# About Us: Faculty Research Interests

Nir Avni
*PhD, Hebrew University*

Group Theory - Algebraic and arithmetic groups.

Keith Burns
*PhD, Warwick University (UK)*

Keith Burns works on geometrical and dynamical problems connected with geodesics in manifolds with nonpositive curvature. Recent work has concentrated on partial hyperbolicity and stable ergodicity.

Frank Calegari
*PhD, University of California, Berkeley*

Calegari's work in number theory includes the study of Galois representations, in particular their connection to modular forms. He also studies the geometry and analysis of p-adic modular forms. Another interest is the application of arithmetic to questions in geometric topology.

Kevin Costello
*PhD, University of Cambridge*

Kevin Costello works on mathematical aspects of quantum field theory and string theory, as well as on related topics in algebra and geometry.

John Francis
*PhD, Massachusetts Institute of Technology*

Algebraic Topology

John Franks
*PhD, University of California, Berkeley*

Franks's recent topological work has focused on actions of groups on surfaces -- in particular, lattice actions which leave invariant a Borel probability measure on the surface. Some of his earlier work concerned knots that occur as closed orbits of flows, index arguments using the zeta function, subshifts of finite type, attractors, and descriptions of possible dynamics.

Ezra Getzler
*PhD, Harvard University*

Getzler has interests ranging over index theory of Dirac operators, Malliavin calculus, cyclic homology, the theory of operads and Gromov-Witten invariants (which lie at the intersection of algebraic geometry, symplectic geometry and theoretical physics).

Paul Goerss
*PhD, Massachusetts Institute of Technology*

Goerss's work is in homotopy theory; in particular. he is interested in combining the algebraic geometry of formal groups and the cohomology of profinite groups to studying stable homotopy theory.

Elton P. Hsu
*PhD, Stanford University*

Hsu's research on stochastic analysis emphasizes applications to parabolic equations in a geometric setting.

Kate Juschenko
*PhD, Texas A&M*

Functional Analysis - Operator algebras, group theory.

Bryna Kra
*PhD, Stanford University*

Kra works in dynamical systems and ergodic theory, with an emphasis on applications to combinatorial number theory.

Aaron Naber
*PhD, Princeton*

Geometry of Manifolds, Ricci Curvature, Geometric Flows, Harmonic Analysis

Mark Pinsky
*PhD, Massachusetts Institute of Technology*
(Emeritus)

Pinsky has worked in probability theory, specifically diffusion processes on manifolds. In recent years he has become interested in classical Fourier analysis, investigating the various implications of the "Pinsky phenomenon".

R.Clark Robinson
*PhD, University of California, Berkeley *
(Emeritus)

Dynamical Systems: His research in dynamical systems uses analytic and geometric methods. Recent work includes questions relating to attractors, correctly aligned windows to get invariant sets, applications of stable manifold theory to celestial mechanics, and Melnikov method applied to higher dimensional Hamiltonian systems to insure horseshoes.

Michael R. Stein
*PhD, Columbia University*

Stein's current research focuses on certain extensions and generalizations of braid groups. He is also interested in questions concerning central extensions and Schur multipliers of Chevalley groups and Kac-Moody groups over commutative rings.

Andrei Suslin
*PhD, Leningrad State University*

Suslin has made fundamental contribution to algebraic K-theory with special emphasis on higher algebraic K-theory and its connections with and applications to algebraic geometry and algebraic number theory. Suslin is the Board of Trustees Professor of Mathematics.

Dmitry Tamarkin
*PhD, Penn State University*

Algebra and homological algebra; specifically, Tamarkin works on operad theory, non-commutative differential geometry, and the applications of these fields to mathematical physics.

Valentino Tosatti
*PhD, Harvard University*

Differential Geometry and PDE on complex, Kahler and symplectic manifolds, with emphasis on Calabi-Yau manifolds, complex Monge-Ampere equations and geometric flows, as well as applications of these transcendental techniques to problems in Algebraic Geometry.

Boris Tsygan
*PhD, Moscow University*

Boris Tsygan's interests include non-commutative geometry, index theory, microlocal analysis, deformation quantization, mathematical physics, and sympletic geometry.

Kari Vilonen
*PhD, Brown University*

Kari Vilonen works on representation theory and the geometric Langlands program using methods from topology and algebraic geometry.

Ben Weinkove
*PhD, Columbia University*

My research focuses on partial differential equations and complex geometry. This includes the study of canonical Kahler metrics, the Kahler-Ricci flow and the complex Monge-Ampere equation and its generalizations.

Jared Wunsch
*PhD, Harvard University*

Wunsch studies linear partial differential equations,with emphasis on spectral theory and propagation of singularities for operators on singular spaces.

Zhihong Jeff Xia
*PhD, Northwestern University*

Xia's research is in the areas of Newtonian n-body problem, Hamiltonian dynamics and general hyperbolic and partially hyperbolic dynamical systems.

Sandy Zabell
*PhD, Harvard University*

Professor Zabell's research interests are in the fields of probability and statistics. He is particularly interested in legal applications of statistics, forensic science, and communications security. He is also interested in mathematical logic and the history of mathematics.

Eric Zaslow
*PhD, Harvard University (Physics)*

Mathematical Physics. Zaslow studies mathematical aspects of string theory, most recently focusing on mirror symmetry of Calabi-Yau threefolds. This involves enumerative invariants, minimal submanifolds, and the equivalence of D-brane categories.

Steve Zelditch
*PhD, University of California, Berkeley *

Steve Zelditch works in spectral and scattering theory for Laplacians on Riemannian manifolds, especially the asymptotics of eigenfunctions (quantum chaos) and the inverse spectral problem (Can you hear the shape of a drum). He also works on asymptotics problems on Bergman kernels and metrics that arise in Kaehler geometry, Gaussian random waves and random metrics.

**Seminar Homepages**

Dynamical Systems

Geometric Langlands Program