# Algebra, Geometry & Topology

# Algebra, Geometry, and Topology

Algebraic geometry, combinatorics, commutative algebra, complex manifolds, Lie groups and algebra, low-dimensional topology, mathematical physics, representation theory, singularity theory.

## Algebraic Geometry

The algebraic side of algebraic geometry addresses the study of varieties and schemes, both over the field of complex numbers and other fields. Schemes also provide a link with algebraic number theory. Members of this group are interested in connections with representation theory, with arithmetic algebraic geometry, and with complex algebraic geometry.

- Prakash Belkale
- Ivan Cherednik
- Olivia Dumitrescu
- Shrawan Kumar
- Richard Rimanyi
- Andrey Smirnov
- Alexander Varchenko

## Commutative Algebra

One important role of commutative algebra is in the foundations of algebraic geometry, through rings of functions on a variety, and generalizations, incorporating nilpotent elements, and also sheaves of rings lying over such a variety (or scheme). Other issues in commutative algebra, such as factorization, also directly relate to number theory. Members of this group study these issues and others, such as the structure of commutative rings.

- Olivia Dumitrescu

## Representation Theory

Groups arise as sets of symmetries of various structures, perhaps geometric, or physical, or algebraic or analytic. Representation theory deals with how these symmetries give rise to families of operators on a vector space. Associated to groups are Lie algebras, group algebras, and other algebras. The study of representations of these structures arises sometimes from the group setting, and in addition can take a life of its own. This central subject connects with many areas of mathematics, in analysis, geometry, and mathematical physics. Members of our faculty do research on topics in Lie algebras and Lie groups, Kac-Moody algebras, quantum groups, geometric methods in representation theory, Lie combinatorics, and special functions.

- Prakash Belkale
- Ivan Cherednik
- Jiuzu Hong
- Shrawan Kumar
- Lev Rozansky
- Richard Rimanyi
- David Rose
- Andrey Smirnov
- Philip Tosteson
- Alexander Varchenko

## Combinatorics

A central theme in combinatorics is to count how many objects there are in a certain structure. Extending this, one seeks to produce a bijective correspondence between two given structures. Accomplishing this may bring in various algebraic techniques, involving symmetries for example, though beyond any general collection of algebraic techniques, combinatorics has its own domain. Members of our department do research in combinatorial aspects of representations of Lie groups, entities for enumeration, characters, and special functions, matroid theory, and finite geometries.

- Ivan Cherednik
- Andrey Smirnov
- Philip Tosteson
- Alexander Varchenko

## Mathematical Physics

Algebraic methods in modern mathematical physics have been influenced by efforts to understand the roles of symmetries in quantum field theory, and particularly by efforts to produce completely integrable systems (a notable example being the Seiberg-Witten equations). The symmetries behind such integrability tend to be hidden, and require sophisticated techniques for exposure. Members of our faculty are engaged in the study of such algebraic methods, including the representation theory of the Virasoro algebra and other infinite dimensional Lie algebras, which yield insights into modern mathematical physics, especially conformal field theory and string theory.

- Ivan Cherednik
- Olivia Dumitrescu
- Lev Rozansky
- Andrey Smirnov
- Alexander Varchenko

## Riemannian Geometry

Created by Carl Friedrich Gauss to take account of the curvature of the Earth in surveys of large areas in Germany, differential geometry with its notion of curvature was extended to spaces of arbitrary dimension by Riemann, and found significant application in dimension 4 in Einstein’s general relativity. It has also developed many connections with complex analysis, algebraic geometry, PDE, and other areas of mathematics. Members of our department study curvature and the Ricci tensor, geometry of symmetric spaces and compact 2-step nilmanifolds, rough metric tensors, Gromov-Hausdorff convergence, and connections with PDE.

- Michael Taylor
- Justin Sawon

## Topology

A primary theme of topology is to associate algebraic structures to a topological space with the goal of determining if such structures allow one to decide whether two topological spaces are equivalent. One goal is to classify the topological structures of manifolds. Topologists in our department do research in characteristic classes, low dimensional topology, especially 3-manifolds, foliations of manifolds, connections with dynamical systems, and Lie groups.

- David Rose
- Lev Rozansky
- Justin Sawon
- Philip Tosteson

## Singularity Theory

This is the study of the images of smooth spaces under smooth maps whose derivatives are not everywhere injective, therefore giving rise to singular images. Precise studies of the nature of these singularities connect to topics such as the behavior of caustics of waves and catastrophes. Members of this group do research on the structure of singularities and stratified spaces. There is also have a joint project with members of the Computer Science Department, on connections with computer vision.

- Richard Rimanyi
- Alexander Varchenko