-]

MATH 224A Mathematical Methods for the Physical Sciences 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green's function. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves. Sequence begins fall. Mathematical Methods for the Physical Sciences: Read More [+]

Prerequisites: Graduate status or consent of instructor

Instructors: 112 or 113C; 104A and 185, or 121A-121B-121C, or 120A-120B-120C.

Mathematical Methods for the Physical Sciences: Read Less [-]

MATH 224B Mathematical Methods for the Physical Sciences 4 Units

Terms offered: Spring 2015, Spring 2014, Spring 2013 Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green's function. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves. Sequence begins fall. Mathematical Methods for the Physical Sciences: Read More [+]

MATH 225A Metamathematics 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Metamathematics of predicate logic. Completeness and compactness theorems. Interpolation theorem, definability, theory of models. Metamathematics of number theory, recursive functions, applications to truth and provability. Undecidable theories. Sequence begins fall. Metamathematics: Read More [+]

Prerequisites: 125A and (135 or 136)

Metamathematics: Read Less [-]

MATH 225B Metamathematics 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Metamathematics of predicate logic. Completeness and compactness theorems. Interpolation theorem, definability, theory of models. Metamathematics of number theory, recursive functions, applications to truth and provability. Undecidable theories. Sequence begins fall. Metamathematics: Read More [+]

MATH 227A Theory of Recursive Functions 4 Units

Terms offered: Spring 2021, Fall 2015, Fall 2013 Recursive and recursively enumerable sets of natural numbers; characterizations, significance, and classification. Relativization, degrees of unsolvability. The recursion theorem. Constructive ordinals, the hyperarithmetical and analytical hierarchies. Recursive objects of higher type. Sequence begins fall. Theory of Recursive Functions: Read More [+]

Prerequisites: Mathematics 225B

Instructor: 225C.

Theory of Recursive Functions: Read Less [-]

MATH 228A Numerical Solution of Differential Equations 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Ordinary differential equations: Runge-Kutta and predictor-corrector methods; stability theory, Richardson extrapolation, stiff equations, boundary value problems. Partial differential equations: stability, accuracy and convergence, Von Neumann and CFL conditions, finite difference solutions of hyperbolic and parabolic equations. Finite differences and finite element solution of elliptic equations. Numerical Solution of Differential Equations: Read More [+]

Prerequisites: 128A

Instructor: 128A-128B.

Numerical Solution of Differential Equations: Read Less [-]

MATH 228B Numerical Solution of Differential Equations 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Ordinary differential equations: Runge-Kutta and predictor-corrector methods; stability theory, Richardson extrapolation, stiff equations, boundary value problems. Partial differential equations: stability, accuracy and convergence, Von Neumann and CFL conditions, finite difference solutions of hyperbolic and parabolic equations. Finite differences and finite element solution of elliptic equations. Numerical Solution of Differential Equations: Read More [+]

MATH 229 Theory of Models 4 Units

Terms offered: Spring 2019, Spring 2015, Spring 2013 Syntactical characterization of classes closed under algebraic operations. Ultraproducts and ultralimits, saturated models. Methods for establishing decidability and completeness. Model theory of various languages richer than first-order. Theory of Models: Read More [+]

Prerequisites: 225B

Theory of Models: Read Less [-]

MATH 235A Theory of Sets 4 Units

Terms offered: Fall 2024, Spring 2024, Fall 2018 Axiomatic foundations. Operations on sets and relations. Images and set functions. Ordering, well-ordering, and well-founded relations; general principles of induction and recursion. Ranks of sets, ordinals and their arithmetic. Set-theoretical equivalence, similarity of relations; definitions by abstraction. Arithmetic of cardinals. Axiom of choice, equivalent forms, and consequences. Sequence begins fall. Theory of Sets: Read More [+]

Prerequisites: 125A and 135

Instructor: 125A and 135.

Theory of Sets: Read Less [-]

MATH 236 Metamathematics of Set Theory 4 Units

Terms offered: Fall 2021, Fall 2014, Fall 2010 Various set theories: comparison of strength, transitive, and natural models, finite axiomatizability. Independence and consistency of axiom of choice, continuum hypothesis, etc. The measure problem and axioms of strong infinity. Metamathematics of Set Theory: Read More [+]

Prerequisites: 225B and 235A

Metamathematics of Set Theory: Read Less [-]

MATH 239 Discrete Mathematics for the Life Sciences 4 Units

Terms offered: Spring 2011, Fall 2008, Spring 2008 Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry. Discrete Mathematics for the Life Sciences: Read More [+]

Prerequisites: Statistics 134 or equivalent introductory probability theory course, or consent of instructor

Discrete Mathematics for the Life Sciences: Read Less [-]

MATH C239 Discrete Mathematics for the Life Sciences 4 Units

Terms offered: Spring 2013 Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry. Discrete Mathematics for the Life Sciences: Read More [+]

Also listed as: MCELLBI C244

MATH 240 Riemannian Geometry 4 Units

Terms offered: Fall 2022, Fall 2021, Fall 2019 Riemannian metric and Levi-Civita connection, geodesics and completeness, curvature, first and second variations of arc length. Additional topics such as the theorems of Myers, Synge, and Cartan-Hadamard, the second fundamental form, convexity and rigidity of hypersurfaces in Euclidean space, homogeneous manifolds, the Gauss-Bonnet theorem, and characteristic classes. Riemannian Geometry: Read More [+]

Riemannian Geometry: Read Less [-]

MATH 241 Complex Manifolds 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2021 Riemann surfaces, divisors and line bundles on Riemann surfaces, sheaves and the Dolbeault theorem on Riemann surfaces, the classical Riemann-Roch theorem, theorem of Abel-Jacobi. Complex manifolds, Kahler metrics. Summary of Hodge theory, groups of line bundles, additional topics such as Kodaira's vanishing theorem, Lefschetz hyperplane theorem. Complex Manifolds: Read More [+]

Prerequisites: 214 and 215A

Complex Manifolds: Read Less [-]

MATH 242 Symplectic Geometry 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2021 Basic topics: symplectic linear algebra, symplectic manifolds, Darboux theorem, cotangent bundles, variational problems and Legendre transform, hamiltonian systems, Lagrangian submanifolds, Poisson brackets, symmetry groups and momentum mappings, coadjoint orbits, Kahler manifolds. Symplectic Geometry: Read More [+]

Symplectic Geometry: Read Less [-]

MATH C243 Seq: Methods and Applications 3 Units

Terms offered: Spring 2015, Spring 2014 A graduate seminar class in which a group of students will closely examine recent computational methods in high-throughput sequencing followed by directly examining interesting biological applications thereof. Seq: Methods and Applications: Read More [+]

Prerequisites: Graduate standing in Math, MCB, and Computational Biology; or consent of the instructor

Additional Format: <br/>

Instructor: Pachter

Also listed as: MCELLBI C243

Seq: Methods and Applications: Read Less [-]

