# Events This Week

Week of October 22 - October 18

### Friday, October 22

Applied and Computational Mathematics

Topic: Modeling Immunity to Malaria with an Age-Structured PDE Framework

Zhuolin Qu | Department of Mathematics, University of Texas at San Antonio

Abstract: Malaria is one of the deadliest infectious diseases globally, causing hundreds of thousands of deaths each year. It disproportionately affects young children, with two-thirds of fatalities occurring in under-fives. Individuals acquire protection from disease through repeated exposure, and this immunity plays a crucial role in the dynamics of malaria spread. We develop a novel age-structured PDE model of malaria specifically tracking acquisition and loss of immunity across the population. Using our analytical calculation of the basic reproduction number (R0), we study the role of vaccination and immunity feedback on severe disease and malaria incidence. Using demographic and immunological data, we parameterized our model to simulate realistic scenarios. Thus, via a combination of analytic and numerical investigations, our work sheds new light on the role of acquired immunity in malaria dynamics and the impact on vaccination strategies in the presence of demographic effects.

This is a joint work with Lauren Childs, Christina Edholm, Denis Patterson, Joan Ponce, Olivia Prosper, and Lihong Zhao.

Zoom access:
Time: 2:00

### Thursday, October 21

MATH FOR ALL

Topic: Want to learn how to make a math or science poster?

Kalina Mincheva – Tulane University

Abstract

Location: Boggs 105
Time: 7:00pm

### Monday, October 18

Algebraic Geometry and Geometric Topology Seminar

Topic: Quasimorphisms, diffeomorphism groups of surfaces, and L^p-metrics.

Michał Marcinkowski | University of Wrocław (Poland)

Abstract: Quasi-(homo)morphisms are real functions on a group that pretend to be homomorphisms. On many groups there is plenty of interesting quasimorphisms. I will ilustrate this notion with simple geometric and combinatoric examples. In particular I will describe how using braids one can construct quasimorphisms on $Diff_0(S,\omega)$, the group of area preserving diffeomorphis of surface S. These quasimorphisms are generalisations of the Calabi invariant. In our recent work with M. Brandenbursky and E. Shelukhin we showed that there exist many quasimorphisms on $Diff_0(S,\omega)$ that are Lipschitz with respect to the $L^p$-norm, $p \geq 1$. The proof uses the compactification of the configuration space of $S$. This allows to show e.g., that right-angled Artin groups can be embedded quasi-isometrically into $Diff_0(S,\omega)$ with the $L^p$ norm. I will explain these notions and show the idea of the proof.

Location: Gibson Hall 325
Zoom access:
Time:
3:00

Week of October 15 - October 11

### Friday, October 15

Applied and Computational Mathematics

Topic: Digital Twins and Efficient Clinical Trials

David Li-Bland | Unlearn.AI

Abstract: Clinical trials for a novel medical treatment measure the effect of the treatment by estimating the change in disease progression between a group of patients who receive the treatment, and a control group of patients who instead receive a placebo. One typically needs a large group of patients to accurately measure this treatment effect, which makes it very difficult to explore new treatments for rare diseases or diseases with a low quality of life (such as Alzheimers, MS, or ALS) where patients face practical challenges when participating in clinical trials.

In this talk, I will describe how we use machine learning to generate Digital Twins of patients. Such Digital Twins significantly reduce the number of patients needed for a clinical trial, and yet we can provide stastical guarantees that their use does not increase the odds that an ineffective or unsafe treatment would be wrongly approved.

Zoom access:
Time: 2:00

### Thursday, October 14

Colloquium

Topic: Convergence results for Yule's "nonsense correlation" using stochastic analysis

Frederi Viens - Michigan State (Host: Glatt-Holtz)

Abstract: We provide an analysis of the empirical correlation of two independent Gaussian processes in two cases: pure diffusion and mean-reverting diffusion. Included are an explicit formula for the variance in the former in discrete time, and some convergence theorems and numerical results in the long-time horizon and in-fill-asymptotics regimes.

