The research projects I give my students are designed to be
multi-dimensional and interdisciplinary.
I will require you combining techniques from various areas of mathematics ( convex & discrete geometry,
combinatorics, algebra, topology, optimization, etc.) My projects are likely inspired or motivated by applications
in computer science, game theory, machine learning, biological imaging, social choice fair-division problems,
assignment/logistic problems, etc.
Here is a sample of research questions I have assigned to students:
- What are good ways to use computers in Mathematics?
- What is the average behavior of an algebraic variety?
- How to select the best choice of parameters for an algorithm?
- What can be said about the geometry of n-player games?
- Can one demonstrate the efficiency of the simplex method?
- Can one develop practical clustering algorithms with provable performance in data?
- Can we develop efficient algorithms for solving special non-linear optimization problems?
Talk to me if any of these sound interesting.
If you are a graduate student considering working with me, I propose you do the following:
A) If you are only in your first year, and you have not passed the
prelims, ask me to audit my weekly CACAO seminar.
This will let you check the type of research I do and give you an idea of how I work with students. I have my own
peculiarities and style. E.g., I am fairly demanding with all my students and I expect a weekly report of your research.
Note: I do not take on Ph.D students until after they have passed their prelims exams. I only mentor students if they passed
the prelims by the end of the Spring of their second year at the latest.
B) If you have passed both of your prelims, contact me asap to
enroll in my seminar CACAO for credit.
Please do so soon after passing your prelims. Preferably contact me no later than Fall of your second year,
but you must for sure contact me by the Spring of your second year. I will then assign you a small task/project
(usually involving some reading, some thinking, some computing) and will ask you to give one or two presentations
in CACAO. Depending on the outcome, we will both decide how to proceed from there.
If you are an undergraduate student, who wants to do research
or write a senior thesis, please contact me at beginning of
your junior year. My requirements are that (a) you are doing well in your upper division courses (>3.3 GPA), (b) you have
completed several upper division math courses completed with A- or better, and (c) that you enjoy mathematics and computers.
a) Lectures on Polytopes, by G. Ziegler, Graduate texts in Mathematics,
vol 152, Springer, Berlin 1995
b) A course on Convexity, by A. Barvinok, Graduate studies in Mathematics, vol. 54, AMS, Providence, 2002
c) Lectures on Discrete Geometry (Springer, May 2002), by Matousek.
a) Enumerative Combinatorics vols. I and II, by R. P. Stanley, Cambridge University Press, Cambridge, 1997.
b) Combinatorics and Commutative Algebra, by R. P. Stanely, second ed., Progress in Mathematics 41, Birkhauser, 1996.
c) Grobner bases and Convex Polytopes, by B. Sturmfels, University texts, AMS, Providence, 1995.
d) Using the Borsuk-Ulam theorem (Lectures on topological methods in combinatorics and geometry) (Springer, April 2003), by Matousek
a) Ideals, Varieties and Algorithms, by D. Cox, J. Little, and D. O'Shea,
Undergraduate texts in Mathematics, Springer, second edition 1996.
b) Modern computer Algebra, by J. von zur Gathen and J. Gerhard., Cambridge Univ. Press, Cambridge, 1999.
c) A=B, by M. Petkovsek, H. Wilf, and D. Zeilberger, AK Peters, Wellesley MA, 1996.
d) Computers and Intractability: A Guide to the Theory of NP-Completeness, by M. R. Garey and D. S. Johnson, W. H. Freeman, 1979.
a) Theory of linear and integer programming, by A. Schrijver, Wiley 1998
b) Geometric Algorithms and Combinatorial Optimization, by M. Groetschel, Lovasz, A. Schrijver, Springer-Verlag, 1993.
c) Nonlinear Discrete Optimization, by S. Onn, Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2010.
Younger students, if you want to learn about my research, you can access an elementary exposition of past research at the level of a smart high school student, with some activities even suitable for younger kids. In particular, if you are really curious, you can find there an explanation of the logo appearing in my main web page. As you can see there even in geometry my taste is discrete (as in finite). Indeed, sometimes I can be a VERY discrete mathematician.