# Colloquium Abstracts Fall 2021

Abstracts will be posted here for the colloquium talks when they are available.

## Freddy Bouchet

CNRS and ENS de Lyon, France

September 16, 2021

*Path Large Deviations for Kinetic Theories: Beyond the Boltzmann, the Landau and the Lenard—Balescu Kinetic Equations*

In many physical systems one seeks to describe effectively mesoscopic or macroscopic variables. Kinetic theories and kinetic equations are examples where the average mesoscopic dynamics is obtained through very clear theoretical procedures and can possibly lead to mathematical proofs. A few works go beyond the average evolution and describe, for instance, Gaussian fluctuations. However, for many physical systems, rare event can be of importance, and Gaussian fluctuations are not relevant. This is the case for instance if one wants to understand the irreversibility paradox associated to the kinetic equations, or to understand the dynamics that leads to rare but important events.

The aim of this presentation is to describe recent results where we derived explicitly the functional that describes the path large deviations for the empirical measure of dilute gases, plasma and systems of particles with long range interactions. The associated kinetic equations (the average evolution) are then either the Boltzmann, the Landau or the Balescu--Lenard—Guernsey equations. After making the classic assumptions in theoretical physics textbooks for deriving the kinetic equation, our derivation of the large deviation functional is exact. I will explain how we plan to generalize these results to turbulence problem.

These path large deviation principles give a very nice and transparent new interpretation of the classical irreversibility paradox. This new explanation is fully compatible with the classical one, but it gives a deeper insight.

**References:**

For the large deviations associated to the Boltzmann equation (dilute gazes), and a general introduction (published in J. Stat. Phys. in 2020): F. Bouchet, 2020, Is the Boltzmann equation reversible? A large deviation perspective on the irreversibility paradox and the Boltzmann equation, Journal of Statistical Physics, 181, 515–550.

For the large deviations associated to the Landau equation (plasma below the Debye length, accepted for publication in J. Stat. Phys. in March 2021): O. Feliachi and F. Bouchet, 2021, Dynamical large deviations for plasma below the Debye length and the Landau equation, Journal of Statistical Physics, 183, 42.

For the large deviations associated to the Balescu—Guernsey--Lenard equation (plasma and systems with long range interactions, submitted to publication in J. Stat. Phys. in March 2021): O. Feliachi and F. Bouchet, 2021, Dynamical large deviations for plasma and other systems with long range interactions associated to the Lenard-Balescu-Guernsey equation, submitted to Journal of Statistical Physics, *https://arxiv.org/abs/2105.05644.*

## Gautam Iyer

Carnegie Mellon University

September 23, 2021

*Dissipation Enhancement, Mixing and Blow-up Suppression*

Diffusion and mixing are two fundamental phenomena that arise in a wide variety of applications. In this talk we quantitatively study the interaction between diffusion and mixing in the context of problems arising in fluid dynamics. The first question we address is how fast the energy can decay in the advection diffusion equation. Even though this is a simple linear equation, the energy decay rate is intrinsically related to the mixing properties of the advecting velocity field, and there are many unresolved open questions. I will present a few recent results involving both upper and lower bounds, and then consider applications to studying the long time dynamics of a few model non-linear equations.

## Bo Luo and Zijun Yao

Electrical Engineering and Computer Science

University of Kansas

September 30, 2021

*A Brief Introduction to Deep Learning: Techniques, Challenges, Security and Privacy*

Deep learning has transformed many data analytic applications, such as speech recognition, computer vision, and natural language processing. It changed the data analytic modeling profoundly from expert-driven feature engineering to data-driven feature construction (i.e., models with multiple layers of neural networks). Over the past few years, an increasing body of state-of-the-art (SOTA) deep learning architectures have been developed for different domain tasks and various types of data. In this talk, we will briefly introduce some of the fundamental techniques in deep learning and the SOTA developed based on them (e.g., Transformer network). Beyond that, we will briefly discuss some of the open challenges of deep learning, such as model explanability. Meanwhile, a broad spectrum of cyber-attacks against deep learning systems has been proposed recently. Such attacks aim to break the integrity or confidentiality of the models. They can be roughly grouped into three categories: the evasion attacks, the exploratory attacks, and the backdoor/poisoning attacks. In this talk, we will also introduce the representative attacks against deep neural networks and the SOTA defense mechanisms. Through this talk, we hope to show a general idea of what deep learning is and what its security and privacy issues are, which may be helpful for audience to identify the opportunities and challenges in their own research fields.

