phd in pure mathematics topics

Group 1 一类大学

 Grade requirement
均分要求75%  

Group 2 二类大学

 grade requirement
均分要求80% 

软科中国大学排名2022（总榜）或软科中国大学排名2023（总榜）排名前100的大学

非‘985工程’的其他 院校

以及以下两所大学：

University of Chinese Academy of Sciences 中国科学院大学
University of Chinese Academy of Social Sciences 中国社会科学院大学

Group 3 三类大学

 grade requirement
均分要求85% 

软科中国大学排名2022（总榜）或 软科中国大学排名2023（总榜）101-200位的大学

Group 1 一类大学

Grade requirement
均分要求75%  

Group 2 二类大学

grade requirement
均分要求80% 

Group 3 三类大学

grade requirement
均分要求85% 

Group 1 一类大学

 Grade requirement
均分要求75%  

Group 2 二类大学

grade requirement
均分要求80% 

Group 3 三类大学

grade requirement
均分要求85% 

Group 1 一类大学

Grade requirement
均分要求75%  

Group 2 二类大学

grade requirement
均分要求80% 

Group 3 三类大学

grade requirement
均分要求85% 

Group 4 四类大学

We will consider students from these institutions ONLY on a case-by-case basis with minimum 85% if you have a relevant degree and very excellent grades in relevant subjects and/or relevant work experience.

来自四类大学的申请人均分要求最低85%，并同时具有出色学术背景，优异的专业成绩，以及（或）相关的工作经验，将酌情考虑。

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What areas of pure mathematics research are best for a post-PhD transition to industry?

I have a student who is looking to start a PhD in pure mathematics. She is talented and motivated, and will do quite well. She is still in a phase of her development where she is still open to the possibility of working in a broad range of research areas. In order to offset the risk of not finding an academic job post-PhD she would like to write a dissertation that will give her increased likelihood of finding work in industry. She wants to do research in pure mathematics however, i.e. prefers proving theorems to writing code or testing models.

Question: What areas of pure mathematics research are best for a post-PhD transition to industry? Please be as specific as possible.

Answers to the questions here and here are certainly relevant, but these questions are obviously distinct from my question. I think this question is useful to the pure mathematics community in that it addresses the fact that there are so many qualified academic job applicants in recent years. (For this reason, I hope the community gives the question a chance.)

• soft-question
• 7 $\begingroup$ While I have to agree with the top answers, I would also recommend a few computationally relevant subjects with pure origins and applications: harmonic analysis (of the signal processing sort), algebraic topology (with an eye towards, e.g., persistent homology), some generalized blob in algorithms/theoretical CS/logic (many real-world offshoots), cryptologic mathematics, and last but not least: mathematical statistical physics (though this shares much with both probability and information theory, both in answers below). $\endgroup$ –  Steve Huntsman Commented Apr 2, 2014 at 13:15
• 6 $\begingroup$ For the internet industry, being able to write solid code in Python, Ruby, C, C++, Java, Javascript, etc., will go much further---and having coding training is an absolute must. That said: the discipline of thinking logically and thoroughly about a problem that mathematical research teaches us to do, is what proves valuable when combined with coding skills. $\endgroup$ –  Suvrit Commented Apr 6, 2014 at 14:40
• 1 $\begingroup$ Pure probability is very "analysis" and I doubt it will give you an advantage for real-world problems. $\endgroup$ –  Rafael Prince Commented Jun 22, 2017 at 4:24

12 Answers 12

Probability. A strong background in probability will permit to qualify for jobs in statistics and financial math. See AMS Notices where a lot of statistics about the employment of recent PhD is published yearly. And a salary survey.

EDIT. My second guess was combinatorics/coding theory or PDE. But my friend explained me that PURE combinatorics is not so hot in the (industrial) job market, coding theory is not pure math, and pure PDE is very different from numerical PDE, the last thing is of course in great demand.

EDIT2. Reply on Peter Shor's comment. The distinction between "pure" and "applied" math is not sharp. On my opinion, if a problem arises from a "real world application", it is applied math (like math physics, control theory, coding theory). If a problem arises from the inner logic of development of math then it is pure math (like Fermat's last theorem). But of course, the difference is fuzzy, and one can trace almost all math problems to the "real world". Frequently it happens like this: a problem arises in the real world, mathematicians like it, and start working on it, and then work and work, forgetting its real world origin. (Example: constructions with compass and ruler, etc.) Probability was also applied math at its origin. And it becomes pure math. Similar things we see in coding theory, math physics and computer science. A lot of "computer science" is pure math.

• 3 $\begingroup$ Excellent. I hadn't thought about looking at the AMS Notices employment statistics. Thanks! $\endgroup$ –  Jon Bannon Commented Apr 2, 2014 at 0:34
• 1 $\begingroup$ I agree. Even a pretty theoretical thesis in probability can open the door in the finance and insurance industry. My last student to graduate easily obtained a good position in an insurance company. $\endgroup$ –  Fabrice Baudoin Commented Apr 2, 2014 at 1:42
• $\begingroup$ statistics is a big deal in big data , datamining these days & also has strong ties to machine learning .... $\endgroup$ –  vzn Commented Apr 2, 2014 at 15:23
• 4 $\begingroup$ The question was about "pure mathematics". Statistics is not usually considered pure mathematics. So I answered the question as precisely as I could. A PhD in probability will qualify you for a job in statistics. $\endgroup$ –  Alexandre Eremenko Commented Apr 2, 2014 at 21:07
• 2 $\begingroup$ What do you mean coding theory is not pure math? Coding theory spans a spectrum from extremely pure math to extremely applied. Of course, you do run into the problem that pure coding theory is somewhat different from applied, but if you can find an advisor in the math department who does coding theory, I think this would be an excellent area. $\endgroup$ –  Peter Shor Commented Apr 6, 2014 at 12:07

I'd like to push back a bit on one of the presuppositions behind the question, namely that the best way to "hedge your bets" is to choose a topic whose mathematical content is most likely to be needed in whatever industrial job you might end up taking. At least in the United States, the dominant factor in getting a job in industry is the impression you make on whoever makes the hiring decision. For a job where your Ph.D. matters to the employer, chances are that you'll be asked to give a talk about your research, just as you would for an academic job. Based on my experience, I would say that the key things people look for in your talk are the following:

Can they understand what problem you worked on and why it's interesting?

Do you seem to really know your stuff and be able to solve problems that others would not be able to solve?

Are you enough of a generalist that you can communicate effectively with people outside your specialty and pick up new areas quickly?

Note that points 1 and 2 are not that different from what an academic employer is looking for, except that for an audience in industry you will typically need to work harder to achieve them.

With the above in mind, the advice I would give is this: First and foremost, you have to pick an area that you are very interested in. That gives you the best chance that you will get significant results and be able to communicate them with enthusiasm. Secondarily, it helps if you can work closely with someone who is not a mathematician, and ideally someone who has experience in industry. For example, if your advisor has collaborators in other fields then you may want to get one of them on your thesis committee, and get their perspective on what sorts of questions are interesting in their field. This doesn't even have to be a field that you're hoping to eventually get a job in; the key thing is to accumulate some experience with talking to people outside mathematics and figuring out what kinds of things they find appealing, as well as how to pitch your work in a way that gets their attention.

Of course, professors in so-called applied areas of math are more likely to have the right kinds of connections, so by following my advice you have an elevated probability of winding up in one of the areas that other responders have mentioned. But my point is that your mindset shouldn't be, "I must learn specific mathematical facts that I will need to know later"; rather, it should be, "I must learn how to communicate with non-mathematicians and demonstrate that I can solve whatever problems they need solved."

• 1 $\begingroup$ Right on the money! I wish I could upvote this $k$ times for a largeish $k$. $\endgroup$ –  Felix Goldberg Commented Apr 6, 2014 at 4:59

If you know coding you can even escape without even having any expertise (or having minimal knowledge) in probability (like me). I didn't end up into things that involved lots of probability theory simply because I didn't liked it (I got offers from finanical sectors but I rejected the job and preferred to be more into technical area of the industry). If you know how to program then fields such as coding theory, number theory and even algorithmic and algebraic number theory does help. PDE and Analysis obviously helps a lot in the areas of engineering.

