Abstract: I will report an ongoing joint work with Tong Liu, concerning the structure of a certain submodule inside prismatic cohomology of a smooth proper scheme over a p-adic ring of integers. I will explain how this part of prismatic cohomology causes various pathologies, then say a few corresponding consequences of our structural result. If time permits, I shall also mention an interesting example, which negatively answers a question of Breuil.
Abstract: The power spectrum of proteins at high frequencies is remarkably well described by the flat Wilson statistics. Wilson statistics therefore plays a significant role in X-ray crystallography and more recently in cryo-EM. Specifically, modern computational methods for three-dimensional map sharpening and atomic modeling of macromolecules by single particle cryo-EM are based on Wilson statistics. In this talk we use certain results about the decay rate of the Fourier transform to provide the first rigorous mathematical derivation of Wilson statistics. The derivation pinpoints the regime of validity of Wilson statistics in terms of the size of the macromolecule. Moreover, the analysis naturally leads to generalizations of the statistics to covariance and higher order spectra. These in turn provide theoretical foundation for assumptions underlying the widespread Bayesian inference framework for three-dimensional refinement and for explaining the limitations of autocorrelation based methods in cryo-EM.
Abstract: We will present the theory behind and new results on the cohomology of super-reflexive Banach G-modules X, where G is a countable discrete group. In particular, we shall show how the cohomology is controlled by the FC-centre of G, that is, the subgroup of elements having finite conjugacy classes. For example, using purely cohomological tools, we show that when X is an isometric super-reflexive Banach G-module so that X has no almost invariant unit vectors under the action of the FC-centre, then the associated cochain complex is split exact. Further connections to the work of Bader-Furman-Gelander-Monod, Nowak, and Bader-Rosendal-Sauer will be presented.
Abstract: The goal of the talk is to describe a bi-Lipshitz embedding of the quotient space of matrices modulo row permutation and of quotient space of symmetric square matrices modulo row and column permutation. In the talk we will see an example of a bi-Lipshitz embedding b_A created by sorting after multiplying a key matrix A. For a fixed matrix A, the set of matrices X, such that b_A(X)= b_A(Y) iff Y is a row permutation of Y are called the separation set of A. We will talk about properties of separated sets, and then explain under what conditions the map b_A is an Bi-Lipschitz injection of this quotient space.
Abstract: HALT is undecidable. But what if you could make 5 queries to HALT? Then what could you compute? Could you computer more than if you could only make 4? What about other undecidable sets? Come and find out!
Abstract: Poroelasticity has many applications in energy, environmental engineering, and even biomedicine. When linear, it simulates the transient flow of a single-phase fluid in a deformable linear elastic porous medium. However, when the medium is brittle and fractured, linear elasticity is not applicable, but new nonlinear implicit models of elasticity lead to a good description of the phenomenon. I shall introduce for beginners the simplest model of poroelasticity, then describe some new implicit nonlinear models and explain their mathematical and numerical issues.
Collaborators: Tameem Almani, Andrea Bonito, Saumik Dana, Benjamin Ganis, Maria Gonzalez Taboada, Diane Guignard, Frederic Hecht, Kundan Kumar, Xueying Lu, Marc Mear, Kumbakonam Rajagopal, Gurpreet Singh, Endre Suli, Mary Wheeler.
Abstract: Gibbs measures describing directed polymers in random potential are tightly related to the stochastic Burgers/KPZ/heat equations. One of the basic questions is: do the local interactions of the polymer chain with the random environment and with itself define the macroscopic state uniquely for these models? We establish and explore the connection of this problem with ergodic properties of an infinite-dimensional stochastic gradient flow. Joint work with Hong-Bin Chen and Liying Li.
Abstract: Counting problems for closed geodesics on hyperbolic surfaces have been extensively studied since the 1950s. I will discuss a new quantitative estimate with a power saving error term for the number of filling closed geodesics of a given topological type and length at most L on an arbitrary closed, orientable hyperbolic surface. This estimate solves an open problem alluded to in work of Mirzakhani and advertised by Wright. The proof relies on recent developments on the theory of effective mapping class group dynamics.
Abstract: The task of sampling from a probability distribution with known density arises almost ubiquitously in the mathematical sciences, from Bayesian inference to computational chemistry. The most generic and widely-used method for this task is Markov chain Monte Carlo (MCMC), though this method typically suffers from extremely long autocorrelation times when the target density has many modes that are separated by regions of low probability. We present several new methods for sampling that can be viewed as addressing this common problem, drawing on techniques from MCMC, graphical models, and tensor networks.
Abstract: Nowadays, we are living in the era of “Big Data.” A significant portion of big data is big spatial data captured through advanced technologies or large-scale simulations. Explosive growth in spatial and spatiotemporal data emphasizes the need for developing new and computationally efficient methods and credible theoretical support tailored for analyzing such large-scale data. Parallel statistical computing has proved to be a handy tool when dealing with big data. In general, it uses multiple processing elements simultaneously to solve a problem. However, it is hard to execute the conventional spatial regressions in parallel. This talk will introduce a novel parallel smoothing technique for generalized partially linear spatially varying coefficient models, which can be used under different hardware parallelism levels. Moreover, conflated with concurrent computing, the proposed method can be easily extended to the distributed system. Regarding the theoretical support of estimators from the proposed parallel algorithm, we first establish the asymptotical normality of linear estimators. Secondly, we show that the spline estimators reach the same convergence rate as the global spline estimators. The proposed method is evaluated through extensive simulation studies and an analysis of the US loan application data.
Abstract: Let G be a special parahoric group scheme of twisted type, excluding the absolutely special case for ramified odd unitary group. Using the methods and results of Zhu, we prove a duality theorem for G: there is a duality between the level one twisted affine Demazure modules and the function rings of certain torus fixed point subschemes in the affine Schubert varieties of G. Along the way, we also establish the duality theorem for untwisted E_6. As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of G. In particular, this confirms a conjecture of Haines and Richarz. This talk is based on the joint work with Marc Besson.