This empirical correlation $\rho_n$, defined for two related series of data of length $n$ using the standard Pearson correlation statistic which is appropriate for i.i.d. data with two moments, is known as Yule's "nonsense correlation" in honor of the statistician G. Udny Yule. He described in 1926 the phenomenon by which random walks and other time series are not appropriate for use in this statistic to gauge independence of data series. He observed empirically that its distribution is not concentrated around 0 but diffuse over the entire interval (−1,1). This well-documented effect was roundly ignored by many scientists over the decades, up to the present day, even sparking recent controversies in important areas like climate-change attribution. Since the 1960s, probability theorists wanted to close any possible ambiguity about the issue by computing the variance of the continuous-time version $\rho$ of Yule's nonsense correlation, based on the paths of two independent Brownian motions. This problem eluded the best minds until it was finally closed by Philip Ernst and two co-authors 90 years after Yule's observation, in a paper published in 2017 in the Annals of Statistics.

The more practical question of what happens with $\rho_n$ in discrete time remained. We address it here by computing its moments in the case of Gaussian data, the second moment being explicit, and by estimating the speed of convergence of the second moment of $\rho - \rho_n$, which we find tends to zero at the rate $1/n^2$. The latter is an important result in practice since it could help justify using statistical properties of $\rho$ when devising tests for pairs of time series of moderate length. We also investigate what remains of the diffuse behavior of $\rho$ and $\rho_n$ in long-time asymptotics. The asymptotic self-similarity of pure random walks means that the distribution of $\rho$ is insensitive to the time scale in the pure-diffusion case, but this is far from being true in mean-reverting cases, as we show when the two processes are Ornstein-Uhlenbeck OU processes observed in discrete or continuous time. In that case, $\rho$ concentrates as time $T$ increases, and has Gaussian fluctuations: the asymptotic variance of $T^{1/2} \rho$ is the inverse of the rate of mean reversion, and we establish a Berry-Esseen-type result for the speed of this normal convergence in Kolmogorov distance, modulo a log correction. We prove that these results for the OU processes also hold for discete-observation case under sufficiently high-frequency assumptions.

In this presentation, we provide ideas of the tools used to prove these results, which come from two seemingly orthogonal directions: algebraically tractable trivariate moment-generating functions for the three components of $\rho$, leading to integro-differential representation formulas for the moments of $\rho$; and applications of the connection between Analysis on Wiener space and Stein's method to access the Kolmogorov distance between $\rho$ and a normal law. Time permitting, we will attempt to explain why these two apparently dissimilar mathematical methodologies are intimately connected because the three components of $\rho$ belong to the so-called second Wiener chaos, which has a remarkable Hilbert-space structure. We conjecture that the speed $1/n^2$ which we found for the convergence of the variance of $\rho$ in in-fill asymptotics only applies because of the random-walk structure (independence of increments), while for other types of time series, such as mean-reverting ones, the speed increases to $1/n$; this would be consistent with our Berry-Esseen result.

This work is partially supported by the US National Science Foundation award DMS-1811779, the Office of Naval Research award N00014-18-1-2192, and a US Fulbright Dissertation Scholarship. It is joint work with Soukaina Douissi (Cadi Ayyad University, Marrakech, Morocco), Philip Ernst and Dongzhou Huang (Rice University, Houston, TX, USA), and Khalifa es-Sebaiy (Kuwait University, Kuwait).

Zoom access: Contact mbrown2@math.tulane.ed
Time: 3:30pm

### Wednesday, October 13

AMS/AWM

Topic: An Illustration of Energy Methods

Kyle K. Zhao | Tulane University

Abstract: Energy methods is an important tool in the study of qualitative properties of partial differential equations. Charlie Doering (Professor of Mathematics and Professor of Physics, University of Michigan - Ann Arbor) once said: "Give me integration by parts and Cauchy-Schwarz inequality, I can conquer the world”. This talk will give an illustration of how these elementary tools could be utilized to prove non-trivial mathematical problems arising in applied science.

