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Week of May 2 - April 28
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Thursday, May 1,
Honors Thesis Defense
Topic: Spectral Analysis of Multiscale Sample Covariance Matrices
Geng Chen - University of Kansas Host: (Scott McKinley)
Abstract: Random matrix theory, originally developed in the 1950s, provides a powerful framework for modeling high-dimensional systems, revealing universal behaviors with applications in physics, probability, and statistics. This thesis examines the spectral behavior of sample covariance matrices, fundamental objects in multivariate analysis, across both low and high dimensional regimes. We first revisit foundational results from classical probability and random matrix theory, including Gaussian and GOE-type asymptotics in low dimensions, and the emergence of the Marchenko-Pastur and Tracy-Widom laws in high dimensions. We then extend the analysis to multiscale sample covariance matrices, introducing a lag parameter to model variance structures across scales and to incorporate dependence through processes such as fractional Brownian motion. Our results show that dependence fundamentally reshapes the spectral behavior of sample covariance matrices: in certain regimes, the top eigenvalues become asymptotically independent and Gaussian; in others, they exhibit level repulsion characteristic of β-Hermite ensembles. Through a combination of theoretical results and Monte Carlo simulations, we characterize new universality classes driven by the interplay between dimensionality and dependence.
Location: Gibson 414
Time: 2:00 pm
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Wednesday, April 30
Probability & Statistics
Topic: Composite likelihood approaches to phylogenetic inference under the multispecies coalescent
Laura Kubatko - Ohio State University Host: (Xiang Ji)
Abstract: Species-level phylogenetic inference under the multispecies coalescent model remains challenging in the typical inference frameworks (e.g., the likelihood and Bayesian frameworks) due to the dimensionality of the space of both gene trees and species trees. Algebraic approaches intended to establish identifiability of species tree parameters have suggested computationally efficient inference procedures that have been widely used by empiricists and that have good theoretical properties, such as statistical consistency. However, such approaches are less powerful than approaches based on the full likelihood. Methods based on composite likelihood are a compromise between these two approaches that enable computationally efficient inference while maximizing use of the available sequence data. In this talk, I’ll describe the relationship between these two approaches, highlighting the strengths and weaknesses of each and providing directions for future work.
Location: Dinwiddie Hall 102
Time: 4:00 pm
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Wednesday, April 30
Algebra and Combinatorics
Topic: Integral Riemann-Roch for Q
Katia Consani - Johns Hopkins University
Abstract: I will present a joint result with A. Connes showing the existence of an integral, Riemann-Roch formula for the number field Q. At the root of this result is the characterization of the integers as a polynomial ring $\mathbb{F}_1[X]$, where the `variable' X fulfills the relation $1+1=X+X^2$.
Location: Gibson Hall 310
Time: 3:00 pm
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Tuesday, April 29
Graduate Student Colloquium
Topic: Order conditions for Rosenbrock methods with stage value
Speaker: Truc T Dang - Tulane University
Abstract: We first review the fundamentals of time integration and the classical Runge–Kutta methods, including their formulation and order conditions. We then explore the correspondence between Runge–Kutta order conditions and the Jacobian‐based stages of Rosenbrock integrators. In this talk, we are interested in deriving order conditions for Rosenbrock schemes: rather than relying on the traditional stage-derivative formulation, we employ stage values, introduce a general ansatz, and apply more than two Newton iterations within each stage.
Location: MA 101
Time: 3:30 PM
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Week of April 25 - April 21
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Monday, April 23
Probability & Statistics
Topic: MCMC Importance Sampling via Moreau-Yosida Envelopes
Eric Chi - Rice University Host: (Xiang Ji)
Abstract: Markov chain Monte Carlo (MCMC) is the workhorse computational algorithm employed for inference in Bayesian statistics. Gradient-based MCMC algorithms are known to yield faster converging Markov chains. In modern parsimonious models, the use of non-differentiable priors is fairly standard, yielding non-differentiable posteriors. Without differentiability, gradient-based MCMC algorithms cannot be employed effectively. Recently proposed proximal MCMC approaches, however, can partially remedy this limitation. These approaches employ the Moreau-Yosida (MY) envelope to smooth the nondifferentiable prior enabling sampling from an approximation to the target posterior. In this work, we leverage properties of the MY envelope to construct an importance sampling paradigm to correct for this approximation error. We establish asymptotic normality of the importance sampling estimators with an explicit expression for the asymptotic variance which we use to derive a practical metric of sampling efficiency. Numerical studies show that the proposed scheme can yield lower variance estimators compared to existing proximal MCMC alternatives.
