# Mathematics

### Social Networks

•

I have included this section in my homepage in case you are considering working with me. This can mean any of the following:

• you are thinking taking a course with me;
• you are considering me as an adviser;
• you are considering me as a mentor for a post-doc;

In any of these cases, you should read first the two sections below on my mathematical interests and teaching philosophy, and then refer to the appropriate section that applies to your situation.

## What are my mathematical interests?

My mathematical interests concentrate in geometry, an area of mathematics that roughly speaking studies the shape of space. Geometry has many incarnations and is related with many other areas of mathematics and the physical sciences. You probably have a rough idea of what geometry is, or you can google many sites that will give you some nice overviews of many aspects of geometry (try, for example, to search geometry in MathWorld or PlanetMath, using the facility on the top of this page).

I am specially interested in Poisson geometry. This is a branch of geometry that grew out of the study of mechanical systems, such as the solar system, spinning tops, etc. If you have a basic knowledge of manifolds, vector fields and differential forms, you can get an idea of what Poisson geometry is by reading a brief introduction (under construction). This also contains some references to a few other subjects that interest me, such as integrable systems and Lie theory, as well as some bibliographic references.

## What is my teaching philosophy?

I believe that mathematics is, in the first place, solving problems. Note that by solving I don't mean applying some known algorithm that someone else developed (the person that created that algorithm is the one that has solved a problem!). I would say that a "big theorem" is one that solves a deep problem, and identifying (deep) problems is an important part of mathematics. Note, also, that deep does not necessarily mean difficult. In fact, my experience tells me that when something looks difficult is because we have not understood the problem properly.

This said, let me emphasize right away that teachers are not dispensable, quite the opposite (I have many doubts about such things as e-learning, etc.). Going to the lectures and attending classes is an essential part of learning mathematics. In the lecture you can witness a certain way of looking at a problem, you can experience the first difficulties in solving a problem, and you can learn how others have solved problems (yes, mathematics have been around for quite sometime!).

The role of a teacher and an adviser, in my opinion, is to point the student to interesting problems and lead him through the problems. It is not the role of the advisor to give him clues on how to solve the problems (and much less to solve the problems). Mathematics can only progress when the students end up knowing more math than their teachers.

## Taking a course with me...

If you are planning to take a course with me be prepared to solve a lot of problems! If you are taking a basic linear algebra or calculus course and you just want to learn some algorithms, you will be loosing your time going to my classes. I believe it is important to understand things. Also, if you just like "the theory" and you don't like to fight with problems, don't take a class with me. Finally, beware that I don't like students who cannot meet deadlines.

## Doing a small research project with me...

If you are an undergraduate student and you would like to do a small research project with me, to experience what research is all about, you should:

• read the sections above about my teaching philosophy and my mathematical interests. The project that I will offer you will reflect this.
• be prepared to learn about some significant result in geometry (most likely, you will have to read a book or a research paper).
• be prepared to write an original project: if it does not necessarily contain original mathematics, it should have, at least, an original presentation.
• set up an appointment with me ().
Here is a list of research projects that some students did under my supervision:

## Becoming a PhD student with me...

A person should only pursue a PhD in Mathematics if he loves mathematics. Getting a good job in Mathematics can be quite hard. Teaching mathematics and doing research in mathematics require dedication and hard work. There are better paid jobs around which require less efforts... Also, at some point one should think what one is doing in mathematics. If you are wondering about this question you may want to read the article What is good mathematics? by Fields Medalist winner Terence Tao.

At Illinois you have the chance to study mathematics in a top level university, with a PhD degree in Mathematics of high international standards. This means that our PhD program gives both an in depth and in breath preparation in Mathematics. Be ready to take preliminary and qualifying exams. If you become a student of mine, be prepared to take coursework in areas not directly related to you thesis project. I will force you to do so, even if you are not required to!

A PhD student should have a good thesis problem. What do I mean by this? I have two main properties in mind:

• It should be an interesting problem. This means that it should have many connections with other (interesting) problems both in the field and outside the field. At the same time, it should involve a nice piece of mathematics that already exists out there (and that it will prove useful for the student later in his life).
• It should be a flexible problem. This means that the problem should admit partial solutions, rather than a Yes/No answer, and leaving a lot of room for further work. Also, it should be possible to test it, saying by looking at interesting examples.

My job as an adviser, besides identifying a good thesis problem, is to let the student know what is going on in the field, and to the let the people in the field know what is going on with the student (for this, the student will have to do something, of course!). If the student does a good job, this will give him a good start.

I am a mentor to both predoctoral and doctoral students in the National Alliance for Doctoral Studies in the Mathematical Sciences

Current and previous PhD students:

## Doing a postdoc with me...

If you are considering a post-doctoral position at UIUC, possibly under my supervision, you should start by taking a look at the web pages of the department (look at what the other faculty is doing, at seminars being offered, and other research activities in the Math Department). Remember, a post-doc is to help your own research career take off. It is not a way to get another advisor for 3 more years!

Current and previous post-doctoral fellows:

• Florian Zeiser (Poisson Geometry), J. L. Doob Research Assistant Professor at UIUC
• Ivan Contreras (Mathematical Physics), currently at Amherst College, Massachusetts
• Ioan Marcut (Poisson geometry), currently at Radboud University Nijmegen, The Netherlands
• Matias del Hoyo (Algebraic topology and Lie groupoids), currently at IMPA, Brazil
• Florian Schaetz (Poisson geometry), currently at the Centre for Quantum Geometry of Moduli Spaces, Aarhus, Denmark
• Olivier Brahic (Poisson geometry), currently at Universidade Federal do Paraná, Curitiba, Brazil
• Philippe Monnier (Poisson geometry), currently at Université de Toulouse, France
• Iakovos Androulidakis (Lie algebroids and Lie groupoids), currently at University of Athens, Greece
• David Martinez Torres (symplectic, contact and Poisson geometry), currently at Universidad Politécnica de Madrid, Spain