MATH 245A General Theory of Algebraic Structures 4 Units

Terms offered: Fall 2017, Fall 2015, Spring 2014 Structures defined by operations and/or relations, and their homomorphisms. Classes of structures determined by identities. Constructions such as free objects, objects presented by generators and relations, ultraproducts, direct limits. Applications of general results to groups, rings, lattices, etc. Course may emphasize study of congruence- and subalgebra-lattices, or category-theory and adjoint functors, or other aspects. General Theory of Algebraic Structures: Read More [+]

Prerequisites: Math 113

General Theory of Algebraic Structures: Read Less [-]

MATH 249 Algebraic Combinatorics 4 Units

Terms offered: Fall 2024, Spring 2024, Spring 2023 (I) Enumeration, generating functions and exponential structures, (II) Posets and lattices, (III) Geometric combinatorics, (IV) Symmetric functions, Young tableaux, and connections with representation theory. Further study of applications of the core material and/or additional topics, chosen by instructor. Algebraic Combinatorics: Read More [+]

Prerequisites: 250A or consent of instructor

Algebraic Combinatorics: Read Less [-]

MATH 250A Groups, Rings, and Fields 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Group theory, including the Jordan-Holder theorem and the Sylow theorems. Basic theory of rings and their ideals. Unique factorization domains and principal ideal domains. Modules. Chain conditions. Fields, including fundamental theorem of Galois theory, theory of finite fields, and transcendence degree. Groups, Rings, and Fields: Read More [+]

Prerequisites: 114 or consent of instructor

Groups, Rings, and Fields: Read Less [-]

MATH 250B Commutative Algebra 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Development of the main tools of commutative and homological algebra applicable to algebraic geometry, number theory and combinatorics. Commutative Algebra: Read More [+]

Prerequisites: 250A

Commutative Algebra: Read Less [-]

MATH 251 Ring Theory 4 Units

Terms offered: Fall 2021, Fall 2016, Spring 2013 Topics such as: Noetherian rings, rings with descending chain condition, theory of the radical, homological methods. Ring Theory: Read More [+]

Ring Theory: Read Less [-]

MATH 252 Representation Theory 4 Units

Terms offered: Fall 2021, Fall 2020, Fall 2015 Structure of finite dimensional algebras, applications to representations of finite groups, the classical linear groups. Representation Theory: Read More [+]

Representation Theory: Read Less [-]

MATH 253 Homological Algebra 4 Units

Terms offered: Spring 2023, Fall 2016, Fall 2014 Modules over a ring, homomorphisms and tensor products of modules, functors and derived functors, homological dimension of rings and modules. Homological Algebra: Read More [+]

Homological Algebra: Read Less [-]

MATH 254A Number Theory 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall. Number Theory: Read More [+]

Prerequisites: 250A for 254A; 254A for 254B

Instructor: 250A.

Number Theory: Read Less [-]

MATH 254B Number Theory 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall. Number Theory: Read More [+]

Prerequisites: 254A

MATH 255 Algebraic Curves 4 Units

Terms offered: Fall 2022, Spring 2019, Fall 2014 Elliptic curves. Algebraic curves, Riemann surfaces, and function fields. Singularities. Riemann-Roch theorem, Hurwitz's theorem, projective embeddings and the canonical curve. Zeta functions of curves over finite fields. Additional topics such as Jacobians or the Riemann hypothesis. Algebraic Curves: Read More [+]

Prerequisites: 250A-250B or consent of instructor

Algebraic Curves: Read Less [-]

MATH 256A Algebraic Geometry 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall. Algebraic Geometry: Read More [+]

Prerequisites: 250A-250B for 256A; 256A for 256B

Algebraic Geometry: Read Less [-]

MATH 256B Algebraic Geometry 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall. Algebraic Geometry: Read More [+]

Prerequisites: 256A

MATH 257 Group Theory 4 Units

Terms offered: Spring 2021, Spring 2018, Spring 2014 Topics such as: generators and relations, infinite discrete groups, groups of Lie type, permutation groups, character theory, solvable groups, simple groups, transfer and cohomological methods. Group Theory: Read More [+]

Group Theory: Read Less [-]

MATH 258 Harmonic Analysis 4 Units

Terms offered: Fall 2023, Fall 2021, Fall 2020 Basic properties of Fourier series, convergence and summability, conjugate functions, Hardy spaces, boundary behavior of analytic and harmonic functions. Additional topics at the discretion of the instructor. Harmonic Analysis: Read More [+]

Prerequisites: 206 or a basic knowledge of real, complex, and linear analysis

Harmonic Analysis: Read Less [-]

MATH 261A Lie Groups 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall. Lie Groups: Read More [+]

Instructor: 214.

Lie Groups: Read Less [-]

MATH 261B Lie Groups 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall. Lie Groups: Read More [+]

MATH 270 Advanced Topics Course in Mathematics 2 Units

Terms offered: Spring 2024, Fall 2023, Spring 2023 This course will give introductions to research-related topics in mathematics. The topics will vary from semester to semester. Advanced Topics Course in Mathematics: Read More [+]

Repeat rules: Course may be repeated for credit when topic changes.

Fall and/or spring: 15 weeks - 1.5 hours of lecture per week

Additional Format: One and one-half hours of lecture per week.

Grading: Offered for satisfactory/unsatisfactory grade only.

Advanced Topics Course in Mathematics: Read Less [-]

MATH 272 Interdisciplinary Topics in Mathematics 1 - 4 Units

Terms offered: Fall 2023, Spring 2019 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Interdisciplinary Topics in Mathematics: Read More [+]

Repeat rules: Course may be repeated for credit without restriction.

Fall and/or spring: 15 weeks - 3-3 hours of lecture per week

Interdisciplinary Topics in Mathematics: Read Less [-]

MATH 273 Topics in Numerical Analysis 4 Units

Terms offered: Spring 2022, Spring 2016, Spring 2014 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Numerical Analysis: Read More [+]

Topics in Numerical Analysis: Read Less [-]

MATH 274 Topics in Algebra 4 Units

Terms offered: Fall 2024, Fall 2023, Spring 2023 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Algebra: Read More [+]

Topics in Algebra: Read Less [-]

MATH 275 Topics in Applied Mathematics 4 Units

Terms offered: Spring 2024, Spring 2023, Fall 2021 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Applied Mathematics: Read More [+]

Topics in Applied Mathematics: Read Less [-]

MATH 276 Topics in Topology 4 Units

Terms offered: Spring 2021, Fall 2017, Spring 2016 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Topology: Read More [+]

Topics in Topology: Read Less [-]

MATH 277 Topics in Differential Geometry 4 Units

Terms offered: Spring 2023, Fall 2022, Fall 2021 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Differential Geometry: Read More [+]

Topics in Differential Geometry: Read Less [-]

MATH 278 Topics in Analysis 4 Units

Terms offered: Fall 2024, Spring 2024, Fall 2021 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Analysis: Read More [+]

Topics in Analysis: Read Less [-]

MATH 279 Topics in Partial Differential Equations 4 Units

Terms offered: Fall 2023, Spring 2023, Fall 2022 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Partial Differential Equations: Read More [+]

Topics in Partial Differential Equations: Read Less [-]

MATH 290 Seminars 1 - 6 Units

Terms offered: Spring 2017, Spring 2015, Fall 2014 Topics in foundations of mathematics, theory of numbers, numerical calculations, analysis, geometry, topology, algebra, and their applications, by means of lectures and informal conferences; work based largely on original memoirs. Seminars: Read More [+]

Fall and/or spring: 15 weeks - 0 hours of seminar per week

Additional Format: Hours to be arranged.

