Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, February 18, 2020.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, February 18, 2020

2:00 pm in 243 Altgeld Hall,Tuesday, February 18, 2020

#### Progress towards Nash-Williams' Conjecture on Triangle Decompositions

###### Michelle Delcourt (Ryerson University)

Abstract: Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently large, triangle divisible graph on $n$ vertices with minimum degree at least $0.75n$ admits a triangle decomposition. In the light of recent results, the fractional version of this problem is of central importance. A fractional triangle decomposition is an assignment of non-negative weights to each triangle in a graph such that the sum of the weights along each edge is precisely one.

We show that for any graph on $n$ vertices with minimum degree at least $0.827327n$ admits a fractional triangle decomposition. Combined with results of Barber, Kühn, Lo, and Osthus, this implies that for every sufficiently large triangle divisible graph on n vertices with minimum degree at least $0.82733n$ admits a triangle decomposition. This is a significant improvement over the previous asymptotic result of Dross showing the existence of fractional triangle decompositions of sufficiently large graphs with minimum degree more than $0.9n$. This is joint work with Luke Postle.

3:00 pm in 245 Altgeld Hall,Tuesday, February 18, 2020

#### Mixture of Experts Regression Models for Insurance Ratemaking and Reserving

###### Tsz Chai "Samson" Fung (University of Toronto)

Abstract: Understanding the effect of policyholders' risk profile on the number and the amount of claims, as well as the dependence among different types of claims, are critical to insurance ratemaking and IBNR-type reserving. To accurately quantify such features, it is essential to develop a regression model which is flexible, interpretable and statistically tractable. In this presentation, I will discuss a highly flexible nonlinear regression model we have recently developed, namely the logit-weighted reduced mixture of experts (LRMoE) models, for multivariate claim frequencies or severities distributions. The LRMoE model is interpretable as it has two components: Gating functions to classify policyholders into various latent sub-classes and Expert functions to govern the distributional properties of the claims. The model is also flexible to fit any types of claim data accurately and hence minimize the issue of model selection. Model implementation is illustrated in two ways using a real automobile insurance dataset from a major European insurance company. We first fit the multivariate claim frequencies using an Erlang count expert function. Apart from showing excellent fitting results, we can interpret the fitted model in an insurance perspective and visualize the relationship between policyholders' information and their risk level. We further demonstrate how the fitted model may be useful for insurance ratemaking. The second illustration deals with insurance loss severity data that often exhibits heavy-tail behavior. Using a Transformed Gamma expert function, our model is applicable to fit the severity and reporting delay components of the dataset, which is ultimately shown to be useful and crucial for an adequate prediction of IBNR reserve. This project is joint work with Andrei Badescu and Sheldon Lin.

4:00 pm in 341 Altgeld Hall,Tuesday, February 18, 2020

#### Julia Robinson and Hilbert's Tenth Problem (film)

Abstract: Julia Robinson was the first woman elected to the mathematical section of the National Academy of Sciences, and the first woman to become president of the American Mathematical Society. While tracing Robinson's contribution to the solution of Hilbert's tenth problem, the film illuminates how her work led to an unusual friendship between Russian and American colleagues at the height of the Cold War.

4:00 pm in 314 Altgeld Hall,Tuesday, February 18, 2020

#### In Transition - Mathematics and Art

###### Kirsi Peltonen   [email] (Aalto University, Finland)

Abstract: This is a talk for the general public and academic audience interested in possibilities for enhancing interaction between contemporary mathematics and arts. What are the needs for this dialogue in different levels of education, research and broader in the society? Recent multidisciplinary activities challenging the traditions and communication of mathematics and arts at Aalto University in Finland have given a new type of platform to share the beauty of mathematics systematically and open accessible layers to a useful interplay. Many outcomes and byproducts of our up-to-date experiments are perfect for applications in digital technologies such as programming, CAD, 3D printing, virtual and augmented reality. Some scenarios for the future development are presented.