Department of Mathematics

A short meeting to decide the Algebra Seminar schedule for the Spring semester. Please show up or e-mail mihai.fulger@uconn.edu if you have a speaker suggestion.

I will describe a variety of nonlocal equations describing several phenomena related to turbulence, quantum mechanics, fluid dynamics or statistical physics. By many means, nonlocal terms are more relevant in these areas than local ones. On the other hand, the fractional laplacian is a well-known object in mathematical analysis such as harmonic analysis, geometric analysis or probabilities.

I will give an overview of the current state of the art in a bunch of selected topics, emphasizing on differences between local and nonlocal phenomena.

I will describe a singular limit of a Ginzburg-Landau system involving the half-laplacian. Weak solutions of the system converge in some suitable sense to an harmonic map with free boundary. this solves a question raised by Da Lio and Riviere. Connections with minimal surfaces with free boundary will be emphasized.

Abstract: The generalized linear model (GLM) is a statistical model which has been widely used in actuarial practice, especially for insurance ratemaking. Due to the inherent longitudinal property of P&C insurance claim dataset, there have been some trials of incorporating unobserved heterogeneity of each policyholder from the repeated measurements. To achieve this goal, random effects model has been proposed but there was less theoretical discussion on the methods to test the presence of random effects in GLM framework.

In this article, the concept of Bregman divergence is explored, which has some good properties for statistical modeling and can be connected to diverse model selection diagnostics as in Goh and Dey (2014). We can apply model diagnostics derived from the Bregman divergence for testing robustness of priors both on the naive model, which assumes that random effect has point mass as its prior density, and the proposed model, which assumes a continuous prior density of random effect. This approach provides the insurance companies concrete framework for testing the presence of random effects in both claim frequency and severity and furthermore appropriate hierarchical model which can explain both observed and unobserved heterogeneity of the policyholders for insurance ratemaking. Both models are calibrated using a claim dataset from the
Wisconsin Local Government Property Insurance Fund which includes both observed claim counts and amounts from a portfolio of policyholders.

In this talk, the Gauss image problem for convex bodies will be introduced. This problem includes the Aleksandrov problem as its special case. The essense of the problem is an attempt to a deeper understanding of the Gauss image map.

In the smooth category, the Gauss image problem reduces to an equation of Monge-Ampère type. However, in this talk, we shall focus on the problem in the general setting (rather than imposing any non-necessary smoothness assumptions). Existence and uniqueness results will be presented.

This is a joint work with Károly Böröczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang.