I am supervising Bachelor's, Master's, and PhD theses related to my research interests for students in Mathematics and in the Teacher Training Program. Here you find some general information on my field of research and some practical guidelines on choosing a field for your thesis.
My field of research belongs to the larger area of Analysis and, more specifically, Partial Differential Equations (PDEs). However, as opposed to several groups in Vienna working on the applied side of PDEs (modelling, asymptotic analysis, numerical simulation), my research is theoretical and rather “pure” than “applied”. In simple terms, my goal is to prove theorems.
The field of dispersive PDEs as a mathematical discipline has its origins in the 1950s. One of the first major results was the proof of local well-posedness for the Einstein equation by Yvonne Choquet-Bruhat. This already underlines the strong ties to mathematical physics and general relativity. The field reached a first climax in the 1980s and is rapidly developing ever since. It is very active and competitive and became a cornerstone of modern PDE theory. At least two Fields medalists (Jean Bourgain and Terence Tao) can be directly assigned to the area of dispersive PDEs. As a consequence, papers in dispersive PDEs are regularly published in the most prestigious international mathematics journals like Annals of Mathematics, Inventiones Mathematicae, Acta Mathematica or Journal of the American Mathematical Society. The field is prominently represented at virtually every major research university in the US (e.g. Princeton, Berkeley, Chicago, UCLA, etc.) and similarly in France or Japan. In the closer neighborhood the situation is also constantly improving with research groups in e.g. Bonn, Bielefeld, Karlsruhe, Zurich, Lausanne, Basel, Rome, Pisa, and, of course, Vienna. This offers exciting career perspectives after the PhD.
Needless to say, the most important selection criterion is your personal taste and talent. Having sorted out this, you should look up the possible advisors and their research interests in the respective area of specialization. If your goal is to pursue an academic career, the choice of a good field/advisor can be crucial (not so much on the Bachelor level but certainly from the Master’s thesis onwards). If you have an idea in which direction you want to go, you should ask yourself the following questions.
As in any science, there are “hot topics” in mathematics and fields that are largely inactive. This does not mean that the mathematics produced in a very active and fashionable area is necessarily better. However, in view of future job opportunities it is important to enter an active field. In order to get a first impression what is considered a hot topic, you can check out the Fields medalists of the last 20 years or so and their fields of interest. You may also browse through the research webpages of the mathematics departments at the top US universities. Furthermore, you can have a look at recent papers in the respective field and see how old the papers are on average that are cited. In an active field there should be a good amount of references that were published in the last 15 years.
In mathematics there are “mainstream” fields that are well established and typically very competitive. Substantial results in these areas have a realistic chance of getting published in one of the top journals. On the other hand, there are certain fields where the best you can hope for is to publish in specialized journals, regardless of the mathematical quality of the work (“specialized” is often code for “not so good”). This can be a serious disadvantage. You have to bear in mind that at some point in your career you will face a hiring committee consisting of people from diverse areas of math or even outside of math. They will have a hard time assessing the quality of a certain specialized journal they have never heard of.
So how can you tell whether a given journal is “good”? Well, this is a hard question. A first and necessarily incomplete picture is provided by journal rankings. What I find quite useful is eigenfactor.org. Check out the article influence number (AI). Everything above 90 is very good. Another rule of thumb is that general math journals are better than specialized journals with a comparable AI since the latter focus on a particular (possibly narrow) field of research only. You can easily tell from the title and the scope of the journal to which class it belongs.
The most important databases for math publications are MathSciNet and zbMATH. Unfortunately, these services require a subscription but you can access them from the university network. I very much prefer MathSciNet and zbMATH over freely accessible alternatives like Google Scholar because they are much more accurate and checked by hand for correctness. For every researcher you can get a complete list of publications and citations. Compare your prospective advisor to other researchers in the same field but do not forget to take into account the academic age (it takes a long time to gather citations in math so even outstanding young mathematicians may have a ridiculously small number of citations compared to older, well-established colleagues). Note also that the total number of publications is not necessarily a good measure. There are huge differences between the different mathematical disciplines. The important thing is that the advisor publishes regularly and in good journals.
Mathematics is a highly international endeavor. Check out the advisor’s international connections. If there are any, he/she will list them on his webpage or they will be obvious from the publication list. It is very important to be part of a network in order to have a realistic chance of getting a postdoc position at a good university after the PhD.
Third-party projects are good for universities because they bring in external money. But they can also be a measure for the quality of the research and the role of the field on an international level because the evaluations of grant applications are typically very competitive and serious (it’s about money after all). In addition, if third-party money is available, you will most likely get paid for working on your PhD. On a national level the main source for third-party funding in theoretical math is the Austrian Science Fund (FWF). Check out the webpage if the advisor has or recently had a project funded by the FWF. In addition, there are the extremely prestigious ERC grants by the European Research Council. If your potential advisor has one of these, he/she belongs to the top researchers in his/her field and if he/she is willing to take you as a student, you should just go for it.
It is a good idea to check out former students of the advisor. Were they able to start a successful career in academia? Where did they get postdoc positions? Did they get permanent or tenure-track positions eventually? Again, you have to take into account the academic age of the advisor. It may well be that he/she is too young to have many or even any successful former student. You can also look at postdocs the advisor was supervising in the past (if any). Where did they come from? Where did they go to? When assessing this you should keep in mind that it is hard but not impossible to get a temporary postdoc position at one of the top institutions (there are also national fellowships for that purpose). What is really tough is getting a (potentially) permanent position just somewhere. Consequently, a permanent or tenure-track position at a seemingly weaker institution counts much more than a temporary postdoc in one of the top places.