Actions of Lie groups and Lie algebras on manifolds

Speaker: 

Professor Moe Hirsch

Institution: 

University of Wisconsin

Time: 

Friday, May 13, 2011 - 9:00am

Location: 

NS2 1201

Given a Lie group G with Lie algebra g, and a manifold M^n of dimension n >0, what invariants determine whether there is an effective action A of G, or g, on M^n? If A exists what can be said about fixed points? Can A be analytic? If not, what can be said about its kernel?

Typical results:

1. The identity component of the group of upper triangular n-by-n real matrices has effective smooth actions on every M^n. But such an action cannot be analytic, because the fixed point set of some 1-dimensional central subgroup has nonempty interior.

2. Assume for some X in g that ad X has m>0 eigenvalues whose imaginary parts are linearly independent over the rationals. Let A be
an effective analytic action of g on M^n.

(a) If n < 2m then A(X) has no fixed points.

(b) Assume n=2m and M^n is compact. Then the number of fixed points of A(X) equals the Euler characteristic Char(M), which is therefore nonnegative.

EXAMPLES: If M^n is compact and Char(M)

Machine Learning Approaches for Genomic Medicine

Speaker: 

Associate Director and Chief Informatics Officer Jill Mesirov

Institution: 

Broad Institute of MIT &amp; Harvard

Time: 

Thursday, May 12, 2011 - 4:00pm

Location: 

NS2 1201

The sequencing of the human genome and the development of new methods for acquiring biological data have changed the face of biomedical research. The use of mathematical and computational approaches is taking advantage of the availability of these data to develop new methods with the promise of improved understanding and treatment of disease.

I will describe some of these approaches as well as our recent work on a Bayesian method for integrating high-level clinical and genomic features to stratify pediatric brain tumor patients into groups with high and low risk of relapse after treatment. The approach provides a more comprehensive, accurate, and biologically interpretable model than the currently used clinical schema, and highlights possible future drug targets.

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.

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