# Mathematics 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**: Nikolai Markarov, Nets Katz, Maksym Radziwill, Eric Rains, Omer Tamuz

### 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, Yi Ni

### Mathematical and Computational Linguistics

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

**Affiliated Faculty**: Matilde Marcolli

### Mathematical and Machine Learning

This new and fast growing area at the interface of pure mathematics and data science includes applications of AI to some of the most challenging math problems, as well as applications of mathematical ideas to the theory of machine learning.

**Affiliated Faculty**: Sergei Gukov

### Mathematical Logic

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

**Affiliated Faculty**: Alexander Kechris

### Mathematical Physics

Research covers Schrödinger operators, random matrices.

**Affiliated Faculty**: Sergei Gukov, Alexei Kitaev, Eric Rains, Barry Simon

### 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

### 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

### 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

### 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

### 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

### 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: Tom Graber, Sergei Gukov, Anton Kapustin, Hirosi Ooguri, Tony Yue Yu