# Colloquium Abstracts Spring 2022

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

## Xin Zhou

Cornell University

February 3, 2022

*Variational Theory and Mean Curvature*

Mathematical models of soap bubbles and capillary surfaces are described by a class of prescribing mean curvature (PMC) equations. Minimal surfaces are a special class of PMC surfaces. In variational theory, PMC surfaces are stationary points of the area functional plus a volume-related term. In this talk, we will survey recent developments of the existence theory of such surfaces, including Yau's conjecture of minimal surfaces, the Morse theory for area functional, and the min-max theory for PMC surfaces. In particular, we will explain our recent result which made a connection between the PMC theory and the Multiplicity One Conjecture for minimal surfaces.

## Philip Engel

University of Georgia

February 24, 2022*Looijenga's Cusp Conjecture and Triangulations of the Sphere*

A "cusp singularity" is an isolated surface singularity whose

minimal resolution is a cycle of smooth rational curves. Cusp singularities

come naturally in "dual pairs," whose resolutions are called "dual cycles."

Looijenga proved in 1981 that if a cusp singularity can be smoothed, the

dual cycle is an anticanonical divisor of a rational surface. He conjectured

the converse. Gross, Hacking, Keel proved Looijenga's conjecture in

2011, and I gave a new proof in 2014. I will discuss this alternative proof,

and some of the interesting research directions it has led to since then.

## Daniel Sanz-Alonso

University of Chicago

March 3, 2022*Auto-differentiable Ensemble Kalman Filters*

Data assimilation is concerned with sequentially estimating a temporally-evolving state. This task, which arises in a wide range of scientific and engineering applications, is particularly challenging when the state is high-dimensional and the state-space dynamics are unknown. In this talk I will introduce a machine learning framework for learning dynamical systems in data assimilation. Our auto-differentiable ensemble Kalman filters (AD-EnKFs) blend ensemble Kalman filters for state recovery with machine learning tools for learning the dynamics. In doing so, AD-EnKFs leverage the ability of ensemble Kalman filters to scale to high-dimensional states and the power of automatic differentiation to train high-dimensional surrogate models for the dynamics. Numerical results using the Lorenz-96 model show that AD-EnKFs outperform existing methods that use expectation-maximization or particle filters to merge data assimilation and machine learning. In addition, AD-EnKFs are easy to implement and require minimal tuning. This is joint work with Yuming Chen and Rebecca Willett.

## Izzet Coskun

University of Illinois Chicago

March 10, 2022*Points in the Projective Plane*

In this talk, I will discuss the geometry of the Hilbert scheme of n points in the projective plane, which is a smooth compactification of the configuration space of n points. I will focus on the question: What is the most special codimension one position that n points can lie in? For example, three points are typically not collinear, but in codimension one they can be collinear. This simple question will lead us to a tour of some fun mathematics ranging from moduli spaces of stable sheaves on the plane to fractal curves and palindromic numbers. This talk is based on joint work with Jack Huizenga and Matthew Woolf.

## Yuchen Liu

Northwestern University*Moduli Space of Fano Varieties*

Fano varieties are complex algebraic varieties admitting positive Ricci curvature metrics. They form one of the three fundamental building blocks of algebraic varieties, thus their classification problem is important. However, it is known that moduli spaces of all Fano varieties have pathological behaviors. In this talk, I will explain that if we impose K-stability on Fano varieties, an algebraic condition arising from the study of Kähler-Einstein metrics, then we indeed get a compact moduli space. Based on joint works with H. Blum, D. Halpern-Leistner, C. Xu, and Z. Zhuang.

## Tim Davis

Texas A&M University

April 21, 2022*Math, Matrices and Music*

We all know what music sounds like, but if you could see a whole piece of

music drawn in a single artistic visualization, what would it look like?

In his day job, Davis creates mathematical algorithms for solving huge sparse matrix problems. His solvers are widely used in industry, academia, and government labs. For example, every photo on the planet in Google StreetView is placed in its proper position by his codes. He also curates a vast collection of matrices, so that he and others in his field can test their

methods on real-life problems.