Location: Dinwiddie Hall 102
Time: 4:00 pm
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Wednesday, April 23,
Applied and Computational Math
Topic: Central limit theorems in classical Hamiltonian dynamics
Alexander Moll - Reed College Host: (Aikaterini Gkogkou)
Abstract: In work with Robert Chang (Rhodes College), we derive certain central limit theorems in the context of classical Hamiltonian dynamical systems with random initial data. In this talk, I'll state our general CLT in the framework of Gaussian measures, then illustrate it in two concrete examples: (i) in the simple linear harmonic oscillator and (ii) in the nonlinear inviscid Burgers equation on the circle. This second example of our CLT implies a new relationship between fractional Brownian motions on the circle and certain Gaussian processes on the line found by Kerov in 1993.
Location: Dinwiddie (DW) Hall 102
Time: 3:00 pm
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Wednesday, April 23
Algebra and Combinatorics
Topic: A modular framework for generalized Hurwitz class numbers
Andreas Mono - Vanderbilt University Host: (Olivia Beckwith)
Abstract: We discover a neat linear relation between the mock modular generating functions of the level 1 and level N Hurwitz class numbers. This relation gives rise to a holomorphic modular form of weight 32 and level 4N for N > 1 odd and square-free. We extend this observation to a non-holomorphic framework and obtain a higher level analog of Zagier’s Eisenstein series as well as a preimage under the ξ-operator. All of these observations are deduced from a more general inspection of the weight 12 Maass–Eisenstein series of level 4N at its spectral point s = 34 . This idea goes back to Duke, Imamo¯glu and Tóth in level 4 and relies on the theory of so-called sesquiharmonic Maass forms. Furthermore, we connect the aforementioned results to a regularized Siegel theta lift as well as a regularized Kudla–Millson theta lift for odd prime levels, which builds on earlier work by Bruinier, Funke and Imamo¯glu.
This is joint work with Olivia Beckwith.
Location: Gibson Hall 310
Time: 3:00 pm
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Tuesday, April 22,
Colloquium
Topic: Resonances
Toan T Nguyen - Penn State University
Abstract: Of great interest is to establish the large time dynamical behavior of a fluid or plasma in a non-equilibrium state, whether transitioning to turbulence or relaxing to neutrality, a resolution of which requires a deep understanding of resonances (mainly between particles and waves). This talk aims to provide a roadmap towards resolving the problem, plus the current state of the art.
Location: Tilton Hall, Room 305
Time: 3:30 pm
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Tuesday, April 22
Algebraic Geometry
Topic: A Comparison of Two Supercharacter Theories
Speaker: Julianne Bilen Rainbolt - Saint Louis University Host: ( Mahir Bilen Can )
Abstract: In 2008, P. Diaconis and I.M. Isaacs introduced a generalization of classical character theory, which they called supercharacter theory. Let X be a partitioning of the irreducible characters of a finite group G such that the trivial character is in its own block of this partition. Let Y be a partitioning of the conjugacy classes of G such that identity element of G is in its own block of this partition. Define the supercharacters of G to be the sums of the irreducible characters in each block in X, weighted by their degree. Define the superclasses of G to be the unions of the elements in each block in Y. If the values of the supercharacters are constant on the superclasses, this is called a supercharacter theory of G. As a supercharacter is determined by these partitions, the structure of a group allows for multiple supercharacter theories. In this talk we will compare the construction of two different supercharacter theories, one based on the degrees of the irreducible characters of G and one based on the size of the conjugacy classes of G. In particular, we will demonstrate necessary and sufficient conditions that will force these two supercharacter theories to coincide when the groups considered are semidirect products of cyclic groups.
