# Colloquia

In 2022-2023, our colloquia will be in-person talks in B239 unless otherwise stated.

## September 9 , 2022, Friday at 4pm Jing Tao (University of Oklahoma)

(host: Dymarz, Uyanik, WIMAW)

**On surface homeomorphisms**

In the 1970s, Thurston generalized the classification of self-maps of the torus to surfaces of higher genus, thus completing the work initiated by Nielsen. This is known as the Nielsen-Thurston Classification Theorem. Over the years, many alternative proofs have been obtained, using different aspects of surface theory. In this talk, I will overview the classical theory and sketch the ideas of a new proof, one that offers new insights into the hyperbolic geometry of surfaces. This is joint work with Camille Horbez.

## September 23, 2022, Friday at 4pm Pablo Shmerkin (University of British Columbia)

(host: Guo, Seeger)

**Incidences and line counting: from the discrete to the fractal setting**

How many lines are spanned by a set of planar points?. If the points are collinear, then the answer is clearly "one". If they are not collinear, however, several different answers exist when sets are finite and "how many" is measured by cardinality. I will discuss a bit of the history of this problem and present a recent extension to the continuum setting, obtained in collaboration with T. Orponen and H. Wang. No specialized background will be assumed.

## September 30, 2022, Friday at 4pm Alejandra Quintos (University of Wisconsin-Madison, Statistics)

(host: Stovall)

**Dependent Stopping Times and an Application to Credit Risk Theory**

Stopping times are used in applications to model random arrivals. A standard assumption in many models is that the stopping times are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. In the first part of the talk, we use a modified Cox construction, along with the bivariate exponential introduced by Marshall & Olkin (1967), to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We also present a series of results exploring the special properties of this construction.

In the second part of the talk, we present an application of our model to Credit Risk. We characterize the probability of a market failure which is defined as the default of two or more globally systemically important banks (G-SIBs) in a small interval of time. The default probabilities of the G-SIBs are correlated through the possible existence of a market-wide stress event. We derive various theorems related to market failure probabilities, such as the probability of a catastrophic market failure, the impact of increasing the number of G-SIBs in an economy, and the impact of changing the initial conditions of the economy's state variables. We also show that if there are too many G-SIBs, a market failure is inevitable, i.e., the probability of a market failure tends to one as the number of G-SIBs tends to infinity.

## October 7, 2022, Friday at 4pm Daniel Litt (University of Toronto)

(host: Ananth Shankar)

**The search for special symmetries**

What are the canonical sets of symmetries of n-dimensional space? I'll describe the history of this question, going back to Schwarz, Fuchs, Painlevé, and others, and some new answers to it, obtained jointly with Aaron Landesman. While our results rely on low-dimensional topology, Hodge theory, and the Langlands program, and we'll get a peek into how these areas come into play, no knowledge of them will be assumed.

## October 14, 2022, Friday at 4pm Andrew Sageman-Furnas (North Carolina State)

(host: Mari-Beffa)

**Constructing isometric tori with the same curvatures**

Which data determine an immersed surface in Euclidean three-space up to rigid motion? A generic surface is locally determined by only an intrinsic metric and extrinsic mean curvature function. However, there are exceptions. These may arise in a family like the isometric family of vanishing mean curvature surfaces transforming a catenoid into a helicoid.

For compact surfaces, Lawson and Tribuzy proved in 1981 that a metric and non-constant mean curvature function determine at most one immersion with genus zero, but at most two compact immersions (compact Bonnet pairs) for higher genus. In this talk, we discuss our recent construction of the first examples of compact Bonnet pairs. It uses a local classification by Kamberov, Pedit, and Pinkall in terms of isothermic surfaces. Moreover, we describe how a structure-preserving discrete theory for isothermic surfaces and Bonnet pairs led to this discovery.

The smooth theory is joint work with Alexander Bobenko and Tim Hoffmann and the discrete theory is joint work with Tim Hoffmann and Max Wardetzky.

## October 20, 2022, Thursday at 4pm, VV911 Simon Tavaré (Columbia University)

(host: Kurtz, Roch)

*Note the unusual time and room!*

**An introduction to counts-of-counts data**

Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers *C _{1}, C_{2}, …* of species represented once, twice, … in a sample of size

*N = C _{1} + 2 C_{2} + 3 C_{3} + ^{….}* containing

*S = C*species; the vector

_{1}+ C_{2}+^{…}*C =*

*(C*gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.

_{1}, C_{2}, …)In this talk I will outline some of the stochastic models used to model the distribution of *C,* and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of *S* in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.

