# 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

### 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, Eric Rains, Omer Tamuz

### Combinatorics

Research covers combinatorial designs and matrix theory, coding theory, extremal set theory.

**Affiliated Faculty**: 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**: Vladimir Markovic, Yi Ni

### 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 set theory and its interactions with analysis; combinatorics, dynamical systems, and model theory.

**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, Dinakar Ramakrishnan

### Mathematical Physics

Research covers Schrödinger operators, random matrices.

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

### 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