Schedule

• Feb 5
  Thursday

  Christophe Garban (Universite Lyon 1 & Courant Institute)

  One-arm exponents of the high-dimensional Ising model

  Abstract: In a joint work with Diederik van Engelenburg, Romain Panis and Franco Severo, we study the probability that the origin is connected to the boundary of the box of size $n$ (the one-arm probability) in several percolation models related to the Ising model. We prove that different universality classes emerge at criticality and that the FK-Ising model has upper-critical dimension equal to 6, in contrast to the Ising model, where it is known to be (less or) equal to 4. I will start the talk with a short introduction on the Ising model on Z^d.

• Feb 9

  Applied Math Colloquium by Joel Tropp

• Feb 16

  President's Day

• Feb 23

  Ron Nissim (MIT)

  Area Law and Mass Gap for Lattice Yang-Mills at Strong Coupling

  Abstract: This talk is based on three recent papers, including two with Scott Sheffield and Sky Cao, on lattice Yang-Mills theory. These papers extend the regime of parameter for which two properties of lattice Yang-Mills theory known as mass gap and area law hold to the 't Hooft regime. As Yang-Mills theory is a mathematical model for particle physics, these properties (mass gap and area law) have real physical implications such as quark confinement. The goal of the talk will be to explain the mechanisms which make the 't Hooft regime tractable from two points of view. One point of view is based on the master loop equation and string trajectories which can be seen as a surface exploration, while the other point of view is based on pairing the recent Langevin dynamic approach with classical arguments from the 70s and 80s.

• Mar 2

  Shrey Aryan (MIT)

  Entropic Selection Principle for Monge's Optimal Transport

  Abstract: Regularization is a standard way to obtain unique minimizers for variational problems. A natural question is to understand the behavior of minimizers as the regularization parameter tends to zero. When the unregularized problem admits a unique minimizer, the regularized minimizers converge to it. On the other hand, when the cost function is not strictly convex, the unregularized problem may have multiple minimizers. It is therefore not clear a priori which one is selected in the limit. In this talk, I will discuss this selection problem in the context of optimal transport with the distance cost in dimensions $d\geq 2$. While the unregularized problem typically admits infinitely many minimizers, its entropic regularization has a unique solution. In joint work with Ghosal, we give a complete characterization of the selected minimizer as the entropic regularization parameter tends to zero.

• Mar 9

  Simons Lectures

• Mar 16

  Vilas Winstein (UC Berkeley)

  Metastable wells in exponential random graph models

  Abstract: Exponential random graph models (ERGMs) are exponential tilts of Erdos-Renyi models where higher-order interactions promote the presence of small subgraphs like triangles. These models exhibit metastable behavior at low temperatures (strong interaction), and decompose as mixtures of phase measures or metastable wells. We present a variety of novel results for these metastable wells, including quantitative CLTs and sharp bounds on the Wasserstein distance between an ERGM in a metastable well and a corresponding Erdos-Renyi model. A common thread in these results is the careful analysis of relevant quantities under a natural Markov chain sampling algorithm.

• Mar 23

  Spring Break

• Mar 30

  Mike Douglas (Harvard)

  Formalizing Quantum Field Theory

  Abstract: Quantum field theory (QFT) is the theoretical framework underlying the Standard Model of particle physics. There are many open questions in the mathematics of QFT, including the Yang-Mills Millennium prize problem, and making a QFT proof of mirror symmetry.
  In this talk we survey these problems and then discuss ongoing work to formalize QFT, joint with Sarah Hoback, Anna Mei, Ron Nissim, Xi Yin and others.

• Apr 6

  Shivam Dhama (BU)

  Quantitative fluctuation analysis of stochastic gradient descent in continuous-time via Malliavin Calculus

  Abstract: We study a Quantitative Central Limit Theorem (QCLT) for the Stochastic Gradient Descent in Continuous Time (SGDCT) algorithm, which involves parameter updates governed by a stochastic differential equation. We establish a rate at which the SGDCT updates converge to a critical point of the objective function in the Wasserstein metric. This convergence rate is primarily influenced by the magnitude of the learning rate; as the learning rate magnitude decreases, for a fixed convexity constant of the objective function, the rate of convergence becomes slower. To obtain the specified rate, we employ techniques from Malliavin calculus. We use the second-order Poincaré inequality and compute the explicit rates for both first- and second-order Malliavin derivatives separately. The computation for the second-order derivative involves several intricate calculations and requires meticulous decompositions to achieve tight bounds. We illustrate our result through several numerical examples. This is a joint work with Prof. Konstantinos and Prof. Solesne.

• Apr 15
  Wednesday, Room 2-449

  Marcus Michelen (Northwestern)

• Apr 20

  Patriot's Day

• Apr 27

  Simons Lectures

• May 4

  Marianna Russkikh (Notre Dame)

• May 6
  Wednesday

  Matthew Nicoletti (Stanford)

• May 11

  Ron Nissim (MIT)