Location: Richardson Memorial - RM-304
Time: 11:00 AM
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Week of April 18 - April 14
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Week of April 11 - April 7
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Friday, April 11,
Applied and Computational Math
Topic: How fluid rheology shapes micororganism swimming gait
Robert Guy - UC Davis Host: (Lisa Fauci)
Abstract: Many microorganisms propel through complex media by deformations of their flagella. The beat is thought to emerge from interactions between forces of the surrounding fluid, passive elastic response from deformations of the flagellum, and active forces from internal molecular motors. The beat varies in response to changes in the fluid rheology, including elasticity, but there is limited data on how systematic changes in rheology alters the beat. In this talk we develop mathematical and computational models in order to understand observations from experimental data of the flagellar gait and swimming speed of the bi-flagellated alga Chlamydomonas reinhardtii in different fluids
In the first part of the talk we use shape mode analysis to quantify changes in the flagellar beat, and we use computational models to vary the stroke and fluid rheology independently. This analysis shows that changes in the mean shape are primarily responsible for the observed change in swimming speed and the variations of the beat around the mean change very little with fluid rheology. In the second part of the talk we examine how fluid elasticity affects the emergent frequency of the beat. We analyze the emergence of beating of an elastic planar filament driven by a follower force at the tip in a viscoelastic fluid. This analysis predicts that the frequency increases with the fluid relaxation time, and using numerical simulations, the model predictions are compared with experimental data. The model shows the same trends in response to changes in both fluid viscosity and fluid elasticity, and thus provides a possible mechanistic explanation of the experimental observations.
Location: Gibson Hall 325
Time: 3:00 pm
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Wednesday, April 9
AMS/AWM
Topic: The Surprising Utility of Noise in Understanding Chaos
Samuel Punshon-Smith - Tulane University
Abstract: Many physical and mathematical systems exhibit chaotic behavior, characterized by extreme sensitivity to initial conditions and seemingly unpredictable dynamics. Proving the existence of chaos, often quantified by positive Lyapunov exponents, can be notoriously difficult for purely deterministic models. This talk will explore how introducing noise, or stochasticity, into dynamical systems can paradoxically make the analysis more tractable. We will discuss key concepts from dynamical systems and ergodic theory, illustrating how random perturbations can help establish chaotic properties and provide insights into the long-term behavior of complex systems, with examples potentially drawn from areas like fluid dynamics.
Location: Gibson Hall 310
Time: 4:15 pm
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Wednesday, April 9
Algebra and Combinatorics
Topic: Tropical Methods in Motivic Enumerative Geometry
Andrés Jaramillo Puentes - University of Tübingen
Abstract: Over the complex numbers the solutions to enumerative problems are invariant: the number of solutions of a polynomial equation or polynomial system, the number of lines or curves in a surface, etc. Over the real numbers such invariance fails. However, the signed count of solutions may lead to numerical invariants: Descartes' rule of signs, Poincaré-Hopf theorem, real curve-counting invariants. Since many of this problems have a geometric nature, one may ask the same problems for arbitrary fields.
Motivic homotopy theory allows to do enumerative geometry over an arbitrary base, leading to additional arithmetic and geometric information. The goal of this talk is to illustrate a generalized notion of sign that allows us to state a movitic version of classical theorems like the Bézout theorem, the tropical correspondence theorem and a wall-crossing formula for curve counting invariants for points in quadratic field extensions.
Location: Gibson Hall 310
Time: 3:00 pm
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Thursday, April 8,
Colloquium
Topic: The inviscid limit from Navier-Stokes to compressible Euler equations involving shock waves.
Geng Chen - University of Kansas Host: (Scott McKinley)
Abstract: The solutions of compressible Euler equations often form discontinuities, i.e. shock waves, in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions from inviscid limits of Navier-Stokes solutions. The mathematical study of this problem is however very difficult because of the destabilization effect of the viscosities.
