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## Mathematics Colloquium

- 9/16
*Mathematics Colloquium*

An Application Of Quiver Theory To Machine Learning

Siu-Cheong Lau (Boston University)#### Mathematics Colloquium

Thursday, September 16th, 2021

An Application Of Quiver Theory To Machine Learning

Siu-Cheong Lau (Boston University)

3:30 PM - 4:30 PM

Storrs Campus

OnlineQuiver representation theory is an important tool in Lie theory and

mathematical physics. It has produced a lot of interesting examples in

noncommutative geometry.

Deep learning shares a common setup with quiver theory. Namely, they both

concern about representations of a directed graph by vector spaces. On the

other hand, non-linear activation functions play a key role in deep

learning, but they were not found in quiver theory. It is natural to ask

for a unified point of view towards the two subjects.

In this talk, I will explain how to run deep learning over the moduli space

of quiver representations. Using uniformization of metrics, we will see

that the usual Euclidean setup is a special instance of our formulation.

Contact Information: Kyu-Hwan Lee More - 9/23
*Mathematics Colloquium*

Gauss's Class Number Problem

Ken Ono (University Of Virginia)#### Mathematics Colloquium

Thursday, September 23rd, 2021

Gauss's Class Number Problem

Ken Ono (University Of Virginia)

3:30 PM - 4:30 PM

Storrs Campus

OnlineIn 1798 Gauss wrote Disquisitiones Arithmeticae, the first rigorous text in number theory. This book laid the groundwork for modern algebraic number theory and arithmetic geometry. Perhaps the most important contribution in the work is Gauss's theory of integral quadratic forms, which appears prominently in modern number theory (sums of squares, Galois theory, rational points on elliptic curves,L-functions, the Riemann Hypothesis, to name a few).

Despite the plethora of modern developments in the field, Gauss’s first problem about quadratic forms has not been optimally resolved. Gauss's class number problem asks for the complete list of quadratic form discriminants with class number h. The difficulty is in effective computation, which arises from the fact that the Riemann Hypothesis remains open. To emphasize the subtlety of this problem, we note that the first case, where h=1, remained open until the 1970s. Its solution required deep work of Heegner and Stark, and the Fields medal theory of Baker on linear forms in logarithms. Unfortunately, these methods do not generalize to the cases where h>1.

In the 1980s, Goldfeld, and Gross and Zagier famously obtained the first effective class number bounds by making use of deep results on the Birch and Swinnerton-Dyer Conjecture. This lecture will tell the story of Gauss’s class number problem, and will highlight new work by the speaker and Michael Griffin that offers new effective results by different (and also more elementary) means.

Contact Information: Kyu-Hwan Lee More - 10/7
*Mathematics Colloquium*

Policy Evaluation And Temporal-Difference Learning In Continuous Time And Space: A Martingale Lens

Xunyu Zhou (Columbia University)#### Mathematics Colloquium

Thursday, October 7th, 2021

Policy Evaluation And Temporal-Difference Learning In Continuous Time And Space: A Martingale Lens

Xunyu Zhou (Columbia University)

3:30 PM - 4:30 PM

Storrs Campus

OnlineWe propose a unified framework to study policy evaluation (PE) and the associated temporal difference (TD) methods for reinforcement learning in continuous time and space. Mathematically, PE is to devise a data-driven Feynman--Kac formula without knowing any coefficients of a PDE. We show that this problem is equivalent to maintaining the martingale condition of a process. From this perspective, we present two methods for designing PE algorithms. The first one, using a ``martingale loss function", interprets the classical gradient Monte-Carlo algorithm. The second method is based on a system of equations called the ``martingale orthogonality conditions". Solving these equations in different ways recovers various classical TD algorithms, such as TD, LSTD, and GTD. We apply these results to option pricing and portfolio selection. This is a joint work with Yanwei Jia.

Contact Information: Kyu-Hwan Lee More

Past Talks | Contact: Kyu-Hwan Lee