Seminars: Read Less [-]

MATH 295 Individual Research 1 - 12 Units

Terms offered: Summer 2016 10 Week Session, Spring 2016, Fall 2015 Intended for candidates for the Ph.D. degree. Individual Research: Read More [+]

Fall and/or spring: 15 weeks - 1-12 hours of independent study per week

Summer: 3 weeks - 5 hours of independent study per week 6 weeks - 2.5-30 hours of independent study per week 8 weeks - 1.5-60 hours of independent study per week

Grading: The grading option will be decided by the instructor when the class is offered.

Individual Research: Read Less [-]

MATH N295 Individual Research 0.5 - 5 Units

Terms offered: Summer 2022 8 Week Session, Summer 2021 8 Week Session, Summer 2006 10 Week Session Intended for candidates for the Ph.D. degree. Individual Research: Read More [+]

Summer: 8 weeks - 1-5 hours of independent study per week

MATH N297 General Academic Internship 0.5 Units

Terms offered: Prior to 2007 This is an independent study course designed to provide structure for graduate students engaging in summer internship opportunities. Requires a paper exploring how the theoretical constructs learned in academic courses were applied during the internship. General Academic Internship: Read More [+]

Summer: 8 weeks - 2.5 hours of independent study per week

Additional Format: Two and one-half hours of independent study per week for 8 weeks.

General Academic Internship: Read Less [-]

MATH 299 Reading Course for Graduate Students 1 - 6 Units

Terms offered: Fall 2018, Fall 2017, Fall 2016 Investigation of special problems under the direction of members of the department. Reading Course for Graduate Students: Read More [+]

Fall and/or spring: 15 weeks - 0 hours of independent study per week

Summer: 6 weeks - 1-5 hours of independent study per week 8 weeks - 1-4 hours of independent study per week

Reading Course for Graduate Students: Read Less [-]

MATH 301 Undergraduate Mathematics Instruction 1 - 2 Units

Terms offered: Fall 2018, Spring 2018, Fall 2017 May be taken for one unit by special permission of instructor. Tutoring at the Student Learning Center or for the Professional Development Program. Undergraduate Mathematics Instruction: Read More [+]

Prerequisites: Permission of SLC instructor, as well as sophomore standing and at least a B average in two semesters of calculus. Apply at Student Learning Center

Repeat rules: Course may be repeated for credit up to a total of 4 units.

Fall and/or spring: 15 weeks - 3 hours of seminar and 4 hours of tutorial per week

Additional Format: Three hours of Seminar and Four hours of Tutorial per week for 15 weeks.

Subject/Course Level: Mathematics/Professional course for teachers or prospective teachers

Grading: Offered for pass/not pass grade only.

Undergraduate Mathematics Instruction: Read Less [-]

MATH 302 Teaching Workshop 1 Unit

Terms offered: Summer 2002 10 Week Session, Summer 2001 10 Week Session Mandatory for all graduate student instructors teaching summer course for the first time in the Department. The course consists of practice teaching, alternatives to standard classroom methods, guided group and self-analysis, classroom visitations by senior faculty member. Teaching Workshop: Read More [+]

Summer: 8 weeks - 1 hour of lecture per week

Additional Format: One hour of Lecture per week for 8 weeks.

Teaching Workshop: Read Less [-]

MATH 303 Professional Preparation: Supervised Teaching of Mathematics 2 - 4 Units

Terms offered: Spring 2017, Spring 2016, Fall 2015 Meeting with supervising faculty and with discussion sections. Experience in teaching under the supervision of Mathematics faculty. Professional Preparation: Supervised Teaching of Mathematics: Read More [+]

Prerequisites: 300, graduate standing and appointment as a Graduate Student Instructor

Fall and/or spring: 15 weeks - 2-4 hours of independent study per week

Additional Format: No formal meetings.

Professional Preparation: Supervised Teaching of Mathematics: Read Less [-]

MATH 600 Individual Study for Master's Students 1 - 6 Units

Terms offered: Summer 2006 10 Week Session, Fall 2005, Spring 2005 Individual study for the comprehensive or language requirements in consultation with the field adviser. Individual Study for Master's Students: Read More [+]

Prerequisites: For candidates for master's degree

Credit Restrictions: Course does not satisfy unit or residence requirements for master's degree.

Fall and/or spring: 15 weeks - 1-6 hours of independent study per week

Summer: 8 weeks - 1.5-10 hours of independent study per week

Subject/Course Level: Mathematics/Graduate examination preparation

Individual Study for Master's Students: Read Less [-]

MATH 602 Individual Study for Doctoral Students 1 - 8 Units

Terms offered: Fall 2019, Fall 2018, Fall 2016 Individual study in consultation with the major field adviser intended to provide an opportunity for qualified students to prepare themselves for the various examinations required for candidates for the Ph.D. Course does not satisfy unit or residence requirements for doctoral degree. Individual Study for Doctoral Students: Read More [+]

Prerequisites: For qualified graduate students

Fall and/or spring: 15 weeks - 1-8 hours of independent study per week

Additional Format: One to Eight hour of Independent study per week for 15 weeks.

Individual Study for Doctoral Students: Read Less [-]

Contact Information

Department of mathematics.

970 Evans Hall

Phone: 510-642-6550

Department Chair

Martin Olsson

953 Evans Hall

Phone: 510-642-4129

[email protected]

Vice-Chair for Graduate Affairs

Thomas Scanlon

723 Evans Hall

[email protected]

Graduate Student Affairs Officer - Academic Advising

Clay Calder

910 Evans Hall

Phone: 510-642-0665

[email protected]

Graduate Student Affairs Officer - Funding & Employment

Christian Natividad

914 Evans Hall

[email protected]

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Mathematics, PhD

Zanvyl krieger school of arts and sciences.

The goal of our PhD program is to train graduate students to become research mathematicians. Each year, an average of five students complete their theses and  go on to exciting careers  in mathematics both inside and outside of academia.

Faculty research interests  in the Johns Hopkins University Department of Mathematics are concentrated in several areas of pure mathematics, including analysis and geometric analysis, algebraic geometry and number theory, differential geometry, algebraic topology, category theory, and mathematical physics. The department also has an active group in data science, in collaboration with the  Applied Math Department .

The Department values diversity among its members, is committed to building a diverse intellectual community, and strongly encourages applications from all interested parties.

A brief overview of our graduate program is below. For more detailed information, please see the links at the right.

Program Overview

All students admitted to the PhD program receive full tuition fellowships and teaching assistantships. Teaching assistant salaries for the 2022-2023 academic year are $33,000, and exceptional applicants are also considered for supplementary fellowships. Students making satisfactory progress can expect to be supported for six years.

