Upcoming Seminars & Events
Qianxiao Li: Free-energy-like concepts for nonequilibrium systems; Cheng Tai: Video Super-Resolution via Adaptive Wavelet Frames and Elastic Deformations
Abstract: Qianxiao Li - The extension of the concept of free energy to fully nonequilibrium systems is an important problem in statistical mechanics. I will discuss how a straightforward generalization, using the nonequilibrium steady-state distribution in place of the canonical ensemble, yields a free energy that is not sufficient in capturing the dynamics of general nonequilibrium steady states. One can overcome this problem by defining the "free action", which is like a trajectory-space free energy. Through a representative example, I will discuss the conceptual and practical usefulness of the free action in quantifying the dynamics of nonequilibrium steady states. If time allows, I will also discuss how one can employ similar ideas to treat systems with deterministic dynamics and random initial conditions. An application to the classical laminar-turbulent transition problem will be presented. Abstract: Cheng Tai - We present a novel approach to the video super-resolution(SR) problem based on adaptive wavelet frames and elasticity theory. By considering the elastic deformations and the blurring effects together, the proposed is able to model much more realistic object motions in the scene. Adaptive wavelet frame is introduced as a novel regularization in SR problem. It is multi-scale and is adapted to data. Hence it is more flexible than pre-defined sparse regularizations. Both novelties contribute to the improved quality of SR construction results.
Proof of Grothendieck-Serre conjecture on principal bundles over regular local rings containing a field
Small aerosols drift down a temperature or turbulence gradient since faster particles fly longer distances before equilibration. That fundamental phenomenon is known since Maxwell and it was universally believed that particles moving down the kinetic energy gradient must concentrate in minima (say, on walls in turbulence). Here, I show that this is incorrect: escaping minima is possible for inertial particles whose time of equilibration is longer than the time to reach the minimum. "The best way out is always through": particles escape by flying through minima or reflecting from walls. I present the analytical solution of this problem, which has surprising analogies with multiple phenomena, from Anderson localization to non-equilibrium steady states and modified fluctuation-dissipation theorem. I shall also describe the related localization-delocalization phase transition upon the change of elasticity of wall reflections.
The random K-SAT model gives a model of random Boolean formulas and is perhaps the canonical random constraint satisfaction problem. The Satisfiability Conjecture posits that the probability of a satisfying assignment undergoes a sharp transition at a critical density of constraints. Heuristics developed in statistical physics predict the location of the transition as well as much more. I will survey what is known and predicted and describe recent progress establishing the conjecture for large enough K. This is joint work with Jian Ding and Allan Sly.
In this talk we will discuss substitution systems that have the property of minimal self-joinings. Then we will focus our attention on self-similar interval exchange transformations and their associated substitutions. We will show that 3-IETs have MSJ. This is joint with Giovanni Forni.
We will introduce two invariants called the arithmetic Tutte polynomial and the Tutte quasi-polynomial, and we will survey their several applications, including: - generalization of graph colorings and flows to cellular complexes of higher dimension; - combinatorial and topological invariants of toric arrangements; - Ehrhart polynomials of zonotopes; - Hilberts series of some modules related to the vector partition function. The talk is partially based on joint work of the speaker with Petter Brändén, Michele D'Adderio and Alex Fink.
In this talk I will review CM theory of complex projective K3 surfaces, and show how it can be used to construct K3 surfaces over finite fields. I will discuss work-in-progress where this is applied to describing: (1) the collection of zeta functions of K3 surfaces over a finite field, and (2) the category of ordinary K3 surfaces over a finite field. These are similar to theorems of Honda and Tate resp. Deligne on abelian varieties over finite fields.
We study transverse representatives of an oriented topological knot type K in the tight contact 3-sphere, where the classical invariant of transverse isotopy introduced by Bennequin is the self-linking number. The botany problem for transverse knots, namely deciphering which transverse isotopy classes exist at a fixed self-linking value, has remained open in general, although for certain knot types a number of interesting botanical features have been shown to exist. In this talk we present partial solutions of the botany problem that hold for arbitrary knot types K.
In this talk, we discuss our work in progress about how degeneration of the divisor at infinity into a normal crossing divisor affects the symplectic homology of an affine variety. From an anti-surgery picture, by developing an anti-surgery formula for symplectic homology similar to work by Bourgeois-Ekholm-Eliashberg, we show that essentially, the change in symplectic homology is reflected by the Hochschild invariants of the Fukaya category of a collection of Lagrangian spheres on the smooth divisor.
I will discuss refinements to the Bombieri-De Giorgi-Giusti minimal graph and three applications: 1. De Giorgi's Conjecture for Allen-Cahn equation; 2. Caffarelli-Berestycki-Nirenberg Conjecture for overdetermined problem on epigraphs; 3. Translating graphs of mean curvature flows.
Let A be an n×n matrix, X be an n×p matrix and Y = AX. A challenging and important problem in data analysis, motived by dictionary learning, is to recover both A and X, given Y. Under normal circumstances, it is clear that the problem is underdetermined. However, as showed by Spielman et. al., one can succeed when X is sufficiently sparse and random. In this talk, we discuss a solution to a conjecture raised by Spielman et. al. concerning the optimal condition which guarantees efficient recovery. The main technical ingredient of our analysis is a novel way to use the ε-net argument in high dimensions for proving matrix concentration, beating the standard union bound. This part is of independent interest. Joint work with K. Luh (Yale).
We will present a simple setup in which one can make sense of a renormalization flow for FK-percolation models in terms of a simple Markov process on a state-sace of discrete weighted graphs. We will describe how to formulate the universality conjectures in this framework (in terms of stationary measures for this Markov process), and how to prove this statement in the very special case of the two-dimensional uniform spanning tree (building on existing results on this model). This is partly based on joint work with Stéphane Benoist and Laure Dumaz.
The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well-studied codes, like Reed-Solomon or Reed-Muller codes. The Reed-Muller code F(n,d) for a finite field F is defined by n-variate degree-d polynomials over F. As far as we know, the list decoding radius of Reed-Muller codes can be as large as the minimal distance of the code. This is known in a number of cases: --The Goldreich-Levin theorem and its extensions established it for d=1 (Hadamard codes); --Gopalan, Klivans and Zuckerman established it for the binary field; --and Gopalan established it for d=2 and all fields. In this work (joint with Lovett), we prove the conjecture for all constant prime fields and all constant degrees. In this talk, I will present the proof for our first main theorem which states that the list decoding radius is exactly equal to the minimum distance of the Reed-Muller code for all constant prime fields and constant degrees. For ease of presentation, we will only see the case when d < char(F). The proof relies on several new ingredients, including higher-order Fourier analysis which I will introduce along the way.
I will describe a spectral sequence that starts at reduced odd Khovanov homology and converges to a version of instanton homology for double branched covers.
The question of global regularity vs. finite time blow-up remains open for many fluid equations. In this talk, I will discuss an active scalar equation which is an interpolation between the 2D Euler equation and the surface quasi-geostrophic equation. We study the patch dynamics for this equation in the half-plane, and prove that the solutions can develop a finite-time singularity. This is a joint work with A. Kiselev, L. Ryzhik and A. Zlatos.