MenuEvents for 03/29/2024 from all calendars
Stochastic Processes Seminar
Time: 10:00AM - 11:00AM
Location: ZOOM
Speaker: Nhu Nguyen, University of Rhode Island
Title: Stochastic Approximation with Discontinuous Dynamics, Differential Inclusions, and Applications
Abstract: This talk develops new results for stochastic approximation algorithms. The emphasis is on treating algorithms and limits with discontinuities. The main ingredients include the use of differential inclusions, set-valued analysis, non-smooth analysis, and stochastic differential inclusions. Under broad conditions, it is shown that a suitably scaled sequence of the iterates has a differential inclusion limit. In addition, it is shown for the first time that a centered and scaled sequence of the iterates converges weakly to a stochastic differential inclusion limit. The results are then used to treat several application examples, including the Markov decision process, Lasso algorithms, Pegasos algorithms, support vector machine classification, and learning.
Noncommutative Geometry Seminar
Time: 10:00AM - 11:00PM
Location: BLOC 306
Speaker: Zhengwei Liu , Tsinghua University
Title: Alterfold Topological Quantum Field Theory
Abstract: We will introduce the 3-alterfold Topological Quantum Field Theory (TQFT). It is a 3D TQFT with space-time boundary, which encodes Jones' theory of planar algebras as a local theory on the 2D boundary. Both Turaev-Viro (TV) TQFT and the Reshetikhin-Turaev (RT) TQFT can be naturally embedded in the alterflold TQFT through blow-up procedures. We provide 3D topologization of various key concepts, such as the Drinfeld center, connections, Frobenius-Schur indicators, etc. Many remarkable results become apparent in this approach, including the equivalence between TV TQFT and RT TQFT. This is recent work joint with Shuang Ming, Yilong Wang and Jinsong Wu, see arXiv:2307.12284 and arXiv:2312.06477. For lectures in the next week, we will introduce the alterfold theory in all dimensions and construct infinite TQFT. We will discuss its connections with operator algebras, topological orders, higher categories, etc. We propose an approach to Poincare conjecture through renormalizations.
Noncommutative Geometry Seminar
Time: 11:10AM - 12:10PM
Location: BLOC 306
Speaker: Fan Lu , Tsinghua University
Title: Classification of exchange relation planar algebras of rank 5
Abstract: Exchange relation planar algebras are natural generalizations of Kac algebras from skein theoretical point of view. We show that its classification is essentially solving a system of algebraic equations, but too complicated to solve directly. Then we introduce a key concept, the type of the fusion rule, which completely detects exchange relations as forest types. According to types, the system of equations reduces to exponentially many subsystems, which are solvable individually. In addition, we propose new analytic criteria to rule out most types from being subfactor planar algebras. Eventually, we are able to classify exchange relation planar algebras of rank 5. This method recovers the previous classification up to rank 4 of Bisch, Jones, and Liu with quick proofs. This is joint work with Zhengwei Liu.
Mathematical Physics and Harmonic Analysis Seminar
Time: 1:50PM - 2:50PM
Location: BLOC 302
Speaker: Matt Powell, Georgia Institute of Technology
Title: Continuity of the Lyapunov exponent for quasi-periodic Jacobi cocycles
Abstract: Many spectral properties of 1D Schr\"odinger operators have been linked to the Lyapunov exponent of the corresponding Schr\"odinger cocycle. While the situation for one-frequency quasi-periodic operators with analytic potential is well-understood, the multifrequency and non-analytic situations are not. The purpose of this talk is twofold: first, discuss our recent work on multi-frequency analytic quasi-periodic cocycles, establishing continuity (both in cocycle and jointly in cocycle and frequency) of the Lyapunov exponent for non-identically singular cocycles (of which the Jacobi cocycles form a special case), and second, discuss ongoing work extending these results to suitable Gevrey classes. Analogous results for analytic one-frequency cocycles have been known for over a decade, but the multi-frequency results have been limited to either Diophantine frequencies (continuity in cocycle) or SL(2,C) cocycles (joint continuity). We will discuss the main points of our argument, which extends earlier work of Bourgain.
Noncommutative Geometry Seminar
Time: 2:00PM - 3:00PM
Location: BLOC 306
Speaker: Zishuo Zhao, Tsinghua University
Title: Relative entropy between bimodule quantum channels
Abstract: We propose a notion of relative entropy between bimodule quantum channels on finite von Neumann algebras, generalizing the remarkable Pimsner-Popa entropy for subfactors. We will discuss various inequalities of this relative entropy. In particular, the relative entropy of bimodule quantum channels is bounded by the relative entropy of their Fourier multipliers, which is a higher analogue of relative entropy of states. The equality holds if the inclusion of von Neumann algebras admits a downward Jones basic construction.
