Spring 2014 MATH 274 001 LEC

Topics in Algebra
Schedule: 
SectionDays/TimeLocationInstructorCCN
001 LECTuTh 11-1230P 740 EVANSFRENKEL, E54447
Units/CreditFinal Exam GroupEnrollment
4NONELimit:36 Enrolled:7 Waitlist:0 Avail Seats:29 [on 05/28/14]
Additional Information: 

 

Prerequisites: Basic Algebra and Group Theory

 

Syllabus: The Langlands Program was launched by Robert Langlands in the late 1960s with the goal of relating Galois representations and Harmonic Analysis. In the last 30 years these ideas propagated to other areas of mathematics, such as geomery and representations of Lie algebras, and then to quantum physics. The goal of this introductionary course is to give a survey of recent developments, focusing on the links between the Langlands Program and quantum fieeld theory.

 

We will start with an overview of the "classical" Langlands correspondence, between representations of Galois groups and automorphic functions. The Galois group here is the group of automorphisms of a number field or the field of functions on a curve over a finite field. In the latter setting, one can reformulate the Langlands correspondence geometrically, using Grothendieck's dictionary linking functions and sheaves. One can then reformulate the geometric version in the case of curves over the complex field (that is, Riemann surfaces). This leads to a mysterious duality between certain categories of sheaves on moduli spaces of (flat) bundles on complex algebraic curves, attached to two Langlands dual groups, which we will discuss.


The geometric form of the Langlands correspondence turns out to be closely related to two different types of quantum field theories: first, 2D conformal field theory, defined on the curve X itself, and second, 4D supersymmetric gauge theory on the product of X and another curve. I will talk about the connections to the 2D and 4D quantum field theories and what we can learn from them. Time permitting, I will also discuss the links between the 2D and 4D theories themselves, which have recently become the focus of extensive research.

 

Required Text:  There is no required text, but some of the lectures will follow closely these two survey articles:

Lectures on the Langlands Program and Conformal Field Theory

Gauge Theory and Langlands Duality

Recommended Reading: My book Love and Math gives an elementary introduction to many of the ideas that will be discussed in this course

Office Hours: Tu 2-3:30