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Events This Week

Week of December 17 - December 13

Tuesday, December 14

Applied and Computational Mathematics

Topic: TBA

Valerio Lucarini | TBA

Abstract: TBA

Location:
Zoom access:
Time: TBA

 

Week of December 10 - December 6

Wednesday, December 8

Algebra and Combinatorics

Topic: Reeb graph metrics from the ground up

Erin Wolf Chambers | Saint Louis University

Abstract: The Reeb graph has been utilized in various applications including the analysis of scalar fields. Recently, research has been focused on using topological signatures such as the Reeb graph to compare multiple scalar fields by defining distance metrics on the topological signatures themselves. In this talk, we will introduce and study five existing metrics that have been defined on Reeb graphs: the bottleneck distance, the interleaving distance, functional distortion distance, the Reeb graph edit distance, and the universal edit distance. This talk covers material from a recent survey paper, which has multiple contributions: (1) provide definitions and concrete examples of these distances in order to develop the intuition of the reader, (2) visit previously proven results of stability, universality, and discriminativity, (3) identify and complete any remaining properties which have only been proven (or disproven) for a subset of these metrics, (4) expand the taxonomy of the bottleneck distance to better distinguish between variations which have been commonly miscited, and (5) reconcile the various definitions and requirements on the underlying spaces for these metrics to be defined and properties to be proven.

Location: GI-310
Time: 3:30pm

 

Week of December 3 - November 29

Wednesday, December 1

A working seminar on Modular Form

Topic: Algebraic relations between modular functions

Armin Straub  | Armin Straub, University of Southern Alabama

Abstract: We will continue the discussion of modular functions and show that any two are algebraically related.  This leads to the observation that modular functions like the j-function take algebraic values at quadratic irrationalities.  This is the tip of an impressive iceberg, and hints at the beginning of topics including complex multiplication and class field theory.

Location: GI-310
Time: 3:00pm

Tuesday, November 30

Graduate Student Colloquium

Topic: An Introduction to Quantum Computing

Zachary Bradshaw | Tulane University

Abstract: As computer components gets smaller, the effects of quantum physics must be considered, and Heisenberg's uncertainty principle presents us with an obstacle. In the past fifty years, a new regime for computing built on the principles of quantum mechanics has risen. This regime, known as quantum computing, is fundamentally different from its classical computing counterpart. Rather than operating on bits of information, it operates on a quantum state known as a qubit. The state of the qubit lives in a two-dimensional Hilbert space, allowing us to consider linear combinations of the two basis states. In contrast, a bit has only two possible states. This feature of the qubit, along with quantum mechanical properties like entanglement, provide us with a powerful computational framework, which was put on display by Peter Shor when he famously constructed his algorithm for efficiently factoring integers, effectively making the common RSA encryption obsolete if a large enough quantum computer can be built. In this talk, I will give a short introduction to the rising field of quantum computation. If time permits, we will see how a quantum state can be teleported.

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

Monday, November 29

Joint Algebraic Geometry and Geometric Topology Seminar

Topic: Homology representations of compactified configurations on graphs

Claudia Yun | Brown University

Abstract: The n-th ordered configuration space of a graph parametrizes ways of placing distinct and labeled particles on that graph. The homology of the one-point compactification of such configuration space is equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. We give a cellular decomposition of these configuration spaces on which the actions are realized cellularly and thus construct an efficient free resolution for their homology representations. As our main application, we obtain computer calculations of the top weight rational cohomology of the moduli spaces M_{2,n}, equivalently the rational homology of the tropical moduli spaces Delta_{2,n}, as a representation of S_n acting by permuting point labels for all n\leq 10. This is joint work with Christin Bibby, Melody Chan, and Nir Gadish. Our paper can be found on arXiv with ID.

Location: Gibson Hall 325
Time: 3:00pm

 

Week of November 19 - November 15

Friday, November 19

Applied and Computational Mathematics

Topic: Numerical approximation of statistical solutions of hyperbolic systems of conservation laws

Franziska Weber | Carnegie Mellon University

Abstract: Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global solutions for multi-dimensional hyperbolic systems of conservation laws. We present a numerical algorithm to approximate statistical solutions of conservation laws and show that under the assumption of ‘weak statistical scaling’, which is inspired by Kolmogorov’s 1941 turbulence theory, the approximations converge in an appropriate topology to statistical solutions. We will show numerical experiments which indicate that the assumption might hold true.

