The positive line bundles on a projective variety often control its geometry. Positivity can be checked numerically for line bundles, but not for bundles of arbitrary rank, even in the semistable case. One known exception is that of bundles with vanishing discriminant. We give algebraic arguments for this. We also address a recent question of S. Misra.

This talk will be held on WebEx. Contact the organizer for the link.

One of the most powerful techniques for solving problems in enumeration is to use a power series, called a generating function, whose coefficients are the terms of a given sequence. A generating function is essentially a "clothesline on which we hang the terms of a sequence" (H. Wilf). Algebraic and analytic techniques can then be applied to help us understand many properties of a sequence -- often to the point of deriving an explicit formula. We will look at several examples, including famous sequences associated with Fibonacci, Euler, and Catalan.


Note: The talk starts at 5:40.

Abstract: The Kakeya (maximal) conjecture concerns how collections of long, thin tubes which point in different directions can overlap. Such geometric problems underpin the behaviour of various important oscillatory integral operators and, consequently, understanding the Kakeya conjecture is a vital step towards many central problems in harmonic analysis. In this talk I will discuss work with K. Rogers and R. Zhang which apply tools from the theory of semialgebraic sets to yield new partial results on the Kakeya conjecture. Also, more recent work with J. Zahl has used these methods to improve the range of estimates on the Fourier restriction conjecture.

Abstract: Linear-quadratic (LQ) framework is widely studied in the literature of stochastic control, game theory, and mean-field analysis due to its simple structure, tractable solution, and local approximation power to nonlinear control problems. In this talk, we discuss several theoretical results of the policy gradient (PG) method, a popular reinforcement learning algorithm, for several LQ problems where agents are assumed to have limited information about the stochastic system. In the single-agent setting, we explain how the PG method is guaranteed to learn the global optimal policy. In the multi-agent setting, we show that (a modified) PG method could guide agents to find the Nash equilibrium solution provided there is a certain level of noise in the system. The noise can either come from the underlying dynamics or carefully designed explorations from the agents. Finally when the number of agents goes to infinity, we propose an exploration scheme with entropy regularization that could help each individual agent to explore the unknown system as well as the behavior of other agents. The proposed scheme is shown to be able to speed up and stabilize the learning procedure.

The numerical performance of PG methods is demonstrated with two examples, one is the optimal execution problem under the single-agent setting and the other one is the institutional negotiation/bargaining problem under the multi-agent setting.

This talk is based on several projects with Xin Guo (UC Berkeley), Ben Hambly (U of Oxford), Huining Yang (U of Oxford), and Thaleia Zariphopoulou (UT Austin).

Webex Meeting Link: https://uconn-cmr.webex.com/uconn-cmr/j.php?MTID=m06c7ca472c3d561ce67c29e8696f4473
(Meeting number: 2620 883 9987
Password: UConn)

Speaker's short bio: Renyuan is currently a WiSE Gabilan Assistant Professor in the Epstein Department of Industrial and Systems Engineering at the University of Southern California. Before joining USC, she spent two years as a Hooke Research Fellow in the Mathematical Institute at the University of Oxford mentored by Professor Rama Cont. She holds a PhD from UC Berkeley (2019) and a B.S. from the University of Science and Technology of China (2014). Her research interests include mathematical finance, stochastic analysis, game theory and machine learning. Please visit her website for more information: https://renyuanxu.github.io/index.html

TBD

In 2014, Sourav Chatterjee demonstrated for a class of Gaussian models of disorder that three phenomena occur together if any one of them occurs. These effects are superconcentration, in which the fluctuation of a random variable is smaller than that present in the classical context of a central limit theorem; multiple valleys, in which many geometrically distinct rivals nearly attain an extreme statistic for the system; and chaos, in which the system is very sensitive to perturbation by random noise. We will discuss Levy's construction of Brownian motion, and a natural dynamical enhancement of Brownian motion involving Ornstein-Uhlenbeck processes; and we will see how these examples may be interpreted in Chatterjee's theory. We will review significant examples of chaotic behaviour when statistical mechanical systems are enhanced by random dynamics. These examples include such enhancements of critical percolation, ensembles of random matrices and last passage percolation.

Abstract: TBA

Abstract: Random ordinary differential equations (RODEs) are pathwise ordinary differential equations that contain a stochastic process in their vector field functions.
They have been used for many years in a wide range of applications, but have been very much overshadowed by stochastic ordinary differential equations (SODEs).
The stochastic process could be a fractional Brownian motion, but when it is a diffusion process there is a close connection between RODEs and SODEs through the Doss-Sussmann transformation and its generalisations, which relate a RODE and an SODE with the same (transformed) solutions. RODEs play an important role in the theory of random dynamical systems and random attractors. They are also useful in biology.

Classical numerical schemes such as Runge-Kutta schemes can be used for RODEs but do not achieve their usual high order since the vector field does not inherit enough smoothness in time from the driving process. It will be shown how, nevertheless, Taylor expansions of the solutions of RODES can be obtained when the stochastic process has Holder continuous sample paths and then used to derive pathwise convergent numerical schemes of an arbitrarily high order. RODEs with Ito noise will also be considered as well as RODEs with affine structure and Poisson noise. Applications
to biology in will be given.

Speaker's short bio: Professor Peter E. Kloeden has wide interests in the applications of mathematical analysis, numerical analysis, stochastic analysis and dynamical systems. He is the coauthor of several influential books on nonautonomous dynamical systems, numerical dynamics and metric spaces of fuzzy sets, and in particular Numerical Solution of Stochastic Differential Equations (with E. Platen) published by Springer in 1992, Random Ordinary Differential Equations and their Numerical Solution (with Xiaoying Han) published by Springer in 2017, and An Introduction to the Numerical Simulation of Stochastic Differential Equations (with Des Higham) published by SIAM in 2021.

Professor Kloeden is a Fellow of the Society of Industrial and Applied Mathematics and was awarded the W.T. \& Idalia Reid Prize from Society of Applied and Industrial Mathematics in 2006. He is the retired Professor of Applied and Instrumental Mathematics at the Goethe University in Frankfurt am Main. He was a Thousand Expert Professor at The Huazhong University of Science and Technology in 2015-2017 and is currently a guest researcher at the University of Tubingen and an Affiliated Professor at Auburn University. His current interests focus on nonautonomous and random dynamical systems and their applications in the biological sciences.

TBD

Random tilings are an important area in probability, combinatorics, and statistical mechanics.

Unfortunately even enumeration of tilings of regions is typically NP-hard except in very special cases.

We propose a variant of the usual random tiling model, called the multinomial tiling model,
which is both very general and also tractable in the sense of allowing exact computation of growth rates.

We can also compute Gaussian scaling limits, find crystallization phenomena, and even random quasicrystals.