Research problems and thesis advising
Current Graduate Students
Experience as a graduate student
I started my graduate course by taking the Part III of the Mathematical Tripos in Cambridge, UK in 1994-1995. This was a one-year graduate course similar to the first year of a Phd course in Austin. I completed it with five three hours written exams (a bit like the prelim exams) in: Algebraic Topology, Differential Geometry, Functional Analysis, Knot Theory, Yang-Mills theory on Riemann surfaces together with an essay titled: The moment map. (This essay forms the first part of my dissertation.) Then Nigel Hitchin became my advisor who supervised my research, which resulted in my thesis in 1998.
Past work with graduate students
I ran a graduate study seminar in Berkeley where I worked with graduate students to learn the basics of the geometry of Nakajima's quiver varieties. The activity started in this seminar finally lead to research papers like this and this. The second of these was written by two graduate students Megumi and Nick .Research competence
My main interest is application of geometrical ideas in various fields in mathematics and physics. The fields in geometry which I am interested in could be listed as the following 6 (=3x2) subjects: algebraic, combinatorial and differential, geometry and topology. The subjects of application include so far combinatorics, commutative algebra, global analysis, mathematical physics, string theory, number theory and representation theory of finite groups of Lie type. Further possibilities I have in mind are conformal field theory and low dimensional topology. My main project is the "Quaternionic Geometry of Everything".Books to read
Depending on the project the graduate student should be familiar with standard books in the particular subject, examples are:
Research projects
I could most efficiently advise graduate students in projects which are related to the five research projects of mine which appear in my October 2001 research proposal (see also the list below). These all relate to hyperkähler (quaternionic, N=4 supersymmetric) gauge theories in various dimensions. For a chart of these theories look at this Zoo. Check out my list of publications for my latest papers related to these projects.Project 0 Integration (index) theory on circle-compact manifolds (joint with András Szenes and Nicholas Proudfoot)
This is one of my most recent research projects: it concerns an integration theory on hyperkähler manifolds with a proper Hamiltonian action of a circle (symplectic manifolds which have such a circle action are called: circle-compact). My hopes are that this theory would help us to build and calculate topological quantum field theories out of hyperkähler gauge theories (possibly leading to new knot and $3$-manifold invariants) on one hand and to be able to characterize all the possible h-vectors of matroids (this is a 25 years old problem of Stanley) on the other (see Project 4 below). This way I hope that this theory is going to unite all my projects below. Suggested readings:Project 1 Mirror symmetry, Langlands duality in the non-Abelian Hodge theory of a curve (joint with Michael Thaddeus and Fernando Rodriguez-Villegas)
This project (my oldest one) is about to find mirror symmetry patterns in the geometry of the Hitchin system for Langlands dual groups. With Michael we have already achieved to show that there is Strominger-Yau-Zaslow mirror pattern for G=SL(n) and G'=PGL(n). We also formulated a conjecture wich claims that these mirror partners have identical Hodge diamond. In order for everything to work out we need to use the pretty language of flat U(1)-gerbes (B-fields for physicists). We proved this conjecture for G=SU(2) and SU(3), for the rest it is open. More recently with Fernando we formulated an analouge conjecture for the SL(n) and PGL(n) character variety of the curve. The attack of this conjecture comes from number theory. Namely we count the points of this variety over a finite field, and this way we can calculate the so-called E-polynomial of the varieties in question. Then we could check our conjecture when n is any prime, or when n=4. This implies the conjecture with Michael on the level of Euler characteristic. Suggested readings:Project 2 Hodge cohomology of non-compact manifolds (joint with Eugenie Hunsicker and Rafe Mazzeo)
We have recently been able to calculate the space of L^2 harmonic forms (Hodge cohomology) on rather general open manifolds (so-called fibred boundary and fibred cusp). This work confirms many predictions arising recently in duality theories in string theory. Our perspective is to find a compactification of the space such that its intersection cohomology would calculate the Hodge cohomology of the original open manifold. However many of the physicists predictions are still open. We are trying to generalize our methods to the more complicated cases. Suggested readings:Project 3 Yang-Mills instantons on ALF gravitational instantons (joint with Gábor Etesi)
In our recent work we have constructed explicit SU(2) Yang-Mills instantons on Gibbons-Hawking multi-Taub-NUT metrics (and also on the A_k ALE spaces). These are the analogues of the classical t'Hooft solutions on R^4. The part of the moduli space we uncover is cornered by reducible solutions, corresponding to L^2 harmonic forms whose existence was first provided in Project 2 above. This project still has many unsolved questions: find the complete moduli space of solutions (even the infinitesimal analysis is non-trivial due to the lack of adequate index theory on these kind of ALF open manifolds). The activity of this project could also lead to general investigations of Donaldson theory on general ALF 4-manifolds. Suggested readings:Project 4 Toric hyperkáhler varieties (joint with Bernd Sturmfels)
We have recently investigated the topology and algebraic geometry of toric hyperkäahler varieties. Our approach finds strong and novel relationships between the geometry of toric hyperkähler varieties and combinatorics of many kind like: convex polyhedra, Lawrence polyhedra, hyperplane arrangements, matroid complexes. My point of view is that our work opens up the possibility to study the h-vectors of matroid complexes (classification theory is open) using the geometry of toric hyperkahler varieties. Most notably index theory and Hodge theory on such manifolds could be advanced in order to get topological restrictions on the underlying manifold and in turn combinatorial restrictions on the h-vectors of matroid complexes. Suggested readings:Project 5 Cohomology ring of quiver varieties (joint with Nicholas Proudfoot)
There has recently been a lot of activity using the cohomology of Nakajima's quiver varieties in representation theory. One interesting application of representation theory ideas give a conjecture, due to Lusztig, of the Betti numbers of certain quiver varieties. This is the main motivation of this project. The quiver variety is obtained as a finite dimensional hyperkähler quotient. Therefore what is needed is an understanding of the cohomology of hyperkähler quotients. This question is well understood in the kähler quotient case, but interestingly the techniques do not generalize to the hyperkähler setting. Therefore new methods should be found. Suggested reading:Research problems
Apart from the problems in the projects above there are some more on the homepages of the seminars I ran in Berkeley. If you have an idea or interest to attack any of these problems, or you are interested in other research problems contact me.