I think the secret is to know how to write a computer program and implement it yourself, because from my experience in the industry: colleagues (esp. from non-mathematical fields) will not believe in your ability until you give them a tangible result and you can only show that something works from a mathematical theory you have in mind if you show them in practice and not if you prove it on paper. This usually involves programming and developing an application that applies your theory, unless you want to end up in a lab and do experiments which is not fun at all. You can almost forget the idea of other colleagues developing an algorithm based on your theories. I remember long time ago I had an idea of a mathematical algorithm that ended up saving my company around 400,000 euros each year but I didn't get any support from my colleagues (who where mostly compute engineers) and so I just programmed my algorithm in my free time and then showed the result when I finally developed the program.

Some of my former colleagues (and friends) working in Monte Carlo theory and Quasi-Monte Carlo theory have successfully moved to the banking/insurance sector after finishing their PhD. The first subject could be seen as a branch of probablity theory, the second as a branch of number theory/numerical mathematics. In both fields you can do "pure" mathematics and still successfully convince people that it is very important for applications.

As someone who had a PhD in one of the more obscure area in math and is working in industry, I have to say in all of my job finding efforts (software, banks, hedgefunds), the math area that was the focus of my PhD has never come into play. What would have helped me get the job easier ? More programming experience, more statistics, but those aren't pure maths. I also find it hard to believe that specializing in one area of pure math will give you some advantage in solving a real world problem over other areas. Most real world problems don't require knowledge that is terribly deep, it shouldn't take more than a few months to equalize whatever knowledge gap one might have. The rest, which is much more important, is the hands on experience. Would a combinatorics Phd has some sort of inherent advantage in tackling a real world algorithm problem over an algebraic geometry student. I don't think so.

Bottom line, spend few hours a week learning programming & statistics (the applied one, not the theory of statistics). Then still specialize in the area of maths that she really likes.

I have thought a bit about this question because I have been contemplating this transition for myself. Here are a few ideas:

-Stochastic analysis: this is a good area of expertise if she hopes to go into finance (though I should include the caveat that recruiters for big finance firms have admitted to me that for recent graduates they filter more by prestige of academic credentials than by research area). There also seem to be many good problems in this area, for instance involving the application of stochastic PDE to differential geometry.

-Machine learning: this is technically an area of computer science, but there are a lot of important theoretical problems and these problems are really math problems (indeed many people working in this area come from a pure math background). You might suggest she take a graduate class in machine learning if her university has one; one doesn't need to penetrate the literature too deeply to find good math problems. This sort of background could help her pursue a career in data science, for instance.

-Mathematical biology: I don't know much this area except that there seem to be lots of jobs for people who know how to model proteins and cells. I've known students who have been jointly advised by a mathematician and a biologist, and this seemed to work out pretty well.

Whatever else happens, if she is serious about this I would strongly encourage her to learn how to code if she hasn't already. I think there was a time when a mathematician could reasonably hope to get a job designing models or algorithms for engineers to implement, but that time seems to have passed. At the very least it will make her a much more flexible and attractive applicant.

• 5 $\begingroup$ I wouldn't define machine learning and mathematical biology "pure mathematics". $\endgroup$ –  Federico Poloni Commented Apr 2, 2014 at 13:47
• 4 $\begingroup$ Stochastic analysis arguably isn't either. The point is that these areas generate lots of pure math problems - e.g. machine learning produces problems in convex geometry and functional analysis and mathematical biology produces problems in dynamical systems and information theory. It is too much to ask for an area of pure math which is oriented towards solving applied math problems (that is what "applied math" means); instead one should look for sources of pure math problems in areas of interest to industry. $\endgroup$ –  Paul Siegel Commented Apr 2, 2014 at 15:47
• $\begingroup$ Machine learning, and now its "bigger cousin", Data Science is extremely hot---it's hard to find data scientists who are not only great at coding, but also have substantial mathematical maturity (but giving the moving nature of "data science" ultimately someone who is flexible is more valuable).. $\endgroup$ –  Suvrit Commented Apr 6, 2014 at 14:43

I once did most of a PhD, and bailed out to take an industry job, and I'm still commercial, although maths is my hobby. That was around twenty years ago.

I'm a contractor/consultant, so I've been to loads of interviews and interview-like situations. As far as I remember, no-one has ever asked what my PhD was in, or even cared about the fact that I never finished it.

What they do care about is the sorts of programs I can write, and occasionally how good I am at solving toy problems. Mostly they look for ability in languages like C or python. When the people who are offering the jobs are themselves computer science graduates or very good programmers, they care about whether I can write in one of the functional languages, say Ocaml, Haskell, or Lisp (any one will do, or anything similar).

Mathematicians are always good at solving the sorts of toy problems you get in interviews. I'd recommend 'Are you smart enough to work at Google' for lots of examples of the sorts of things interviewers ask. She'll read it in an afternoon and enjoy it.

Writing programs tends to be seriously illuminating when you're trying to understand a mathematical idea.

So I'd advise her to do some piece of maths which deals with problems where you can use a computer to get answers. (I'm pretty sure that's all areas of maths, actually, but maybe there are some subjects where a computer would be useless. In which case I'd be worried that they weren't actually about anything.).

And she should try to write programs and make pictures about her field. And she should write these programs in C, python, and lisp, to get the feel for the differences between them. Mathematicians tend to love lisp. My favourite flavours are scheme and clojure. If she has any matrices to multiply, she should try MATLAB/Octave as well.

She should also play with maxima/mathematica/maple if she does a lot of symbolic calculation, and R if she does statistics or likes to make nice graphs.

I think if she does this she'll have a serious advantage over non-programming mathematicians in maths itself, and if she eventually goes into industry, she'll have picked up the crucial skills by magic, and have enjoyed the process.

As far as which area to choose goes, a Maths PhD is such a dreadful experience if you don't love your subject that the only possible advice is 'either do something you're obsessively interested in, or don't do one at all'.

If she has more than one thing she's obsessed about, pick the one that's more amenable to understanding by writing programs.

• 2 $\begingroup$ Thinking about it, various banks in London keep getting in touch with me offering silly money just because I speak Clojure and can do maths. I turn them down because I hate big cities. How employable does she want to be? $\endgroup$ –  John Lawrence Aspden Commented Apr 6, 2014 at 22:44
• $\begingroup$ I can ask her and get back to you! Thanks for the thoughtful answer. $\endgroup$ –  Jon Bannon Commented Apr 6, 2014 at 23:13

I think combinatorics is also good for a post-PhD transition to industry. It is closely related to the computer science. For example, combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of different kinds of algorithms. So when you go to an IT company and you need to write the code or design an algorithm, a good background in combinatorics will help.

And one part of the combinatorics, the graph theory, also has numerous natural connections to other areas. For example, graph theory gives a model for the network. So I think if you go to the industry to seek a job for companies related the network, having some knowledge of graph theory may be helpful.

I have to say that maybe combinatorics are not so close to the industry as the probability, since the theorists of combinatorics are mainly interested in the intrinsic property of a discrete structure, and in industry people often use discrete structure such as graphs to solve problems in reality. Though there is a little work to do for a PhD student to find job in the industry, but a good understanding of combinatorics, I think, will be helpful. There are a few examples of master students and PhD students which transition to industry in our department, so maybe this is a way.

