Colloquia/Spring2024

Spring 2024

date	speaker	title	host(s)
Monday Jan 22 at 4pm in B239	Yingkun Li (Darmstadt Tech U, Germany)	Arithmetic of real-analytic modular forms	Yang
Thursday Jan 25 at 4pm in VV911	Sanjukta Krishnagopal (UCLA/UC Berkeley)	Theoretical methods for data-driven complex systems: from mathematical machine learning to simplicial complexes	Smith
Jan 26	Jacob Bedrossian (UCLA)	Lyapunov exponents in stochastic systems	Tran
Feb 2	William Chen	Orbit problems and the mod p properties of Markoff numbers	Arinkin
Feb 9	No colloquium
Feb 16	Jack Lutz (Iowa State)	Algorithmic Fractal Dimensions	Guo
Feb 23	No colloquium
Mar 1	Per-Gunnar Martinsson (UT-Austin)	Randomized algorithms for linear algebraic computations	Li
Mar 8	Anton Izosimov (U of Arizona)	Incidences and dimers	Gloria Mari-Beffa
Mar 15	Peter Humphries (Virginia)	Equidistribution, Period Integrals of Automorphic Forms, and Subconvexity	Marshall
Monday Mar 18 at 4pm in B239	Cole Graham (Brown)	Invasion in general domains	Albritton, Smith, Tran
Wednesday Mar 20 at 4 pm in B239	Wanlin Li (Washington U St Louis)	Diophantine problem and rational points on curves	Dymarz, GmMaW
Mar 29	Spring break
Apr 5	Ovidiu Savin (Columbia)	Free boundary problems with interacting boundaries	Tran
Friday Apr 12 at 5pm in Tripp Commons	Mikayla Kelley (U Chicago Philosophy)	Math And... seminar: Accuracy and the Patterns of Rational Credence	Ellenberg, Marshall
Apr 19	Yanyan Li (Rutgers)	Liouville theorems for conformally invariant equations	Tran
Apr 26	Chris Leininger (Rice)	Billiards, symbolic coding, and geometry	Uyanik
May 3	Jin-Yi Cai (UW-Madison)	Shor's Quantum Algorithm Does Not Factor Large Integers in the Presence of Noise	Yang

Abstracts

Monday, January 22. Yingkun Li

Arithmetic of real-analytic modular forms

Modular form is a classical mathematical object dating back to the 19th century. Because of its connections to and appearances in many different areas of math and physics, it remains a popular subject today. Since the work of Hans Maass in 1949, real-analytic modular form has found important applications in arithmetic geometry and number theory. In this talk, I will discuss the amazing works in this area over the past 20 years, and give a glimpse of its fascinating future directions.

Thursday, January 25. Sanjukta Krishnagopal

Theoretical methods for data-driven complex systems: from mathematical machine learning to simplicial complexes

In this talk I will discuss some aspects at the intersection of mathematics, machine learning, and networks to introduce interdisciplinary methods with wide application.

First, I will discuss some recent advances in mathematical machine learning for prediction on graphs. Machine learning is often a black box. Here I will present some exact theoretical results on the dynamics of weights while training graph neural networks using graphons - a graph limit or a graph with infinitely many nodes. I will use these ideas to present a new method for predictive and personalized medicine applications with remarkable success in prediction of Parkinson's subtype five years in advance.

Then, I will discuss some work on higher-order models of graphs: simplicial complexes - that can capture simultaneous many-body interactions. I will present some recent results on spectral theory of simplicial complexes, as well as introduce a mathematical framework for studying the topology and dynamics of multilayer simplicial complexes using Hodge theory, and discuss applications of such interdisciplinary methods to studying bias in society, opinion dynamics, and hate speech in social media.

Friday, January 26. Jacob Bedrossian

Lyapunov exponents in stochastic systems

In this overview talk we discuss several results regarding positive Lyapunov exponents in stochastic systems. First we discuss proving "Lagrangian chaos" in stochastic fluid mechanics, that is, demonstrating a positive Lyapunov exponent for the motion of a particle in the velocity field arising from the stochastic Navier-Stokes equations. We describe how this chaos can be used to deduce qualitatively optimal almost-sure exponential mixing of passive scalars. Next we describe more recently developed methods for obtaining strictly positive lower bounds and some quantitative estimates on the top Lyapunov exponent of weakly-damped stochastic differential equations, such as Lorenz-96 model or Galerkin truncations of the 2d Navier-Stokes equations (called "Eulerian chaos" in fluid mechanics). Further applications of the ideas to the chaotic motion of charged particles in fluctuating magnetic fields and the non-uniqueness of stationary measures for Lorenz 96 in degenerate forcing situations will be discussed if time permits. All of the work except for the charged particles (joint with Chi-Hao Wu) is joint with Alex Blumenthal and Sam Punshon-Smith.

