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The Department of Pure Mathematics and Mathematical Statistics (DPMMS) at Cambridge has a large number of faculty, postdocs and graduate students, with active research groups across the spectrum of mathematics. Strengths include:

• algebra (group theory, representation theory);
• algebraic geometry (higher-dimensional algebraic geometry, algebraic cycles, abelian varieties, mirror symmetry, geometric aspects of representation theory);
• analysis (geometric analysis, PDEs, mathematical relativity, discrete analysis);
• category theory (logic, proof theory, topos theory, higher-dimensional category theory)
• combinatorics (random structures, combinatorial number theory, Ramsey theory, graph theory)
• differential geometry (dynamics, low-dimensional topology and knot theory, gauge theory, Riemannian geometry, symplectic topology);
• number theory (algebraic number theory, Iwasawa theory, computational number theory);
• probability and statistics (including financial mathematics, operational research, stochastic methods, transport and networking problems).

More detailed information can be found on other departmental webpages. If you have interests in these areas, particularly in the areas of expertise of departmental members, consider coming to Cambridge for a postdoc. (Related research groups at Imperial College, London and at Oxford are close, and there are close links -- including shared participation in some seminars -- with these departments, which adds to the richness of the local mathematical community.) There are various kinds of postdoc available, suitable for people at different levels, and requiring different application procedures. Some more information on each is available below.

• HERCHEL-SMITH POSTDOCTORAL RESEARCH FELLOWSHIPS: A Fellowship in Mathematics is advertised as part of a University wide scheme. We expect to appoint one or more positions each year. The postdocs are tenable for 3 years and are on a competitive salary scale. The positions carry no teaching obligations, and are purely intended for research, although opportunities to give Masters-level or graduate-level courses are available. Applications from all areas of pure mathematics will be considered. Applicants should be towards the end of their PhD or within the first 2-3 years of postdoctoral research. Further particulars will be given on the departmental vacancies page when applications are open, which is typically around late October.
• COLLEGE RESEARCH FELLOWSHIPS: These are prestigious postdocs, funded directly by the Cambridge Colleges. Holders of the positions are fully integrated into DPMMS, with office space in the department as well as their college. The precise details of these positions vary from college to college, and their availability varies from year to year. Typically, colleges offer up to 4 positions, for a tenure of 3 or 4 years. All positions are research positions, with few or minimal teaching obligations; the positions are associated with membership of a college, and typically come with accommodation and other perquisites. In some but not all cases there may be criteria of eligibility (e.g. citizen of an EU member state, or graduand of a UK university, etc). All such positions are advertised in the Cambridge University Reporter: www.admin.cam.ac.uk/reporter/current/weekly/ ; check regularly to see if there are openings in Mathematics. Note that although these positions are competitive across subjects, mathematicians are often successful: in a typical year there are 3 or more pure mathematicians appointed to such positions from the various colleges. For a further view see Professor Körner's unofficial
• Guide for Mathematicians Seeking a Cambridge Research Fellowship
• 1851 ROYAL COMMISSION RESEARCH FELLOWSHIPS: 2-year postdoctoral fellowships in mathematics and biological/physical sciences, which can be held at any UK university. Applicants should recently have completed or be about to complete a PhD: there are strict eligibility criteria (UK or Commonwealth national, or citizen of Ireland or Pakistan). Further information and application deadlines can be found here.
• MARIE-CURIE POST-DOCTORAL FELLOWSHIPS: 2 year research-only postdoctoral positions, funded by the EU (to a generous salary), available to EU nationals to work in a country which is neither the country of their PhD nor their home country. Open calls for applications -- which are often but not always present -- can be found with some effort via the website ; follow links for Individual Researchers. Marie-Curie grant applications are made jointly with the department, and are directly linked to the research of a specific member of faculty. The first stage in applying is to make contact with such a potential supervisor and discuss possible projects and establish whether or not there is support for and scope for a plausibly successful application.
• ROYAL SOCIETY UNIVERSITY RESEARCH FELLOWSHIPS: Unique 8 year research-only fellowships. These are suitable for outstanding young mathematicians who have completed their PhD and already done one postdoc. Applications must have emphatic support from DPMMS; if you are interested in applying, contact a member of the department in a related subject. Further information is available from their website :     The deadline is typically in very early November of the preceeding year. Results are only available in April, but who else is going to give you 8 years money with no obligations?

POSSIBILITIES NEARBY: Imperial College London is only an hour from Cambridge, also has outstanding research groups in various fields (including the geometry group led by Simon Donaldson), and we often travel to and from one another's seminars. Oxford is slightly further but by no means inaccessible. Why not try applying there too? More information can be found on the websites https://www.imperial.ac.uk/mathematics/research/opportunities/ , and www.maths.ox.ac.uk/ . Example opportunities include:

• Chapman Fellowships at Imperial
• Colleges at the University of Oxford also offer Junior Research Fellowships, details are usually available via the Gazette www.ox.ac.uk/gazette/ .