• 5 $\begingroup$ I find this a bit of a stretch. $\endgroup$ –  Anthony Quas Commented Apr 2, 2014 at 4:53
• 3 $\begingroup$ @Anthony Quas: In our department, for PhD students doing research in the pure math, combinatorics is one of easiest subject for them to find jobs in the industry, so I proposed this answer. Maybe my explanation is not sufficient, so I've added a few words to express my understanding. $\endgroup$ –  Siming Tu Commented Apr 2, 2014 at 5:41
• 3 $\begingroup$ @SimingTu I can confirm your suggestion; I work in the routing business and there are several challenging problems from graph theory, combinatorial optimization. What a good advice is, also depends on the character of the person. If someone likes to figure out hidden laws, then statistics is certainly a good recommendation, if she however likes to bring things to perfection, then (combinatorial) optimization would be a better suggestion. $\endgroup$ –  Manfred Weis Commented Apr 2, 2014 at 6:18
• 2 $\begingroup$ I have been working on a project involving graph clustering for a company that specializes in social media analysis. They have been very receptive to this, in part because there has been a lot of demand lately for network analysis tools from their clients. $\endgroup$ –  Paul Siegel Commented Apr 2, 2014 at 10:36
• 4 $\begingroup$ As someone who works as a software engineer, I find the claim that having PhD-level training in combinatorics will help you analyze the complexity of algorithms and therefore write better code flat out wrong. There are anecdotal examples where it might work out, but it's not playing the averages. I wouldn't get a PhD in combinatorics to learn how to write good software - I would spend 5 or 6 years writing software. $\endgroup$ –  James Kingsbery Commented Apr 3, 2014 at 14:06

If your definition of pure math is "proving theorems", she can work in the following (math related to the following):

1). Control theory (this can mean doing dynamical systems, PDEs, etc): PhDs hired by several leading companies such as Ford, Bosch, Honda, United Technologies, Boeing, NASA

2). Information theory and networks : Internet companies

3). Optimization and/or nonlinear programming: PhDs hired by numerous companies

4). Communication theory: PhDs hired by communication giants

5). Stochastic geometry: Same as 4

6). Probability: Finance/banks/internet

I suggest looking at the top journals of each of these fields, it will give you an idea of the type of mathematics that goes on in there.

• 2 $\begingroup$ +1 for control theory. See this document by JM Coron for some interesting real world applications of the theory, such as regulating rivers. $\endgroup$ –  Willie Wong Commented Apr 2, 2014 at 14:39

I am not answering your question directly, but will address another consideration which is what is the proper industrial environment to work in. I have strong opinions:

• Never work in a situation where your superior does not have a degree equal to or greater than yours. This occurs often in industry wherein your manager has a BS or possibly MS. The managers are well intended, but they have a significantly different mindset. Not only are they motivated only by money and time but they have very little conception of research and development and the creativity of original thinking. In this situation you will very possibly be considered no more than just another engineer.
• Once into industry, it will be very difficult to enter academia. The longer you stay out of academia, the more difficult. The only exception is if you are one of the especially lucky/gifted scientists who are able to continue publishing and have demonstrated raising grant money.
• If you are not an especially social and outward going person, I would not recommend going into industry. It is just a matter of human nature that people feel more comfortable around someone they socialize and converse with. Promotion in industry has a lot to do with your superiors feeling comfortable with you. Introverts do not get promoted no matter how much superior work they do.

Having said that, it is sad to see how academia has degenerated into a self-protecting tenured focused institution. So maybe industry is a viable alternative. But please take my recomendations into serious consideration.

• 4 $\begingroup$ I am not sure if I would agree with all things you write here. But I do agree on the first one, because of my own experience. I get bosses from those who cannot count to those who believe if you double the PCs in the office the algorithms will converge faster. I partially agree in 3. There is more unfairness in the industry where power-struggle is almost a must (some industry are fairer and some aren't, but these things changes very fast in the industry). $\endgroup$ –  Jose Capco Commented Apr 7, 2014 at 10:12

A PhD at the interface between logic and theoretical informatics (for example Lambda calculus ) is pure mathematics, and surely interests fashionable companies like Google or others...

• 4 $\begingroup$ I am not 100% sure - this might be a bit too theoretical for them. $\endgroup$ –  Felix Goldberg Commented Apr 2, 2014 at 11:20
• 2 $\begingroup$ I know at least one person with a PhD in computability theory who now works at Google. $\endgroup$ –  Benedict Eastaugh Commented Apr 2, 2014 at 13:08
• 1 $\begingroup$ Sure, if you can convert it to Haskell code. $\endgroup$ –  Tom LaGatta Commented Apr 2, 2014 at 23:46
• 2 $\begingroup$ I don't think a PhD in lambda calculus would be any more use than a PhD in number theory for a programming job. $\endgroup$ –  anon Commented Apr 3, 2014 at 7:30

You say that your student "... would like to write a dissertation that will give her increased likelihood of finding work in industry. She wants to do research in pure mathematics however ..." . The assumption is that she isn't really motivated to work outside academia, and would only take an industry job as a last resort.

That attitude/approach is just not appealing to employers. It's possible to get a job in industry with a mathematics PhD, but you have to want to do it in the first place. If an interviewer asks why you applied for the job, and you say "Well, I couldn't get a postdoc" then it's game over.

For the rest, I agree with Timothy Chow's answer.

• $\begingroup$ Having interests in both pure mathematics research and work in industry is possible. And even if industry is a second option, there's nothing wrong with that. The right thing to do in that case is not to disclose that to potential employers, but instead for example create the impression you are no longer interested in academia and more interested in "real work". $\endgroup$ –  spin Commented Jun 22, 2017 at 12:47
• $\begingroup$ @spin Unfortunately the ability to dissemble is not one that's generally included in a typical pure mathematics PhD program. In any case, an experienced interviewer is likely to probe beneath the faked enthusiasm for "real work". $\endgroup$ –  Phil Harmsworth Commented Jun 23, 2017 at 5:16

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

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Pure Mathematics has a long tradition of excellence at Manchester. The 1920s and 30s saw Manchester become one of the world's leading centres for number theory, with Louis Mordell and Kurt Mahler holding chairs here. In 1945 Max Newman arrived from code-breaking work at Bletchley Park and ensured the growth in eminence of the department, recruiting stars such as the logician Alan Turing , often considered to be the father of artificial intelligence, and the topologist Frank Adams . Manchester also has a long tradition in algebra, through the work of leading figures such as Bernhard Neumann , Hanna Neumann and Brian Hartley .

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In the Department of Mathematics we offer a range of scholarships, studentships and awards to support UK and overseas postgraduate researchers.

Funding is also available at university and faculty level and can be viewed on our funding page . Alternatively, you can use our funding database to find scholarships, studentships and awards you may be eligible for.

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Mathematics PhD theses

A selection of Mathematics PhD thesis titles is listed below, some of which are available online:

2023   2022   2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Reham Alahmadi - Asymptotic Study of Toeplitz Determinants with Fisher-Hartwig Symbols and Their Double-Scaling Limits

Anne Sophie Rojahn –  Localised adaptive Particle Filters for large scale operational NWP model

Melanie Kobras –  Low order models of storm track variability

Ed Clark –  Vectorial Variational Problems in L∞ and Applications to Data Assimilation

Katerina Christou – Modelling PDEs in Population Dynamics using Fixed and Moving Meshes  

Chiara Cecilia Maiocchi –  Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

Samuel R Harrison – Stalactite Inspired Thin Film Flow

Elena Saggioro – Causal network approaches for the study of sub-seasonal to seasonal variability and predictability

Cathie A Wells – Reformulating aircraft routing algorithms to reduce fuel burn and thus CO 2 emissions  

Jennifer E. Israelsson –  The spatial statistical distribution for multiple rainfall intensities over Ghana

Giulia Carigi –  Ergodic properties and response theory for a stochastic two-layer model of geophysical fluid dynamics

André Macedo –  Local-global principles for norms

Tsz Yan Leung  –  Weather Predictability: Some Theoretical Considerations

Jehan Alswaihli –  Iteration of Inverse Problems and Data Assimilation Techniques for Neural Field Equations

Jemima M Tabeart –  On the treatment of correlated observation errors in data assimilation

Chris Davies –  Computer Simulation Studies of Dynamics and Self-Assembly Behaviour of Charged Polymer Systems

Birzhan Ayanbayev –  Some Problems in Vectorial Calculus of Variations in L∞

Penpark Sirimark –  Mathematical Modelling of Liquid Transport in Porous Materials at Low Levels of Saturation

Adam Barker –  Path Properties of Levy Processes

Hasen Mekki Öztürk –  Spectra of Indefinite Linear Operator Pencils

Carlo Cafaro –  Information gain that convective-scale models bring to probabilistic weather forecasts

Nicola Thorn –  The boundedness and spectral properties of multiplicative Toeplitz operators

James Jackaman  – Finite element methods as geometric structure preserving algorithms

Changqiong Wang - Applications of Monte Carlo Methods in Studying Polymer Dynamics

Jack Kirk - The molecular dynamics and rheology of polymer melts near the flat surface

Hussien Ali Hussien Abugirda - Linear and Nonlinear Non-Divergence Elliptic Systems of Partial Differential Equations