Friday, February 2. William Chen

Orbit problems and the mod p properties of Markoff numbers

Markoff numbers are positive integers which encode how resistant certain irrational numbers are to being approximated by rationals. In 1913, Frobenius asked for a description of all congruence conditions satisfied by Markoff numbers modulo primes p. In 1991 and 2016, Baragar, Bourgain, Gamburd, and Sarnak conjectured a refinement of Frobenius’s question, which amounts to showing that the Markoff equation x^2 + y^2 + z^2 - xyz = 0 satisfies “strong approximation”; that is to say: they conjecture that its integral points surject onto its mod p points for every prime p. In this talk we will show how to prove this conjecture for all but finitely many primes p, thus reducing the conjecture to a finite computation. A key step is to understand this problem in the context of describing the orbits of certain group actions. Primarily, we will consider the action of the mapping class group of a topological surface S on (a) the set of G-covers of S, where G is a finite group, and (b) on the character variety of local systems on S. Questions of this type have been related to many classical problems, from proving that the moduli space of curves of a given genus is connected, to Grothendieck’s ambitious plan to understand the structure of the absolute Galois group of the rationals by studying its action on “dessins d’enfant”. We will explain some of this history and why such problems can be surprisingly difficult.

Friday, February 16. Jack Lutz

Algorithmic Fractal Dimensions

Algorithmic fractal dimensions are computability theoretic versions of Hausdorff dimension and other fractal dimensions. This talk will introduce algorithmic fractal dimensions with particular focus on the Point-to-Set Principle. This principle has enabled several recent proofs of new theorems in geometric measure theory. These theorems, some solving long-standing open problems, are classical (meaning that their statements do not involve computability or logic), even though computability has played a central in their proofs.

Friday, March 1. Per-Gunnar Martinsson

Randomized algorithms for linear algebraic computations

The talk will describe how randomized algorithms can effectively, accurately, and reliably solve linear algebraic problems that are omnipresent in scientific computing and in data analysis. We will focus on techniques for low rank approximation, since these methods are particularly simple and powerful, and are well understood mathematically. The talk will also briefly survey a number of other randomized algorithms for tasks such as solving linear systems, estimating matrix norms, and computing full matrix factorizations.

Friday, March 8. Anton Izosimov

Incidences and dimers

Incidence theorems are statements about points, lines, and possibly higher-dimensional subspaces and their incidences. Examples include classical theorems of Desargues and Pappus. In this talk, we'll discuss a connection between incidence geometry and an archetypal model of statistical physics - the dimer model. The talk will be based on the work of many people, including my ongoing work with Pavlo Pylyavskyy (Minnesota).

Friday, March 15. Peter Humphries

Equidistribution, Period Integrals of Automorphic Forms, and Subconvexity

A fundamental conjecture in number theory is the Riemann hypothesis, which implies the prime number theorem with an optimally strong error term. While a proof remains elusive, many results in number theory can nonetheless be proved using weaker inputs. I will discuss how one such weaker input, subconvexity, can be used to prove strong results on the equidistribution of geometric objects such as lattice points on the sphere. I will also discuss how various proofs of subconvexity reduce to understanding period integrals of automorphic forms.

Monday, March 18. Cole Graham

Invasion in general domains

The sciences teem with examples of invasion, in which one steady state spatially invades another. Mathematically, we can express this phenomenon through reaction-diffusion equations. These are well understood in the free space, but applications call for more complex geometries. In this talk, I will discuss reaction-diffusion invasion in multiple dimensions and general domains.

Wednesday, March 20. Wanlin Li

Diophantine problem and rational points on curves

Diophantine problem asks for integral/rational solutions to polynomial equations. These solutions correspond to rational points on algebraic varieties. The study of Diophantine problems led to many essential developments of modern number theory and arithmetic geometry. Today I will briefly discuss the history of Diophantine problems and introduce various tools developed to study these problems. I will also introduce my joint work with Litt, Salter and Srinivasan on constructing cohomology classes which provide obstruction to the existence of rational points on curves.

Friday, April 5. Ovidiu Savin

Free boundary problems with interacting boundaries

We discuss some energy models for N ordered elastic membranes subject to forcing and boundary conditions. The heights of the membranes are described by real functions u_1, u_2,...,u_N, which minimize an energy functional involving the Dirichlet integral and a potential term depending on (u_1,..,u_N). The minimization problems can be understood as systems of N-1 coupled scalar free boundary problems with interacting free boundaries. I will review the free boundary regularity for the scalar cases, which correspond to the classical obstacle problem or the Bernoulli problem, and discuss the situation when dealing with 3 or more membranes.

