Quiver 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.

In 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.

We 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.