Andrew Gibbs - Numerical methods for high frequency scattering by multiple obstacles (PDF-2.63MB)

Mohammad Al Azah - Fast Evaluation of Special Functions by the Modified Trapezium Rule (PDF-913KB)

Katarzyna (Kasia) Kozlowska - Riemann-Hilbert Problems and their applications in mathematical physics (PDF-1.16MB)

Anna Watkins - A Moving Mesh Finite Element Method and its Application to Population Dynamics (PDF-2.46MB)

Niall Arthurs - An Investigation of Conservative Moving-Mesh Methods for Conservation Laws (PDF-1.1MB)

Samuel Groth - Numerical and asymptotic methods for scattering by penetrable obstacles (PDF-6.29MB)

Katherine E. Howes - Accounting for Model Error in Four-Dimensional Variational Data Assimilation (PDF-2.69MB)

Jian Zhu - Multiscale Computer Simulation Studies of Entangled Branched Polymers (PDF-1.69MB)

Tommy Liu - Stochastic Resonance for a Model with Two Pathways (PDF-11.4MB)

Matthew Paul Edgington - Mathematical modelling of bacterial chemotaxis signalling pathways (PDF-9.04MB)

Anne Reinarz - Sparse space-time boundary element methods for the heat equation (PDF-1.39MB)

Adam El-Said - Conditioning of the Weak-Constraint Variational Data Assimilation Problem for Numerical Weather Prediction (PDF-2.64MB)

Nicholas Bird - A Moving-Mesh Method for High Order Nonlinear Diffusion (PDF-1.30MB)

Charlotta Jasmine Howarth - New generation finite element methods for forward seismic modelling (PDF-5,52MB)

Aldo Rota - From the classical moment problem to the realizability problem on basic semi-algebraic sets of generalized functions (PDF-1.0MB)

Sarah Lianne Cole - Truncation Error Estimates for Mesh Refinement in Lagrangian Hydrocodes (PDF-2.84MB)

Alexander J. F. Moodey - Instability and Regularization for Data Assimilation (PDF-1.32MB)

Dale Partridge - Numerical Modelling of Glaciers: Moving Meshes and Data Assimilation (PDF-3.19MB)

Joanne A. Waller - Using Observations at Different Spatial Scales in Data Assimilation for Environmental Prediction (PDF-6.75MB)

Faez Ali AL-Maamori - Theory and Examples of Generalised Prime Systems (PDF-503KB)

Mark Parsons - Mathematical Modelling of Evolving Networks

Natalie L.H. Lowery - Classification methods for an ill-posed reconstruction with an application to fuel cell monitoring

David Gilbert - Analysis of large-scale atmospheric flows

Peter Spence - Free and Moving Boundary Problems in Ion Beam Dynamics (PDF-5MB)

Timothy S. Palmer - Modelling a single polymer entanglement (PDF-5.02MB)

Mohamad Shukor Talib - Dynamics of Entangled Polymer Chain in a Grid of Obstacles (PDF-2.49MB)

Cassandra A.J. Moran - Wave scattering by harbours and offshore structures

Ashley Twigger - Boundary element methods for high frequency scattering

David A. Smith - Spectral theory of ordinary and partial linear differential operators on finite intervals (PDF-1.05MB)

Stephen A. Haben - Conditioning and Preconditioning of the Minimisation Problem in Variational Data Assimilation (PDF-3.51MB)

Jing Cao - Molecular dynamics study of polymer melts (PDF-3.98MB)

Bonhi Bhattacharya - Mathematical Modelling of Low Density Lipoprotein Metabolism. Intracellular Cholesterol Regulation (PDF-4.06MB)

Tamsin E. Lee - Modelling time-dependent partial differential equations using a moving mesh approach based on conservation (PDF-2.17MB)

Polly J. Smith - Joint state and parameter estimation using data assimilation with application to morphodynamic modelling (PDF-3Mb)

Corinna Burkard - Three-dimensional Scattering Problems with applications to Optical Security Devices (PDF-1.85Mb)

Laura M. Stewart - Correlated observation errors in data assimilation (PDF-4.07MB)

R.D. Giddings - Mesh Movement via Optimal Transportation (PDF-29.1MbB)

G.M. Baxter - 4D-Var for high resolution, nested models with a range of scales (PDF-1.06MB)

C. Spencer - A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table.

P. Jelfs - A C-property satisfying RKDG Scheme with Application to the Morphodynamic Equations (PDF-11.7MB)

L. Bennetts - Wave scattering by ice sheets of varying thickness

M. Preston - Boundary Integral Equations method for 3-D water waves

J. Percival - Displacement Assimilation for Ocean Models (PDF - 7.70MB)

D. Katz - The Application of PV-based Control Variable Transformations in Variational Data Assimilation (PDF- 1.75MB)

S. Pimentel - Estimation of the Diurnal Variability of sea surface temperatures using numerical modelling and the assimilation of satellite observations (PDF-5.9MB)

J.M. Morrell - A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations (PDF-7.7MB)

L. Watkinson - Four dimensional variational data assimilation for Hamiltonian problems

M. Hunt - Unique extension of atomic functionals of JB*-Triples

D. Chilton - An alternative approach to the analysis of two-point boundary value problems for linear evolutionary PDEs and applications

T.H.A. Frame - Methods of targeting observations for the improvement of weather forecast skill

C. Hughes - On the topographical scattering and near-trapping of water waves

B.V. Wells - A moving mesh finite element method for the numerical solution of partial differential equations and systems

D.A. Bailey - A ghost fluid, finite volume continuous rezone/remap Eulerian method for time-dependent compressible Euler flows

M. Henderson - Extending the edge-colouring of graphs

K. Allen - The propagation of large scale sediment structures in closed channels

D. Cariolaro - The 1-Factorization problem and same related conjectures

A.C.P. Steptoe - Extreme functionals and Stone-Weierstrass theory of inner ideals in JB*-Triples

D.E. Brown - Preconditioners for inhomogeneous anisotropic problems with spherical geometry in ocean modelling

S.J. Fletcher - High Order Balance Conditions using Hamiltonian Dynamics for Numerical Weather Prediction

C. Johnson - Information Content of Observations in Variational Data Assimilation

M.A. Wakefield - Bounds on Quantities of Physical Interest

M. Johnson - Some problems on graphs and designs

A.C. Lemos - Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts

R.K. Lashley - Automatic Generation of Accurate Advection Schemes on Structured Grids and their Application to Meteorological Problems

J.V. Morgan - Numerical Methods for Macroscopic Traffic Models

M.A. Wlasak - The Examination of Balanced and Unbalanced Flow using Potential Vorticity in Atmospheric Modelling

M. Martin - Data Assimilation in Ocean circulation models with systematic errors

K.W. Blake - Moving Mesh Methods for Non-Linear Parabolic Partial Differential Equations

J. Hudson - Numerical Techniques for Morphodynamic Modelling

A.S. Lawless - Development of linear models for data assimilation in numerical weather prediction .

C.J.Smith - The semi lagrangian method in atmospheric modelling

T.C. Johnson - Implicit Numerical Schemes for Transcritical Shallow Water Flow

M.J. Hoyle - Some Approximations to Water Wave Motion over Topography.

P. Samuels - An Account of Research into an Area of Analytical Fluid Mechnaics. Volume II. Some mathematical Proofs of Property u of the Weak End of Shocks.

M.J. Martin - Data Assimulation in Ocean Circulation with Systematic Errors

P. Sims - Interface Tracking using Lagrangian Eulerian Methods.

P. Macabe - The Mathematical Analysis of a Class of Singular Reaction-Diffusion Systems.

B. Sheppard - On Generalisations of the Stone-Weisstrass Theorem to Jordan Structures.

S. Leary - Least Squares Methods with Adjustable Nodes for Steady Hyperbolic PDEs.

I. Sciriha - On Some Aspects of Graph Spectra.

P.A. Burton - Convergence of flux limiter schemes for hyperbolic conservation laws with source terms.

J.F. Goodwin - Developing a practical approach to water wave scattering problems.

N.R.T. Biggs - Integral equation embedding methods in wave-diffraction methods.

L.P. Gibson - Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model.

A.K. Griffith - Data assimilation for numerical weather prediction using control theory. .

