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Shabnam Akhtari: number theory, Diophantine equations.

My main research interests are in Number Theory, in particular in Diophantine approximation, transcendental number theory and applications of mathematical analysis to Diophantine equations and inequalities.


Arkady Berenstein: quantum groups, representation theory, algebraic combinatorics.

My research interests include Representation Theory of Lie algebras and Coxeter Groups, Hopf Algebras, Algebraic Combinatorics and related aspects. I attach a few links to MathWorld (a good source of information for beginners) containing accesible descriptions of these fields of Mathematics.


Boris Botvinnik: differential topology, positive scalar curvature, Morse theory.

I study algebraic topology and differential geometry, with a focus on conformal geometry and the space of metrics of positive scalar curvature.


Marcin Bownik: harmonic analysis, wavelets, approximation theory.

I work in the area of harmonic analysis and wavelets. More specifically my research areas are:

  • construction of wavelet bases with good time-frequency localization for large classes of dilations;
  • limitations on the existence of such wavelet bases;
  • anisotropic function spaces and their study through wavelet bases, general L^2 theory of wavelets, frame wavelets, and generalized multiresolution analysis.


Jon Brundan: algebraic groups, combinatorial representation theory, Lie superalgebras.

I study representation theory and combinatorics arising from semisimple Lie algebras and algebraic groups, like the Lie algebra gl_n(C) of all n by n matrices over C and the group GL_n(C) of invertible such matrices.


Dan Dugger: motivic homotopy theory, K-theory, homological algebra.

I work in algebraic topology and homological algebra. I study algebraic K-theory and other "motivic" cohomology theories for algebraic varieties, as well as the homotopy theory of differential graded algebras. I'm interested in applications of homotopy-theoretic methods to geometry and commutative algebra.


Peter Gilkey: higher-signature differential geometry, heat trace analysis.

I work in pseudo-Riemannian geometry. I also work studying the asymptotics of the heat equation and their applications to questions in geometry.


Weiyong He: differential geometry and partial differential equations.

  • Complex geometry and Kahler geometry, extremal metrics and Calabi flow.
  • Geometric evolution equations, mean curvature flow.
  • Nonlinear partial differential equations.


Jim Isenberg: mathematical general relativity, Ricci flow, nonlinear PDEs.

I study the behavior of solutions of nonlinear partial differential equations of the sort that arise in Einstein's theory of general relativity, in Ricci flow, and in related problems from physics and geometry.


Alexander Kleshchev: representation theory, Lie theory, group theory.

I study representation theory of Lie algebras, algebraic groups and related objects, such as symmetric groups, Hecke algebras, etc.


David Levin: Markov chains and random walks, multiparameter processes, potential theory.

My research is in probability theory, including: random walks, Markov chains, multiparameter processes, jump processes, and related potential theory. Recently, I am interested in quantatitive estimates on the time for ergodic Markov chains to equilibriate.


Shlomo Libeskind: mathematics education.

I am interested in investigating the following:

  • how middle and high school students become skilled at writing proofs and problem solving; what does and doesn't work to facilitate the acquisition of these skills;
  • developing and testing high school curriculum materials that emphasize the use of Polya-type heuristics in problem solving and in writing proofs;
  • to what extent high school students and teachers possess the ability to generalize; how to intervene in order to foster a greater ability to generalize in different topics in mathematics.


Huaxin Lin: Functional analysis, C*-algebras, dynamical systems.

I am currently interested in the structure of C*-algebras and applications of C*-algebra theory in classical topological dynamical systems and non-commutative dynamical systems.


Peng Lu: geometric analysis, Ricci flow, complex geoemtry.

My research is in geometric analysis. Currently I am working on Ricci flow, a heat type equation which evolves Riemannian metrics by its Ricci curvature. More precisely I am interested in the ancient solutions and the singularity analysis of Ricci flow.


Victor Ostrik: geometric Lie theory, tensor categories, Hopf algebras.

I am currently interested in the categorification of ring theory, that is study of tensor categories and module categories over them, and in geometric representation theory, which means study of representation theoretic questions using tools from algebraic geometry (perverse sheaves and D-modules).


N. Christopher Phillips: C*-algebras, functional analysis, noncommutative geometry.

I study C*-algebras, which are special algebraic structures which arise in analysis. The easiest examples of C*-algebras are C(X), the algebra of all continuous functions on a compact Hausdorff space X, and L(H), the algebra of all continuous linear operators on a Hilbert space H. The combination of strong extra structure and usefulness in applications has made C*-algebras a broad and very active branch of mathematics. For example, the C*-algebra associated to a locally compact group G is connected to the representation theory of the group. More generally, the crossed product C*(G, A) is made from an action of G on a C*-algebra A. When A = C(X), the study of the crossed product connects with dynamical systems. Most of my current research concerns group actions on C*-algebras (often ones of the form C(X)), with emphasis on but not limited to the structure and classification of crossed products. Even when the group is the integers and the C*-algebra is C(X), or when the group is Z/2Z and the C*-algebra is simple, many questions remain open.