## Mark Shoemaker

Colorado State University

October 14, 2021

*Enumerative Geometry and Mirror Symmetry*

The goal of enumerative geometry is to study a geometric space by counting certain subspaces within it. The first result in enumerative geometry is Euclid’s observation that, given 2 distinct points in the plane, there is a single line through these points. A harder question is, given 2 points and 3 random lines in the plane how many conics (degree 2 curves) pass through both points and are tangent to each of the lines. These types of questions have interested geometers since the 1800's and earlier, but they are famously difficult. However, a breakthrough occurred in the 1990’s when a surprising connection was made with physics. It was discovered that techniques and intuitions from string theory could be used to answer longstanding questions in enumerative geometry. The phenomenon behind this remarkable connection came to be known as mirror symmetry. In this talk I will give an introduction to mirror symmetry and its connection to enumerative geometry. At the end of the talk I will mention some current directions of inquiry and open questions.

## Ruixiang Zhang

University of California-Berkeley

November 4, 2021

*A Stationary Set Method for Estimating Oscillatory Integrals*

Given a polynomial *P* of constant degree in *d* variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming *d* is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a "stationary set'' method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry's problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent *p* is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.

## Sarah Pollock

University of Florida

November 18, 2021

*Extrapolation Methods for Eigenvalue Problems*

We will discuss accelerating convergence to numerical solutions of eigenvalue problems using a simple post-processing step applied to standard eigensolver techniques. First we will consider accelerating the standard power iteration by a depth-one extrapolation method which combines the latest and previous updates to the approximate eigenvector to obtain a new iterate at each step. We will see how choosing the extrapolation parameter as the ratio between latest and previous residuals provides asymptotically exponential convergence, and transforms the power iteration into an efficient and robust technique. We will then look at applying a similar technique to a restarted Arnoldi method to boost its performance with little additional computational cost. Numerical examples will illustrate the theory.

## Chi Li

Rutgers University

December 2, 2021

*On the Algebraic Uniqueness of Kahler-Ricci Flow Limits on Fano Manifolds*

Let X be a Fano manifold. The Hamilton-Tian conjecture, which has been studied by Perelman, Tian-Zhang, Chen-Wang, Bamler and others, states that the normalized Kaehler-Ricci flow on X converges in the Gromov-Hausdorff topology to a possibly singular Kaehler-Ricci soliton as the time goes to infinity. Chen-Sun-Wang further conjectured that the limit space does not depend on the initial Kaehler metric but depends only on the algebraic structure of X. I will discuss a joint work with Jiyuan Han, which confirms this uniqueness conjecture. The proof is based on the study of an optimization problem for real valuations on the functional field of X. Our result has applications in identifying limits in concrete examples.

## Tian-Jun Li

University of Minnesota

December 9, 2021

*Symmetries of Symplectic Rational Surfaces*

We study symplectic rational surfaces equipped with a finite symplectomorphism group G. They are the symplectic analogs of (complex) rational G-surfaces studied in algebraic geometry, which are rational surfaces equipped with a holomorphic G-action. These rational G-surfaces played a central role in the classification of finite subgroups of the plane Cremona group, a problem dating back to the early 1880s. Our work shows that a large part of the story regarding the classification of rational G-surfaces can be recovered by techniques from 4-manifold theory and symplectic topology. Furthermore, we also add some new symplectic geometry aspect to the study of rational $G$-surfaces. This is a joint work with Weimin Chen and Weiwei Wu.