Related sites

• Statistical Laboratory
• Faculty of Mathematics
• Isaac Newton Institute for Mathematical Sciences

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Welcome to the Math PhD program at Harvard University and the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Learn more about Harvard’s Math community and our statement on diversity and inclusion.

The Harvard Griffin GSAS Office of Equity, Diversity, Inclusion & Belonging offers student affinity groups for graduate students and many other resources.

The Harvard University Office for Gender Equity has dedicated GSAS Title IX resource coordinators who work with and support graduate students.

open. The application deadline is December 15, 2021. -->

The application deadline for fall 2024 admission has passed. Applications for fall 2025 admission will open in September 2024.

For information on admissions and financial support, please visit the Harvard Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Harvard Griffin GSAS is committed to ensuring that our application fee does not create a financial obstacle. Applicants can determine eligibility for a fee waiver by completing a series of questions in the Application Fee section of the application. Once these questions have been answered, the application system will provide an immediate response regarding fee waiver eligibility.

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Elaine Liu: Charging ahead

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MIT senior Elaine Siyu Liu doesn’t own an electric car, or any car. But she sees the impact of electric vehicles (EVs) and renewables on the grid as two pieces of an energy puzzle she wants to solve.

The U.S. Department of Energy reports that the number of public and private EV charging ports nearly doubled in the past three years, and many more are in the works. Users expect to plug in at their convenience, charge up, and drive away. But what if the grid can’t handle it?

Electricity demand, long stagnant in the United States, has spiked due to EVs, data centers that drive artificial intelligence, and industry. Grid planners forecast an increase of 2.6 percent to 4.7 percent in electricity demand over the next five years, according to data reported to federal regulators. Everyone from EV charging-station operators to utility-system operators needs help navigating a system in flux.

That’s where Liu’s work comes in.

Liu, who is studying mathematics and electrical engineering and computer science (EECS), is interested in distribution — how to get electricity from a centralized location to consumers. “I see power systems as a good venue for theoretical research as an application tool,” she says. “I'm interested in it because I'm familiar with the optimization and probability techniques used to map this level of problem.”

Liu grew up in Beijing, then after middle school moved with her parents to Canada and enrolled in a prep school in Oakville, Ontario, 30 miles outside Toronto.

Liu stumbled upon an opportunity to take part in a regional math competition and eventually started a math club, but at the time, the school’s culture surrounding math surprised her. Being exposed to what seemed to be some students’ aversion to math, she says, “I don’t think my feelings about math changed. I think my feelings about how people feel about math changed.”

Liu brought her passion for math to MIT. The summer after her sophomore year, she took on the first of the two Undergraduate Research Opportunity Program projects she completed with electric power system expert Marija Ilić, a joint adjunct professor in EECS and a senior research scientist at the MIT Laboratory for Information and Decision Systems.

Predicting the grid

Since 2022, with the help of funding from the MIT Energy Initiative (MITEI), Liu has been working with Ilić on identifying ways in which the grid is challenged.

One factor is the addition of renewables to the energy pipeline. A gap in wind or sun might cause a lag in power generation. If this lag occurs during peak demand, it could mean trouble for a grid already taxed by extreme weather and other unforeseen events.

If you think of the grid as a network of dozens of interconnected parts, once an element in the network fails — say, a tree downs a transmission line — the electricity that used to go through that line needs to be rerouted. This may overload other lines, creating what’s known as a cascade failure.

“This all happens really quickly and has very large downstream effects,” Liu says. “Millions of people will have instant blackouts.”

Even if the system can handle a single downed line, Liu notes that “the nuance is that there are now a lot of renewables, and renewables are less predictable. You can't predict a gap in wind or sun. When such things happen, there’s suddenly not enough generation and too much demand. So the same kind of failure would happen, but on a larger and more uncontrollable scale.”

Renewables’ varying output has the added complication of causing voltage fluctuations. “We plug in our devices expecting a voltage of 110, but because of oscillations, you will never get exactly 110,” Liu says. “So even when you can deliver enough electricity, if you can't deliver it at the specific voltage level that is required, that’s a problem.”

Liu and Ilić are building a model to predict how and when the grid might fail. Lacking access to privatized data, Liu runs her models with European industry data and test cases made available to universities. “I have a fake power grid that I run my experiments on,” she says. “You can take the same tool and run it on the real power grid.”