J. Bryans - Denotational semantic models for real-time LOTOS.

I. MacDonald - Analysis and computation of steady open channel flow .

A. Morton - Higher order Godunov IMPES compositional modelling of oil reservoirs.

S.M. Allen - Extended edge-colourings of graphs.

M.E. Hubbard - Multidimensional upwinding and grid adaptation for conservation laws.

C.J. Chikunji - On the classification of finite rings.

S.J.G. Bell - Numerical techniques for smooth transformation and regularisation of time-varying linear descriptor systems.

D.J. Staziker - Water wave scattering by undulating bed topography .

K.J. Neylon - Non-symmetric methods in the modelling of contaminant transport in porous media. .

D.M. Littleboy - Numerical techniques for eigenstructure assignment by output feedback in aircraft applications .

M.P. Dainton - Numerical methods for the solution of systems of uncertain differential equations with application in numerical modelling of oil recovery from underground reservoirs .

M.H. Mawson - The shallow-water semi-geostrophic equations on the sphere. .

S.M. Stringer - The use of robust observers in the simulation of gas supply networks .

S.L. Wakelin - Variational principles and the finite element method for channel flows. .

E.M. Dicks - Higher order Godunov black-oil simulations for compressible flow in porous media .

C.P. Reeves - Moving finite elements and overturning solutions .

A.J. Malcolm - Data dependent triangular grid generation. .

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 125 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.

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

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Graduate Student Issues, math graduate admissions

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Some of our research students are funded via the EPSRC Mathematical Sciences Doctoral Training Partnership, some are funded by University studentships, others are self-funded.

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Choosing a PhD topic [closed]

My supervisor and I recently had a long chat about PhD related stuff. He said something to the extent that your chances of employment after finishing your PhD among other factors depends on the topic of your PhD. The reason he mentioned was that for certain fields there aren't that many open positions. He mentioned that there are many open positions in the fields of Algebraic Geometry and also in Number Theory.

(I will enumerate the questions to make it easier to answer)

The following are my questions:

(1) Now I am wondering are there any other fields where research is comparably active? Concretely, what are the top topic in pure maths apart from AG and number theory that are "in"?

(answer below does not address this question)

(2) And in particular: are differential geometry and differential topology among them?

(answer below does not address this question either)

(3) And which are the fields with the least open positions for lecturers?

I am passionate about more than one field hence I would like to choose the optimal topic from among the topics I am passionate about. I am prepared to move to just about anywhere. To make answers more precise and useful for others too maybe you could include geographical information in your answer.

• soft-question
• career-development

• 1 $\begingroup$ I think the answer depends on country where you reside. $\endgroup$ –  user63181 Commented Apr 12, 2013 at 8:12
• 2 $\begingroup$ I am looking for a Postdoc at the moment. Let me tell you: the pickings are slim! $\endgroup$ –  user1729 Commented Apr 12, 2013 at 9:01
• 3 $\begingroup$ @SamiBenRomdhane: Academics dream of working in their native continent - they dare not dream of working in their native country, for such a dream so rarely comes true...... $\endgroup$ –  user1729 Commented Apr 12, 2013 at 9:03
• 6 $\begingroup$ Questions like this one are asked and answered on Academia regularly. You may want to browse the questions there for more input on this topic. $\endgroup$ –  Ben Norris Commented Apr 12, 2013 at 10:59
• 1 $\begingroup$ @user1729: Slim Pickens is slim! Yeee-hawwww! $\endgroup$ –  Asaf Karagila ♦ Commented Apr 12, 2013 at 12:03

3 Answers 3

If I have any advice for pursuing your Ph.D., choose something about which you are passionate and will maintain that passion over a period of 3-5 years. If being employed after you finish is the main concern, keep this in mind: your success in your work will depend on your ability to learn new things, to finish small projects, to keep your verbal and written commitments, and to communicate with your colleagues.

I really feel that, while your field of expertise is important, you will be best served by working at something that your truly enjoy and will succeed at doing. I have seen too many Ph.D. prospects flounder because such prospects treated their Ph.D. as a professional program . Such people largely end up miserable because a Ph.D. takes too much time and dedication to treat as a hurdle in the way of your career development. Rather, it is an opportunity to pursue something about which you simply want to do for its own sake, because you love doing it. If you are lucky enough to find the right combination of research opportunity and compatible advisor, then the career success will come by itself.

That all said, there is certainly a variation in opportunities over subfields. I am not a professional mathematician and therefore do not have a feel for which positions currently have the best/worst opportunities for lecturing. As far as which subfields in pure math are the "hottest," keep in mind that some of the heat is driven by non-academic considerations. Many mathematicians I know who are not in academia pursue computational geometry (believe it or not, but there are quite a few opportunities in semiconductors and electronics) and number theory (network security). Lecturing opportunities in these fields are likely the best because of the employment demand (which goes against what I said above, sadly). I am in the United States, so there may be different opportunities in other countries.

It sounds like you are interested in differential geometry and topology. Your advisor is correct: there are few positions in academia. But as I said above, if you focus on what you really want to do in graduate school and publish great work and make a name for yourself, you will find that, if you have geographical flexibility, you will find good work post-graduation.

• 5 $\begingroup$ On reading your answer your second time it contains no helpful or useful information pertaining to my question. I apologise as I am about to downvote. Feel free to edit and I will gladly remove the down vote. $\endgroup$ –  tom b. Commented Apr 12, 2013 at 8:44
• 18 $\begingroup$ @tomb.: I am sorry if the answer is not what you had in mind. I am not a professional mathematician and do not have access to employment statistics for various subfields in mathematics. I do, however, have experience in getting a Ph.D. in a technical field, as well as in getting and keeping jobs and hiring Ph.D.'s. I am in close contact with lecturers and professors with whom I studied and I feel my advice is relevant to your question, rather than being a "disaster of an entirely unhelpful answer." Perhaps if you had a more open mind and were less rude, you might find more advice coming. $\endgroup$ –  Ron Gordon Commented Apr 12, 2013 at 8:52
• 5 $\begingroup$ @tomb. you have a small time window to revert your vote to prevent accidental misclicks. The system is not designed to handle situations where a user votes before actually carefully reading the answers. Take it as a lesson learned and be more careful with your votes in the future. $\endgroup$ –  Willie Wong Commented Apr 12, 2013 at 8:53
• 15 $\begingroup$ @tomb. Are you serious?! $\endgroup$ –  Cocopuffs Commented Apr 12, 2013 at 9:10
• 22 $\begingroup$ @tomb.: Calling my answer a "disaster" is not exactly diplomatic. How was it a disaster? Did I mislead you, or injure you? Was I rude? You could have told me simply that I didn't really answer your question, and could I please elaborate on this or that point. Look, I am not angry and am happy to help you; I am here because I love being here. But, as I said, effective communication is a key component to being a successful mathematician (or physicist, engineer, etc.), so reflect on what you write before clicking "add." I wish you only the very best of luck and success. $\endgroup$ –  Ron Gordon Commented Apr 12, 2013 at 9:13

I will not tell you the answer to any of your questions, since I do not feel comfortable drawing conclusions in broad strokes from the data available to me . But I'll show you some data, and tell you how to find some more, so you can make up your mind for yourself.

The American Mathematical Society produces an Annual Survey of mathematical sciences, and among it contains information on the fields of studies of new doctoral recipients and their hiring statistics . If you go to this page you will find "Supplemental Table E.3" showing last year's employment statistics of new PhDs by field of specialty. From there you see that

• Applied maths is much more employed over all, with specifically biostatistics/statistics leading the field, this undoubtedly having to do with its industry applications.
• Of the traditional pure maths, the most popular is grouped under "Algebra/number theory", and in second place is "geometry/topology" (good for you!).
• You also see that pretty much across the board for the pure fields, around 10% of the students were still seeking employment at the time of the survey.

(This last point actually brings up something interesting: while your advisor is right that certain fields are more popular than others in terms of having more research interest and more jobs available, be aware that this also means that those fields usually have more people competing for those jobs. On average you are much better off trying to find a subject you enjoy and are good at, instead of finding a subject that has more absolute number of jobs. This is sort of the standard advice that you would get everywhere.)

