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Title: Mixing times for the TASEP on the circle
Abstract: The exclusion process is one of the best-studied examples of an interacting particle system. In this talk, we consider simple exclusion processes on finite graphs. We give an overview over some recent results on the mixing time of the totally asymmetric simple exclusion process (TASEP). In particular, we provide bounds on the mixing time of the TASEP on the circle, using a connection to periodic last passage percolation. This talk is based on joint work with Allan Sly (Princeton).
Title: Shift invariance of half space integrable models
Abstract: I'll discuss recent work on shift invariance in a half space setting. These are non-trivial symmetries allowing certain observables of integrable models with a boundary to be shifted while preserving their joint distribution. The starting point is the colored stochastic six vertex model in a half space, from which we obtain results on the asymmetric simple exclusion process, as well as for the beta polymer through a fusion procedure, both in a half space setting. An application to the asymptotics of a half space analogue of the oriented swap process is also given.
Title: Matrices with random perturbation: theory and applications
Abstract: The classical perturbation theory studies the changes of matrix parameters (such as leading eigenvectors or eigenvalues) under a small noise to the matrix. In modern applications, the noise is random, and thus it is of fundamental interest to rewrite the theory with this assumption.
I am going to survey our works on this project (starting from late 2000s) and relevant results from both random matrix theory and data science. Next, I am going to discuss several applications in various fields, such as matrix completion and fast randomized computation.
(join work with S. O'rourke, K. Wang, P. Tran, A. Bhardwaj).
Biography: Van Vu obtained his PhD at Yale (1998) under the supervision of L. Lovasz. After spells at IAS, MRS, UCSD and Rutgers, he returned to Yale as a P. F. Smith professor in Mathematics in 2011. Dr. Vu is a Sloan fellow, and recieved the Polya prize (SIAM, 2008) and Fulkerson prize (AMS, 2012) for his works in combinatorics and probability. In the same year, he was a Medallion lecturer at the 8th World congress in probability and statistics (Instanbul). He gave an invited talk at ICM 2014 (Seoul). By Mathscinet statistics, he ranks third among all mathematicians with a 1998 PhD (behind E. Candes and C. Villani).
Title: Correlation decay for hard spheres via Markov chains
Abstract: The hard sphere model is a well-known statistical physics model of monatomic gases. A central open question is whether the model exhibits a phase transition in terms of the fugacity parameter. If there is a phase transition, then below the critical fugacity a sampled packing would almost surely have a crystalline-like pattern, whereas above the critical fugacity a sampled packing would almost surely look disordered. We improve upon all known lower bounds on the critical fugacity and critical density of the hard sphere model in dimensions two and higher. As the dimension tends to infinity our improvements are by factors of 2 and 1.7, respectively. We make these improvements by utilizing techniques from theoretical computer science to show that a certain Markov chain for sampling from the hard sphere model mixes rapidly at low enough fugacities. We then prove an equivalence between optimal spatial and temporal mixing for hard spheres to deduce our results. This is joint work with Tyler Helmuth and Will Perkins. After discussing this work, I will overview some of my current research directions in computational biology.
Title: Invertibility of inhomogeneous heavy-tailed matrices
Abstract: We will show the sharp estimate on the behavior of the smallest singular value of random matrices under very general assumptions. One of the key steps in the proof is a result about the efficient discretization of the unit sphere in an n-dimensional euclidean space. The proof of the result will be outlined. Partially based on the joint work with Tikhomirov and Vershynin.
Title: Random graphs and the Kahn-Kalai conjecture
Abstract: Sparse random graphs are central objects in probability and combinatorics. An important aspect of random graphs that has been extensively studied since early works of Erdos and Renyi is the threshold phenomenon. The threshold of an increasing graph property is the density at which a random graph transitions from unlikely satisfying to likely satisfying the property. Kahn and Kalai conjectured that this threshold is always within a logarithmic factor of the expectation threshold, a natural lower bound to the threshold which is often much easier to compute. In probabilistic combinatorics and random graph theory, the Kahn-Kalai conjecture directly implies a number of difficult results, such as Johansson, Kahn and Vu’s breakthrough work on Shamir’s problem on hypergraph matchings and graph factors. I will discuss joint work with Jinyoung Park that resolves the Kahn-Kalai conjecture.
Interestingly, our proof of the Kahn-Kalai conjecture is closely related to our resolution of a conjecture of Talagrand on suprema of certain stochastic processes driven by sparse Bernoulli random variables (known as selector processes), and a question of Talagrand on suprema of general positive empirical processes. These results give the first steps towards Talagrand’s last ``Unfulfilled dreams’’ in the study of suprema of general empirical processes.
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