In collaboration with Yifan Hu at Yahoo! Labs, these matrices are converted

into images via a physics simulation. The primary purpose of these

visualizations is to understand the relationships in the matrix and how various solvers behave on different problems. But the images also happen to be stunningly beautiful, and they have caught the eye of the popular press (Geeky Science Problems Double as Works of Art, FastCompany,

https://www.fastcodesign.com/1662136/geeky-science-problems-double-as-w…).

The matrices in Davis' collection have nothing to do with music, but in 2013

the organizers of the London Electronic Arts Festival came across the

images and gave him a challenge: "These pictures are amazing ... could you create similar images from sound bites?"

Davis' first thought was "Music?! That's crazy; these are matrices, not

music." Taking up the challenge, however, Davis constructed a mathematical algorithm for converting an entire piece music into a sparse matrix, capturing time and frequency, and remapping them into a new domain of space and color. His method captures a visual essence of an entire piece of music in a single image. The heavy regular beat of jazz and electronic music converts into simple elegant meshes, like a fish net. Complex orchestral works convert into dazzlingly complex fuzzy structures. Davis' art appeared on billboards around London as the theme art for the Festival.

In this seminar, Davis will present his music artwork and how it came to be created. You can preview his portfolio at notesartstudio.com, and browse his sparse matrices at http://sparse.tamu.edu.

## Jesús De Loera

University of California, Davis

April 14, 2022*On the (Discrete) Geometric Principles of Machine Learning and Statistical Inference*

In this talk I explain the fertile relationship between inference and learning to classical combinatorial geometry. My presentation contains several powerful situations where famous theorems in discrete geometry answered natural questions from machine learning and statistical inference. In this tasting tour, I will include the problem of deciding the existence of Maximum likelihood estimator in multiclass logistic regression, the variability of behavior of k-means algorithms with distinct random initializations and the shapes of the clusters, and the estimation of the number of samples in chance-constrained optimization models. These obviously only scratch the surface of what one could do with extra free time. Along the way we will see fascinating connections to the coupon collector problem, topological data analysis, measures of separability of data, and to the computation of Tukey centerpoints of data clouds (a high-dimensional generalization of median). All new theorems are joint work with subsets of the following wonderful folks: T. Hogan, D. Oliveros, E. Jaramillo-Rodriguez, and A. Torres-Hernandez.

## Aaron Yip

Purdue University

April 28, 2022*A Revisit of Homogenization of Motion by Mean Curvature*

Despite a long history and many results on motion by mean curvature and homogenization, there are still many interesting questions when these two problems are put together. This is a typical question of gradient flow on wiggly energy landscapes. The talk will revisit the problem of motion by mean curvature in heterogeneous medium, in both the periodic and random case, reviewing some earlier and recent results.

## Maria Helena Noronha

California State University

May 5, 2022

Russell Bradt Undergraduate Colloquium*How We Get Students PUMPed into PhDs*

Underrepresented minority (URM) and first-generation college math majors, as well as those with financial constraints abound in several parts of the country and in particular in Southern California. Some of these students are unaware

of the many opportunities available to them, that a PhD degree can boost their careers or, even worse, lack confidence that they can succeed in graduate school.

In this talk I will describe how my col- laborators and I have been mentoring these math majors at some campuses of the California State University sys- tem. I will also talk about how we ex- tended this work to local Community Colleges. Our work is changing the culture of our departments, that is, students are inspired, build self-con- fidence, and raise their aspirations. We work in the project named PUMP: Preparing Undergraduates through Mentoring towards PhDs and RE-C^2: Research Experiences in Community Colleges. I will also talk about lessons that we have learned and challenges to be faced.

## Tamas Hausel

Institute of Science and Technology Austria

May 6, 2022*Ubiquity of Systems of Homogenous Polynomial Equations with a Unique Solution*

Following Macaulay we will analyse systems of equations as in the title leading to marvelous properties of its multiplicity algebra. Examples include isolated surface singularities, equivariant cohomology and fixed point sets of group actions as well as the Hitchin integrable system on very stable upward flows.