Departmental Colloquia
Date Time |
Location | Speaker | Title – click for abstract | |
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01/14 4:00pm |
Bloc 117 | Ye He | Beyond Log-Concavity: Sampling Challenges and Advances in Multimodal and Heavy-Tailed Distributions
Sampling from non-log-concave distributions poses significant challenges in a variety of fields, from Bayesian inference to computational physics and machine learning. Unlike log-concave distributions, which offer theoretical guarantees for efficient sampling, non-log-concave distributions often feature complex landscapes, including multimodality and heavy tails, that hinder standard algorithms from exploring the state space effectively. In this talk, I will discuss key obstacles and recent advances in sampling algorithms for non-log-concave distributions. First, I will explore the behavior of classical methods, such as Langevin Monte Carlo (LMC) and Proximal Sampler in the presence of multiple modes and heavy-tailed behaviors, highlighting issues like metastability and slow mixing. I will then introduce techniques designed to overcome these challenges, including using denoising diffusion and novel modifications to the Gaussian noise. This presentation aims to shed light on how these innovations bridge the gap between theory and practice, offering a more nuanced understanding of sampling in complex, high-dimensional spaces. By addressing these fundamental challenges, we can deepen our insight into the behavior of advanced sampling algorithms in non-log-concave regimes. |
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01/16 4:00pm |
Bloc 117 | Adrian van Kan | From numerical simulations of rotating Rayleigh-Bénard convection at very low Ekman numbers to stochastic dynamics in quasi-two-dimensional turbulence
Rapidly rotating Rayleigh-Bénard convection (RRRBC) provides a paradigm for direct numerical simulations (DNS) of geo- and astrophysical fluid flows, but the accessible parameter space, despite great computational efforts, has remained restricted to moderately fast rotation (Ekman numbers $Ek \gtrsim 10^{-8}$), while realistic values of $Ek$ for applications are orders of magnitude smaller. Reduced equations of motion, the non-hydrostatic quasi-geostrophic equations (NHQGE) describing the leading-order behavior in the limit of rapid rotation ($Ek\to 0$) cannot capture finite rotation effects. This leaves the physically most relevant part of parameter space with small but finite $Ek$ currently inaccessible. I will describe the rescaled incompressible Navier-Stokes equations (RiNSE) [1,2] – a reformulation of the Navier-Stokes-Boussinesq equations informed by the scaling laws valid for $Ek\to 0$. I present the first fully nonlinear DNS of RRRBC at unprecedented rotation strengths down to $Ek=10^{-15}$ and below, showing numerically that the RiNSE predicts statistics which agree favorably with the NHQGE at very low $Ek$. This work opens the door to the exploration of a large region in the parameter space of rotating convection.
Beyond the stiffness of the Navier-Stokes equations in the presence of a small parameter such as the Ekman number, the chaotic nature of turbulence also presents a significant challenge. The Navier-Stokes equations in two dimensions (2D) differ significantly from three dimensions (3D) due to additional conservation laws. Solving the 3D Navier-Stokes equations in a thin-layer geometry, there is a critical layer height $H$ below which rigorous bounding arguments show that 3D modes decay due to viscosity, leading to 2D flow. Close to this critical threshold, the energy contained in 3D modes exhibits highly intermittent dynamics. In the second part of this talk, motivated by numerical simulations of this phenomenon, I will describe stochastic dynamics of a single mode in the vicinity of a b |
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01/24 4:00pm |
Bloc 117 | Genming Bai | On the convergence of parametric finite element methods for geometric flows
In this talk, we present our recent major progress in parametric finite element
methods by providing the first-ever convergence proof for two long-standing open problems
using a novel framework of projection error. Dziuk’s method and the Barrett–Garcke–Nurnberg
(BGN) method are the two most fundamental finite element methods for discretizing geometric
flows, including mean curvature flow, surface diffusion, and two-phase flow, among others.
However, the rigorous justification of their convergence has remained open since they were
first proposed in 1990 and 2007 respectively. The main difficulty in Dziuk’s method for mean
curvature flow lies in the loss of H1 parabolicity structure in the error equation. Surprisingly,
within the framework of projection error, the intrinsic orthogonality structure helps us recover
H1 positive definiteness, thereby ensuring overall convergence. This framework is expected
to be a new powerful tool which would help to design and analyse robust and convergent
algorithms where our analysis of a stabilized version of the BGN method is the first example
of this kind. We will also discuss applications such as shape optimization, simulations of
bubbles and biomembranes, and front tracking of a surface in complex fluid environments. The
methodologies and treatments developed in this series of works are hopeful to become standard
in the future.