PhD candidates take two or three courses per semester over the first several years of the program. These are a mix of required and intermediate-level graduate courses, independent studies, and special topics classes offered by our faculty.

By the beginning of their second year, students are asked to demonstrate competency in algebra and in analysis by passing written qualifying exams in these two broad areas. Students are then expected to choose an advisor, who will supervise their dissertation and also administer an oral qualifying exam to be taken in the second or third year. More specifics about all these requirements are described on the  requirements page .

All graduate students are invited to attend  weekly research seminars in a variety of topic areas  as well as regular department teas and a weekly wine and cheese gathering attended by many junior and senior members of the department. A graduate student lunch seminar series provides an opportunity for our students to practice their presentation skills to a general audience.

PhD students will gain teaching experience as a teaching assistant for undergraduate courses. Most of our students lead two TA sections per week, under the supervision of both the faculty member teaching the course and the director of undergraduate studies. Students wanting more classroom experience (or extra pay) can teach their own sections of summer courses. First-year students are given a reduced TA workload in the spring semester, in preparation for the qualifying exams.

In addition to their stipend, each student is awarded an annual travel allowance to enable them to attend conferences for which limited funding is available or visit researchers at other institutions.

Financial Aid

Students admitted to the Ph.D. program receive teaching assistantships and full tuition fellowships. Exceptional applicants become candidates for one of the university's George E. Owen Fellowships.

William Kelso Morrill Award

The William Kelso Morrill Award for excellence in the teaching of mathematics is awarded every spring to the graduate student who best exemplifies the traits of Kelso Morrill: a love of mathematics, a love of teaching, and a concern for students.

Excellence in Teaching Awards

Three awards are given each year to a junior faculty member and graduate student teaching assistants who have demonstrated exceptional ability and commitment to undergraduate education.

Admission Requirements

Admission to the PhD program is based on primarily on academic records, letters of recommendation, and a personal statement. The Department of Mathematics values diversity among its members, is committed to building a diverse intellectual community, and strongly encourages applications from all interested parties.

Via the online application , applicants should submit:

• A Statement of Purpose
• An optional Personal Statement
• Transcripts from all institutions attended
• Three letters of recommendation
• Official GRE scores for both the general and the subject test
• Official TOEFL scores (if English is not your first language)

The required Statement of Purpose discusses your academic interests, objectives, and preparation. The optional Personal Statement describes your personal background, and helps us create a more holistic understanding of you as an applicant. If you wish you may also discuss your personal background in the Statement of Purpose (e.g. if you have already written a single essay addressing both topics), instead of submitting separate statements.

Application fee waivers are available based on financial need and/or participation in certain programs .

Many frequently asked questions about the graduate admission process are answered here .

No application materials should be mailed to the department. All application materials are processed by the Graduate Admissions Office .

Undergraduate Background

The following is an example of what the math department would consider a good background for a student coming out of a four-year undergraduate program at a college or university in the U.S. (assuming a semester system):

• Calculus in one variable (two semesters, or AP credits)
• Multivariable Calculus (one semester)
• Linear Algebra (one semester)
• Complex analysis (one semester)
• Real analysis (two semesters)
• Abstract algebra (two semesters)
• Point-set topology (one semester)

Many admitted students have taken upper-level undergraduate mathematics courses or graduate courses. Nevertheless, the department does admit very promising students whose preparation falls a little short of the above model. In such cases, we strongly recommend that the student start to close the gap over the summer, before arriving for the start of the fall semester.

Financial Support   

Students admitted to the PhD program receive full tuition fellowships and teaching assistantships. Teaching assistant salaries for the 2022–2023 academic year are $33,000. Students making satisfactory progress can expect to be supported for six years. Exceptional applicants are considered for supplementary fellowships of $6,000 each year for three years.

Students from underrepresented groups may be eligible for other university-wide supplemental fellowships. Summer teaching is available for students seeking extra income.

Additional Information for International Students

Student Visa Information:  The Office of International Services at Homewood  will assist admitted international students in obtaining a student visa.

English Proficiency: Johns Hopkins University requires students to have adequate English proficiency for their course of study. Students must be able to read, speak, and write English fluently upon their arrival at the university. Applicants whose native language is not English must submit proof of their proficiency in English before they can be offered admission and before a visa certificate can be issued. Proficiency can be demonstrated by submitting results from either the Test of English as a Foreign Language (TOEFL) or the IELTS . Johns Hopkins prefers a minimum score of 100 on the TOEFL or a Band Score of 7 on the IELTS. Results should be sent to Johns Hopkins directly by TOEFL or IELTS. Applicants taking the IELTS must additionally upload a copy of their score through the application system. However, do not send the student copy or a photocopy of the TOEFL.

Program Requirements

Course requirements.

Mathematics PhD candidates must show satisfactory work in Algebra (110.601-602), Real Variables (110.605), Complex Variables (110.607), and one additional non-seminar mathematics graduate course in their first year. The first-year algebra and analysis requirement can be satisfied by passing the corresponding written qualifying exam in September of the first year; these students must complete at least two courses each semester. In addition, PhD candidates must take Algebraic Topology (110.615) and Riemannian Geometry (110.645) by their second year. Students having sufficient background can substitute an advanced topology course for 110.615, or an advanced geometry course for 110.645 with the permission of the instructor.

Candidates must show satisfactory work in at least two mathematics graduate courses each semester of their second year, and if they have not passed their oral qualifying exam, in the first semester of their third year.

Qualifying Exams

Candidates must pass written qualifying exams by the beginning of their second year in Analysis (Real & Complex) and in Algebra. Exams are scheduled for September and May of each academic year, and the dates are announced well in advance.

Candidates must pass an oral qualifying examination in the student’s chosen area of research by April 10 of the third year. The topics of the exam are chosen in consultation with the faculty member who has agreed (provisionally) to be the student’s thesis advisor, who will also be involved in administering the exam.

PhD Dissertation

Candidates must produce a written dissertation based upon independent and original research. After completion of the thesis research, the student will defend the dissertation by means of the  Graduate Board Oral exam . The exam must be held at least three weeks before the Graduate Board deadline the candidate wishes to meet.

Our PhD program does not have a foreign language requirement.

• University of Pennsylvania
• School of Arts and Sciences
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Department of Mathematics

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Graduate Program

Thinking of applying to graduate school in mathematics.

Penn was ranked 8th among all US universities in a leading national study , and our mathematics graduate program was recently highest in a study of graduate programs in arts and sciences at Penn. We have a very active and involved mathematics faculty , diverse course offerings and a broad seminar schedule , with a variety of research projects and strengths in algebra, analysis, geometry-topology, combinatorics, logic, probability, and mathematical physics. We have a supportive atmosphere, with personal attention from the faculty and extensive interaction among graduate students. Our grad students can take courses not only in the Mathematics Department but also elsewhere at Penn, and the wide resources of the university are available. Our former graduate students have gone on to mathematical careers both in academia and in industry.

Our full-time Ph.D. students receive a generous and competitive support package including

• five years of funding with a combination of fellowships and teaching assistantships;
• a stipend and a full tuition scholarship;
• no teaching responsibilities for at least two years (generally including the first and fourth year);
• health insurance coverage provided at no cost to the student.

We invite you to learn about our graduate program, our math department, our university and living in Philadelphia, a cosmopolitan city and a true mathematical hub, with easy access to nearby mathematics departments and research institutes.

We are looking for interested, mathematically talented and dedicated students to be a part of our group of excellent future mathematicians. Consider applying to Penn for your graduate education. Questions?

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Ph.D. Program in Mathematics

Degree requirements.

A candidate for the Ph.D. degree in mathematics must fulfill a number of different departmental requirements.

NYU Shanghai Ph.D. Track

The Ph.D. program also offers students the opportunity to pursue their study and research with Mathematics faculty based at NYU Shanghai. With this opportunity, students generally complete their coursework in New York City before moving full-time to Shanghai for their dissertation research. For more information, please visit the  NYU Shanghai Ph.D. page .

Sample course schedules (Years 1 and 2) for students with a primary interest in:

Applied Math (Math Biology, Scientific Computing, Physical Applied Math, etc.)

Additional information for students interested in studying applied math is available here .

Probability

PDE/Analysis

The Written Comprehensive Examination

The examination tests the basic knowledge required for any serious mathematical study. It consists of the three following sections: Advanced Calculus, Complex Variables, and Linear Algebra. The examination is given on three consecutive days, twice a year, in early September and early January. Each section is allotted three hours and is written at the level of a good undergraduate course. Samples of previous examinations are available in the departmental office. Cooperative preparation is encouraged, as it is for all examinations. In the fall term, the Department offers a workshop, taught by an advanced Teaching Assistant, to help students prepare for the written examinations.

Entering students with a solid preparation are encouraged to consider taking the examination in their first year of full-time study. All students must take the examinations in order to be allowed to register for coursework beyond 36 points of credit; it is recommended that students attempt to take the examinations well before this deadline. Graduate Assistants are required to take the examinations during their first year of study.

For further details, consult the page on the written comprehensive exams .

The Oral Preliminary Examination

This examination is usually (but not invariably) taken after two years of full-time study. The purpose of the examination is to determine if the candidate has acquired sufficient mathematical knowledge and maturity to commence a dissertation. The phrase "mathematical knowledge" is intended to convey rather broad acquaintance with the basic facts of mathematical life, with emphasis on a good understanding of the simplest interesting examples. In particular, highly technical or abstract material is inappropriate, as is the rote reproduction of information. What the examiners look for is something a little different and less easy to quantify. It is conveyed in part by the word "maturity." This means some idea of how mathematics hangs together; the ability to think a little on one's feet; some appreciation of what is natural and important, and what is artificial. The point is that the ability to do successful research depends on more than formal learning, and it is part of the examiners' task to assess these less tangible aspects of the candidate's preparation.

The orals are comprised of a general section and a special section, each lasting one hour, and are conducted by two different panels of three faculty members. The examination takes place three times a year: fall, mid-winter and late spring. Cooperative preparation of often helpful and is encouraged. The general section consists of five topics, one of which may be chosen freely. The other four topics are determined by field of interest, but often turn out to be standard: complex variables, real variables, ordinary differential equations, and partial differential equations. Here, the level of knowledge that is expected is equivalent to that of a one or two term course of the kind Courant normally presents. A brochure containing the most common questions on the general oral examination, edited by Courant students, is available at the Department Office.

The special section is usually devoted to a single topic at a more advanced level and extent of knowledge. The precise content is negotiated with the candidate's faculty advisor. Normally, the chosen topic will have a direct bearing on the candidate's Ph.D. dissertation.

All students must take the oral examinations in order to be allowed to register for coursework beyond 60 points of credit. It is recommended that students attempt the examinations well before this deadline.

The Dissertation Defense

The oral defense is the final examination on the student's dissertation. The defense is conducted by a panel of five faculty members (including the student's advisor) and generally lasts one to two hours. The candidate presents his/her work to a mixed audience, some expert in the student's topic, some not. Often, this presentation is followed by a question-and-answer period and mutual discussion of related material and directions for future work.

Summer Internships and Employment

The Department encourages Ph.D. students at any stage of their studies, including the very early stage, to seek summer employment opportunities at various government and industry facilities. In the past few years, Courant students have taken summer internships at the National Institute of Health, Los Alamos National Laboratory, Woods Hole Oceanographic Institution, Lawrence Livermore National Laboratory and NASA, as well as Wall Street firms. Such opportunities can greatly expand students' understanding of the mathematical sciences, offer them possible areas of interest for thesis research, and enhance their career options. The Director of Graduate Studies and members of the faculty (and in particular the students' academic advisors) can assist students in finding appropriate summer employment.

Mentoring and Grievance Policy

For detailed information, consult the page on the Mentoring and Grievance Policy .

Visiting Doctoral Students

Information about spending a term at the Courant Institute's Department of Mathematics as a visiting doctoral student is available on the Visitor Programs  page.

Mathematics Education, PHD

On this page:, at a glance: program details.

• Location: Tempe campus
• Second Language Requirement: No

Program Description

Degree Awarded: PHD Mathematics Education

This transdisciplinary PhD program in mathematics education accommodates students from a variety of academic backgrounds. It provides students with a solid foundation in graduate-level mathematics as well as research skills and perspectives that enable them to incorporate mathematics into such core educational areas as:

Conducting individual and collaborative research in the learning and teaching of mathematics is an integral part of the program.

Degree Requirements

84 credit hours, a written comprehensive exam, an oral comprehensive exam, a prospectus and a dissertation

Required Core (12 credit hours) MTE 501 Research in Undergraduate Mathematics Education I (3) MTE 502 Research in Undergraduate Mathematics Education II (3) MTE 503 Research in Undergraduate Mathematics Education Ill (3) MTE 504 Research in Undergraduate Mathematics Education IV (3)

Electives (42 credit hours)

Area Courses (12 credit hours)

Research (6 credit hours) MTE 792 Research (6)

Culminating Experience (12 credit hours) MTE 799 Dissertation (12)

Additional Curriculum Information Four to five graduate-level (500 and above) elective courses from mathematics, cognitive science, psychology, educational technology, philosophy or research should be taken as approved by the advisor.

For the area courses, students are required to take four graduate-level courses from the following areas of interest: mathematics, applied mathematics or statistics. Students should see the academic unit for the approved course list.

Students should see the school's website for information about qualifier and comprehensive examinations based on math coursework.

The doctoral dissertation culminating experience consists of a dissertation prospectus, oral dissertation defense and the submission of a final revised, formatted dissertation document to the Graduate College. Dissertations are composed under chair- and committee-supervised research, including literature review, research, data collection and analysis, and writing.

When approved by the student's supervisory committee and the Graduate College, up to 30 credit hours from a previously awarded master's degree may be used for this program. If students do not have a previously awarded master's degree, the remaining coursework is made up of electives and research.

Admission Requirements

Applicants must fulfill the requirements of both the Graduate College and The College of Liberal Arts and Sciences.