(I am also slightly surprised to see that the statistics have not improved since the economic bubble of 2008; for the more applied fields it seems the situation has slightly improved, though this is drawing on just 4 data points.)

In addition to the AMS data, an imperfect proxy for research interest is the number of arXiv postings per period of time. Of course you will need to adjust by typical length of paper, and other field-dependent cultural aspects to make the numbers really meaningful.

For employment availability, however, a simple way is to browse through the job postings at MathJobs . I do not know if there is a way to quickly filter by fields and such, but at worst you just have to read through every single job posting and categorize them yourself.

• 1 $\begingroup$ Biostatistics - forgot about that one. Very popular option right now, I see lots of movement there in academia. $\endgroup$ –  Ron Gordon Commented Apr 12, 2013 at 9:37

My advisor told me, twice, before accepting me (although he said that it is just to keep him with a clear conscience) that it is very hard to find a position with a Ph.D. in set theory today, and he remarked that my interest in the extremely pathological properties of models without the axiom of choice would make it even harder.

He added that if you're really good then you won't have an actual problem, but it's difficult to be that good. I am aware of the problems set theorists have in finding a position, especially in Israel nowadays where set theory is in decline.

You need to consider two things, I believe:

Are you willing to move to another country? Or do you want to go back to your home land, or even home town? Perhaps to your current university? If you are willing to uproot it might be easier to find positions in your preferred topic.

How good do you think you can write about a topic you are less passionate about? I am currently in the midst of choosing the exact questions I will work on in my dissertation, but the process began with my advisor and me compiling a list of five possible topics, and a sixth one which I told him I may be interested in working on (as a topic that did not come up in the conversation).

Regardless to the fact that all the topics we discussed about are very interesting to me, I kept drifting back to my own idea, and after a long period of two months where I was trying to fight it I gave in and decided that I really have to pursue the things I want to pursue, or else I will give up in the middle of the work.

Of course, understanding all the things I want to understand I am bound to work on the other topics we had originally discussed, but my aim is solve another problem, and that's fine.

• $\begingroup$ The first two paragraphs are very much in the direction of what I am asking for. Although set theory is not among the things I'm passionate about it is useful to know that set theory is not as high in demand as AG and number theory. $\endgroup$ –  tom b. Commented Apr 12, 2013 at 9:01
• 2 $\begingroup$ @tom: The topic is irrelevant. And I don't know how it's useful to know that set theory is in less demand than AG... $\endgroup$ –  Asaf Karagila ♦ Commented Apr 12, 2013 at 10:45
• 1 $\begingroup$ +1: Thanks for sharing this bit of background. Your interest in set theory and the axiom of choice definitely shows up here... $\endgroup$ –  A.P. Commented Apr 12, 2013 at 12:38

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PhD positions in pure mathematics

We are looking for motivated PhD students in the following areas of pure mathematics.

• Algebraic and arithmetic geometry
• Random matrices,  complex analysis
• Differential geometry
• Functional analysis and operator algebras
• Group theory  and Lie algebras

Special attention will be given to applicants motivated to build bridges between two or more of these areas. A more detailed description of the main research topics of the pure mathematics research groups at KU Leuven can be found here . All PhD students will have a limited amount of teaching duties. We strive to improve the gender balance and diversity in the pure mathematics research groups and encourage candidates from underrepresented groups to apply.

Master degree in mathematics, or equivalent, with a strong interest in research, as demonstrated by your master thesis. Master students who will obtain their master degree before the fall of 2024 can apply. Candidates should be fluent in English. No Dutch knowledge is required.

We offer a competitive four year PhD stipend, a budget for conference participation and short research stays in other institutions, health insurance and social benefits for the PhD student and their accompanying partner and/or children. The starting date of the position is flexible, but preferably before 1 October 2024.

About the research environment

Financed by the long term Methusalem funding, the KU Leuven research groups in pure mathematics collaborate within the research project Classification, symmetries and singularities at the frontiers of algebra, analysis and geometry , encompassing the expertise of the different research groups involved and focusing on some of the most challenging problems in algebra, analysis and geometry. In this large scale research project, several PhD students and postdocs are employed. We organize colloquia, seminars and advanced courses. In this stimulating environment, all researchers are exposed to a broad spectrum of mathematical research with several collaboration opportunities.

How to apply

• The application deadline is 12 February 2024.
• You have to apply via the online tool available here .
• You have to upload exactly the following three pdf-files.
• Motivation letter.
• CV. Include the names of three references. These references do not have to send a reference letter now. We will contact the references of shortlisted candidates on 21 February 2024 with a request to submit their reference letter before 13 March 2024. In your CV, also mention the topic of your master thesis and your (tentative) graduation date. If you have any preprints or publications, also mention these. The master thesis itself and the publications themselves should not be included.
• Transcripts of your bachelor and master studies.
• Interviews of shortlisted candidates will take place on 14 March 2024 and can be online or in person, depending on the candidate’s preference.

• Colloquia, Seminars & Lectures
• Outreach Activities
• Publications
• Research Topics
• Job Opportunities
• Conferences

Duality of cones for Mori dream spaces (Elisa Postinghel)

Distance reducing markov bases (oliver clarke).

PhD projects

Several School members offer supervision for PhD research projects in the School of Mathematics and Statistics.

Navigate via the tabs below to view project offerings by School members in the areas of Applied Mathematics, Pure Mathematics and Statistics. (This list was updated September 2022.)

Please note that this is not an exhaustive list of all potential projects and supervisors available in the School. 

Information about PhD research offerings and potential supervisors can be found in various locations. It's worth browsing the current research students list to see what research our PhD students are currently working on, and with whom.

There is also a past research students list which provides links to the theses of former students and the names of their supervisors. 

It's also recommended to browse our Staff Directory , where our staff members' names are linked to their research profiles which provide details about their areas of research and often include the topics they are open to supervising students in.

We host PhD information sessions in the School of Mathematics and Statistics twice a year. Keep an eye on our events page for session information. 

