#### What do the mathematical physicists do?

The mathematical physicists work as mathematicians on important problems of modern physics or, more generally, of natural sciences. These can be problems of quantum physics and wave propagation, statistics and stability of matter, elementary particle physics and bacterial chemotaxis in the ocean.

#### Why mathematicians have to study problems of physics?

- Without using and developing powerful mathematical methods, one can no more analyze or even discover many phenomena that are important from the point of view of physics.
- The most intriguing problems of physics necessarily lead to the most interesting mathematical problems. The most beautiful mathematical structures are inevitably related to the fundamental questions of physics.

#### Why the mathematical physicists can deal with problems from very different fields?

Often one and the same mathematical construction arises in course of mathematical investigations in very different fields, and so, often one and the same mathematical method allows to attack very different problems.

#### Why do we do mathematical physics?

Most of us graduated from the Faculty of Physics. All of us are inspired by physics, but we are working as mathematicians. We like when our problems and results can be formulated using physics terms, when our results describe new physical effects, and when one can use physical intuition to solve a mathematical problem. At the same time, we appreciate the internal beauty of mathematical constructions and like the mathematical way of thinking.

#### You should choose the mathematical physics as your future profession if

- when entering the university, you liked physics with its formulas and calculations;
- after the first year in the university, the lectures in mathematics became at least as interesting for you as the lectures in physics;
- you are not afraid of the mathematical problems suggested to you when studying at the faculty.

#### The traditional fields of research at our department (honest advertisement)

It is rather difficult to describe in detail our areas of research: one can easily do it only when discussing with colleagues. So, we begin by singling out two our traditional fields of research:

- Spectral theory. This is the mathematical basement of all the quantum physics. Without the spectral theory, it is difficult to study modern problems of solid state physics, statistical physics, soliton theory, nuclear physics and so on. Without the spectral theory, a theoretical physicist becomes an engineer!
- Wave propagation theory. If you believe that this theory is completely overtimed, you can consider changing your future profession: there exist not only the sea waves, but also electromagnetic, acoustic, gravitational, quantum and many other waves. All the photons, phonons, gravitons and other -ons are waves… Interactions of elementary particles (birth and death) can be regarded as interactions of solitons, i.e., localized nonlinear waves.

#### A bit more details on our interests.

Here, we list our colleagues interested in collaboration with students and describe their fields of interests. We strongly recommend the students interested in mathematical physics to speak with other members of our department. They also can decide to work with you, and they can point toward another colleague that could be interested in such a collaboration. As for the list below, you have to be careful: if, having looked at it, you decide that all is clear, then, most probably, either you are wrong, or you know already too much…

#### Full professors:

- Mikhail Belishev – classical inverse problems
- Alexander Blagoveshchenskii – classical inverse problems
- Aleksei Kiselev – diffraction and wave propagation
- Evgeny Korotyaev – spectral theory, quantum inverse problems and scattering theory, integrable systems
- Mikhail Lyalinov – diffraction and wave propagation, difference equations on the complex plane, integral transformations, asymptotic methods
- Boris Plamenevskii – pseudodifferential operators, boundary problems, mathematical problems of the waveguide theory
- Nataliya Smorodina – probability theory, path integrals
- Tatiana Suslina – spectral theory of differential operators, periodic operators of mathematical physics, homоgenization theory
- Ludvig Faddeev – quantum field theory, quantum integrable models, quantum inverse problems and scattering theory
- Alexander Fedotov – spectral theory, quasi-periodic operators, wave propagation, quasi-classical asymptotics, difference equations on the complex plane

#### Associated professors:

- Andrey Badanin – quantum inverse problems, periodic differential operators
- Alexander Budylin – quasi-classical asymptotics, pseudodifferential equations
- Sergey Levin – many-particle quantum scattering theory
- Maria Perel – applications of the wavelet theory to the study of wave propagation
- Oleg Sarafanov – pseudodifferential operators, mathematical problems of the waveguide theory
- Vladimir Sloushch – spectral theory of differential and integral operators
- Vladimir Sukhanov – quantum inverse problems and integrable nonlinear equations
- Nikolai Filonov – spectral theory, Maxwell operator, periodic operators, quantum field theory

#### Do we have contacts with “pure mathematicians” and “pure physicists”?

We often have discussions with “pure mathematicians”, and we try to drag their attention to mathematical physics. Among our partners, there are specialists working in a very wide range of fields from the analytic number theory to the algebraic geometry. Speaking about our contacts with physicists, the members of our department can be divided into two groups: the colleagues who already collaborate with physicists and the colleagues who would like to have contacts with them. Actually, for any mathematical physicist, it is a great pleasure to get a result that is interesting for physicists, and one that physicists themselves would like to get. Now, M.Belishev, A.Blagoveshchenskii, L.Faddeev, S.Levin, A.Kiselev, and B.Plamenevskii have active contacts with physicists.

