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12:00pm - zoom - Probability and Analysis Webinar Eric Carlen - (Rutgers) TBA |
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1:00pm to 2:00pm - RH 440R - Dynamical Systems Victor Kleptsyn - (CNRS, University of Rennes 1, France) From the percolation theory to Fuchsian equations and Riemann-Hilbert problem Consider the critical percolation problem on the hexagonal lattice: each of (tiny) hexagons is independently declared «open» or «closed» with probability (1/2) — by a fair coin tossing. Assume that on the boundary of a simply connected domain four points A,B,C,D are marked. Then either there exists an «open» path, joining AB and CD, or there is a «closed» path, joining AD and BC (one can recall the famous «Hex» game here). Cardy’s formula, rigorously proved by S. Smirnov, gives an explicit value of the limit of such percolation probability, when the same smooth domain is put onto lattices with smaller and smaller mesh. Though, a next natural question is: what if more than four points are marked? And thus that there are more possible configurations of open/closed paths joining the arcs? In our joint work with M. Khristoforov we obtain the answer as an explicit integral for the case of six marked points on the boundary, passing through Fuchsian differential equations, Riemann surfaces, and Riemann-Hilbert problem. We also obtain a generalization of this answer to the case when one of the marked points is inside the domain (and not on the boundary).
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2:00pm - 510R Rowland Hall - Combinatorics and Probability Tingwei Meng - (UCLA) Overcoming the curse of dimensionality for solving high-dimensional Hamilton-Jacobi partial differential equations or optimal control problems using neural networks Hamilton-Jacobi PDEs and optimal control problems are widely used in many practical problems in control engineering, physics, financial mathematics, and machine learning. For instance, controlling an autonomous system is important in everyday modern life, and it requires a scalable, robust, efficient, and data-driven algorithm for solving optimal control problems. Traditional grid-based numerical methods cannot solve these high-dimensional problems, because they are not scalable and may suffer from the curse of dimensionality. To overcome the curse of dimensionality, we developed several neural network methods for solving high-dimensional Hamilton-Jacobi PDEs and optimal control problems. This talk will contain two parts. In the first part, I will talk about SympOCNet method for solving multi-agent path planning problems, which solves a 512-dimensional path planning problem with training time of less than 1.5 hours. In the second part, I will show several neural network architectures with solid theoretical guarantees for solving certain classes of high-dimensional Hamilton-Jacobi PDEs. By leveraging dedicated efficient hardware designed for neural networks, these methods have the potential for real-time applications in the future. These are joint works with Jerome Darbon, George Em Karniadakis, Peter M. Dower, Gabriel P. Langlois, and Zhen Zhang. |
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3:00pm to 4:00pm - RH 306 - Number Theory Shamil Asgarli - (Santa Clara University) Blocking sets arising from plane curves over finite fields Let F_q be a finite field, and consider the set P^2(F_q) of all F_q-points in the projective plane. A subset B of P^2(F_q) is called a blocking set if B meets every line defined over F_q. Given an algebraic plane curve C in P^2, when does the set of F_q-rational points on C form a blocking set? We will see that curves of low degree do not give rise to blocking sets. As an example of this principle, we will show that cubic plane curves defined over F_q do not give rise to blocking sets whenever q is at least 5. On the other hand, we will describe explicit constructions of smooth plane curves (of large degree) that do give rise to blocking sets. Finding blocking curves of optimal degree over a given finite field remains open. This is joint work with Dragos Ghioca and Chi Hoi Yip. |