Alexander Polischuk: algebraic geometry, noncommutative tori, mathematics related to string theory.

My general area of research is algebraic geometry. More specifically, recently I work with problems involving derived categories of coherent sheaves on algebraic varieties, noncommutative geometry and higher homotopy structures (such as A-infinity algebras) appearing in algebraic geometry.


Nicolas Proudfoot: symplectic algebraic geometry, matroid theory, quivers.

I study symplectic geometry in the category of algebraic varieties, and most of the spaces with which I work are motivated by either representation theory or combinatorics.


Hal Sadofsky: stable homotopy theory, homological algebra, complex cobordism.

I work primarily in stable homotopy theory which is the part of algebraic topology concerned with properties of maps which are preserved after ``suspending'' (cross X with the unit interval, and identify X x 0 to a point and X x 1 to a point). At the moment the questions I'm working on concern understanding how to compute generalized homology theories (functors from spaces to graded groups which obey most of the axioms of homology theory) on certain types of spaces arising from limit constructions, and their applications.


Brad Shelton: non-commutative algebraic geometry, ring theory, homological algebra.

I study noncommutative ring theory and homological algebra, including noncommutative algebraic geometry, Koszul algebras and generalized Koszul algebras.


Chris Sinclair: random matrix theory, heights of polynomials.

I am interested in the statistics of eigenvalues of random matrices and the roots of random polynomials. I am also interested in measures of complexity of polynomials (heights) and the distribution of roots of polynomials with low height.


Dev Sinha: rational homotopy theory, knot theory, operads.

Most of my work has applied algebraic topology to answer geometric questions, in particular about group actions on manifolds and knot theory. I also have a strong interest in configuration spaces. My current projects focus on rational homotopy theory (simplifying topology by ignoring torsion, so that for example one cannot tell the difference between an odd-dimensional sphere and the corresponding projective space), and I'm starting to work on group cohomology (where there is nothing but torsion!).


Bartek Siudeja: probability, differential equations.

In probability, I am interested in symmetric stable processes and their potential theory, subordinate Brownian motions, and the "Hot-spots" conjecture. On the differential equations side, I am interested in Dirichlet and Neumann eigenvalue problems for elliptic operators, and the spectral theory of Laplacians.


Arkady Vaintrob: algebraic geometry, knot theory, mathematical physics, Lie theory.

I study algebra and geometry motivated by physics. My current interests involve algebraic geometry, in particular orbifolds. My past interests have included knot theory and representation theory.


Marie Vitulli: commutative algebra, closure operations, valuation theory, algebraic geometry.

I have recently worked on three projects in commutative algebra:

  • Properties of affine semigroup rings and Rees algebras including questions like when is an affine semigroup ring regular in codimension k and when is the Rees algebra of an ideal in a special class of monomial ideals normal. The polynomial ring K[X_1, ... ,X_n] is an example of an affine semigroup ring.
  • Exploring connections between the weak subintegral closure of an ideal and reductions of ideals and developing a theory for weak subintegral closure of modules, including a valuative theory. The weak subintegral closure (also called weak normalization) of an integral domain R is a subring R of the integral closure of K in its quotient field (also called the normalization of R) whose prime spectrum is in one-to-one correspondence with the prime spectrum of R. The connections I am looking at have counterparts in complex analytic spaces.
  • Investigating the core of monomial ideals. The core of an ideal in a Noetherian ring is the intersection of all reductions of the ideal.


    • Vadim Vologodsky: algebraic geometry, homological algebra.

    I work in Algebraic geometry, Number theory, and Homological Algebra. I study the p-adic Hodge theory, motives and algebraic K-theory. I'm interested in applications of homotopy-theoretic methods to Algebra.


    Hao Wang: superprocesses, mathematical finance, stochastic PDEs.

    My current research interest is in the area of measure-valued processes or superprocesses that come out as limits in distribution of a sequence of branching particle systems.


    Yuan Xu: numerical analysis, orthogonal polynomials, approximation theory, harmonic analysis.

    I work in several directions in analysis, applied math, and numerical analysis: orthogonal polynomials, approximation theory, Fourier analysis, numerical integration, as well as computed tomography (CT) and Radon transforms.


    Ben Young: algebraic and enumerative combinatorics, perfect matchings.

    I study algebraic and enumerative combinatorics. My main specialty is the theory of perfect matchings on planar bipartite graphs, domino tilings, and plane partitions. As such my work has application to random matrix theory, exactly solvable statistical mechanics, and certain parts of algebraic geometry; I enjoy discovering and solving combinatorial problems which arise in those and other areas. Finally, I use experimental mathematics and, when applicable, high-performance computing heavily in my work. One of my goals is to improve upon the computational toolbox of the discrete mathematician.


    Sergey Yuzvinsky: algebraic combinatorics, hyperplane arrangements, algebraic geometry and topology.

    I study hyperplane arrangements, an area of mathematics where methods of algebra, geometry, topology, and combinatorics interplay. It is a good area to apply your passion for (or deepen your skill in) practically any field of math.