The geometry of the Kapustin-Witten Equations

Speaker: 

Professor and Sid W. Richardson Regents Chairholder Karen Uhlenbeck

Institution: 

University of Texas, Austin

Time: 

Thursday, May 12, 2011 - 10:30am

Location: 

NS2 1201

The self-dual Yang Mills equations are equations for a connection in a principal bundle on a 4-manifold with a real structure group such as SU(2). They have been a source of immeasurable geometric, analytic and topological interest since they were introduced to the mathematics community in the l960's. It is natural to define a complex connection with the same structure group; Kapustin and Witten have introduced a one parameter family of equations for this complexified connection which are related in a natural way to the Yang-Mills equations. We carefully review some of the geometry and topology of the self-dual Yang Mills connections and describe how the Kapustin-Witten equations are related to these older self-dual and anti self-dual equations. After touching briefly on interesting aspects of complex geometry which arise for these complex equations over a real four manifold, we finish by describing the basic unsolved question of the existence of global estimates.

Minimal numbers in topological fixed point and coincidence theory

Speaker: 

Professor Ulrich Koschorke

Institution: 

Universitat Siegen

Time: 

Thursday, May 12, 2011 - 9:00am

Location: 

NS2 1201

Coincidences of two given maps f , g (between smooth manifolds M and N, of dimensions m and n, resp.) are points x in M where f(x) = g(x). The obstruction to removing such coincidences can be measured by certain minimum numbers. In this lecture we compare them to four distinct types of Nielsen numbers. These agree with the classical Nielsen number when m = n (e. g. in the fixed point setting where M = N and one of the maps is the identity map). However, in higher codimensions m - n > 0 their definitions and computations involve distinct aspects of differential topology and homotopy theory. We develop tools which help us 1.) to decide when a minimum number is equal to a Nielsen number ("Wecken theorem"), and 2.) to determine Nielsen numbers. Failures of the Wecken property can have very interesting geometric consequences. The selfcoincidence case where the two maps are homotopic turns out to be particularly illuminating. We give many concrete applications in special settings where M or N are spheres, spherical space forms, projective spaces, tori, Stiefel manifolds or Grassmannians. Already in the simplest examples an important role is played e. g. by Kervaire invariants, all versions of Hopf invariants (`a la James, Hilton, Ganea,. . . ), and the elements in the stable homotopy of spheres defined by invariantly framed Lie groups.

Witt Vectors

Speaker: 

Chris Davis

Institution: 

UC Irvine

Time: 

Thursday, May 12, 2011 - 2:00pm

Location: 

RH 440R

(p-typical) Witt vectors provide a way to go from characteristic p to characteristic zero. For instance, if we apply the Witt vector functor to the field with p elements, we get the ring of p-adic integers. A whole
talk could be devoted to the p-adic integers and why they are important, but instead I will focus on general properties of Witt vectors and techniques for working with them.

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