Join us:
Zoom access:
Time:
2:00

Thursday, November 18

Colloquium

Topic: Mock Theta Functions: Yesterday and Today

Frank Garvan - University of Florida (Host: Beckwith)

Abstract: We discuss historical and modern developments of Ramanujan's mock theta functions. This includes a cast of characters. We present a number of recent discoveries and identities found through computer experiments.

Join us:
Zoom access:
Contact mbrown2@math.tulane.edu
Time: 3:30pm

Wednesday, November 17

AMS/AWM

Topic: Landen transformations

Victor Moll | Tulane University

Abstract: The evaluation of the length of an ellipse was one of the interesting problems of the $19^{th}$ century. Landen developed some formulas that lead to the efficient numerical evaluation of these integrals. The talk will include some modern approaches and extensions to this problem.

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

Wednesday, November 17

Algebra and Combinatorics

Topic: Newton-Okounkov Bodies, Rees Algebra, and Analytic Spread of Graded Systems of Monomial Ideals

Thai Nguyen | Tulane University

Abstract: Newton-Okounkov bodies are convex sets associated to algebro-geometric objects, that was first introduced by Okounkov in order to show the log-concavity of the degrees of algebraic varieties. In special cases, Newton-Okounkov bodies associated to graded systems of ordinary powers and symbolic powers of a monomial ideal are Newton polyhedron and symbolic polyhedron of the ideal. Studying these polyhedra can be beneficial to the study of relation between ordinary powers, integral closure powers and symbolic powers of a monomial ideal as well as its algebraic invariants. In this talk, I will survey some known results in this subject and present our results on computing and bounding the analytic spread of a graded system of monomial ideals and some related invariants through the associated Newton-Okounkov body. This is joint work with Tài Huy Hà.

Location: GI-310
Time: 3:00pm

Wednesday, November 17

A working seminar on Modular Form

Topic: Modular functions and the inevitable j-function

Armin Straub  | Tulane University

Abstract: We will review previous results on modular forms and the j-function, with the goal of (re)introducing the j-function as inevitable and natural.  We will discuss some of its properties as well as its fundamental role as a modular function, and might indicate extensions to the higher level case.

Location: GI-310
Time: 3:00pm

Tuesday, November 16

Graduate Student Colloquium

Topic: Synchronization in nature and mathematics

Dana Ferranti | Tulane University

Abstract: The emergence of synchronization is a common theme in nature at a variety of scales. The opening of the London Millennium Bridge is a fascinating example of how crowd synchronization can have profound consequences in engineering. At much smaller scales, we observe microorganisms coordinating the beating of their flagella or cilia. The reason for this synchronization is unclear and in recent years there have been a number of studies probing for mechanisms underlying the phenomenon. This talk will discuss these ideas in an accessible way and present ongoing work on a minimal model that predicts a hydrodynamic mechanism behind synchronization for ciliates and flagellated organisms.

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

Monday, November 15

Joint Algebraic Geometry and Geometric Topology Seminar

Topic: Quantitative conditions for right-handedness

Umberto Hryniewicz | RWTH Aachen University

Abstract: We present numeric conditions for a dynamically convex Reeb flow on the 3-sphere, in the sense of Hofer-Wysocki-Zehnder, to be right-handed, in the sense of Ghys. Once right-handedness is checked, the following interesting conclusions about the dynamics can be deduced: (a) every link of periodic orbits is a fibered link, and (b) every finite collection of periodic orbits spans a global surface of section for the flow. As an application, we find an explicit pinching constant 0 < d < 0.7225 such that if a Riemannian metric on the 2-sphere is pinched by at least d then its geodesic flow lifts to a right-handed flow on the 3-sphere. This is joint work with Anna Florio.

