Mathematics

Einstein metrics with prescribed conformal infinity

Xinran Yu (UIUC)

Abstract: Given a conformal class of metrics on the boundary of a manifold, one can ask for the existence of an Einstein metric whose conformal infinity satisfies the boundary condition. In 1991, Graham and Lee studied this boundary problem on the hyperbolic ball. They proved the existence of metrics sufficiently close to the round metric on a sphere by constructing approximate solutions to a quasilinear elliptic system. In his monograph (2006), Lee discussed the boundary problem on a smooth, compact manifold-with-boundary. Using a similar construction, he proved the existence and regularity results for metrics sufficiently close to a given asymptotically hyperbolic Einstein metric. The proof is based on a linear theory for Laplacian and the inverse function theorem.

An introduction to p-adic analysis

Robert J. Dicks (UIUC)

Abstract: This talk is meant to be an introduction p-adic number and p-adic analysis. These arise via completing the rational numbers using a metric (relative to a prime number p) which is very different from the ordinary euclidean metric. The first part of the talk will introduce this number system and focus on ways in which it differs from our "Euclidean" intuition. We then will discuss p-adic integration. We will then define the p-adic zeta function. The goal will be to describe (without proof) how these are related to the Riemann zeta function. This talk is aimed at 1st year graduate students.

Dimension distortion and applications

Efstathios Konstantinos Chrontsios Garitsis (UIUC Math)

Abstract: Since Hausdorff dimension was first introduced in 1918, many different notions of dimension have been defined and used throughout many areas of Mathematics. An interesting topic has always been the distortion of said dimensions of a given set under a specific class of mappings. More specifically, Gehring and V\"ais\"al\"a proved in 1973 a theorem concerning the distortion of Hausdorff dimension under quasiconformal maps, while Kaufman in 2000 proved the analogous result for Box-counting dimension. In this talk, an introduction to the different types of dimensions will be presented, along with the results of of Gehring, V\"ais\"al\"a and Kaufman. We will then proceed to discuss analogous theorems we proved for the Assouad dimension and spectrum, which describe how K-quasiconformal maps change these notions of a given subset of $\mathbb{R}^n$. We will conclude the talk by demonstrating how said theorems can be applied to fully classify polynomial spirals up to quasiconformal equivalence. Contact: ryanm12@illinois.edu for Zoom coordinates.

Dimension distortion and applications

Efstathios Konstantinos Chrontsios Garitsis (UIUC Math)

Abstract: Since Hausdorff dimension was first introduced in 1918, many different notions of dimension have been defined and used throughout many areas of Mathematics. An interesting topic has always been the distortion of said dimensions of a given set under a specific class of mappings. More specifically, Gehring and V\"ais\"al\"a proved in 1973 a theorem concerning the distortion of Hausdorff dimension under quasiconformal maps, while Kaufman in 2000 proved the analogous result for Box-counting dimension. In this talk, an introduction to the different types of dimensions will be presented, along with the results of of Gehring, V\"ais\"al\"a and Kaufman. We will then proceed to discuss analogous theorems we proved for the Assouad dimension and spectrum, which describe how K-quasiconformal maps change these notions of a given subset of $\mathbb{R}^n$. We will conclude the talk by demonstrating how said theorems can be applied to fully classify polynomial spirals up to quasiconformal equivalence. Contact: ryanm12@illinois.edu for Zoom coordinates.

Traveling wave solutions for discrete and nonlocal diffusive Lotka-Volterra system

Ting-Yang Hsiao (UIUC)

Abstract: We will introduce a biological model called Lotka-Volterra competition system. We construct an N-shaped constrained region. By using this region we give an a priori estimate for this system and show that the total population of this 2-species system has a nontrivial lower bound. Finally, we know that one of the important issues in ecology is biodiversity. By using this estimate, we find necessary conditions for coexistence of 3-species Lotka-Volterra system.

Index theorems and boundary conditions

Gayana Jayasinghe (UIUC)

Abstract: The study of the Fredholm Index of operators is of great interest in analysis. The 60's saw the development of the theory of the index of elliptic operators as a topological invariant on compact manifolds, a vast generalization of the Gauss Bonnet theorem. Now there are generalizations of all sorts of spaces including spaces with boundary, singularities and in non commutative geometry. I will begin by introducing Index theory, motivating the key ideas with the example of the Gauss Bonnet theorem, on manifolds with boundary. The index in this example has a topological contribution as well as a contribution coming from the asymmetry of the boundary, which can be described in terms of geodesic curvature. I will then describe how for general operators, this asymmetry can be expressed using boundary conditions for the operator, and in the case of spectral boundary conditions, one deals with the spectral asymmetry. I won't assume any prior knowledge on Index theory, and it should be accessible for analysts

Graduate Analysis Seminar

Aric Wheeler (Indiana University)

Abstract: Generalizing results of Matthews-Cox/Sukhtayev for a model reaction-diffusion equation, we derive and rigorously justify weakly nonlinear amplitude equations governing general Turing bifurcation in the presence of conservation laws. In the nonconvective, reaction-diffusion case, this is seen similarly as in Matthews-Cox, Sukhtayev to be a real Ginsburg-Landau equation weakly coupled with a diffusion equation in a large-scale mean-mode vector comprising variables associated with conservation laws. In the general, convective case, by contrast, the amplitude equations consist of a complex Ginsburg-Landau equation weakly coupled with a singular convection-diffusion equation featuring rapidly-propagating modes with speed $\sim 1/\epsilon$ where $\epsilon$ measures amplitude of the wave as a disturbance from a background steady state. Applications are to biological morphogenesis, in particular vasculogenesis, as described by the Murray-Oster and other mechanochemical/hydrodynamical models. This work is joint with Kevin Zumbrun.