Events
Department of Mathematics and Statistics
Texas Tech University
Capacity of sets and condensers in Euclidean spaces has been an object of study by researchers working in several areas of Analysis and Geometry due to its applications and the important role it plays in physics.
In the first part of this talk, we will present some basic facts about conformal and analytic capacity and its connection and applications
to holomorphic functions. A distortion theorem will be examined and the asymptotic behavior of capacity under covering maps.
In the second part we will introduce the modulus metric and examine a conjecture of J. Ferrand, G. Martin and M. Vuorinen from 1991 that every isometry in the modulus metric is a conformal mapping. We will discuss the tools we used and the method we followed to solve the conjecture in the recent papers [1] and [2].
References:
[1] Dimitrios Betsakos and Stamatis Pouliasis, ''Isometries for the modulus metric are quasiconformal mappings'', Trans. Amer. Math. Soc. Published electronically: November 21, 2018.
[2] Stamatis Pouliasis and Alexander Yu. Solynin, ''Infinitesimally small spheres and conformally invariant metrics'', submitted, 2018. https://arxiv.org/abs/1812.04651
In this talk, we will overview the recent developments in the theory of minimal surfaces, and constant mean curvature surfaces in 3-manifolds.
In particular, we will discuss the asymptotic Plateau problem in $ H^3 $ and $ H^2 \times R $, the Calabi-Yau conjecture, and minimal surfaces in hyperbolic 3-manifolds.

Conformal geometry is the study of spaces where we can define the angles of crossing curves, but not their lengths. One of the most successful programs for studying conformal geometry has been the holographic program initiated by Fefferman and Graham, which proceeds by realizing the conformal space as the "boundary at infinity" of a complete Riemannian Einstein manifold of one higher dimension. I will describe this program, and then discuss my project to extend it to conformal spaces which themselves have boundaries. This entails relating the conformal space to an Einstein manifold with a "corner at infinity" of codimension two, and the presence of the corner leads to several interesting analytic complications arising from doubly singular PDEs.
 | Monday
Feb. 11
3:00 PM
MATH 110
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Lars Christensen
Department of Mathematics and Statistics, Texas Tech University
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