# Math Research

Research in the field of mathematics covers a broad range of areas, including

### Algebra

Research covers finite group theory, algebraic groups, representation theory, symmetric functions, algebraic K-theory.

**Affiliated Faculty**:** **Matthias Flach, Dinakar Ramakrishnan, Xinwen Zhu

### Algebraic Geometry

Research covers moduli spaces, birational geometry, Hodge theory, Calabi-Yau varieties, arithmetic geometry.

**Affiliated Faculty**: Tom Graber, Elena Mantovan, Eric Rains, Dinakar Ramakrishnan, Xinwen Zhu

### Analysis

Research covers classical real and complex analysis, harmonic analysis, functional analysis and operator theory, orthogonal polynomials; complex, smooth, and random dynamical and Hamiltonian systems, fractal, integrable systems, partial differential equations.

**Affiliated Faculty**: Rupert L. Frank, Nikolai Markarov, Nets Katz, Vladamir Markovic, Maksym Radziwill, Eric Rains, Omer Tamuz, Lu Wang

### Combinatorics

Research covers
extremal and probabilistic combinatorics, additive combinatorics and Ramsey
theory.

**Affiliated Faculty**: David Conlon, Nets Katz, Eric Rains

### Geometry and Topology

Research covers low-dimensional topology, hyperbolic geometry, geometric group theory and foliations; symplectic geometry and topology, topological gauge theory, knot theory, and their interface with theoretical physics.

**Affiliated Faculty**: Sergei Gukov, Vladimir Markovic, Yi Ni, Lu Wang

### Noncommutative Geometry

Research covers noncommutative differential geometry and its applications to physics and number theory andnoncommutative algebraic geometry and the theory of noncommutattive motives.

**Affiliated Faculty**: Matilde Marcolli, Eric Rains

### Mathematical Logic

Research covers mathematical logic and its interactions with analysis,
dynamical systems and combinatorics.

**Affiliated Faculty**: Alexander Kechris

### Number Theory

Research covers algebraic number theory, automorphic forms, Shimura varieties, Galois representations, and L-functions.

**Affiliated Faculty**:** **Matthias Flach, Elena Mantovan, Maksym Radziwill, Dinakar Ramakrishnan, Xinwen Zhu

### Mathematical Physics

Research covers Schrödinger operators, random matrices.

**Affiliated Faculty**: Rupert L. Frank, Sergei Gukov, Alexei Kitaev, Eric Rains, Barry Simon

### Probability

Research covers stochastic processes, statistical mechanics, random geometry, and relations to mathematical physics, complex analysis and ergodic theory.

**Affiliated Faculty**: Thomas Hutchcroft, Omer Tamuz, Nick Makarov

### String Theory (Mathematical Aspects)

Research covers enumerative geometry, mirror symmetry, gauge theory and its applications to low-dimensional topology and the geometric representation theory.

**Affiliated Faculty**:** **Sergei Gukov, Anton Kapustin, Hirosi Ooguri

### Mathematical **and Computational Linguistics**

Research focuses on mathematical models of acquisition and evolution of natural languages.

**Affiliated Faculty**: Matilde Marcolli

### Mathematical **Structures in Quantum Physics**

Research covers algebraic and geometric structures arising in quantum field theory, string theory, and other high-energy physics models.

**Affiliated Faculty**: Matilde Marcolli

### Quantum Topology

Research covers various constructions of topological invariants of knots and 3-manifolds based on representation theory of quantum groups and vertex algebras.

Affiliated Faculty: Sergei Gukov