Mathematics

Mathematics of collective behavior and self-organized dynamics

Roman Shvydkoy (University of Illinois at Chicago)

Abstract: We discuss a new class of hydrodynamic models arising in studies of self-organized dynamics of agents. These models belong to the class of fractional parabolic equations with (critical) drift and present unique challenges from the analytical prospective. We will show how these models can be used to prove two observable phenomena: alignment (such as consensus of opinions) and flocking (such as swarming of animals).

An Extreme Value Approach to the Pricing and Basis Risk Characterization of ILS

Zhongyi Yuan (Pennsylvania State University Smeal College of Business)

Abstract: Insurance-Linked Securities (ILS) as a channel to transfer catastrophe risks to the capital market have been widely used by insurers to enhance their risk bearing capacity. They have developed from covering one single area/peril to multiple, and in the meantime, while traditional ILS are typically linked to natural catastrophe risks only, recent innovations have introduced ILS that are also linked to broader financial risks. We propose a general pricing framework with a pricing measure that combines a risk-neutral measure, which prices the financial risks, and a distorted measure, which prices the natural catastrophe risks. We then use Catastrophe (CAT) bonds as an example to discuss their pricing. Furthermore, since the hedging by ILS may not be a perfect one for insurers, we propose two models to characterize the hedging basis risk, using dual-triggered Industry Loss Warranties (ILW) as an example. In our analysis we employ an extreme value approach to approximate the distribution of the ILS triggers. Finally, we show a few numerical examples to illustrate the results.

Disjoint cycles in graphs with bounded independence number

Michael Santana (GVSU Math)

Abstract: In 1963, Corràdi and Hajnal showed that for every positive integer $k$, each graph on at least $3k$ vertices with minimum degree at least $2k$ must contain $k$ vertex-disjoint cycles. This result is sharp in terms of both the number of vertices and the minimum degree. Recently, Kierstead, Kostochka, and Yeager characterized the sharpness examples of this statement, in which the main family of sharpness examples contains a 'large' independent set. Thus, it natural to believe that one can further lower the minimum degree of Corràdi-Hajnal and still obtain $k$ vertex-disjoint cycles, provided the indepence number of graph is not too big. Recent work by Balogh et al. has shown this to be true when the graph is 'dense' (i.e., $k \approx n/3$). In this talk, we will explore a sparse version of this result, and present several other possible extensions to chorded cycles as well as a 'mixed' version.

Irreducible components of exotic Springer fibers

Daniele Rosso (Indiana University Northwest)

Abstract: The Springer resolution is a resolution of singularities of the variety of nilpotent elements in a reductive Lie algebra. It is an important geometric construction in representation theory, but some of its features are not as nice if we are working in Type C (Symplectic group). To make the symplectic case look more like the Type A case, Kato introduced the exotic nilpotent cone and its resolution, whose fibers are called the exotic Springer fibers. We give a combinatorial description of the irreducible components of these fibers in terms of standard Young bitableaux and obtain an exotic Robinson-Schensted correspondence. This is joint work with Vinoth Nandakumar and Neil Saunders.

Abstract: The purpose of this initial conversation is to identify things which are going well in our large classes, things which are not going well, and ways in which we could make these classes easier to teach and better for students. Anyone with experience or interest in large lecture teaching is encouraged to attend.

The SOA Curriculum Changes

Stuart Klugman, FSA, CERA, PhD (SOA Senior Staff Fellow, Education)

Abstract: With a unique perspective on the exams and modules, Stuart will discuss changes to the SOA curriculum. Stuart Klugman (FSA, CERA, PhD) is a Senior Staff Fellow in the Education Department at the SOA, a position he has held since 2009. From 1974-2009 he was a professor at The University of Iowa and Drake University, retiring from Drake as the Principal Financial Group Distinguished Professor of Actuarial Science. He has co-authored/edited three books used on SOA exams, published numerous research papers, and served two terms on the SOA Board of Directors. He co-chaired the committee that developed the 2005-2007 exam changes and provided staff support for the 2018 changes. He is a two-time recipient of the SOA’s Presidential Award.