Applicants are eligible to apply to the program if they have earned a bachelor's or master's degree in mathematics or a closely related area, with exceptionally high grades in advanced coursework in mathematics, from a regionally accredited institution.

Applicants must have a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in the last 60 hours of their first bachelor's degree program or a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in an applicable master's degree program.

All applicants must submit:

• graduate admission application and application fee
• official transcripts
• statement of education and career goals
• writing sample
• three letters of recommendation
• proof of English proficiency

Additional Application Information An applicant whose native language is not English must provide proof of English proficiency regardless of their current residency.

At least two of the letters of recommendation must be from faculty.

Next Steps to attend ASU

Learn about our programs, apply to a program, visit our campus, application deadlines, learning outcomes.

• Able to complete original research in applied mathematics.
• Able to incorporate mathematical concepts into novel teaching methods.
• Address an original research question in mathematics education.

Career Opportunities

Graduates of the doctoral program in mathematics education have opportunities in Arizona, the U.S. and internationally. Opportunities are typically at research universities and liberal arts colleges, community colleges, and education consulting firms and in roles such as:

• faculty-track academic
• education consultant or analyst
• mathematics professor, instructor or researcher

Program Contact Information

If you have questions related to admission, please click here to request information and an admission specialist will reach out to you directly. For questions regarding faculty or courses, please use the contact information below.

• [email protected]
• 480/965-3951

PhD Degree in Mathematics

Prerequisites: .

To enter the Ph.D. program a student should hold at least a Bachelor's degree in mathematics.  The academic record of a student applying to the Ph.D. program should contain substantial evidence that the student will succeed in the doctoral program.  In reviewing an applicant's folder, the Graduate Committee gives substantial weight to the applicant's transcripts and letters of recommendation.

Requirements: 

The Ph.D. degree has no rigid course requirement beyond the residency requirement (however, breadth and depth of knowledge are strongly encouraged). 

It does require:

• 1. Passing written and oral qualifying examinations.
• 2. Writing a dissertation embodying the results of original research which is acceptable to the   student's dissertation committee.
• 3. A final oral defense of the dissertation. A student's progress towards the Ph.D. degree is initially supervised by a three-person committee, increasing to four or five members following the written qualifying exams. The student's faculty advisor chooses this committee and is its chair.

The Ph.D. Qualifying Examination System consists of two parts. The first part consists of four Written Qualifying Exams and the second consists of an Oral Qualifying Exam.

Written Qualifying Exams  are offered every year in August before the start of Fall semester classes and in January before the start of Spring semester classes.   Study guides  and copies of  previous qualifying exams  are available on the Graduate Program website for students to use in preparing for their Written Qualifying Exams.

Written qualifying exams are offered in algebra, complex analysis, numerical analysis, probability, real analysis and topology. 

There are three possible grades on each exam: pass, master's pass or fail.  Each PhD candidate is required to either: (i) attain pass grades on three written qualifying exams or (ii) attain pass grades on two written qualifying exams and master's pass grades on two written qualifying exams. 

The choice of which three or four exams to apply to meet these requirements from the six available exams must be approved by the student's Preliminary Advisory Committee.

The Oral Qualifying Exam is based on the student's anticipated area of specialization. In it, the student is expected to present material from a research paper and to answer general questions about their area of specialization. It is typical for students to take their oral exam within 1 year of their passing the Written Qual requirements. (Students who passes Written Quals early will sometimes take additional time to pass the Oral Qual.) To begin preparing for the Oral Qual, a committee of four or five is chosen (including the student's thesis advisor). The student prepares by reading research papers in the area, and the student, advisor, and committee agree upon a body of material for which the student will be responsible. The exam consists of a presentation on the prepared research papers, followed by a question period covering the presentation and the agreed upon body of material.

Graduate Guidebook

Need more Info?

D avid Gay Graduate Program Coordinator Laura Rider Graduate Admissions Coordinator Lucy Barerra Graduate Admissions Coordinator

Please direct questions about our graduate program to: [email protected]

We appreciate your financial support. Your gift is important to us and helps support critical opportunities for students and faculty alike, including lectures, travel support, and any number of educational events that augment the classroom experience.  Click here to learn more about giving .

Every dollar given has a direct impact upon our students and faculty.

Book series

Graduate Texts in Mathematics

About this book series.

• Patricia Hersh,
• Ravi Vakil,
• Jared Wunsch

Book titles in this series

Fundamentals of fourier analysis.

• Loukas Grafakos
• Copyright: 2024

Available Renditions

Random Walks on Infinite Groups

• Steven P. Lalley
• Copyright: 2023

An Invitation to Mathematical Logic

• David Marker

Drinfeld Modules

• Mihran Papikian

An Introduction to Automorphic Representations

With a view toward trace formulae

• Jayce R. Getz
• Heekyoung Hahn

Publish with us

Abstracted and indexed in.

• Mathematical Reviews
• Norwegian Register for Scientific Journals and Series

• Academics /

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At Harvard Extension School, our courses are the cornerstone of our academic offerings.

You may choose to take a single course — perhaps to build a new skill, explore a passion, or prepare for graduate school. Or you may decide to take courses in pursuit of a degree or certificate . The choice is yours.

Our courses are open enrollment, requiring no application to enroll. Whatever your goal, you’ll find courses that balance academic rigor with flexibility and value.

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Visit the CARC website to explore all of these valuable resources and more.

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Emporia State School of Science + Mathematics

Master of Science, Mathematics – Statistics and Applied Mathematics Concentration

Level up your mathematical skills.

Emporia State University’s Master of Science, Statistics and Applied Mathematics program is flexible so you can reach your personal goals in the way that will best fit your needs. Our classes are available both on-campus and online, allowing you to decide when and how you take your courses.

Up to nine credit hours of graduate courses from other universities may also be transferred for the program with approval from ESU. Transfer courses must be completed within a 7-year window that ends at the date when the program is completed.

On-campus students are eligible for teaching assistantships in the math department which pay a stipend and include a full tuition waiver.

For more information about the mathematics program, contact [email protected] . For questions about the ESU Graduate School application, tuition and fees, admission or other general ESU questions, visit emporia.edu/grad or contact [email protected] .

Additional information

Students must complete:

MA 701: Mathematical Proofs

MA 735 Advanced Calculus I, 3 hours

MA 728 Vector Spaces, 3 hours

MA 532 Mathematical Statistics I, 3 hours

MA 733 Mathematical Statistics II, 3 hours

MA 810 Seminar (Non-Thesis option) or MA 850 (Thesis)

In addition, students must take at least twelve credit hours from the following Applied Mathematics and Statistics courses:

MA 537 Financial Mathematics, 3 hours

MA 731 SAS, 3 hours

MA 732 Categorical Data Analysis, 3 hours

MA 738 Applied Differential Equations, 3 hours

MA 739 Applied Analysis, 3 hours

MA 758 Wavelets, 3 hours

MA 760 Numerical Analysis, 3 hours

MA 762 Optimization Techniques, 3 hours

MA 763 Simulation Techniques, 3 hours

MA 764 Regression Analysis, 3 hours

MA 765 Numerical Linear Algebra, 3 hours

MA 766 Nonparametric Statistics, 3 hours

MA 767 Multivariate Data Analysis, 3 hours

MA 768 Design of Experiments, 3 hours

MA 769 Spatial Data Analysis, 3 hours

MA 770 Bayesian Data Analysis, 3 hours

EC 710 Econometrics I, 3 hours

EC 711 Econometrics II, 3 hours

Other Notes

* These courses are considered non-core courses since they are focused on math education. A maximum of 6 credit hours of non-core courses can be applied toward the completion of the M.S. Math program. (None may be applied to the Graduate Certificate programs.)