Location: Gibson Hall 400A
Zoom Access:  N/A
Time
: 4:15

### Wednesday, October 13

Algebra and Combinatorics

Topic: How do modular forms appear from the parametrization of cubic curves

Olivia Beckwith | Tulane University

Abstract: For Part 2 of my introduction to my research, I will focus on my other favorite area: quadratic number fields. First I will define a class of real-analytic modular forms. Then I will show how they can be used in the study of class numbers of imaginary quadratic number fields, as well as Hecke series for real quadratic number fields. This includes joint ongoing work with Gene Kopp.

Location: GI-310

Time: 3:00

### Wednesday, October 13

A working seminar on Modular Form

Topic: The First Appearance of Modular Forms

Victor H. Moll | Tulane University

Abstract: TBA

Location: Gibson 310
Zoom access:
Time:
2:00

### Tuesday, October 12

Topic: On the Empirical Spectral Distribution for Random Matrices with Independent Rows

Oliver Orejola | Tulane University

Abstract: Empirical Spectral Distributions (ESDs) are a central object of study in random matrix theory. Often, we are interested in the eigenvalue behavior of matrices.  This talk will introduce modern results concerning random matrices with independent rows. We begin with classic results including Wigner's Semi Circle theorem and Marchenko-Pastur theorem. Then, we depart from $i.i.d$ entries and explore some recent results concerning independent rows.

Location: Gibson 126A
Zoom access:
Time:
5:00pm

### Monday, October 11

Algebraic Geometry and Geometric Topology Seminar

Topic: Vector bundles and affine Nash groups

Mahir Can | Tulane University

Abstract: In this talk, we will make a gentle introduction to the theory of (the vector bundles on affine) Nash manifolds. We will introduce a special family of affine Nash groups. Then we will announce a classification theorem related to these new Nash groups.

Location: Gibson Hall 325
Zoom access:
Time
: 3:00

Week of October 8 - October 4

### Friday, October 8

Applied and Computational Mathematics

Topic: Non-conservative H^{1/2-} weak solutions of the incompressible 3D Euler equations

Matthew Novack | Institute for Advanced Study

Abstract: We will discuss the motivation and techniques behind a recent construction of non-conservative weak solutions to the 3D incompressible Euler equations on the periodic box. The most important feature of this construction is that for any positive regularity parameter β < 1/2, it produces infinitely many solutions which lie in C^0_t H^β_x . In particular, these solutions have an L^2-based regularity index strictly larger than 1/3, thus deviating from the scaling of the Kolmogorov-Obhukov 5/3 power spectrum in the inertial range.

This is joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol.

Zoom access:
Time: 2:00

### Wednesday, October 6

AMS/AWM

Topic: An Introduction to Homogeneous Spaces

Mahir Can | Tulane University

Abstract: In this talk we will explain how two important fields of mathematics, combinatorics, and algebraic geometry, meet. The interplay between the ideas of these two fields lead to remarkable results in representation theory.

Location: Gibson Hall 126A
Zoom Access:
Time: 4:15

### Wednesday, October 6

Algebra and Combinatorics

Topic: Modular forms and divisibility properties of partition numbers

Olivia Beckwith | Tulane University

Abstract: My research focuses on elliptic modular forms and their connections to different areas of number theory. Two of my favorite areas are the study of integer partitions and quadratic number fields. For Part 1 of this series, I will start with a brief crash course defining modular forms. Then I will describe some of my work studying the divisibility properties of numbers which count integer partitions. This includes joint work with Scott Ahlgren and Martin Raum, and may briefly mention ongoing work with Jack Chen, Maddie Diluia, Oscar Gonzales, and Jamie Su.

Location: GI-310

Time: 3:00

### Wednesday, October 6

A working seminar on Modular Form

Topic: Modular forms as coefficients of the P-function

Victor H. Moll | Tulane University

Abstract: The study of polynomial equations is one of the basic problems in Number Theory. In this first talk we will show that “modular forms”, the subject of this seminar, appear in a natural manner in the study of the curve “quadratic in y = cubic in x”. The details are completely elementary.

Location: Gibson 310
Zoom access:
Time
: 2:00

### Tuesday, October 5

Topic: A naive introduction to affine schemes

Nestor F. Diaz Morera | Tulane University

Abstract: I will be sailing over some algebraic sea such that at some point I reach a piece of geometric land. Hopefully, the (Zariski) topological moon will help me out.