*References*

[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943

[2] Arratia R, Barbour AD & Tavaré S. *Logarithmic Combinatorial Structures,* EMS, 2002

[3] Ewens WJ. Theoret Popul Biol, 3, 1972

[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)

## October 21, 2022, Friday at 4pm Ngoc Mai Tran (Texas)

(host: Rodriguez)

**Forecast science, learn hidden networks and settle economics conjectures with combinatorics, geometry and probability.**

In many problems, one observes noisy data coming from a hidden or complex combinatorial structure. My research aims to understand and exploit such structures to arrive at an efficient and optimal solution. I will showcase a few successes, achieved with different tools, from different different fields: networks forecasting, hydrology, and auction theory. Then I will outline some open questions in each field.

## October 28, 2022, Friday at 4pm Qin Li (UW)

**Multiscale inverse problem, from Schroedinger to Newton to Boltzmann**

Inverse problems are ubiquitous. We probe the media with sources and measure the outputs, to infer the media information. At the scale of quantum, classical, statistical and fluid, we face inverse Schroedinger, inverse Newton’s second law, inverse Boltzmann problem, and inverse diffusion respectively. The universe, however, expects a universal mathematical description, as Hilbert proposed in 1900. In this talk, we discuss the connection between these problems. We will give arguments for justifying that these are the same problem merely represented at different scales. It is a light-hearted talk, and I will mostly focus on the story instead of the derivation. PDE background is appreciated but not necessary.

## November 7, 2022, Monday at 4pm Kristen Lauter (Facebook)

Distinguished lectures

(host: Yang).

**Private AI: Machine Learning on Encrypted Data**

As the world adopts Artificial Intelligence, the privacy risks are many. AI can improve our lives, but may leak our private data. Private AI is based on Homomorphic Encryption (HE), a new encryption paradigm which allows the cloud to operate on private data in encrypted form, without ever decrypting it, enabling private training and private prediction. Our 2016 ICML CryptoNets paper showed for the first time that it was possible to evaluate neural nets on homomorphically encrypted data, and opened new research directions combining machine learning and cryptography. The security of Homomorphic Encryption is based on hard problems in mathematics involving lattices, recently standardized by NIST for post-quantum cryptography. This talk will explain Homomorphic Encryption, Private AI, and explain HE in action.

## November 8, 2022, Tuesday at 4pm Kristen Lauter (Facebook)

Distinguished lectures in VV911. *Note: unusual room.*

(host: Yang).

**Artificial Intelligence & Cryptography: Privacy and Security in the AI era**

How is Artificial Intelligence changing your life and the world? How do you expect your data to be kept secure and private in the future? Artificial intelligence (AI) refers to the science of utilizing data to formulate mathematical models that predict outcomes with high assurance. Such predictions can be used to make decisions automatically or give recommendations with high confidence. Cryptography is the science of protecting the privacy and security of data. This talk will explain the dynamic relationship between cryptography and AI and how AI can be used to attack post-quantum cryptosystems.

The first talk is based on my 2019 ICIAM Plenary Lecture and the second one is based on my 2022 SIAM Block Prize Lecture.

## November 11, 2022, Friday at 4pm Joel Tropp (Caltech)

This is the Annual LAA lecture. See this for its history.

(host: Qin, Jordan)

**Rocket-propelled Cholesky: Addressing the challenges of large-scale kernel computations**

Kernel methods are used for prediction and clustering in many data science and scientific computing applications, but applying kernel methods to a large number of data points N is expensive due to the high cost of manipulating the N x N kernel matrix. A basic approach for speeding up kernel computations is low-rank approximation, in which we replace the kernel matrix A with a factorized approximation that can be stored and manipulated more cheaply. When the kernel matrix A has rapidly decaying eigenvalues, mathematical existence proofs guarantee that A can be accurately approximated using a constant number of columns (without ever looking at the full matrix). Nevertheless, for a long time designing a practical and provably justified algorithm to select the appropriate columns proved challenging.

Recently, we introduced RPCholesky ("randomly pivoted" or "rocket-propelled" Cholesky), a natural algorithm for approximating an N x N positive semidefinite matrix using k adaptively sampled columns. RPCholesky can be implemented with just a few lines of code; it requires only (k+1)N entry evaluations and O(k^2 N) additional arithmetic operations. In experiments, RPCholesky matches or improves on the performance of alternative algorithms for low-rank psd approximation. Moreover, RPCholesky provably achieves near-optimal approximation guarantees. The simplicity, effectiveness, and robustness of this algorithm strongly support its use for large-scale kernel computations.

Joint work with Yifan Chen, Ethan Epperly, and Rob Webber. Available at arXiv:2207.06503.

