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Angela Gibney, Rutgers University
In this talk I will give a tour of recent results and open problems about vector bundles on the moduli space of curves constructed from the representation theory of affine Lie algebras. I will discuss how these questions fit into the context of some of the open problems about the birational geometry of the moduli space.
José González Llorente, Universidad Autónoma de Barcelona
The Mean Value Property for harmonic functions is at the crossroad of Potential Theory, Geometric Function Theory and Probability. In the last years substantial efforts have been made to build up stochastic models for certain nonlinear PDE's like the $p$-laplacian or the infinity-laplacian and the key is to figure out which are the corresponding (nonlinear) mean value properties. After introducing a "natural" nonlinear mean value property related to the $p$-laplacian we will focus on functions satisfying the so called one-radius mean value property. We will review some classical results in the linear case ($p=2$) and then recent nonlinear versions in the more general context of metric measure spaces.
Eduardo Teixeira, University of Central Florida
The development of modern free boundary theory has promoted major knowledge leverage across pure and applied disciplines and in this talk I will provide a panoramic overview of such endeavor. The goal of lecture, however, will be to explicate how geometric insights and powerful analytic tools pertaining to free boundary theory can be imported to investigate regularity issues in nonlinear diffusive partial differential equations. This new systematic approach has been termed non-physical free boundaries, and in the past few years has led us to a plethora of unanticipated results.
Konrad Koerding, University of Pennsylvania
Narek Hovsepyan, Temple University
It is shown that the eigenvalues of an analytic kernel on a finite interval go to zero at least as fast as $R^{ - n} $ for some fixed $R < 1$. The best possible value of R is related to the domain of analyticity of the kernel. The method is to apply the Weyl–Courant minimax principle to the tail of the Chebyshev expansion for the kernel. An example involving Legendre polynomials is given for which R is critical.
Reference - G. Little, J. B. Reade, Eigenvalues of analytic kernels , SIAM J. Math. Anal., 15(1), 1984, 133–136.
Growth in groups via linear algebra
Abstract: Finite groups are often studied using basic combinatorics and number theory, as in a first course in Abstract Algebra. For infinite groups, however, many of these techniques are unavailable. Since so many infinite groups play an important role in geometry and topology, different methods need to be developed for their study.
In this talk, I’ll introduce the growth of a group, which is perhaps the most basic notation of `size’ when the group is infinite. As we shall see, for certain classes of groups, growth can be studied using properties of directed graphs and basic linear algebra. We will assume no prerequisites beyond the definition of a group.
There are no conferences this week.