• Applied mathematics
• Pure mathematics
• Real world problem solving using dynamical systems, stochastic modelling and queueing theory for stochastic transport and signalling in cells. 
• Real & Computational Algebraic Geometry: Possible subjects include nonnegativity of real polynomials, polynomial system solving, semialgebraic sets, and algorithmic aspects of real algebraic & convex geometry.
• Polynomial & Convex Optimization: Potential topics include convex relaxations, designing algorithms, exploiting structure (e.g. sparsity), and applications in science & engineering.
• Dynamical Systems and Ergodic Theory: Projects that combine techniques from nonlinear dynamics, ergodic theory, functional analysis, differential geometry, or machine learning and can range from pure mathematical theory through to numerical techniques and applications (including ocean/atmosphere/fluids/blood flow), depending on the student.
• Optimisation: Projects are occasionally available in optimisation, mainly using either techniques from mixed integer programming to solve applied problems (e.g. transport, medicine,…) or mathematical problems arising from dynamics.
• Modelling and analysis of ocean biogeochemical cycles including isotope dynamics, inverse modelling of hydrographic data to detect climate-driven circulation changes, and analysis of large-scale ocean transport. PhD students should be highly motivated, have a strong background in applied mathematics and/or theoretical physics, and will have the opportunity to contribute to shaping their project.
• Data-Driven Multi-stage Robust Optimization: The aim of this study is to develop mathematical principles for multi-stage robust optimization problems, which can identify true optimal solutions and can readily be validated by common computer algorithms, to design associated  data-driven numerical methods to locate these solutions and to provide an advanced optimization framework to solve a wide range of real-life optimization models of multi-stage technical decision-making under evolving uncertainty.
• Semi-algebraic Global Optimization: The goal of this study is to examine classes of semi-algebraic global optimization problems, where the constraints are defined by polynomial equations and inequalities. These problems have numerous locally best solutions that are not globally best. We develop mathematical principles and numerical methods which can identify and locate the globally best solutions.
• Detection and cloaking of surface water waves created by submerged objects
• Decomposition of ocean currents into wave-like and eddy-like components
• Theory and application of Quasi-Monte Carlo methods: for high dimensional integration, approximation, and related problems.
• Computational Mathematics: with specialised topics in radial basis functions, random fields, uncertainty quantification, partial differential equations on spheres and manifolds, stochastic partial differential equations. 
• Discrete Integrable Systems: These are birational maps with particularly ordered dynamics and their study is a nice motivation for using algebraic geometry, symmetry, ideal theory and number theory in the study of dynamical systems.
• Arithmetic Dynamics:  This field is the study of iterated rational maps over the integers or rationals or over finite fields, rather than the complex or real numbers. I am particularly interested in how the usual structures present in dynamical systems over the continuum manifest themselves over discrete spaces.
• Convex geometry: Focused on the study of the facial structure of convex sets and the relations between the geometry of convex optimisation problems and performance of numerical methods. The project can be oriented towards convex algebraic geometry, experimental mathematics or classical convex analysis.
• Algebraic and Geometric Aspects of Integrable Systems: The ubiquitous nature of integrable systems is reflected in their (apparent or disguised) presence in a wide range of areas in both mathematics and (mathematical) physics. Projects focus on the algebraic and/or geometric aspects of discrete and/or continuous integrable systems, depending on the individual student's background and preferences. 
• Analysis of multiscale problems in stochastic systems: These projects will involve an analytical study of certain multiscale problems arising in Markov chains and stochastic differential equations. These projects are suited for those interested in both analysis and probability, and will employ tools from differential equations, functional analysis and stochastic processes.
• Numerical methods for sampling constrained distributions: These projects are aimed at sampling problems arising in molecular dynamics. They will deal with designing and analysing numerical schemes to sample constrained probability distributions using stochastic differential equations.  
• Fluid flow in channels with porous walls
• Mathematics education
• Nonlinear differential equations
• Difference equations
• Dynamic equations on time scales
• How many oceans are there? Using novel statistical and machine learning techniques to characterise oceanic zones and provide a blueprint for quantifying the ocean's role in a changing climate.
• How does heat get into the ocean? An investigation of the physical mechanisms that control the ocean's uptake of heat and its effect on climate.
• Making climate models work better: Developing new methods to validate and improve the inner workings of numerical climate models and improve their projections of global warming and its impacts.
• Will it mix? New perspectives on turbulence in rotating fluid flows and how we estimate mixing from observations. 
• Combinatorics
• Graph theory
• Coding theory
• Extremal set theory
• Operator algebras (von Neumann algebras)
• Mathematical physics (quantum field theory)
• Group theory
• Jones subfactor theory
• Vaughan Jones' connection between conformal field theory, Richard Thompson's groups and knot theory.
• Noncommutative algebra
• Algebraic geometry
• Quantum groups/supergroups
• The Schur-Weyl duality
• Representation Theory
• Random graphs
• Asymptotic enumeration
• Randomized combinatorial algorithms

Extremal and probabilistic combinatorics: Possible subjects therein include Ramsey theory, random graphs, positional games and hypergraphs.

• Unlikely Intersection in Number Theory and Diophantine Geometry: These are problems of showing that arithmetic “correlations" between specialisations of algebraic functions are rare unless there is some obvious reason why they happen. These “correlations” may refer to common values or to values factored into essentially the same set of prime ideals and similar. 
• Arithmetic Dynamics: This area is concerned with algebraic and arithmetic aspects of iterations of rational functions over domains of number theoretic interest. 
• Isometries, conformal mappings, and other special mappings on metric Lie groups
• Complex structures on Lie groups and their Lie algebras
• Counting integral and rational solutions to Diophantine equations and congruences. The goal is to obtain upper bounds on the number of integer solutions to some multivariate equations and congruences in variables from a given interval [M, M+N]. Similarly, for rational solutions one restricts both numerators and denominators to certain intervals.
• Kloostermania: Kloosterman and Salie sums and their applications. A classical direction in analytic number theory where the goal is to obtain new bounds on bilinear sums of Kloosterman and Salie sums and apply them to various arithmetic problems, such as the Dirichlet divisor problem in progressions.

Exponential sums and applications. This topic is about understanding the behaviour (e.g. extreme and typical values) of some most important exponential sums, in particular of Weyl sums.  

• Non-commutative functional analysis and its applications to non-commutative geometry, particularly those related to quantised calculus and index theorems.
• Singular (Dixmier) traces and their applications
• Non-commutative integration theory
• Non-commutative probability theory
• Various aspects of Banach space geometry and its applications
• Algebraic geometry (birational geometry and moduli)
• Hodge theory
• Transcendental methods in algebraic geometry 
• Motivic cohomology and algebraic K-theory - an intersection of algebraic geometry and algebraic topology
• Equivariant algebraic topology
• Extreme Value Analysis: Projects available on the modelling of the dependence of multivariate and spatial extremes, spatio-temporal modelling, high-dimensional inference. Interests in environmental/climate applications. 
• Symbolic Data Analysis: Projects available on symbol design, distributional symbols and others. Applications in big and complex data analysis.
• Ancient river systems and landscape dynamics with Bayeslands framework
• Bayesian inference and machine learning for reef modelling 
• Deep learning for the reconstruction of 3D ore-bodies for mineral exploration 
• Bayesian deep learning for protein function detection  
• COVID-19 modelling with deep learning
• Variational Bayes for surrogate assisted deep learning
• Bayesian deep learning for incomplete information
• Computational Statistics
• Event sequence data analysis
• Hidden Markov Models and State-Space Models and their inference and applications
• Financial data analysis and modelling
• Point processes and their inference and applications
• Semi- and non-parametric inference
• Bayesian statistical inference
• Computational statistics and algorithms
• Approximate Bayesian inference
• Quantile regression method
• Statistical text analyses
• Applications to climate science, social science, image analyses

*Yanan Fan is an Adjunct A/Prof in the School and is able to co-supervise students (not as primary supervisor)

• Nonparametric and semiparametric statistics: Nonparametric dependence modelling (copulas) and nonparametric functional data analysis.
• Social network analysis for epidemiology, social sciences, defence, national security, and other areas
• Statistical models for dependent categorical data
• Survey sampling (design and inference), particularly for network data
• Statistical computing, particularly MCMC-based methods
• Dependence measures
• Complex-valued random variables
• Goodness-of-fit tests
• Machine learning (with potential applications in medical imaging)
• Time series analysis
• Real-time analytics with the Raspberry Pi
• For some examples of my current projects, have a look at my  personal webpage .
• Fast and efficient model selection for high-dimensional data
• Development of efficient estimation and sampling algorithms for random graphs and spatial point processes
• Development of model compression methods for deep neural networks.

Topics include regression to the mean, interrupted time series, meta-analysis, and population attributable fractions.

Monte Carlo and Uncertainty Quantification 

• Projects on the stochastic analysis and development of modern Monte Carlo methods for uncertainty quantification, sequential Bayesian inference, high dimensional sampling, particle based Variational Inference (knowledge/experience with stochastic analysis and SDE & PDE theory highly desired).

Machine learning and generative modelling 

• Projects with a focus on (but not restricted to) medical imaging and machine learning methods for uncertainty quantification of image segmentation
• Theoretical analysis of modern machine learning methods (knowledge/experience with functional and stochastic analysis highly desired).

Mathematics of sustainability 

• Projects on stochastic games, agent based models, network science and their applications in sustainability science.
• Automating data analyses via natural language queries
• Bayesian statistics, algorithms and applications
• Building software tools, services and packages
• Data privacy and synthetic data
• Data science, theory and application
• Defence applications (nationality restrictions may apply)
• Extreme value theory and applications
• Machine learning 
• Symbolic data analysis
• Developing statistical methods for point processes
• Financial data modeling
• Computational statistics
• Analysis of capture-recapture data
• Estimation of animal abundance
• Measurement error modelling
• Model selection for multivariate data
• Non-parametric smoothing
• Statistical ecology
• High-dimensional data analysis
• Simulation-based inference
• Eco-Stats project ideas

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getting a job with a PhD in (pure) mathematics

Probably this concern has been addressed several times here but I will give a try to get other people's perspective from my situation. I am feeling sort of hopeless these times.

I am in my last year of my PhD in mathematics and doing my dissertation soon, my research was mainly devoted to algebraic topology and I made some progress in my advisor's particular area. However, I faced some problems during my studies (personal issues) and I never built a strong relationship with my advisor; so I could not finish all the initial research proposal intended for my PhD thesis.

