Department of

# Mathematics

Seminar Calendar
for events the day of Monday, February 17, 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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Monday, February 17, 2020

2:00 pm in 241 Altgeld Hall,Monday, February 17, 2020

#### From Polaris Variable Annuities to Regression-based Monte Carlo

###### Zhiyi (Joey) Shen (University of Waterloo)

Abstract: In this talk, I will first discuss the no-arbitrage pricing of Polaris variable annuities (VAs), which were issued by the American International Group in recent years. Variable annuities are prevailing equity-linked insurance products that provide the policyholder with the flexibility of dynamic withdrawals, mortality protection, and guaranteed income payments against a market decline. The Polaris allows a shadow account to lock in the high watermark of the investment account over a monitoring period that depends on the policyholder’s choice of his/her first withdrawal time. This feature makes the insurer’s payouts depend on policyholder’s withdrawal behaviours and significantly complicates the pricing problem. By prudently introducing certain auxiliary state variables, we manage to formulate the pricing problem into solving a convoluted stochastic optimal control framework and developing a computationally efficient algorithm to approach the solution. Driven by the challenges from the pricing Polaris VAs, in the second part of the talk, I will introduce a regression-based Monte Carlo algorithm, which we propose to solve a class of general stochastic optimal control problems numerically. The algorithm has three pillars: a construction of auxiliary stochastic control model, an artificial simulation of the post-action value of state process, and a shape-preserving sieve estimation method. The algorithm enjoys many merits, including obviating forward simulation and control randomization, eliminating in-sample bias, evading extrapolating the value function, and alleviating the computational burden of the tuning parameter selection. This talk is based on two joint works with Chengguo Weng from the University of Waterloo.

3:00 pm in 441 Altgeld Hall,Monday, February 17, 2020

#### A geometric perspective on the foundations of modern homotopy theory

###### Brian Shin (Illinois Math)

Abstract: Homotopy theorists have always been interested in studying spaces. However, the meaning of the word space'' has evolved over the years. Whereas one used to say space to mean a topological space, it seems the modern stance is to view a space as an $\infty$-groupoid. In this expository talk, I would like to connect the modern stance back to geometry. In particular, I will demonstrate how the $\infty$-category of spaces can be built out of the category of manifolds. As an application, we will use this connection to give a geometric perspective on infinite loop space theory.

3:00 pm in 243 Altgeld Hall,Monday, February 17, 2020

#### Prequantization, Differential Cohomology and the Genus Integration

###### Rui Loja Fernandes (Illinois)

Abstract: In recent joint work with Ivan Contreras, we have introduced the genus integration of a Lie algebroid. In this talk, I will explain the relationship between the genus integration of a canonical Lie algebroid associated with a closed 2-form and the classical prequantization condition. As a by product, one recovers the usual classification of principal $S^1$-bundles with connection in terms of differential cohomology. Time permitting, I will also discuss the first steps in extending these results to higher algebroids and higher degree differential cohomology.

5:00 pm in Altgeld Hall,Monday, February 17, 2020

Abstract: CEB3