GRG Permanent Faculty

Geometry Group Faculty

Daniel Allcock <allcock@math.utexas.edu>
My favorite objects are groups, and how they act on all sorts of objects (hyperbolic space, finite geometries, algebraic varieties,...). At the moment I am most interested in the algebra-geometric applications of discrete groups--a surprising number of moduli spaces from algebraic geometry can be described as quotients of symmetric spaces (like the complex ball) by discrete groups. I also think about the topological consequences for spaces of this sort.
David Ben-Zvi <benzvi@math.utexas.edu>
My research interests concern the interaction of algebraic geometry, representation theory and mathematical physics (or put more linearly, the geometric realization of algebraic structures arising from physics). In geometry, I am interested in parameter (or ``moduli'') spaces for algebraic curves and surfaces and various structures on them, such as bundles and differential operators, and most recently noncommutative algebraic geometry. In representation theory, I am interested in the structure of vertex algebras and related infinite--dimensional Lie algebras, and the geometric Langlands program. In mathematical physics, I am interested in the basic structures of conformal field theory, integrable Hamiltonian systems, soliton equations and lately matrix models.
Gavril Farkas <gfarkas@math.utexas.edu>
I do research in algebraic geometry. Among other things I am interested in the moduli space of curves (global aspects and intersection theory on the moduli space), enumerative geometry, abelian varieties and their moduli (Schottky type problems, stratifications of the moduli space, linear series on abelian varieties), vector bundles on curves (the global geometry of the moduli space of bundles, Brill-Noether type questions, degenerations), K3 surfaces and syzygies of varieties.
Dan Freed <dafr@math.utexas.edu>
Research interests: global issues in geometry, topology, and linear analysis; geometric questions in quantum field theory and string theory.
Bob Gompf <gompf@math.utexas.edu>
Research interests: Low-dimensional topology, especially the differential topology of 4-manifolds (eg. Kirby calculus, Lefschetz pencils, exotic smooth structures on R^4). Symplectic and contact topology, particularly in low dimensions, and the topology of Stein surfaces. (For more details on any of this, see my book, AMS Grad Studies in Math #20, 1999.)
Tamás Hausel <hausel @math.utexas.edu>
My main interest is application of geometrical ideas in various fields in mathematics and physics; my main project is called the "Quaternionic Geometry of Everything". 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, finite group theory, global analysis, mathematical physics, number theory, representation theory, string theory. Further possibilities I have in mind are conformal field theory and low dimensional topology.
Sean M. Keel <keel@math.utexas.edu>
Research interests: Algebraic Geometry, particularly Mori's program, GIT, moduli problems, and intersection theory.
Daniel Knopf <danknopf@math.utexas.edu>
My research focuses on using geometric evolution (e.g. Ricci flow or mean curvature flow) to find canonical or optimal geometries. The guiding heuristic is that such flows acquire very special geometries when they develop singularities. Understanding their singularity formation can show how a geometric object may be simplified and improved, possibly with a change in its topology. These ideas have exciting applications to topology and complex (Kaehler) geometry. I'm also interested in developing combinatorial flows and in exploring connections between geometric flows and quantum field theories in physics.
Lorenzo Sadun <sadun@math.utexas.edu>
My interests lie in diverse areas where geometry and physics meet. Most of my recent work has been on the topology and dynamics of aperiodic tilings in 2 and 3 dimensions. I am also studying differential geometric structures in quantum mechanics and solid state physics, especially adiabatic quantum transport. I have also supervised students working on the geometry and thermodynamics of DNA supercoiling, and on singularity formation in nonlinear wave equations, such as arise in gauge theory. If a subject is geometric and physical I may or may not know much about it, but I'm bound to be interested.
Karen Uhlenbeck <uhlen@math.utexas.edu>
My research interests at this time are in integrable systems, paricularly questions which come from geometry and theoretical physics. I have also been looking at both the geometric non-linear Schrodinger equation and the wave map equation in 2+1 dimensions, which is the dimension in which the energy for both equations is scale invariant. I am particularly interested in the question of blow-up, as well as well-posedness questions. Most of my earlier work was in elliptic partial differential equations, particularly scale invariant systems such as the harmonic map equation in two dimension and the Yang-Mills equations in four dimensions. Many of my students have worked in gauge theory.

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