Week of December 17 - December 13
Applied and Computational Mathematics
Topic: Dynamical Landscape and Multistability of the Earth’s Climate
Valerio Lucarini | University of Reading
Abstract: For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in the past our planet flipped between these two states, and possibly additional ones. Here, we explore the global stability properties of the system first by investigating the properties of the competing attractors and of the edge state situated in the basin boundary, and then by introducing random Gaussian perturbations that modulate the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attraction. In the weak noise limit, large deviation laws define the invariant measure, the statistics of escape times, and typical escape paths called instantons. Indeed, the system lives in an energy-like landscape with valleys and mountain ridges defined by the Graham's quasipotential. For low (high) values of the solar irradiance, the zero-noise limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first-order phase transition in the system. We then compare the results obtained using the theory of quasipotentials with what can be obtained using a bottom-up approach. Harnessing techniques from data science, specifically manifold learning, we characterize the data landscape to find climate states and basin boundaries within a fully agnostic and unsupervised framework. Both approaches show remarkable agreement, and reveal, apart from the well known warm and snowball earth states, a third intermediate, new stable state in one of the two climate models we consider. The combination of our approaches allows to identify how macroscopic, physical properties of the climate system - the role of the ocean heat transport and of the hydrological cycle - drastically change the topography of the dynamical landscape of Earth's climate. Finally, we show how considering more general classes of noise laws than the Gaussian one leads to a fundamental change in the framework given above. The framework we propose seems of general relevance for the study of complex multistable systems with multiple scales of motions.
Location: Newcomb Hall - NH-025
Zoom access:
Time: 2:00
Week of December 10 - December 6
Applied and Computational Mathematics
Topic: Intermittency in turbulence and the 3D Navier-Stokes regularity problem
Aseel Farhat | Florida State University
Abstract: We describe several aspects of an analytic/geometric framework for the three-dimensional Navier-Stokes regularity problem, which is directly inspired by the morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of three-dimensional turbulence. Among these, we present our proof that the scaling gap in the 3D Navier-Stokes regularity problem can be reduced by an algebraic factor within an appropriate functional setting incorporating the intermittency of the spatial regions of high vorticity.
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Time: 2:00
Colloquium
Topic: Random matrix theory for high-dimensional time series
Alex Aue - The Chair of the Statistics Department at UC Davis
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
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: 3:00pm
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: 2:00pm
Joint Algebraic Geometry and Geometric Topology Seminar
Topic: An optimal numerical algorithm for solving polynomial systems
Elise Walker | Texas A&M (over Zoom)
Abstract: Numerical homotopy continuation is a useful numerical algorithm for computing the solutions of a system of polynomial equations. Such solution sets are sometimes known as varieties. Homotopies compute varieties by tracking paths from the solutions of a similar, pre-solved system. Generally, homotopies may track extraneous paths, which wastes computational resources. A homotopy is optimal if paths are smooth and there are no extraneous paths. Embedded toric degenerations are one source for optimal homotopy algorithms. In particular, if a variety has a toric degeneration, then there is an optimal homotopy for computing linear sections of that variety. There is a toric degeneration for any variety which has an associated finite Khovanskii basis. This work provides the appropriate embeddings for the Khovanskii toric degeneration and gives the corresponding optimal homotopy algorithm for computing a linear section of the variety. This is joint work with Michael Burr (Clemson University) and Frank Sottile (Texas A&M University).
Location: Gibson Hall 325
Time: 3:00pm
Week of December 3 - November 29
Applied and Computational Mathematics
Topic: Invariance of the Gibbs measures for the periodic generalized KdV equations
Andreia Chapouto | University of California, Los Angeles
Abstract: In this talk, we consider the periodic generalized Korteweg-de Vries equations (gKdV). In particular, we study gKdV with the Gibbs measure initial data. The main difficulty lies in constructing local-in-time dynamics in the support of the measure. Since gKdV is analytically ill-posed in the L2-based Sobolev support, we instead prove deterministic local well-posedness in some Fourier-Lebesgue spaces containing the support of the Gibbs measure. New key ingredients are bilinear and trilinear Strichartz estimates adapted to the Fourier-Lebesgue setting. Once we construct local-in-time dynamics, we apply Bourgain's invariant measure argument to prove almost sure global well-posedness of the defocusing gKdV and invariance of the Gibbs measure. Our result completes the program initiated by Bourgain (1994) on the invariance of the Gibbs measures for periodic gKdV equations.
This talk is based on joint work with Nobu Kishimoto (RIMS, University of Kyoto).
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Zoom access:
Time: 2:00
Colloquium
Topic: Gravitational Collapse for Newtonian Stars
Juhi Jang - USC
Abstract: In astrophysical fluid dynamics, stars are considered as isolated fluid masses subject to self-gravity. A classical model to describe the dynamics of Newtonian stars is given by the gravitational Euler-Poisson system, which admits a wide range of star solutions that are in equilibrium or expand for all time or collapse in a finite time or rotate. In particular, using numerics, the Euler-Poisson system in the super-critical regime has been widely used in astrophysics literature to describe the gravitational collapse, but its rigorous proof has been established only recently. The main challenge comes from the pressure, which acts against gravitational force. In this talk, I will discuss some recent progress on Newtonian dust-like collapse and self-similar collapse.
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Zoom access: Contact mbrown2@math.tulane.edu
Time: 3:30pm
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
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
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.
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Time: 2:00
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.
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Zoom access: Contact mbrown2@math.tulane.edu
Time: 3:30pm
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
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
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
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
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
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.
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Zoom access:
Time: 2:00
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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Time: 2:00
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
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
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
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.
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Time: 2:00
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
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
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.
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Time: 2:00
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).
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Zoom access: Contact mbrown2@math.tulane.ed
Time: 3:30pm
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
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
A working seminar on Modular Form
Topic: The First Appearance of Modular Forms
Victor H. Moll | Tulane University
Abstract: TBA
Location: Gibson 310
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Time: 2:00
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
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Time: 5:00pm
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
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Time: 3:00
Week of October 8 - October 4
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
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
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
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
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
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
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
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
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Time: 2:00
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
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
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
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
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
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.
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Zoom access: Contact mbrown2@math.tulane.ed
Time: 3:30pm
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
Applied and Computational Mathematics
Topic: TBA
Beskos ? | TBA
Abstract: TBA
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Time: 3:30
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).
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Zoom access: Contact mbrown2@math.tulane.edu
Time: 3:30pm
