Mathematics

Orders from $\widetilde{PSL_2(\mathbb{R})}$ Representations and Non-examples.

Xinghua Gao   [email] (University of Illinois)

Abstract: Let $M$ be an integer homology 3-sphere. One way to study left-orderability of $\pi_1(M)$ is to construct a non-trivial representation from $\pi_1(M)$ to $\widetilde{PSL_2(\mathbb{R})}$. However this method does not always work. In this talk, I will give examples of non L-space irreducible integer homology 3-spheres whose fundamental groups do not have nontrivial $\widetilde{PSL_2(\mathbb{R})}$ representations.

One-loop Effective Potentials via Quasinormal Modes.

Cindy Keeler (Niels Bohr Institute)

Abstract: We present a method of calculating one-loop effective potentials for quantum fluctuations from quasinormal modes. We will demonstrate the method for a harmonic oscillator and a scalar in De Sitter space. After briefly discussing its application in Anti-de Sitter space, we will speculate on the physical meaning of this mathematical trick as well as its extension to fields with higher spin, to product spaces, and to spaces with non-Dirichlet boundary conditions.

The Aviles Giga functional. A history, a survey and some new results

Andrew Lorent (University of Cincinnati)

Abstract: The Aviles-Giga functional $I_{\epsilon}(u)=\int_{\Omega} \frac{\left|1-\left|\nabla u\right|^2\right|^2}{\epsilon}+\epsilon \left|\nabla^2 u\right|^2 \; dx$ is a well known second order functional that models phenomena from blistering to liquid crystals. The zero energy states of the Aviles-Giga functional have been characterized by Jabin, Otto, Perthame. Among other results they showed that if $\lim_{n\rightarrow \infty} I_{\epsilon_n}(u_n)=0$ for some sequence $u_n\in W^{2,2}_0(\Omega)$ and $u=\lim_{n\rightarrow \infty} u_n$ then $\nabla u$ is Lipschitz continuous outside a locally finite set. This is essentially a corollary to their theorem that if $u$ is a solution to the Eikonal equation $\left|\nabla u\right|=1$ a.e. and if for every "entropy" $\Phi$ function $u$ satisfies $\nabla\cdot\left[\Phi(\nabla u^{\perp})\right]=0$ distributionally in $\Omega$ then $\nabla u$ is locally Lipschitz continuous outside a locally finite set. In recent work with Guanying Peng we generalized this result by showing that if $\Omega$ is bounded and simply connected and $u$ satisfies the Eikonal equation and if $$ \nabla\cdot\left(\Sigma_{e_1 e_2}(\nabla u^{\perp})\right)=0\text{ and }\nabla\cdot\left(\Sigma_{\epsilon_1 \epsilon_2}(\nabla u^{\perp})\right)=0\text{ distributionally in }\Omega, $$ where $\Sigma_{e_1 e_2}$ and $\Sigma_{\epsilon_1 \epsilon_2}$ are the entropies introduced by Ambrosio, DeLellis, Mantegazza, Jin, Kohn, then $\nabla u$ is locally Lipschitz continuous outside a locally finite set. Most of the talk will be an elementary introduction to the Aviles Giga functional, why it is important, why the $\Gamma$-convergence conjecture is so interesting. The final third will motivate and very briefly indicate some of the methods used in the proof of the above result. We will finish with some open problems.

Organizational meeting

Pizzas and Toric surfaces with Kazhdan-Lusztig atlases

Balazs Elek (Cornell University)

Abstract: A Bruhat atlas, introduced by He, Knutson and Lu, on a stratified variety is a way of modeling the stratification locally on the stratification of Schubert cells by opposite Schubert varieties. They described Bruhat atlases on many interesting varieties, including partial flag varieties and on wonderful compactifications of groups. We will discuss some results toward a classification of varieties with Bruhat atlases, focusing on the 2-dimensional toric case. In this case, the answer may be stated in terms of the moment polygon of the toric surface, which one should first slice up, then put toppings on, much like one would do while preparing a pizza.

Risk sharing and risk aggregation via risk measures

Haiyan Liu (University of Waterloo)

Abstract: In this talk, we discuss two problems in risk management using the tools of risk measures. In the first part of the talk, we address the problem of risk sharing among agents using a two-parameter class of quantile-based risk measures, the so-called Range-Value-at-Risk (RVaR), as their preferences. We first establish an inequality for RVaR-based risk aggregation, showing that RVaR satisfies a special form of subadditivity. Then, the risk sharing problem is solved through explicit construction. Comonotonicity and robustness of the optimal allocations are investigated. We show that, in general, a robust optimal allocation exists if and only if none of the risk measures is a VaR. Practical implications of our main results for risk management and policy makers will be discussed. In the second part of the talk, we study the aggregation of inhomogeneous risks with a special type of model uncertainty, called dependence uncertainty, evaluated by a generic risk measure. We establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with uncertainty in the dependence structure, a relevant situation for risk management practice.

IGL Kickoff Meeting

Abstract: Spring 2017 organizational meeting for the Illinois Geometry Lab