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Keywords: Hyperbolic PDEs, Hamilton-Jacobi, applications to vehicular traffic

In this course, we will study the theory of scalar conservation laws with application to traffic modeling. Fluid-dynamic models of traffic, based on conservation laws, have been intensively investigated in the last twenty years. The idea of modeling unidirectional car traffic on a single road in terms of the scalar conservation law tu+∂xf(u) = 0, was first proposed in the seminal papers by Lighthill, Whitham, and Richards (LWR model). Here, the unknown u = u(t, x) denotes the traffic density taking values in a compact interval, and the flux has the form f(u) = uv(u), where v(u) is the average velocity of cars which is assumed to depend on the density alone. In the first part of the course, we will study weak solutions for scalar conservation laws, discuss various entropy admissibility conditions, and prove the existence and uniqueness of solutions with bounded total variation. Then we will explore the duality between conservation laws and Hamilton-Jacobi equations. These tools will be used to analyze the LWR model on a single stretch of road and on a network. In the last part, we will discuss recent developments and the challenges of control problems for traffic regulation at fixed locations (traffic lights, traffic signals, pay tolls, etc.).

Keywords: Temporal difference learning, non-convex optimization, sample complexity

This minicourse provides a mathematical introduction to reinforcement learning, one of the cornerstone subjects in machine learning. Reinforcement learning aims at approximating optimal decisions within unknown environments by relying on experimentation and sampling. Our curriculum adopts a contemporary perspective, leveraging recent algorithmic breakthroughs in non-convex optimization. Our take will require background knowledge on probability theory and time-varying Markov chains, as well as a sound understanding of convergence of properties of stochastic gradients flows in non-convex landscape, which we review. The course also touches upon vital aspects of optimal control theory, especially Markov decision processes, Bellman's optimality principle, and policy optimization strategies. A critical and challenging aspect of reinforcement learning is estimating value functions which we do not have access to; this is classically done through temporal difference learning, a significant focus of our study. By proving results on sample complexity, we will show how the combination of temporal difference learning and policy gradients has become an effective method that supports many recent developments in machine learning.

Keywords: Polynomial solutions to polynomial equations, genera, elliptic curves

This minicourse is a study of the topology of maps between Riemann surfaces and some applications. We will begin with the Riemann-Hurwitz theorem, the basic equation governing the topology of such maps, and see applications to the question of computing the genus of degree d plane curves, finding the group law on an elliptic curve, and whether polynomial equations in two variables have a polynomial solution. Then we will study the monodromy representation of such maps, relating such covers to actions of the fundamental group on finite sets, and thus to enumerative questions in the symmetric group. Time permitting we will study the solutions to some of the enumerative questions, and indicate their connections with solutions to differential equations coming from physics.