Location: Gibson Hall 325
Time: 3:00pm

 

Week of November 12 - November 8

Friday, November 12

Applied and Computational Mathematics

Topic: Mean field analysis and scaling limits for neural networks: typical events and fluctuations.

Konstantinos Spiliopoulos | Boston University

Abstract: Machine learning, and in particular neural network models, have revolutionized fields such as image, text, and speech recognition. Today, many important real-world applications in these areas are driven by neural networks. There are also growing applications in finance, engineering, robotics, and medicine. Despite their immense success in practice, there is limited mathematical understanding of neural networks. Our work shows how neural networks can be studied via stochastic analysis and develops approaches for addressing some of the technical challenges which arise. We analyze both multi-layer and one-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations. In the case of single layer neural networks, we rigorously prove that the empirical distribution of the neural network parameters converges to the solution of a nonlinear partial differential equation. In addition, we rigorously prove a central limit theorem, which describes the neural network's fluctuations around its mean- field limit. The fluctuations have a Gaussian distribution and satisfy a stochastic partial differential equation. For multilayer neural networks we rigorously derive the limiting behavior of the neural networks output. We also prove convergence to the global minimum under appropriate conditions. We demonstrate the theoretical results in the study of the evolution of parameters in the well known MNIST and CIFAR10 data sets.

Join us:
Zoom access:
Time:
2:00

Wednesday, November 10

Algebra and Combinatorics

Topic: Completions of degenerations of toric varieties.

Netanel Friedenberg | Tulane University

Abstract: After reviewing classical results about existence of completions of varieties, I will talk about a class of degenerations of toric varieties which have a combinatorial classification - normal toric varieties over rank one valuation rings. I will then discuss recent results about the existence of equivariant completions of such degenerations. In particular, I will show a new result about the existence of normal equivariant completions of these degenerations. Prior knowledge of toric varieties will not be necessary for understanding this talk.

Location: GI-310
Time: 3:00pm

Wednesday, November 10

A working seminar on Modular Form

Topic: The j-function

Victor Moll | Tulane University

Abstract: We give a description of this important function and explain its role in connection to modular functions.

Location: GI-310
Zoom access: https://tulane.zoom.us/j/94370467366 
Time: 2:00pm

Tuesday, November 9

Graduate Student Colloquium

Topic: Arithmetic properties of the sum of divisors

Vaishavi Sharma | Tulane University

Abstract: Given a prime p, the p-adic valuation of a number x is defined as the highest power of p that divides x. In this talk, I will discuss the p-adic valuation of the sum of divisors of an integer n. I will present a closed-form formula for νp(σ(n)) and discuss some upper bounds. This is joint work with Tewodros Amdeberhan, Victor Moll and Diego Villamizar.

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

Monday, November 8

Algebraic Geometry and Geometric Topology Seminar

Topic: A Classification of One-dimensional Nash Supergroups.

Mahir Can | Tulane University

Abstract: In this second part of our seminar on semialgebraic geometry, we will continue to explain our new categories related to Nash manifolds. In particular, in this talk we will present our classification theorem of one-dimensional Nash supergroups.

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

 

Week of November 5 - November 1

Thursday, November 4

Colloquium

Topic: Stochastic analysis and geometric functional inequalities

Maria Gordina - UConn

Abstract: We will survey different methods of proving functional inequalities for hypoelliptic  diffusions and the corresponding heat kernels. Some of these methods rely on geometric methods such as curvature-dimension inequalities (due to Baudoin-Garofalo), and some are probabilistic such as couplings or Lyapunov function techniques. One of the recent applications is to ergodicity for Langevin dynamics. This is based on joint work with F. Baudoin, B. Driver, D. Herzog, T. Melcher, Ph. Mariano et al.

Join us:
Zoom access:
Contact mbrown2@math.tulane.edu
Time: 3:30pm

Wednesday, November 3

AMS/AWM

Topic:  Linking and knotting in vector fields

Rafal Komendarczyk | Tulane University

Abstract: I will talk about knots and links, their invariants and how they can be applied to problems in fluid dynamics and magnetic relaxation.

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

Wednesday, November 3

Algebra and Combinatorics

Topic: Saturation bounds for smooth varieties.

