# Algebra: Number Theory, Topology, and Vertex Operators

**Representation Theory and Number Theory**

**Robert Boltje** and his students work in the representation theory of finite groups. They are primarily involved with the conjectures of Alperin, Broué and Dade. Current research interests include canonical induction formulas, biset functors, fusion systems, block theory of group algebras, and equivalences between such blocks. Professor Boltje has also worked in the area of algebraic number theory, where he has developed functorial methods to understand Galois actions on rings of algebraic integers, and other structures associated to number fields.

**Martin Weissman**'s research involves the interaction between representation theory, geometry, and number theory. Specifically, he works on automorphic forms and representations -- the network of theorems and conjectures known as the Langlands program. Within the Langlands program, he is interested in exceptional and metaplectic groups, and broad questions in the representations of p-adic groups. He has also studied connections between arithmetic and Coxeter groups, and the visualization of algebra and number theory.

**Finite Groups of Lie Type and Finite Geometries**

**Bruce Cooperstein*** (emeritus)* studies finite groups of Lie type, in the context of finite geometries (geometries with a finite number of points and lines) and combinatorics. His work has involved classification and characterization results as well of novel constructions, for example, an ovoid in a hyperbolic orthogonal space in ten (linear) dimensions over the finite field GF(5). His work has touched on the theory of error correcting codes, finite algebraic geometry. Papers of his have been cited by physicists studying Quantum Information Theory.

**Algebraic Topology**

**Hirotaka Tamanoi** has worked in homotopy theory, in particular, generalized cohomology theories known as cobordism theories. He is fascinated by mathematical ideas and methods inspired by string theory and conformal field theory in mathematical physics. His work has been centered around algebraic topological aspects of loop spaces such as elliptic genus (which was given a quantum field theoretical interpretation by Witten’s work on string theory), and Sullivan's string topology and its relation to symplectic topology. He has written a book on vertex operator algebras and elliptic genera. More recently, he is interested in topological quantum field theories and their applications.

**Vertex Operator Algebras**

**Chongying Dong** and **Geoffrey Mason** work in the area of vertex operator algebras. This area has its origins in two-dimensional conformal field theory, monstrous moonshine and vertex operator representations of affine Kac-Moody algebras. They are interested in vertex operator algebras, infinite dimensional Lie algebras, Hopf algebras, category theory and mathematical physics. Their current research focuses on the structure and representation theory of vertex operator algebras.

**Number Theory and Algebraic Geometry**

**Laura Escobar Vega**'s research focuses on the interplay between combinatorics and algebraic geometry. She has developed combinatorial models as part of her program to understand the singularities of Schubert varieties and the action of tori on these varieties. This work has led her to construct and study polytopes using tools from symplectic and algebraic geometry. Lately, Escobar has been working with Newton-Okounkov bodies.

**Jesse Kass** studies algebraic geometry and related topics in commutative algebra, number theory, and algebraic topology. He has major projects on moduli spaces of sheaves on singular curves and on counting algebraic curves arithmetically using motivic homotopy theory.

**Junecue Suh** is interested in the arithmetic aspects of algebraic geometry, including the cohomology of Shimura varieties and the zeta function of varieties over finite fields.

**Algebra and Topology**

**Beren Sanders** works in algebra and topology. His research centers on triangulated categories and their applications, especially tensor triangular geometry and examples arising in stable homotopy theory, modular representation theory, and algebraic geometry. Other interests include equivariant homotopy theory, motivic homotopy theory, higher category theory, and the representation theory of groups and associative algebras.

**Topology and Group Theory**

**Kasia Jankiewicz** works in geometric group theory and its connections to geometric topology, especially in low-dimensions. Among the main themes of her research are metric notions of negative and non-positive curvature, group theoretic analogues of topological properties of 3-manifolds, and Coxeter and Artin groups.