** MA 810 is the course that non-thesis students take when they are ready to give their seminar presentation before graduation. MA 850 is for on-campus thesis students only.

A grade point average of not less than 2.5 in the last 60 semester hours of study or an overall grade point average of no less than 3.0 for a completed master’s degree.

Applicants who have completed 24 credit hours of undergraduate mathematics, including at least two semesters of Calculus and at least one course where writing mathematical proofs is a significant part of the content, are viewed favorably.

An undergraduate degree in mathematics, mathematics education, or an area with a significant mathematics requirement is preferred.

In general, success in our programs requires familiarity with the content of the first two semesters of a typical Calculus sequence as well as experience in writing mathematical proofs.

Graduate School Application

Upon filling out the application, one has the choice to choose a concentration to supplement their Master's Degree. If one does not desire to choose a concentration, then leave that section blank on the application.

1. Official transcripts from all colleges/universities attended.

2. Copy of government-issued ID.

Submitting Application Materials

The following items must be provided/completed at the time of application submission through the Application Portal :

*Unofficial bachelor's degree transcript

*Copy of government-issued ID

Official transcripts

Students must submit official bachelor’s degree transcripts containing at least 60 credit hours of coursework and final grades. Any additional transcripts from college credit accumulated after the bachelor’s degree MUST be submitted if you will be using these credits for transfer credit or for last 60 GPA. Transcripts are considered official when they arrive in the Graduate Office in a sealed envelope from the issuing institution or are received through a secure electronic transcript service to [email protected] .

If the transcripts need to be mailed, send to:

Emporia State University Graduate School Campus Box 4003 Emporia State University 1 Kellogg Circle Emporia, KS 66801-5415.

All students must complete their comprehensive exams according to the following guidelines.

Seminar students must take exams from four subjects and pass three of them.

Thesis students (only available for on-campus students) must take exams from three subjects and pass two of them.

Students must take at least one exam in each of the areas of Algebra, Analysis, and Statistics/Applied Mathematics.

Students must choose their subjects and notify the appropriate instructors by the end of the fourth week of the semester in which they are taking the exam and must take the exam itself by the end of the eighth week.

If a student does not pass the necessary number of exams, they can make one more attempt that semester. If the student was one subject away from passing, they may retake only the failed subjects; otherwise, the student must retake all subjects. No more than two attempts per semester are allowed. Each course's exam may be attempted a maximum of three times, after which a different course must be selected for future exams.

Transfer courses, non-core courses, and MA701 Mathematical Proofs are not eligible subjects for the exams.

All exams must be taken at the same time.

Students must be enrolled in at least one class in the semester that the exams are attempted.

Why choose ESU?

At ESU, we know it’s your future, emphasis on the ‘your.’ So tell us what you want to do with your certificate, and we’ll show you an educational track that sets you up for success.

Take action

Apply to the Emporia State University Graduate School.

Request more information

Request more information today. To contact the department, see the contact information at the bottom of this page.

Scholarships

Scholarships are available for Mathematics students at ESU.

Contact the Mathematics Department

620-341-5281

g [email protected]

Science Hall Room 141

Campus Box 4027 1 Kellogg Circle Emporia, KS 66801

by Steve Walentik | Jun 3, 2024

Luis Scheegans will be the first member of his immediate family to graduate from college when he completes his BS in mathematics later this summer. He will begin a doctoral program in geophysics at Stanford University this fall. (Photos by Derik Holtmann)

Luis Schneegans remembers his mother’s excitement and pride when he called home from the Emerging Researchers National Conference in STEM in March in Washington, D.C.

The soon-to-be graduate of the University of Missouri–St. Louis had just gotten the email letting him know he’d been accepted into a doctoral program in geophysics at Stanford University, and he was eager to share the news with his family.

“I could just hear her screaming on the phone,” Schneegans said of his mother. “It was even funnier because she was with my youngest sister, who’s 9 years old. She doesn’t even understand the magnitude of any of this that I’m doing. But she just knew it was a big deal for me. She ran around the house.”

Schneegans felt a strong desire to be around and to set an example for all three of his younger siblings when he became the first member of his immediate family to pursue higher education. That was a big reason he decided to enroll at UMSL in the fall of 2019, though he had many options after graduating from Parkway North High School as part of the National Hispanic Recognition Program .

“They were they were like 4, 7 and 13 at the time when I was making a decision for college,” Schneegans said of his younger sisters and younger brother. “I don’t think a 4-year-old can remember their older brother very often if he’s only coming to visit during holidays.”

Luis Schneegans presents some of his research on “Mathematical Modeling of EGaIn Droplets Sliding Down an Inclined Plane,” during UMSL’s Undergraduate Research Symposium in late April.

By living at home, they got to observe the work it took to finish his degree. With the completion of one elective course later this summer, Schneegans will have earned his Bachelor of Science in mathematics with a minor in philosophy .

“They don’t have to follow in my footsteps at all,” said Schneegans, who participated in a May 11 commencement ceremony for graduates of the College of Arts and Sciences . “They’re not me, and I’m not them. I can’t make them do anything. In the end, if I can just have them acknowledge that it’s possible, that’s fine by me because I didn’t even know this was a possible route in the first place.”

He relied on his own mentors to help him navigate his path to college, through his course work, into opportunities taking part in undergraduate research and through the process of applying to graduate school, all with the encouragement of his parents. But it wasn’t always easy.

Schneegans was helped financially by receiving the Curators Scholarship as well as the Margaret Bush Wilson , Eugene Meehan , Purdy Emerging Leaders and Mathematical Sciences Alumni scholarships and a Bright Flight Scholarship during his time at UMSL. But he had to overcome other challenges along the way to earn his degree.

“It started off pretty rough,” Schneegans said. “My freshman year, I had a go on a break my first semester for two months. It was a mental health break. I feel like every college student goes through. Sometimes personal life and school life jumbles up into one and becomes too difficult. For me, it just so happened to be in my first year when it happened.”

It set back his academic progress and led him to complete 27 hours of course work in the spring semester – the previous fall’s slate of classes plus new ones – as he tried to catch up.

The COVID-19 pandemic hit during that semester, adding another challenge as all his classes shifted online.

“If I didn’t learn how to time manage through that, I don’t know where I would learn time management in the future,” Schneegans said.

He intended to major in physics, but he wound up changing plans to pursue philosophy and later switched his major to mathematics. It better fit his interest in finding solutions to real-world problems, though he credits his study of philosophy in helping him learn to think logically and more critically.