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01/28 4:00pm |
Bloc 117 | Anya Katsevich | Anya Katsevich |
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01/31 4:00pm |
Bloc 117 | Ionut Chifan | Classification of von Neumann algebras associated with property (T) groups
In the mid-thirties, F.\ Murray and J.\ von Neumann introduced a natural way to associate a von Neumann algebra, $L(G)$, to every countable group $G$. Understanding the structural theory of these algebras, particularly the classification of $L(G)$ in terms of $G$, quickly became a central and challenging research theme. This is because these algebras often have a very limited \emph{memory} of the underlying group. A striking illustration of this phenomenon is A.\ Connes’ celebrated 1976 result, which shows that all nontrivial amenable with infinite nontrivial conjugacy classes (icc) groups give rise to isomorphic von Neumann algebras. Thus, in this case, aside from amenability, $L(G)$ retains no additional information about the algebraic structure of $G$.
In the non-amenable case, the classification remains wide open and significantly more complex. However, instances where the von Neumann algebraic structure completely retains algebraic properties of the underlying group have been discovered through Popa’s deformation/rigidity theory. In my talk, I will survey several recent advances in the classification and the structural study of von Neumann algebras arising from property (T) groups. In this context, the famous Connes Rigidity Conjecture (1982) predicts that all icc property (T) groups $G$ are completely recognizable from $L(G)$. In the first part, I will introduce the first (and currently only) known examples of groups satisfying this conjecture. These groups, known as \emph{wreath-like products}, arise naturally in the context of group-theoretic Dehn filling. Next, I will discuss a natural generalization of the Connes Rigidity Conjecture for property (T) central extensions, and introduce a class of groups that satisfy this generalization, constructed using a natural quotienting technique applied to wreath-like products. In the final part, I will explain how wreath-like product groups can also be used to advance other longstanding open problems posed by A.\ Connes, V.F.R.\ Jones and S.\ Popa, concerning |
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02/06 4:00pm |
Bloc 117 | Elad Romanov | On Principal Component Regression in High Dimension
Principal component regression (PCR) is a classical two-step approach to linear regression, where one first reduces the data dimension by projecting onto its leading principal components, and then performs ordinary least squares regression. We study PCR in an asymptotic high-dimensional regression setting, where the number of data points is proportional to the dimension. Our main deliverables are asymptotically exact limiting formulas for the estimation and prediction risks, which depend in a nuanced way on the eigenvalues of the population covariance, the alignment between the population principal components and the true signal, and the number of selected components.
A key challenge in the high-dimensional regime is that the sample covariance matrix is an inconsistent estimate of its population counterpart, and thus sample principal components may fail to capture potential latent low-dimensional structure in the data. We demonstrate this point through several case studies, including that of a spiked covariance matrix. The analysis of (random design) linear regression in high dimension typically builds on powerful results from random matrix theory, such as the Marchenko–Pastur law and deterministic equivalents for the resolvent of a sample covariance matrix. However, these standard tools alone are not sufficient for analyzing the prediction risk of PCR. To that end, we leverage and develop somewhat less standard techniques, which, to our knowledge, have not seen wide use in the statistics literature to date: multi-resolvent traces and their associated eigenvector overlap measures.
The majority of this talk is based on joint work with Alden Green (Stanford). As time permits, I hope to also tell you a bit about other past and ongoing projects within the broader theme of PCA.
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04/11 3:00pm |
BLOC 302 | Victor Reiner University of Minnesota |
Ehrhart theory and a q-analogue
Classical Ehrhart theory begins with this fact: for a convex polytope P whose vertices lie in the integer lattice Z^n, the number of lattice points in the integer dilates mP grow as a polynomial function of m. We will review some highlights of the classical theory, and explain a new "q-analogue": it replaces the number of lattice points in mP by a polynomial in q that specializes to the lattice point count at q=1. There are q-analogues for several classical Ehrhart theory results, some proven, others conjectural. In particular, a certain new commutative algebra, and the theory of Macaulay's inverse systems, play a prominent role.
(Based on arXiv:2407.06511, with Brendon Rhoades)
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04/22 4:00pm |
BLOC 117 | Suncica Canic | More information coming. |