Friday, April 12. Mikayla Kelley

Accuracy and the Patterns of Rational Credence

A credence is a belief-like attitude that encodes one's degree of confidence in some way the world could be. For example, you might be 60% confident that the Democrats will win the presidential election. Some patterns of credence are irrational. Being 90% confident that Goldbach's conjecture is true and 90% confident that Goldbach's conjecture is false seems irrational. This is because it violates the following plausible pattern of rational credence: your credences in p and not p sum to 100%. How do we identify the patterns of rational credence? According to accuracy-first epistemology, we do so by identifying which patterns promote accuracy, where accuracy is represented formally as a real-valued function. In this talk, I will introduce the basics of accuracy-first epistemology and discuss my own work on using accuracy to study the patterns of rational credence when one has infinitely many credences.

Friday, April 19. Yanyan Li

Liouville theorems for conformally invariant equations

A fundamental  theorem of Liouville asserts that positive entire harmonic functions in Euclidean spaces must be  constant. A remarkable Liouville-type theorem of Caffarelli-Gidas-Spruck states that positive entire solutions of $-\Delta u=u^{ \frac {n+2}{n-2} }$, $n\ge 3$,  are unique modulo M\"obius transformations.  Far-reaching extensions were established for general fully nonlinear conformally invariant equations through the works of Chang-Gursky-Yang, Li-Li, Li, and Viaclovsky. In this paper, we derive necessary and sufficient conditions for the validity of such Liouville-type theorems. This leads to necessary and sufficient conditions for local gradient estimates of solutions to hold, assuming a one-sided bound on the solutions, for a wide class of fully nonlinear elliptic equations involving Schouten tensors. A pivotal advancement in proving these Liouville-type theorems is our enhanced understanding of solutions to such equations near isolated singularities. In particular, we utilize earlier results of  Caffarelli-Li-Nirenberg on lower- and upper-conical singularities. For general conformally invariant fully nonlinear elliptic equations, we prove that a viscosity super- (sub-)solution can be extended across an isolated singularity if and only if it is a lower-

 (upper-)conical singularity. We also provide necessary and sufficient conditions for lower- (upper-)conical behavior of a function near isolated singularities in terms of its conformal Hessian. This is a joint work with Baozhi Chu and Zongyuan Li. The talk is accessible to first year graduate students.

Friday, April 26. Chris Leininger

Billiards, symbolic coding, and geometry

A billiard trajectory in a polygon P in the Euclidean plane is the path of a particle inside P, following straight lines until it encounters a side, and then bouncing off so that the angle of reflection equals the angle of incidence. A generic trajectory never encounters the corners in forward or backward time, and so produces a biinfinite symbolic coding: the itinerary of sides encountered by the trajectory. A natural question studied by Bobok and Troubetzkoy about 15 years ago is whether one can recover the shape of the polygon from the set of all itineraries: the symbolic coding. They observed that two polygons with vertical and horizontal sides that differ by an affine transformation have the same codings, and proved that among rational polygons (those whose interior angles are rational multiple of pi), this essentially accounts for the only ambiguity. I'll describe work with Duchin, Erlandsson, and Sadanand where we prove an analogous result with no restrictions on interior angles. Time permitting, I will also describe a companion result for billiards in hyperbolic polygons with Erlandsson and Sadanand, where the exceptional cases are much more robust.

Friday, May 3. Jin-Yi Cai

Shor's Quantum Algorithm Does Not Factor Large Integers in the Presence of Noise

Shor's quantum factoring algorithm is the raison d'être for the field of quantum computing. The security of encryption systems such as RSA depends on the (conjectured) infeasibility of factoring in (classical) polynomial time, but Shor's algorithm can do so in Bounded-error Quantum Polynomial time (BQP). The key ingredient of this algorithm is the so-called Quantum Fourier Transform (QFT). BQP (in particular QFT) assumes infinite precision quantum rotation gates are available. This talk presents the first proof that, if the rotation gates have a vanishingly small level of noise, Shor's algorithm does not factor integers of the form n = pq for a positive density of primes p and q. It also fails with probability 1 - o(1) for random primes p and q. This proof applies to any algorithm that uses QFT. If time permits, I will also discuss my (speculative) view on the suitability of BQP replacing P or BPP in the strong Church-Turing thesis.