Quasi-conformal geometry on the boundaries of word hyperbolic groups

Colloquium
Time: Oct 23 2008 - 4:30pm - 5:30 pm
Location: CH 240
Speaker: Marc Bourdon
Abstract:

An important theme in geometric group theory is to study groups up to coarse equivalences. Quasiisometry is one of the most important examples of such an equivalence.

Boundaries of word hyperbolic groups equipped with their quasi-conformal strutures are full quasi-isometric invariants of groups. This provides a very strong link between geometry and (nonsmooth) analysis, which has been very active recently.

We will describe these objects and discuss the Cannon's conjecture which is one of the main problems inhe subject. We will present a strategy due to Bonk and Kleiner to approach this conjecture.

On Sums of Three Squares

Colloquium
Time: Oct 2 2008 - 4:30pm - 5:30 pm
Location: MA 240
Speaker: J. Cogdell
Abstract:

The question of when an integer can be written  as a sum of squares has a long and venerable history.
More generally, Hilbert's eleventh problem asks (among other things) which integers are integrally represented by a given quadratic form over a number field. The case of quadratic forms in two variables was understood by Hilbert. For forms in four or more variables the situation is quite different and has been understood for some time. The case of three variables remained open.

In collaboration with Piatetski-Shapiro and Sarnak,  we have provided a solution in the three variable  case by analytic means. One consequence is that every sufficiently large square-free integer in a totally real field is a sum of three integral squares iff it is so locally. In this talk I would like to explain the reduction of this theorem to a question in analytic number theory and our solution of this problem. Along the way I will introduce you to quadratic forms,
theta series, modular forms and L-functions and how one translates an algebraic question into an analytic one.

Unitary representations of simple Lie groups

Colloquium
Time: Oct 30 2008 - 4:30pm - 5:30 pm
Location: MA240
Speaker: David Vogan
Abstract:
By 1950, work of Gelfand and others had led to a general program for  "non-commutative harmonic analysis": understanding very general mathematical problems (particularly of geometry or analysis) in the  presence of a (non-commutative) symmetry group G.  A first step in that program is classification of unitary representations - that is, the realizations of G as automorphisms of a Hilbert space.  Despite tremendous advances from the work of Harish-Chandra, Langlands, and others, completing this first step is still some distance away.                

Since functional analysis is not as fashionable now as it was in 1950, I'll explain some of the ways that Gelfand's problem can be related to algebraic geometry (particularly to equivariant K-theory).  I'll also discuss the (closely related) question of whether computers may be  able to help solve these problems.                

Cremona Transformations and Homeomorphisms of Topological Surfaces

Colloquium
Time: Oct 9 2008 - 4:30pm - 5:30 pm
Location: MA240
Speaker: János Kollár
Abstract:

The simplest Cremona transformation of projective 3-space is the
involution
$\sigma: ( x_0 : x_1 : x_2 : x_3 ) \mapsto ( \frac{1}{x_0} : \frac{1}{x_1} : \frac{1}{x_2} : \frac{1}{x_3} ) $ ,
which is a homeomorphism outside of the " coordinate tetrahedron" ($ x_0 x_1 x_2 x_3 = 0 $).
By studying the action of $\sigma$ on real quadric surfaces, we show that
$\sigma$ and its conjugates generate a dense subgroup of Homeo(S^2) , the group of homeomorphisms of the 2-sphere.
Then we show that the same holds if the 2-sphere is replaced by the torus or
by any non-orientable surface and explain why there cannot be similar results
for orientable surfaces of genus $\geq2$.
(Joint work with Frederic Mangolte.)

Narrow Tubes and Quantum Graphs.

Colloquium
Time: May 29 2008 - 4:30pm - 5:30 pm
Location: MA240
Speaker: Leonid Friedlander
Abstract:
Back in the 1930's, Pauling came up with an idea of approximating electron wave functions in complicated molecules by functions on a graph: the atoms are vertexes of this graph, and bonds are the edges. A wave function is an eigenfunction on the Schroedinger operator on the graph, which is a one-dimensional variety, rather than a combinatorial graph. Such a structure got a name of a quantum graph. Recent advances in nano-technology drew attention to the study of systems in narrow tubes, and such system can also be approximated by quantum graphs. In the talk, I will discuss both the approximation problems and the

Model Theory and Exponentiation

Colloquium
Time: May 22 2008 - 4:30pm - 5:30 pm
Location: MA240
Speaker: David Marker
Abstract:
In the 90s model theoretic methods were used by Wilkie to show that sets defined in the real field with exponentiation have many of the good geometric and topological properties of real algebraic varieties. For example, any such set has only finitely many connected components. Complex exponentiation has a very different flavor. The definability of the integers leads to pathologies, but there is still some hope for a reasonable theory of definable sets. In this lecture I will review some of the older work on the real field and discuss Zilber's program for understanding complex exponentiation.

Topological field theory and enumeration problems in the topology of surfaces

Colloquium
Time: May 15 2008 - 4:30pm - 5:30 pm
Location: MA 240
Speaker: Vladimir Turaev
Abstract:
Abstract. A study of Topological Quantum Field Theory in dimension 2 leads to an interesting algebra (group-graded Frobenius algebras) and to interesting topology (counting sections of fiber bundles over surfaces). I will discuss these and related developments. No particular background is required.

Persistent Topology and Data

Colloquium
Time: Apr 10 2008 - 4:30pm - 5:30 pm
Location: MA240
Speaker: Gunnar Carlsson
Abstract:
Abstract: The science and engineering disciplines are producing enormous volumes of data from many different experimental sources. The data comes in many forms, and developing methods for usefully analyzing it is of great importance. In this talk, we will discuss some methods arising out of topology for extracting qualitative information from these data sets. We will discuss persistent homology, topological methods for providing "roadmaps" to the data, and homotopy theoretic methods for analyzing the stability of the road methods.

Recent progress on Riemannian manifolds of positive curvature

Colloquium
Time: Mar 31 2008 - 4:30pm - 5:30 pm
Location: MA240
Speaker: Richard Schoen
Abstract:
We will discuss the geometric problem of classification of compact Riemannian manifolds of positive sectional curvature. In particular, we will describe our recent classification of pointwise 1/4-pinched manifolds. The work is joint with Simon Brendle and employs the Ricci flow.

Spectral gap and effective equidistribution

Colloquium
Time: Mar 6 2008 - 4:30pm - 5:30 pm
Location: MA 240
Speaker: Einsiedler, Manfred
Abstract:
The dynamics on homogeneous spaces has many interesting connections to number theory. One of the main problems here is to understand the distribution of closed orbits for subgroups H of the ambient Lie group G. In joint work with G.Margulis and A.Venkatesh we prove an error rate in the equidistribution for semisimple subgroups H acting on congruence quotients of G. This makes use of spectral gap in the form of property (tau). However, the proof of our theorem can also be used to prove all cases of property (tau) except for groups of type A_1. We will discuss the relationship between spectral gap, effective