Office of Research, UC Riverside
Carl Mautner
Assistant Professor
Mathematics
carlm@ucr.edu
(951) 827-3109


Categories of Sheaves in Representation Theory

AWARD NUMBER
009718-002
FUND NUMBER
33423
STATUS
Active
AWARD TYPE
3-Grant
AWARD EXECUTION DATE
4/18/2018
BEGIN DATE
7/1/2018
END DATE
6/30/2021
AWARD AMOUNT
$167,104

Sponsor Information

SPONSOR AWARD NUMBER
1802299
SPONSOR
NATIONAL SCIENCE FOUNDATION
SPONSOR TYPE
Federal
FUNCTION
Organized Research
PROGRAM NAME

Proposal Information

PROPOSAL NUMBER
18040479
PROPOSAL TYPE
New
ACTIVITY TYPE
Basic Research

PI Information

PI
Mautner, Carl
PI TITLE
Other
PI DEPTARTMENT
Mathematics
PI COLLEGE/SCHOOL
College of Nat & Agr Sciences
CO PIs

Project Information

ABSTRACT

Representation theory is the study of symmetries in algebra. An understanding of symmetry allows us to reduce complicated problems to simpler ones. Algebra can be used to describe a wide range of phenomena and structures throughout mathematics and the real world, and consequently representation theory has many important applications. Sheaves are geometric objects that generalize the usual notion of functions and have proven to be extremely effective in advancing our understanding of representation theory. This research project aims to uncover finer information about sheaves and applications of this information to representation theory.

Parity sheaves were introduced by the PI and his collaborators as a tool for studying the representation theory of reductive groups in positive characteristic. The study of parity sheaves has also suggested the existence of new structures in categories of perverse sheaves. The PI will explore these structures in some special, important, cases and their expected applications in a number of areas including the representations of Hecke algebras and modular representations of finite groups of Lie type. The geometric spaces to be studied are nilpotent cones and their generalizations for symmetric pairs and in gauge theory, as well as (generalized) flag varieties and toric varieties. The proposed methods include utilizing cohomological parity vanishing properties, nearby cycles and hyperbolic localization functors.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
(Abstract from NSF)