Advising

I have included this section in my homepage in case you are considering becoming a student of mine. This can mean any of the following:

• you are thinking taking a course with me;
• you are considering doing a LMAC project with me;
• you are considering doing a Master's thesis with me;
• you are considering me as an adviser for a PhD at IST;
• you are considering me as a mentor for a post-doctoral degree at IST;

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 apropriate 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, that I have written. 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 don't believe in 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 calculus course and you just want to learn some algorithms, you will be loosing your time going to my classes. It is not enough to go the problem solving sessions: if you just do that, most likely you will flank the course. 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.

LMAC students

If you are an undergraduate student (LMAC) and would like to do a 3rd year project with me, 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 paper in English).
• be prepared to write a project which does not necessarily contain original mathematics, but, at least, has an original presentation.
• set up an appointment with me ().

• João Guerreiro, Orbifolds and Wallpaper Patterns
• Pedro Vitória, A Quantização de Kontsevich
• Ricardo Joel Andrade, Esferas Exóticas
• Ricardo Gonçalves, Grupos não-Realizáveis por Transformações Lineares

Master's students

If you are a master's student (MMA) and you would like to do a thesis with me, you should:

• read the sections above about my teaching philosophy and my mathematical interests. The thesis problem that I will offer you will reflect this.
• have completed basic courses in geometry (e.g., Riemannian Geometry and Differential Geometry), topology (e.g., Introduction to Topology, Algebraic Topology) analysis (e.g., Real Analysis, ODE and PDE) and algebra (e.g., Algebra, Commutative Algebra).
• be mature enough to do some original mathematics. Your thesis must have some original math,
• set up an appointment with me ().

Here is a list of previous students of mine and their thesis topics:

• Luís Alexandre Pereira, Integrability of Lie algebroids by proper groupoids
• Ricardo Joel Andrade, The h-principle
• João Boavida, Singular foliations and Lie algebroids
• João Paulo Santos, Integrable systems and algebraic geometry
• António Serra, Quadratic Poisson brackets on Lie groups

PhD students

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.

Mathematics in Portugal is improving but it is still in its infancy when compared with many other western european countries or the USA. If you have the chance to study mathematics in a top level university, in one of those countries, there is no reason to study at IST. This said, we are trying to raise our PhD degree in Mathematics to high international standards. This means a PhD program giving both an in depth and in breath preparation in Mathematics. Be ready for taking qualifying exams and 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 in the field and outside the field and, at the same time, involve a nice piece of mathematics that already exists out there (and that it will prove usefull 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.

Here is a list of PhD students of mine:

• Hassan Najafi Alishah, (current student working on KAM theory and stability)
• Juan Pablo Roggiero, Lie Algebroids over Quotient Spaces (graduated from IST in September 2011)
• Pedro Frejlich, h-principles around Poisson Geometry (graduated from IST in June 2011)
• Ivan Struchiner, The Classifying Lie Algebroid of a Geometric Structure (graduated from UNICAMP, Brazil, in January 2009)

Post-doctoral students

If you are considering a post-doctoral position at IST, 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 carreer take off. It is not a way to get another advisor for 3 more years!

There are two ways to get a post-doctoral position at IST:

• You can apply for a post-doctoral grant from the Center for Mathematical Analysis, Geometry, and Dynamical Systems.
• You can apply for a post-doctoral grant from the Portuguese Science Foundation.

Here is a list of current and previous post-doctoral and pre-doctoral students of mine and their area of expertise:

• Matias del Hoyo (Algebraic topology and Lie groupoids)
• Florian Schaetz (Poisson geometry)
• Olivier Brahic (Poisson geometry)
• Philippe Monnier (Poisson geometry)
• Iakovos Androulidakis (Lie algebroids and Lie groupoids)
• David Martinez Torres (symplectic geometry)