# Algebra: Number Theory, Topology, and Vertex Operators

There are seven faculty actively engaged in research in the field of algebra.

**Algebra 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, Broue and Dade in the theory of "modular representations" of finite groups. Professor Boltje also works in the area of algebraic number theory, where he has developed functorial methods to understand Galois actions on rings of algebraic integers in number fields.

**Bruce Cooperstein** studies finite groups of Lie type, in the context of finite geometries (geometries with a finite number of points and lines) and combinatorics.

**Samit Dasgupta** studies the connections between special values of L-functions, algebraic points on Abelian varieties, and units in number fields. His work uses the theory of p-adic modular forms and Galois representations, and has enabled him to make headway on some central problems in modern number theory. In particular, he has made progress on Stark's conjectures, which are precise versions of Hilbert's 12th problem, and on related conjectures of Gross. These conjectures consist of explicit formulae which express units in number fields in terms of special values of certain L-functions. Dasgupta's work in this area represents significant progress since Stark first made his conjectures in the 1970's. He received a prestigious Research Fellowship from the Alfred P. Sloan Foundation in 2009.

**Martin Weissman's** research involves the interaction between representation theory, geometry, and number theory. Specifically, he works on automorphic forms and representations, and what is generally known as the Langlands program. Within the Langlands program, he is interested in modular forms on exceptional groups, representations of p-adic groups, and L-functions. Presently he is interested in some foundational questions about automorphic L-functions, interactions between algebraic deformation theory and representations of p-adic groups, and some aspects of Iwasawa theory applied to modular forms.

**Algebraic Topology**

**Hirotaka Tamanoi** works on ideas in algebraic topology inspired by constructions in mathematical physics. His work has ranged from elliptic cohomology (which was given a major impetus by the work of Witten on the relation of the elliptic genus to string theory) to Sullivan's string topology.

**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 quantum field theory, and has had important applications to areas of mathematics as far a field as the theory of finite groups and the invariants of knots and of three-manifolds, as well as with topics such as elliptic cohomology, Hopf algebras and tensor categories, Kac-Moody Lie algebras.

They run, with Professor Boltje, the very active Algebra seminar in the department. Professor Mason co-organizes the joint Berkeley-Santa Cruz Lie Groups and Algebra seminar.