Location: Gibson 126A
Zoom access:
Time: 3:00

### Monday, October 4

Algebraic Geometry and Geometric Topology Seminar

Topic: Rigidity theory for Gaussian graphical models

Daniel Bernstein | Tulane University

Abstract: Many modern biological applications require one to fit a statistical model with many parameters to a dataset with relatively few points. This begs the question: for a given model, what is the fewest number of data points needed in order to fit? In this talk, I will discuss this question for the class of Gaussian graphical models, highlighting connections to discrete geometry, convex geometry, classical combinatorics, and rigidity theory.

Location: Gibson Hall 325
Zoom access:
Time: 3:00

Week of October 1 - September 27

### Friday, October 1

Applied and Computational Mathematics

Topic: Parameter Estimation in an SPDE Model for Cell Repolarisation

Josef Janák | University of Potsdam and Humboldt University

Abstract: We propose a stochastic Meinhardt model for cell repolarisation and study how parameter estimation techniques developed for simple linear SPDE models apply in this situation. We pursue estimation of the diffusion term based on continuous time observations which are localised in space. We show asymptotic normality for our estimator as the space resolution becomes finer. We demonstrate the performance of the model and the estimator in numerical and real data experiments.

Zoom access:
Time: 2:00

### Wednesday, September 29

A working seminar on Modular Form

Topic: How do modular forms appear from the parametrization of cubic curves

Victor H. Moll | Tulane University

Abstract: The study of polynomial equations is one of the basic problems in Number Theory. In this first talk we will show that “modular forms”, the subject of this seminar, appear in a natural manner in the study of the curve “quadratic in y = cubic in x”. The details are completely elementary.

Location: Gibson 310
Zoom access:
Time: 2:00

### Tuesday, September 28

Topic: An Introduction to Stokes Flow

Kendall Gibson | Tulane University

Abstract: This talk will be an introduction to incompressible Stokes flow. First, we will get a general idea of some of the properties of Stokes flow and when its applicable. Then we will look at how to solve these equations.

Zoom access:
Time: 5:00

### Monday, September 27

Algebraic Geometry and Geometric Topology Seminar

Topic: Vietoris-Rips thickenings of spheres

Abstract: If a dataset is sampled from a manifold, then as more and more samples are drawn, the persistent homology of the Vietoris-Rips complexes of the dataset converges to the persistent homology of the Vietoris-Rips complexes of the manifold. But little is known about Vietoris-Rips complexes of manifolds. An exception is the case of the circle: as the scale parameter increases, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, ..., until finally they are contractible. The Vietoris-Rips thickenings of the n-sphere first obtain the homotopy type of the n-sphere, and then next the (n+1)-fold suspension of a (topological) quotient of the special orthogonal group SO(n+1) by an alternating group A_{n+2}. Not much is known at later scales, even though (as we will explain) these homotopy types have applications for generalizations of the Borsuk-Ulam theorem, for projective codes (packings in projective space), and (conjecturally) for Gromov-Hausdorff distances between spheres. This is joint work with Michal Adamaszek, Johnathan Bush, and Florian Frick.

Zoom access:
Time: 3:00 CT

Week of September 24 - September 20

### Wednesday, September 22

Algebra and Combinatorics

Topic: Group Algebras of Compact Groups and Enveloping Algebras of Profinite-Dimensional Lie Algebras

Karl H. Hofmann | Tulane University

Abstract:

Zoom access:
Time: 3:00

### Monday, September 20

Algebraic Geometry and Geometric Topology Seminar

Topic: Integrals, trees, and spaces of pure braids and string links

Robin M Koytcheff | TBA

Abstract: The based loop space of configurations in a Euclidean space R^n can be viewed as the space of pure braids in R^{n+1}.  In joint work with Komendarczyk and Volic, we studied its real cohomology using an integration map from a certain graph complex and recovered a result of Cohen and Gitler.  Specifically, the map we studied is a composition of Kontsevich’s formality integrals and Chen’s iterated integrals.  We showed that it is compatible with Bott-Taubes integrals for spaces of 1-dimensional string links in R^{n+1}.  As a corollary, the inclusion of pure braids into string links in R^{n+1} induces a surjection in cohomology for any n>2.  More recently, we showed that the dual to the integration map embeds the homotopy groups of the space of pure braids into a space of trivalent trees.  We also showed that a certain subspace of these homotopy groups injects into the homotopy groups of spaces of k-dimensional string links in R^{n+k} for many values of n and k.