“Luis was a very motivated and hard-working student,” said Associate Teaching Professor David Covert , the undergraduate mathematics director in the Department of Mathematics, Physics, Astronomy and Statistics . “He was not deterred by any setbacks, and he tended to learn from his mistakes. He was very responsive to feedback.”

Schneegans also was intentional about finding learning opportunities outside the classroom, first getting involved in the Missouri Louis Stokes Alliance for Minority Participation , or MOLSAMP, an alliance of nine institutions of higher education in Missouri that have collaborated to try to increase the number of underrepresented minority students statewide completing undergraduate and advanced degrees in STEM fields.

He learned of the program through a friend who received an invitation as a biology major. Though he was then undeclared, he reached out to Professor E. Paulette Isaac-Savage , a co-PI on the project, to see if he could participate and was approved.

The program is funded through a grant from the National Science Foundation . Participating students attend monthly meetings and receive a stipend each semester provided they maintain a minimum GPA and meet other requirements for involvement.

By taking part in MOLSAMP, Schneegans discovered how to pursue NSF-supported Research Experiences for Undergraduates or REUs. That proved invaluable to his own educational experience, and he was quick to share that with others.

“Luis is passionate about learning, which shines through in his coursework and research projects,” said Jamillah Boyd , an associate teaching professor of information systems and technology and the faculty advisor for the MOLSAMP program at UMSL. “He is always open to sharing what he learns in the classroom, at conferences, and at his summer research experiences. He uses his experiences to mentor others, which makes him an invaluable asset to our community.”

Schneegans participated in three REUs during his time at UMSL, spending summers gaining research experience at the University of Missouri–Columbia, Kansas State University and North Carolina State University.

“I was very impressed by his pursual of REUs,” Covert said. “They are one of the best tools students can use to determine what graduate research will be like.”

In the first, he learned about nonlinear dynamics and chaos theory. In the second, he researched a partial differential equation and helped model it for multiple dimensions. In his most recent internship, he worked on a mathematical physics project focused on model liquid metal dynamics for Eutectic gallium-indium , or EGaIn, a liquid metal alloy that’s a combination of gallium and indium.

By the end of his second REU, at Kansas State, he decided he wanted to go to graduate school to pursue a PhD, eventually deciding to focus on geophysics.

“It just so happened that I fell in love with research,” he said. “I fell in love with it due to the struggles of research, rather than the easiness of research.”

He relayed the experience of writing a paper highlighting his findings from a summer spent gathering and analyzing data. He estimated it was about 20 pages long, but because of a critical error made defining terms, about half those pages had to be deleted. He worked with his mentor for several hours trying his best to salvage what he could from the work.

“I didn’t realize I was going to become so passionate about the research,” Schneegans said.

He had multiple chances to present last summer’s work on “ Mathematical Modeling of EGaIn Droplets Sliding Down an Inclined Plane ,” taking part in the Emerging Researchers National Conference and UMSL’s Undergraduate Research Symposium .

Schneegans, who was also active in Math Club throughout his time at UMSL and served as an iMentor in the to freshmen, sophomores and transfer students adjusting to the university, would like to continue doing similar research work as he pursues his doctorate. He’s still deciding where that might lead him in the future.

“I’m leaning more towards the industry-research track,” he said, rather than a position in academia. “Specifically, I would like to work at a national lab. There’s a few national labs and private labs that are specifically lined up with what I want to do – Livermore National Lab , Santa Fe Institute . But I don’t want to limit my options.”

He has time to figure it out, of course, but he’s continuing to show those around him – and himself – just what is possible.

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Steve Walentik

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I appreciated reading Kara Miller’s The Big Idea column “PhD: Pretty heavily disappointed” (Business, May 22), about people with doctoral degrees struggling to build careers in academia. It made me think back to a conversation I had when I was about to graduate from high school.

I happened to run into a former track coach of mine, and as we were reminiscing he asked me what I planned as a major in college. “History,” I responded. He said, “Why don’t you take some computer classes also? It never hurts to be able to do something useful.”

I did not reflect on his motivation at the time, but my track coach was a young guy, and he was probably giving me advice straight from his own life, as a parent trying to raise his own young children. I did take computer classes in college and ultimately received a PhD in chemical engineering. I always remember that conversation as being a kind of turning point.

Earning a doctoral degree is a life commitment of great proportion. It can take, as Miller notes, between four and seven years. If we think of working life as roughly between the ages of 22 and 65, then a PhD requires more than 10 percent of a person’s working life. People need to think carefully about that investment.

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Two powerful arguments in favor of the path of science, technology, engineering, and math are that there tend to be more STEM jobs for PhDs, and many universities’ STEM departments are generous in covering their PhD students’ tuition and cost of studies, including a stipend toward food, rent, and other expenses.

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Not much has changed in 30 years

As I prepared to graduate in 1995 with a doctor of education degree from the Harvard Graduate School of Education, my mother memorably said to me, “Of my four children, you are the one with the most education and the smallest salary.” Apparently not much has changed in 30 years.

I must congratulate these students, however, on following their passion rather than following the money. I can’t help but think that their lives, though stressful, may contain greater happiness.

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Lawyers & electricians & philosophers, oh my!

Kara Miller’s column on the career challenges for people with doctoral degrees generated more than 260 comments on Boston.Globe.com. The following is an edited sample of readers’ reactions:

Lots of law school grads are underemployed as well. (PL)

So true, PL. The market in Massachusetts is flooded with talented lawyers seeking work. (Roforma)

Supply and demand, the market at work. (guk)

Investing in education and research in all fields is the hallmark of a society with staying power. Disinvesting from these endeavors signals decline and decay. (Massachusetts citizen)

Electricians, plumbers, mechanics, and other skilled technical professions have no problems getting $100k jobs with great benefits. (ramsen)

Not enough turnover from tenured professors, leaving little space for new faculty. Although the tenured, well-established professors are needed, it’s the junior faculty who are hungry and with new ideas that help build new programs. The whole graduate program model is a bad model. I worked two jobs, had my tuition and some type of minimal student health insurance and could barely cover the rent with my stipend, and the second job paid for everything else. Though I was working on many faculty projects, it was the faculty who said this would be good for me. Never did they say it was also good for them. (TravelerofNJ2)

I just retired from a tenured faculty position in science. I’m in my early 70s. I have colleagues who are still doing what they do well into their 70s, a couple approaching 80. There is no active incentive from the university to move the older faculty on, to make way for a new generation. (Lola-lola)

The next step is for adjuncts to go on strike across the nation and hold colleges and universities accountable. The current system is completely absurd. (Wordsmith2358)

Universities should be required to release disclosure data about the fate of their PhD graduates. (davidman820)

I knew an attorney who managed a Cheesecake Factory. She had worked in food services through school. As an attorney, she really did not make that much money and was not doing the field of law of her choice. How many real estate closings can you do without dying of boredom? She went into management in the food industry and makes the same salary. (Antietem)

It was always a question and puzzling to me why people study philosophy. (Blazer27)

Globe Opinion