Fri, 03 Jun 2016

16:00 - 17:00
L1

Eigenvectors of Tensors

Bernd Sturmfels
(UC Berkeley)
Abstract

Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalization. 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

Fri, 20 Nov 2015

16:00 - 17:00
L1

Effective behaviour of random media: From an error analysis to elliptic regularity theory

Felix Otto
(Max-Plank-Institute)
Abstract
Heterogeneous media, like a sediment, are often naturally described in statistical terms.  How to extract their effective behaviour on large scales, like the permeability in Darcy's law, from the statistical specifications?  A practioners numerical approach is to sample the medium according to these specifications and to determine the permeability in the Cartesian directions by imposing simple boundary conditions.  What is the error made in terms of the size of this "representative volume element''?  Our interest in what is called  "stochastic homogenization'' grew out of this error analysis.

 

In the course of developing such an error analysis, connections with the classical regularity theory for elliptic operators have emerged. It turns out that the randomness, in conjunction with statistical homogeneity, of the coefficient field (which can be seen as a Riemannian metric) generates large-scale regularity of harmonic functions (w.r.t. the corresponding Laplace-Beltrami operator).  This is embodied by a hierarchy of Liouville properties:
 
   Almost surely, the space of harmonic functions of given but arbitrary growth rate has the same dimension as in the flat (i. e. Euclidean) case.

 

  Classical examples show that from a deterministic point of view, the Liouville property fails already for a small growth rate:

 

  There are (smooth) coefficient fields, which correspond to the geometry of a cone at infinity, that allow for sublinearly growing but non-constant harmonic functions.
 
 
 
Fri, 22 May 2015

16:30 - 17:00
L1

Bott Periodicity and Beyond

Andre Henriques
(Universiteit Utrecht)
Abstract

I will review Bott's classical periodicity result about topological K-theory (with period 2 in the case of complex K-theory, and period 8 in the case of real K-theory), and provide an easy sketch of proof, based on the algebraic periodicity of Clifford algebras. I will then introduce the `higher real K-theory' of Hopkins and Miller, also known as TMF. I'll discuss its periodicity (with period 576), and present a conjecture about a corresponding algebraic periodicity of `higher Clifford algebras'. Finally, applications to physics will be discussed.

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