#### Range of our scientific contacts

It is very wide. We have longlasting and close scientific relations with colleagues from Great Britain (London universities: King’s College, Imperial College and University College; Bath University), France (Paris universities: Pierre et Marie Curie, Paris Nord and Val-de-Marne, Universities of Bordeaux, of Rennes), Germany (in Berlin: Humboldt University and Technische University, Stuttgart University, Gutenberg University in Mainz), Sweden (Stockholm University, Chalmers University in Gothenburg), USA (Universities of Missouri, Alabama, Indiana), Canada (Toronto), Japon (Tokio Gakushuin University), Chile (Santiago University). Colleagues working on problems of diffraction theory have been closely collaborating for many years with Commissariat Energy Atomic (France) and a research division of British Airspace. Many of us have worked for several months in international mathematical centers: Mittag-Leffler Institut and Kungliga Tekniska Högskolan in Sweden, Schrödinger Institute in Wien, Isaac Newton Institute in Cambridge, Fields Institute in Toronto, Technische Universität Berlin, Stuttgart University, Bordeaux University, Harvard University and so on.

#### How to choose a scientific adviser?

One has to speak with different members of the department. By default, it is better to assume that, being a second year student, you still know little about mathematics and so can not choose your scientific adviser using information on his scientific interests. It is natural to ask your possible adviser what kind of work you are supposed to do: to carry out abstract analysis or to work with formulas, to develop an abstract theory and use it to study particular problems or to study possibly deep but particular problems and develop an abstract theory using results obtained for such problems. Both approaches can be fruitful. The choice depends on the personal taste. Also, it is useful to understand which of the already familiar domains of mathematics your future activity will resemble (e.g., algebra or analysis). The choice of a scientific adviser can determine your life for a long period of time. So, it is worth considering your personal compatibility. Having chosen a scientific adviser, you have to see him or her regularly, trying to acquire the maximum of the possible help, information, skill, professionalism. You can “disappear” for a long time only in the case when you do not need any scientific adviser anymore. Usually, a good scientific adviser is quite busy. So, you have to remind about yourself periodically and ask for appointments. Of course, each time you come to see your adviser, you have to bring something new: new questions, new formulas and so on.

#### Is it difficult to be a student at our department?

It is difficult, especially, if you are not motivated. If somebody says that it is easy to become a professional in some scientific area (but that it is difficult to become a mathematical physicists),

he is cunning. There are no miracles: to become successful, one has to work. On the other hand, we all are different: one can be a future great theoretician, and another can become a very good experimenter.

#### Do we like our students?

We do like the students who are motivated, interested, trying and working. We help them to carry out their research, to go to conferences, we try to find for them a financial support. As soon as students arrive to publish his or her first paper, it becomes much easier for us to find money, grants and scholarships for them.

#### What do our students do after graduating from the university?

If, after the university, you are planning to become a researcher, you have to plan to work on your PhD thesis. Therefore, you have to become a postgraduate student in Russia or abroad. We help good young mathematicians to get PhD positions at our department and in the St. Petersburg Division of Steklov Institute. Many of our students found positions in France, Germany, Great Britain, USA and other countries. Having defended a good thesis, you can find a postdoctoral position. After a postdoctoral position, a successful young mathematician finds a job at a university or a research institute. In Saint Petersburg, there are three natural places for that: our department, Steklov Institute, and Chebyshev Laboratory. Abroad, mathematical physicists work at any mathematical and at most of physical faculties (departments) of all important universities, and in most of mathematical and physical research centers.

After the university, you can also teach mathematics and physics. Many of our students work in

non-academic organizations. These can be large telecommunication or car manufacturing companies, large banks or large and small companies making computer simulations and so on.

Here, mathematical physicists work on applied problems, e.g., problems of car stability, financial flow dynamics, underwater acoustics. To study such problems, instead of direct computer calculations, one can first use mathematical methods, and only then turn to computers. This idea often appears to be very fruitful, and the specialists who can choose this way of analysis are highly appreciated.