Liu’s model predicts cascade failures as they evolve. Supply from a wind generator, for example, might drop precipitously over the course of an hour. The model analyzes which substations and which households will be affected. “After we know we need to do something, this prediction tool can enable system operators to strategically intervene ahead of time,” Liu says.

Dictating price and power

Last year, Liu turned her attention to EVs, which provide a different kind of challenge than renewables.

In 2022, S&P Global reported that lawmakers argued that the U.S. Federal Energy Regulatory Commission’s (FERC) wholesale power rate structure was unfair for EV charging station operators.

In addition to operators paying by the kilowatt-hour, some also pay more for electricity during peak demand hours. Only a few EVs charging up during those hours could result in higher costs for the operator even if their overall energy use is low.

Anticipating how much power EVs will need is more complex than predicting energy needed for, say, heating and cooling. Unlike buildings, EVs move around, making it difficult to predict energy consumption at any given time. “If users don't like the price at one charging station or how long the line is, they'll go somewhere else,” Liu says. “Where to allocate EV chargers is a problem that a lot of people are dealing with right now.”

One approach would be for FERC to dictate to EV users when and where to charge and what price they'll pay. To Liu, this isn’t an attractive option. “No one likes to be told what to do,” she says.

Liu is looking at optimizing a market-based solution that would be acceptable to top-level energy producers — wind and solar farms and nuclear plants — all the way down to the municipal aggregators that secure electricity at competitive rates and oversee distribution to the consumer.

Analyzing the location, movement, and behavior patterns of all the EVs driven daily in Boston and other major energy hubs, she notes, could help demand aggregators determine where to place EV chargers and how much to charge consumers, akin to Walmart deciding how much to mark up wholesale eggs in different markets.

Last year, Liu presented the work at MITEI’s annual research conference. This spring, Liu and Ilić are submitting a paper on the market optimization analysis to a journal of the Institute of Electrical and Electronics Engineers.

Liu has come to terms with her early introduction to attitudes toward STEM that struck her as markedly different from those in China. She says, “I think the (prep) school had a very strong ‘math is for nerds’ vibe, especially for girls. There was a ‘why are you giving yourself more work?’ kind of mentality. But over time, I just learned to disregard that.”

After graduation, Liu, the only undergraduate researcher in Ilić’s MIT Electric Energy Systems Group, plans to apply to fellowships and graduate programs in EECS, applied math, and operations research.

Based on her analysis, Liu says that the market could effectively determine the price and availability of charging stations. Offering incentives for EV owners to charge during the day instead of at night when demand is high could help avoid grid overload and prevent extra costs to operators. “People would still retain the ability to go to a different charging station if they chose to,” she says. “I'm arguing that this works.”

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Related links.

• Electric Energy Systems Group
• MIT Energy Initiative
• Department of Electrical Engineering and Computer Science
• Department of Mathematics

Related Topics

• Electrical engineering and computer science (EECS)
• Mathematics
• Laboratory for Information and Decision Systems (LIDS)
• Undergraduate Research Opportunities Program (UROP)
• Renewable energy
• Electricity
• Electric vehicles
• Undergraduate

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The degree of Doctor of Philosophy (PhD) is the University's principal research degree for graduate students and is available in all faculties and departments.

A Cambridge PhD is intellectually demanding and you will need to have a high level of attainment and motivation to pursue this programme of advanced study and research.

In most faculties, a candidate is expected to have completed one year of postgraduate study, normally on a research preparation master's course, prior to starting a PhD.

Completion normally requires three or four years of full-time study, or at least five years of part-time study, including a probationary period.

Terms of research are normally consecutive and, for full-time students, require residency in Cambridge. Not all departments offer part-time research degrees.

Various routes to the PhD are possible and, if you are made an offer of admission, it will be made clear whether you are required to study for a master's degree or certificate in the first instance, or will be admitted directly to the probationary year for the PhD. You are registered for the PhD only after a satisfactory progress assessment at the end of the probationary year (five terms for part-time degrees). The assessment is designed also to focus your mind on the stages necessary for the completion of your research within the normal time limit and to address any structural problems that have arisen during the first year. Students must pass the first year assessment in order to continue their PhD study.

During your PhD, your effort will be focused on writing a dissertation. The word count of the dissertation is dependent on the department and the Student Registry or Educational Student Policy will be able to tell you the maximum word limit. This must represent a significant contribution to learning, for example through the discovery of new knowledge, the connection of previously unrelated facts, the development of a new theory, or the revision of older views, and must take account of previously published work on the subject. Some Cambridge dissertations go on to form the basis of significant publications.

Although you will spend long hours working independently, your department and College will both support you throughout your PhD. You are also able to attend regular seminars in your subject area and could be involved in teaching, perhaps giving seminars or supervising, or in the social life of your department and College.

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