Date: April 8
Speaker: Sergey Fomin (University of Michigan)
Title: Computing without subtracting (and/or dividing)
Abstract:

Algebraic complexity of a rational function can be defined as the minimal number of arithmetic operations required to compute it. Can restricting the set of allowed arithmetic operations dramatically increase the complexity of a given function (assuming it is still computable in the restricted model)? In particular, what can happen if we disallow subtraction and/or division? This is joint work with D. Grigoriev and G. Koshevoy.

Date: April 15
Speaker: Vlad Markovic (Caltech)
Title: Harmonic maps and heat flows on hyperbolic spaces
Abstract:

We prove that any quasi-conformal map of the (n-1)-dimensional sphere, when n > 2, can be extended to a smooth quasi-isometry F of the n-dimensional hyperbolic space such that the heat flow starting with F converges to a quasi-isometric harmonic map. This implies the Schoen-Li-Wang conjecture that every quasi-conformal map of the (n-1)-sphere can be extended to a harmonic quasi-isometry when n > 2. We also prove the corresponding conjecture when n = 2 (which was the original Schoen Conjecture), but this proof does not involve heat flows.

Date: April 22
Speaker: Bernd Sturmfels (University of California, Berkeley)
Title: Eigenvectors of Tensors
Abstract:

Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalisation. The spectral theory of tensors was pioneered by Lim and Qi around 2005. It has numerous applications, and ties in closely with optimization and dynamical systems.

We present an introduction that emphasizes algebraic and geometric aspects.

Date: April 29
Speaker: Bill Duke (University of California, Los Angeles)
Title: Geometric invariants for real quadratic fields
Abstract:

I will describe some joint work with A. Toth and O. Imamoglu on a new geometric invariant, a certain bordered Riemann surface, associated to an ideal class of a real quadratic field.

This surface has the usual modular closed geodesic as its boundary and its area is determined by the length of an associated backward continued fraction. We study its distribution properties on average over a genus. This complements in a natural way the distribution of the closed geodesics themselves. In the process we give an extension of the Katok-Sarnak formula relating certain periods of Maass differentials to Weyl-type sums for the surfaces.