Zoom access:
Time: 3:00  CT

Week of September 17 - September 13

### Friday, September 17

Applied and Computational Mathematics

Topic: Almost-Periodic Schr\"odinger Operators with Thin Spectra

Jake Fillman | Texas Tech

Abstract: The determination of the spectrum of a Schr\"odinger operator is a fundamental problem in mathematical quantum mechanics. We will discuss a series of results showing that almost-periodic Schr\"odinger operators can exhibit spectra that are remarkably thin in the sense of Lebesgue measure and fractal dimensions: the spectrum can be a Cantor set of zero Lebesgue measure and zero Hausdorff dimension.  [joint work with D. Damanik, A. Gorodetski, and M. Lukic]

Zoom access:
Time: 2:00

### Thursday, September 16

Colloquium

Topic: Our Place Among the Infinities

Bill Taber - JPL (Host: Glatt-Holtz)

Abstract: We can see the planets and smaller bodies of the solar system with earth bound telescopes, but telescopes cannot answer the big questions.  How do these bodies “work?”  What is their chemistry, their dynamics, their evolution? Where is there water in the solar system?  Is there now or has there ever been life anywhere but Earth.  To answer the big questions, we cannot do so from the comfort of Earth; we have to go there.  To go there requires machines that did not exist 100 years ago: rockets, ultra-stable oscillators,  deep space communication antennae, computers, etc.  But even more than these machines, it requires mathematics: mathematic to design trajectories from earth to distant bodies; mathematics to navigate the trajectories, mathematics to control the flight of spacecrafts, mathematics to communicate with spacecraft, and mathematics to arrive safely.   This talk will sketch out in broad strokes the mathematics of deep space exploration and how it can help us to know our place among the infinities.

Bill Taber is group supervisor of the Mission Design and Navigation Software Group at NASA’s Jet Propulsion Laboratory in Pasadena, California where he has been since 1983.  He holds the degrees of Masters in Business Administration from the Peter Drucker School of Management of the Claremont Graduate University at Claremont, California, a Ph. D. in Mathematics  M.S. in Mathematic from the University of Illinois at Urbana-Champagne, Illinois, and a B.A. in Mathematics from Eastern Illinois University at Charleston, Illinois.

Zoom access: Contact mbrown2@math.tulane.ed
Time: 3:30pm

### Tuesday, September 14

Topic: Zeros of Orthogonal Polynomials

Victor Bankston | Tulane University

Abstract: We will plot the Krawtchouk polynomials to illustrate the interleaving of zeros of orthogonal polynomials.

Zoom access:
Time: 5:00

Week of September 3 - August 30

### Friday, September 3

Applied and Computational Mathematics

Topic: TBA

Beskos ? | TBA

Abstract: TBA

Zoom access:
Time: 3:30

### Thursday, September 2

Colloquium

Topic: Random matrix theory for high-dimensional time series

Alexander Aue - UC Davis (Host: Didier, Gustavo)

Abstract: This talk is concerned with extensions of the classical Marcenko–Pastur law to time series. Specifically, p-dimensional linear processes are considered which are built from innovation vectors with independent, identically distributed entries possessing zero mean, unit variance and finite fourth moments. Under suitable assumptions on the coefficient matrices of the linear process, the limiting behavior of the empirical spectral distribution of both sample covariance and symmetrized sample autocovariance matrices is determined in the high-dimensional setting for which dimension p and sample size n diverge to infinity at the same rate, enabling the use of results from random matrix theory. The presented theory extends existing contributions available in the literature for the covariance case and is one of the first of its kind for the autocovariance case. Several applications are discussed to highlight the potential usefulness of the results. The talk is based on joint work with Haoyang Liu (New York Fed) and Debashis Paul (UC Davis).