Department of

Mathematics


Seminar Calendar
for events the day of Friday, January 27, 2017.

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Friday, January 27, 2017

3:00 pm in 243 Altgeld Hall,Friday, January 27, 2017

Raindrop. Droptop. Symmetric functions from DAHA.

Josh Wen (UIUC Math)

Abstract: In symmetric function theory, various distinguished bases for the ring of (deformed) symmetric functions come from specifying an inner product on said ring and then performing Gram-Schmidt on the monomial symmetric functions. In the case of Jack polynomials, there is an alternative characterization as eigenfunctions for the Calogero-Sutherland operator. This operator gives a completely integrable system, hinting at some additional algebraic structure, and an investigation of this structure digs up the affine Hecke algebra. Work of Cherednik and Matsuo formalize this in terms of an isomorphism between the affine Knizhnik-Zamolodichikov (KZ) equation and the quantum many body problem. Looking at q-analogues yields a connection between the affine Hecke algebra and Macdonald polynomials by relating the quantum affine KZ equation and the Macdonald eigenvalue problem. All of this can be streamlined by circumventing the KZ equations via Cherednik's double affine Hecke algebra (DAHA). I hope to introduce various characters in this story and give a sense of why having a collection of commuting operators can be a great thing.

4:00 pm in 245 Altgeld Hall,Friday, January 27, 2017

A Marked Cox Model for IBNR Claims: Theory and Application

Dameng Tang (University of Waterloo)

Abstract: Incurred but not reported (IBNR) loss reserving is a very important issue for Property & Casualty (P&C) insurers. To calculate IBNR reserve, one needs to model claim arrivals and then predict IBNR claims. However, factors such as temporal dependence among claim arrivals and exposure fluctuation are often not incorporated in most of the current loss reserving models, which greatly affect the accuracy of IBNR predictions. In this talk, I will present a new modelling approach under which the claim arrival process together with the reporting delays follows a marked Cox process. The intensity function of the Cox process is governed by a hidden Markov chain. I will show that the proposed model is versatile in modeling temporal dependence, can incorporate exposure fluctuation, and can be interpreted naturally in the insurance context. The associated reported claim process and IBNR claim process remain to be a marked Cox process with easily convertible intensity function and marking distribution. The specific structure of the intensity function allows for generating discretely observed claim processes, which is critical for data fitting purposes. Closed-form expressions for both the autocorrelation function (ACF) and the distributions of the numbers of reported claims and IBNR claims are derived. I will then present a generalized expectation-maximization (EM) algorithm to fit the model to data and to estimate the model parameters. The proposed model is examined through simulation studies and is applied to a real insurance claim data set. We compare the predictive distributions of our model with those of the over-dispersed Poisson model (ODP), a stochastic model that underpins the widely used chain-ladder method. The results show that our model can yield more accurate best estimates and more realistic predictive distributions. This is joint work with Andrei Badescu and Sheldon Lin.

4:00 pm in 241 Altgeld Hall,Friday, January 27, 2017

Galois Categories and the Topological Galois Correspondence

Daniel Carmody (UIUC Math)

Abstract: Classical Galois theory for fields gives a correspondence between closed subgroups of the Galois group of a Galois extension and intermediate subfields. The theory of covering spaces in topology gives a correspondence between connected coverings of nice spaces and subgroups of the fundamental group. The purpose of this talk is to explain the relationship between (and generalization) of these two theorems.

4:00 pm in 345 Altgeld Hall,Friday, January 27, 2017

On "Structurable equivalence relations" by R. Chen and A. Kechris: Introduction

Anush Tserunyan (UIUC Math)

Abstract: For a class $\mathcal{K}$ of countable relational structures, a countable Borel equivalence relation $E$ is said to be $\mathcal{K}$-structurable if there is a Borel way to put a structure from $\mathcal{K}$ on each $E$-equivalence class. The paper of Chen and Kechris [arXiv link] studies the global structure (including Borel homomorphisms and reductions) of the classes of $\mathcal{K}$-structurable equivalence relations for various $\mathcal{K}$. In this introductory talk, we will give some background and survey the main results of the paper.