## November 18, 2022, Friday at 4pm Jim Freitag (U of Illinois-Chicago) Zoom link: https://go.wisc.edu/jimfreitag

**Now available:** Recording and Slides

(hosts: Lempp, Andrews)

**When any three solutions are independent**

In this talk, we'll talk about a surprising recent result about the algebraic relations between solutions of a differential equation. The result has applications to functional transcendence, diophantine geometry, and compact complex manifolds.

## November 21, 2022, Monday at 4pm Andrei Negut (MIT) Zoom link: https://go.wisc.edu/andreinegut

Hiring talk.

(hosts: Arinkin, Caldararu)

**From gauge theory to geometric representation theory and back**

We start from the celebrated construction (due to Grojnowski and Nakajima) of a Heisenberg algebra action on the cohomology groups of Hilbert schemes of points on surfaces

- replacing Hilbert schemes with moduli spaces of higher rank sheaves yields a computation of Nekrasov partition functions in 5d supersymmetric gauge theory, and a proof of the deformed Alday-Gaiotto-Tachikawa conjecture.
- replacing cohomology by Chow groups gives a proof of the Beauville conjecture in the hyperkahler geometry of Hilbert schemes of points on K3 surfaces (with Maulik)
- working with derived categories allows us to construct a detailed framework realizing categorical knot invariants in terms of the geometry of Hilbert schemes of points on the affine plane (with Gorsky and Rasmussen)

## December 2, 2022, Friday at 4pm: Promit Ghosal (MIT)

**Fractal Geometry of the KPZ equation**

The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to many important models like random growth processes, Burgers turbulence, interacting particles system, random polymers etc. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi. If time permits, I will also mention an interesting story about the (2+1)-d and (3+1)-d case (work in progress with Jaeyun Yi).

## December 5, 2022, Monday at 4pm: Di Fang (Berkeley)

(host: Qin Li)

**Quantum algorithms for Hamiltonian simulation with unbounded operators**

Recent years have witnessed tremendous progress in developing and analyzing quantum computing algorithms for quantum dynamics simulation of bounded operators (Hamiltonian simulation). However, many scientific and engineering problems require the efficient treatment of unbounded operators, which frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure theory, quantum control and quantum machine learning. We will introduce some recent advances in quantum algorithms for efficient unbounded Hamiltonian simulation, including Trotter type splitting and the quantum highly oscillatory protocol (qHOP) in the interaction picture. The latter yields a surprising superconvergence result for regular potentials. In the end, I will discuss briefly how Hamiltonian simulation techniques can be applied to a quantum learning task achieving optimal scaling. (The talk does not assume a priori knowledge on quantum computing.)

Zoom option: https://go.wisc.edu/difang

## December 7, 2022, Wednesday at 4pm: Benjamin Eichinger (Vienna University of Technology)

(host: Denisov)

**An approach to universality using Weyl m-functions**

In this talk I will present an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl m-function at the point. We show that bulk universality of the Christoffel-Darboux kernel holds for any point where the imaginary part of the m-function has a positive finite nontangential limit. This approach is based on studying this problem in the more general setting of canonical systems and the realization that bulk universality for an associated matrix reproducing kernel at a point is equivalent to the fact that the corresponding m-function has normal limits at the same point. The talk is based on a joint work with Milivoje Lukic and Brian Simanek. If time permits, I will discuss some work in progress with Milivoje Lukic and Harald Woracek on rescaling limits for other

universality classes.

## December 9, 2022, Friday at 4pm: Dallas Albritton (Princeton)

(host: Feldman)

**Non-uniqueness of Leray solutions to the forced Navier-Stokes equations**

In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative within the "forced" category, by exhibiting a one-parameter family of distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo.

## December 12, 2022, Monday at 4pm: Laurel Ohm (Princeton)

(host: Spagnolie, Thiffeault)

**A PDE perspective on the hydrodynamics of flexible filaments**

Many fundamental biophysical processes, from cell division to cellular motility, involve dynamics of thin structures immersed in a very viscous fluid. Various popular models have been developed to describe this interaction mathematically, but much of our understanding of these models is only at the level of numerics and formal asymptotics. Here we seek to develop the PDE theory of filament hydrodynamics. First, we propose a PDE framework for analyzing the error introduced by slender body theory (SBT), a common approximation used to facilitate computational simulations of immersed filaments in 3D. Given data prescribed only along a 1D curve, we develop a novel type of boundary value problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing. Second, we consider a classical elastohydrodynamic model for the motion of an immersed inextensible filament. We highlight how the analysis can help to better understand undulatory swimming at low Reynolds number. This includes the development of a novel numerical method to simulate inextensible swimmers.