Tài Huy Hà | Tulane University

Abstract: Let X be a smooth variety which is ideal-theoretically defined by an ideal J. We discuss linear bounds for the saturation degrees of powers of J in terms of its generating degrees. Our work extends a classical result of Macaulay, and fills the gap between studies on algebraic and geometric notions of the Castelnuovo-Mumford regularities of smooth varieties. This is a joint work with L. Ein and R. Lazarsfeld.

Location: GI-310
Time: 3:00

Wednesday, November 3

A working seminar on Modular Form

Topic: Modular forms on congruence subgroups

Olivia Beckwith  | Tulane University

Abstract: Last week we defined modular forms on the special linear group SL_2(Z). This time we'll replace SL_2(Z) with certain finite index subgroups. We'll also show how to apply these ideas to obtain a formula for the number of ways an integer can be written as a sum of four squares.

Join us: Gibson Hall 310
Zoom access: https://tulane.zoom.us/j/94370467366 
Time: 2:00

Tuesday, November 2

Graduate Student Colloquium

Topic: An Introduction to the Immersed Boundary Method

Sang-Eun Lee | Tulane University

Abstract: The immersed boundary (IB) method is a numerical method to express fluid dynamical phenomena. The IB method was devised firstly by Charles Peskin in 1972 for simulating the blood flow inside a heart of a patient who has heart failure and is currently developing by himself with his academic descendants. Although the IB method cannot simulate electromagnetic flows such as neurotransmitters through synapses, the method uses a complete form of incompressible Navier-Stokes Equations. The classical IB method uses an elastic fiber immersed in a background fluid. The location and the velocity of the fiber are the Lagrangian coordinate, so as to use the Eulerian coordinate, the Navier-Stokes equation takes the Lagrangian coordinate place to the Eulerian coordinate by utilizing the Dirac-Delta force function. In this talk, we discuss the classical IB method and its application to biological fluid dynamics.

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

Monday, November 1

Algebraic Geometry and Geometric Topology Seminar

Topic: Geometric Approaches on Persistent Homology

Baris Coskunuzer | University of Texas, Dallas

Abstract: Persistent Homology is one of the most important techniques used in Topological Data Analysis. In this talk, after giving a short introduction to the subject, we study the persistent homology output via geometric topology tools. In particular, we give a geometric description of the term “persistence”. The talk will be non-technical, and accessible to graduate students. This is joint work with Henry Adams.

Location: TBA
Zoom access:
Time:
3:00pm

 

Week of October 29 - October 25

Wednesday, October 27

AMS/AWM

Topic: Statistical Challenges in the Analysis of Methylation Profiles

Michelle Lacey | Tulane University

Abstract: Cytosine methylation is a fundamental epigenetic process that regulates gene expression. Variation in cytosine methylation at CpG dinucleotides is often observed in genomic regions, and a common objective in epigenetic analysis is the detection of differentially methylated sites or regions that may be associated with variation in gene expression related to disease or other factors such as environmental exposure. For sequencing-based methods, this often involves comparing two groups of samples that contain counts of methylated and unmethylated reads, and classical approaches such as logistic regression are typically employed to identify statistically significant differences. However, these algorithms largely ignore sources of both technical and biological bias and variability that violate key statistical assumptions, consequentially producing unreliable results. In particular, our research demonstrates that methylation patterns in individual molecules exhibit strong evidence of local spatial dependence, and further correlation is induced by variability in read coverage in sequencing experiments. This talk will discuss past and present efforts to develop improved statistical models for methylation sequencing data.

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

Wednesday, October 27

Algebra and Combinatorics

Topic: Algebraic matroids in rigidity theory and matrix completion

Daniel Bernstein | Tulane University

Abstract: A set of quantities is algebraically independent over a field F if they satisfy no polynomial equations with coefficients in F. Matroids are a combinatorial generalization of (algebraic) independence. In this talk, I will give an introduction to matroids from an algebraic perspective, and explain how they arise in rigidity theory and matrix completion. Time permitting, I will introduce tropical geometry and discuss how it can be used to understand particular algebraic matroids.

