Faculty Research Interests
I work in algebraic geometry, mainly using derived categories of coherent sheaves. My interests include compact hyperkähler manifolds, rationality questions, and classical algebraic geometry.
My main research is in Diophantine analysis and the geometry of numbers. I am particularly interested in studying classical Diophantine problems through modern algebraic and geometric tools and applying them to better understand arithmetic structures. I also work on questions related to heights and invariants of algebraic number fields and polynomials.
Quantum Groups, Representation Theory, Algebraic Combinatorics
My research interests include Representation Theory of Lie Algebras, Quantum Groups, Coxeter Groups, Hopf Algebras, Algebraic Combinatorics, Cluster Algebras, Noncommutative Algebra, and related aspects. I attach a few links to MathWorld and Wikipedia containing accessible descriptions of these fields of Mathematics.
- Group representations
- Lie algebra representations
- Quantum Groups
- Coxeter groups
- Hopf algebras
- Cluster algebra
- Noncommutative algebraic geometry
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.
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.
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.
Homotopy Theory, K-theory, Homological Algebra
Most of my work is in motivic and equivariant homotopy theory, these days largely focused on the latter. I study equivariant cohomology theories that are graded by representations rather than by integers.
I work primarily in number theory, especially algebraic number theory. My research is largely driven by questions concerning automorphic forms (a class of functions that includes modular forms) and L-functions (a class of functions that includes the Riemann zeta function). Because they encode significant algebraic and arithmetic information (for example about ideal class groups, elliptic curves, and Galois representations), L-functions play a key role in several major conjectures (including the Birch and Swinnerton-Dyer Conjecture and the Main Conjectures of Iwasawa Theory, which have in turn led to many of the questions I seek to answer in my research). In my work, I use a variety of techniques, including algebraic, p-adic, arithmetic geometric, and analytic.
Representation Theory, Categorification
I study categorical representation theory, a relatively new field that takes representation theory to the next level. The categorification of a ring is a monoidal category whose Grothendieck group is that ring. A categorified module is an action of the monoidal category on another category, whose Grothendieck group is that module. Categorical representation theory has an astounding amount of structure, making it an interesting topic of study in its own right, but it also can be used to study ordinary representation theory, especially in finite characteristic.
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.
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.
Algebraic and Topological Combinatorics
I work in algebraic and topological combinatorics. This includes studying combinatorial structure of partially ordered sets and topological structure of associated simplicial complexes, studying stratified spaces arising out of areas such as representation theory by playing combinatorial and topological structure off of each other, group actions on posets, the development and refinement of combinatorial-topological techniques such as shellability and discrete Morse theory, total positivity theory, Coxeter groups, and Bruhat order.
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.
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.
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.
Low-dimensional and Symplectic Topology
I use techniques from symplectic geometry and, to a lesser extent, algebraic topology and abstract algebra, to study questions about knots and 3- and 4-dimensional manifolds. Most of this falls under the category of Floer homology and pseudo-holomorphic curves.
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.
My current research interests are in theoretical neuroscience. My goal is to understand the neural basis of sensory perception, how it is modulated by expectations leading to behavior. Some of my recent projects address the following questions: Which mechanisms underlie associative learning at the level of synaptic plasticity in local cortical circuits? How does behavior emerge from the temporal dynamics of cortical networks? To study these questions, I use methods from statistical physics, information theory, machine learning, and dynamical systems. I combine statistical analysis of neurophysiological and imaging data from large populations of neurons in behaving animals with theoretical models based on neural networks. My projects rely heavily on numerical analysis and computer simulations of recurrent networks.
My research is in theoretical neuroscience. Our group’s research involves developing mathematical models of brain computations, drawing from dynamical systems theory, stochastic processes, and reinforcement learning. We also draw on recent developments in artificial intelligence, using artificial neural networks such as the ones used in AI to emulate the complex information processing performed by the brain. Finally, our group works together with experimental collaborators, where our role is to use modeling and data analysis to interpret data and to contribute to experimental design by generating predictions that can be tested in future experiments.
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.
I also work on operator algebras on L^p spaces for p different from 2. This is a very new area, with less structure but still quite promising, and there are many problems which nobody has looked at yet. (I have a collection of over 70 problems, varying greatly in difficulty.) Since the area is so new, there is much less prior work to learn about. The methods are less algebraic than for C*-algebras.
Algebraic Geometry, Noncommutative Geometry
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.
Combinatorics and Algebraic Geometry
My work is somewhere in between algebraic geometry, combinatorics, representation theory, and algebraic topology. I work mostly with algebraic varieties that are built using the data of a hyperplane arrangement; examples include hypertoric varieties and various generalizations of (partial compactifications of) configuration spaces. I then study the relationship between the algebraic invariants of these spaces and the combinatorial invariants of the input data. The flow of information goes in both directions. Sometimes I use combinatorics to compute objects of intrinsic geometric interest (categories of sheaves, cohomology rings, etc.) and sometimes I use geometric machinery to prove purely combinatorial theorems.
Mathematical Biology, Evolution, Statistics
I work on probability and statistics as applied to understanding ecology and evolution, in particular developing new stochastic models of biological evolution and using statistical inference and visualization methods to find out what genomes can tell us about biology.
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.
Algebraic Geometry and Mathematical Physics
I am interested in algebraic geometry and mathematical physics. My recent work focus on curve counting theories, such as Gromov-Witten theory and Fan-Jarvis-Ruan-Witten theory, and the mirror symmetry beyond them. These theories have deep connections to complex geometry, number theory, and representation theory.
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.
Algebraic and Geometric Topology
I work in a range of areas in topology, mostly in algebraic topology but some in geometric topology as well. I like to see the geometry which underlies homotopical structures. My interests are broad, with my most recent projects being in cohomology of alternating and symmetric groups, in group theory related to rational homotopy theory, in knot theory, and in neural networks. Configuration spaces are a part of much of what I do, and I like to study topology, geometry, algebra and combinatorics related to them. Some of my research along with other topics of interest in algebraic topology are the subject of a lecture series available at:
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.
Geometric Analysis, Partial Differential Equations
I study Geometric Analysis and Geometric PDE. My most recent work has involved fourth order elliptic minimal surface equations, and other recent work has been on constructions of Ricci curvature with applications to machine learning. In general I study fully nonlinear elliptic PDE such as the Monge-Ampère (sometimes involving optimal transportation) and special Lagrangian equations.
Approximation Theory, Harmonic Analysis, Orthogonal Polynomials, Numerical Analysis
I work in several directions in analysis, applied mathematics. My main research is in approximation theory, Fourier analysis, orthogonal polynomials and special functions, which are really all connected. I’m also interested in constructing minimal numerical integration formulas. Most of my work focuses on multidimensional problems.
Algebraic and Enumerative Combinatorics, Perfect Matchings
I work in enumerative, bijective and algebraic combinatorics. Most of what I am working on at the moment is related to the dimer model, or to Schubert calculus and the combinatorics of reduced words. Lately I’ve also been spending a lot of time thinking about Kazhdan-Lusztig polynomials for matroids. I use computers heavily in my work.