I recently had a sincere talk with my advisor about my future, and he said that my progress is worth for a PhD thesis but it is not anything very surprising that is worth publishing, so I will be graduating with zero publications. He said that my chances of getting a PostDoc positions are really low and that I can not expect a strong recommendation letter from him. My advisor has a few collaborators (from overseas mainly) and none in the local department. He also never supervised another Master or PhD student while I was under his supervision, then I never got a chance to network or engage properly with other classmates. My advisor kindly suggested to explore other options in the job-market besides academia.

I like to explore new things, and in my free time during my studies I attended several teaching workshops, seminars and courses (I got a small degree in "university teaching"), and I am also familiar with Python, R, Matlab and Sage; however, I learn those things for fun and I have never really worked in a big project using any of that.

I do not know what to focus on and what kind of job should I look for after I finish my PhD

I would like to keep doing research in Math, but my area is not very popular lately and I feel that I have no chances of succeed getting a PostDoc; I also feel that I do not have (demonstrable) skills to perform a job with "real-life" applications so probably my chances getting there are also low. And I do not really know the Teaching job-market, I think that those positions are temporary and it would be hard to find a place that is willing to sponsor a work visa (I am not a US-citizen, and the job market in the third-world country that I am from is nonexistent for PhD's in this area)

• mathematics
• career-path

• 4 Learn to program. –  The Dude Commented Jan 25, 2021 at 15:51
• 1 Probably much too late for the original poster, but maybe helpful to some others: have you heard of topological data analysis ? Not saying being good at TDA would get you a job on its own, but could be a stepping stone to applied areas if you already know algebraic topology. –  J W Commented Sep 1, 2021 at 14:06

4 Answers 4

Your supervisor is honest, blunt, and very helpful because by ruling out the academic path he's saving you a lot of time.

The bad news is it sounds like you screwed up your PhD. Not getting strong research results, not having diversified your PhD experience (e.g. lack of teaching experience makes it harder for you to get a teaching job), and not having a firm idea of what you want to do after the PhD means you sound lost right now. The good news is, chances are you learned more than you think you did, and those things are applicable to the job market.

First : don't think you have to do research. It is what you've been doing for the PhD, but it's not what you have to do in the future. Just consider this: if you like research + are good at it, why didn't you do better in the PhD? Why weren't you able to complete the initial research proposals, why weren't you able to get publishable results, why weren't you able to impress your supervisor such that he can write a strong letter of recommendation? You might have good reasons for these, and it's up to you to convince yourself that yes, you still want to do research, and yes, you are talented + motivated enough to succeed in it.

If I were you, I'd start examining the possibility that I am not actually good at research and / or it is not what I want to do. Do some serious soul-searching here. You are making a life-altering decision. If it makes you feel bad that by leaving academia you are "failing", don't worry too much about it: there's a good chance that by leaving academia you'll have a more successful life (in terms of material possessions).

Second : go to your local jobs portal (use Google if you don't know what these are) and search for jobs that require a PhD in mathematics. Do you find anything that catches your fancy? For example searching on indeed.com for "phd math", I get this job among many others. Note the requirements:

Preferred qualifications: MBA, Master's or PhD degree in a quantitative field . Experience with stakeholder management and ability to influence senior stakeholders. Demonstrated knowledge of statistics and data analysis including R programming or other statistical software packages.

I highlighted the most relevant parts. You say you are familiar with R. That means you are in business! You can potentially do this job! If you further have experience with statistics and data analysis (do you?) you're in an even better position.

It's up to you to search the jobs portal for jobs you can do, and then it's up to you to apply. It's true that visa issues might sink your candidacy, but it's also true that because PhDs are relatively rare, you might have skills nobody else will have and therefore the employer is willing to sponsor you. You will not know unless you try.

Do remember that even if you can't find a US employer, you can still work elsewhere. It's a big world out there, and it's not true that third-world countries don't need your skills. Example of such a job in India (admittedly they're not looking for PhD-level candidates, but you could still apply).

Third : once you have an idea about what the options in industry are like, then you can start thinking about what you want to do. Are you sure you still want to do math research? If so I'd talk to your supervisor again about what options are available to you. He's already told you your prospects are not good, so you'll need all the help you can get from him. For example, perhaps you could find a position as a teaching assistant somewhere and do research in your spare time. This will not be easy - doing research while holding a full-time job is very difficult - and it's not likely to be well-paid, but if it's what you want to do, you can try.

On the other hand if you think industry will serve you better, then finish up your thesis & defense and start applying for jobs. Be sure to visit your university's career center as well; they'll be able to help you a lot.

This question is a year old, but it matched my own experience enough (including the specialization in algebraic topology) that I felt obliged to comment. Getting an academic math position is a crapshoot under the best of conditions. The field is glutted with other highly competent applicants (pretty much everyone going into pure math does so with the intention of doing pure math research); the field doesn't have the money of, say, computer science; and there really aren't any opportunites to do pure math research outside of academia and maybe a single-digit number of industry labs. With a great publication record and glowing recommendations from well-established professors, you have an outside shot at it. Without them, it's vanishingly likely to happen. It's unfair, especially if you wind up with a useless or abusive advisor, fall into a field that isn't the cool new thing, choose (or get saddled with) a research project in your limited time to prove yourself that doesn't work out, and so on. What's even worse is that (unlike some other fields) there's really no way to burnish your credentials; there's nothing you can do outside of academia that academic mathematicians would care about.

So, unfortunately, there's probably nothing you can do. It should be relatively easy for you to find some job in industry, though; the tricky part is finding one you like. Just being able to think rigorously and scientifically is a prized skill in industry, while it's taken as given in academia. (Conversely, some interpersonal and organizational skills work the opposite way. On the other hand, there are more opportunities to pick up the latter, whereas there's really no place to pick up pure math skills outside academia.) I don't see anything in your post to suggest that you aren't employable, so at least you should have some breathing room to take a look at what's out there and see if there's anything that might be, if not exciting, at least acceptable.

While a position at a top research institution may not be in your immediate future there are things you can do, up to and including proving that your advisor is wrong.

You don't actually need a faculty position to do research or publish in math but, in my experience, you do need a circle of collaborators. You indicate that you don't have that, but it would be good to start to develop it. Attendance at conferences is a good way in computer science, at least. Meet people, speak with them about ideas, and such.

There are also many teaching colleges around the world that value good math teaching over research. Almost all will require some, but many emphasize teaching the next generation over serious research. The MAA, for example, is full of people who, while doing research, focus much of their efforts on teaching.

And, of course, teaching at pre-college level is also open to you. You may find yourself overqualified for this, but it can make a satisfying life if you value teaching.

But, you can put it together if you have the willingness to do so. A lower level position, while developing mathematical ideas within a circle of collaborators can put you on a path to a better position, proving your advisor wrong in the long term if not immediately.

If your advisor is actually inexperienced at this he may be making a mistake in his analysis and also in his general advising. In particular, his assessment that you can't publish what you've done might just be wrong. You won't know unless you try.

Your life is what you make it.

• 9 Unfortunately, the visa issue is a serious one for US jobs. Academic positions at colleges and universities are not subject to the H-1B quota, but pre-college teaching and industry jobs are, and, even if one can get a position, one has less than 50% chance of getting an H-1B and being able to accept the position. –  Alexander Woo Commented Mar 1, 2019 at 21:42
• 2 The OP should be aware of the "Optional Practical Training" program that allows recent graduates in STEM fields to work in the US for a period of time (up to 3 years as I recall) after completing their degrees. Teaching at a community college on OPT is sometimes a viable option for new PhD's who have acquired enough teaching experience to obtain such a position. –  Brian Borchers Commented Mar 2, 2019 at 0:25

I know it's too late but for someone in the future, there are some folks in machine learning / AI (or more generally applied/engineering) community who are interested in the intersection with pure math, e.g., Topological data science employing the concept of Homology from Algebraic topology or homotopy methods in computational aspects of polynomials.

So, considering these directions and finding positions in industry might be interesting! ** In these directions, people likely do not appreciate the advancement in theoretical direction like the way you used to research on though...

• 3 Forks or folks? –  Moishe Kohan Commented Aug 12, 2023 at 17:21
• 1 @MoisheKohan thank you! –  Rowing0914 Commented Aug 12, 2023 at 21:56

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