Location: GI-310
Time: 3:00

Wednesday, October 27

A working seminar on Modular Form

Topic: Level One Modular Forms

Vaishavi Sharma | Tulane University

Abstract: In this meeting we'll show that the Eisenstein series G_k  studied in the previous lectures live in finite-dimensional vector spaces of functions on the upper half-plane with certain transformation properties. We will also use basic properties of these spaces to prove a surprising identity for the divisor sum functions sigma_k.

Join us: Gibson Hall 310
Zoom access:
Time:
2:00

Tuesday, October 26

Graduate Student Colloquium

Topic: Combinatorial Nullstellensatz

Thai Nguyen | Tulane University

Abstract: Nullstellensatz is “the theorem of zero-set” in German. While Hilbert’s Nullstellensatz is the foundation of algebraic geometry, the Combinatorial Nullstenllensatz, introduced by Noga Alon in 1999, has seen many dramatic applications in additive number theorey, extremal graph theory and combinatorics. I will talk about some simple applications of the Combinatorial Nullstenllensatz, including high school math problems and two classical theorems: Cauchy-Davenport theorem in additive combinatorics and Chevalley-Warning theorem in algebra.

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

Monday, October 25

Algebraic Geometry and Geometric Topology Seminar

Topic: On toric degenerations

Lara Bossinger | National Autonomous University of Mexico (UNAM)

Abstract: In this talk I will give an overview on different constructions of toric degenerations, in particular from valuations and from Gröbner theory. I will show how they are related. By the end of the talk we will explore a possible construction to extend the notion of moment polytope to not-necessarily toric varieties via toric degenerations.

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

 

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.

Join us:
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

Wednesday, October 20

AMS/AWM

Topic: 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. In this talk I will give a short introduction to modular forms, and then I will describe some of my work studying the divisibility properties of numbers which count integer partitions. This includes a discussion of joint work with Scott Ahlgren and Martin Raum, and ongoing joint work with Jack Chen, Maddie Diluia, Oscar Gonzales, and Jamie Su.

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

Wednesday, October 20

Algebra and Combinatorics

Topic: Toric ideals from statistics

Daniel Bernstein | Tulane University

Abstract: Toric ideals are prime polynomial ideals that are generated by monomial differences. They have a rich theory with connections to polyhedral combinatorics, optimization, and statistics, which I will discuss in this talk.

Location: GI-310
Time: 3:00

Wednesday, October 20

A working seminar on Modular Form

Topic: First examples of modular forms: Eisenstein series

Olivia Beckwith | Tulane University

Abstract: Picking right up where we left off last week, we'll examine the Taylor coefficients of the Weierstrass P function in more detail and we'll compute their Fourier expansions. Then we'll define vector spaces which contain these functions and we'll show that they are finite dimensional.

Join us:
Zoom access:
Time:
2:00

Tuesday, October 19

Graduate Student Colloquium

Topic: Hard Instances from Generalized Error Correcting Codes

Victor Bankston | Tulane University

Abstract: We relate the problem of finding hard instances of Independent Set to a problem of coding theory. We propose a method of solution based on a connection with quantum information and an accompanying generalization to the theory of error correcting codes.

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

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 310
Zoom access:
Time:
 2:00

 

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.

Join us:
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).

Join us:
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

Graduate Student Colloquium

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.

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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
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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
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Time
: 2:00

Tuesday, October 5

Graduate Student Colloquium

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
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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
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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.

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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.

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

Tuesday, September 28

Graduate Student Colloquium

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.

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Time: 5:00 

Monday, September 27

Algebraic Geometry and Geometric Topology Seminar

Topic: Vietoris-Rips thickenings of spheres

Henry Adams | Colorado State University

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.

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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: 

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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.

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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]

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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.

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

Tuesday, September 14

Graduate Student Colloquium

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.

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Time: 5:00

 

Week of September 3 - August 30

Friday, September 3

Applied and Computational Mathematics

Topic: TBA

Beskos ? | TBA

Abstract: TBA

Join us:
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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).

Join us:
Zoom access: Contact mbrown2@math.tulane.edu
Time: 3:30pm