Bianchini and Bressan proved the inviscid limit to small BV solutions in one space dimension using the so-called artificial viscosities in 2004. However, until very recently, achieving this limit with physical viscosities remained an open question. In this presentation, the recent advances on the L2 theory of compressible fluid mechanics will be introduced. This method is employed to describe the physical inviscid limit in the context of the barotropic Euler equations, and to solve the Bianchini and Bressan's conjecture. This is a joint work with Moon-Jin Kang and Alexis Vasseur.
Our current research program on the uniqueness and stability of shock wave in mutliple space dimension will be introduced in the end of the presentation.
A Colloquium Reception will be held in the common room at 3 pm BEFORE the colloquium.
Location: Tilton Hall 305
Time: 3:30 pm
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Tuesday, April 8
Algebraic Geometry
Topic: Tensor Products of Leibniz Bimodules and Grothendieck Rings
Speaker: Joerg Feldvoss - University of South Alabama, Mobile
Abstract: Leibniz algebras were introduced by Blo(k)h and Loday as non-anticommutative analogues of Lie algebras. Many results for Lie algebras and their modules have been proven to hold for Leibniz algebras and Leibniz bimodules, but there are also several results that are not true in this more general context. In this talk we will define three different notions of tensor products for Leibniz bimodules. The "natural" tensor product of Leibniz bimodules is not always a Leibniz bimodule. In order to fix this, we will introduce the notion of a weak Leibniz bimodule and show that the "natural" tensor product of weak bimodules is again a weak bimodule. Moreover, it turns out that weak Leibniz bimodules are modules over a cocommutative Hopf algebra canonically associated to the Leibniz algebra. Therefore, the category of all weak Leibniz bimodules is symmetric monoidal and the full subcategory of finite-dimensional weak Leibniz bimodules in addition is rigid and pivotal. On the other hand, we introduce two truncated tensor products of Leibniz bimodules which are again Leibniz bimodules. These tensor products induce a non-associative multiplication on the Grothendieck group of the category of finite-dimensional Leibniz bimodules. In particular, we prove that in characteristic zero for a finite-dimensional solvable Leibniz algebra over an algebraically closed field this Grothendieck ring is an alternative power-associative commutative Jordan ring, but for a finite-dimensional non-zero semi-simple Leibniz algebra it is neither alternative nor a Jordan ring. We also expect it not to be power-associative in the semi-simple case, but at the moment we are neither able to prove nor to disprove this.
This is joint work with Friedrich Wagemann from Nantes Universit\'e.
Location: Lindy Boggs Energy Center - BO-242
Time: 3:00 PM
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Monday, April 7
Integrability and Beyond !!!
Topic: Estimates for the largest eigenvalue of GUE random matrices
John Lopez Santander - Tulane University
Abstract: This is a continuation of the lecture from 2 weeks ago. The speaker will present some estimates regarding the probability that the largest eigenvalue is greater than (2 + \delta) \sqrt{N}.
Location: Gibson Hall 310
Time: 3:00pm
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Week of April 4 - March 31
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Friday, April 4,
Applied and Computational Math
Topic: Solitons + Periodic Traveling Waves = Breathers, Dispersive Shock Waves, Oh My!
Mark Hoefer - University of Colorado, Colorado Boulder Host: (Aikaterini Gkogkou)
Abstract: Completely integrable evolutionary partial differential equations admit a plethora of fascinating, nonlinear-only wave solutions that are realized in a variety of physical systems. Solitons and nonlinear periodic traveling waves TWs are perhaps the most well-known. By nonlinear superposition and modulation, these two basic solution components give rise to new, intriguing solutions and physical phenomena. This talk will use the Darboux transformation on cnoidal TW solutions of the Korteweg-de Vries (KdV) equation to construct exact dark and bright breather solutions. These solutions are then used to describe the transmission and trapping of a soliton by a dispersive shock wave, an expanding modulated nonlinear wavetrain, using multiphase Whitham modulation theory. Finally, experiments in a miscible, viscous core-annular fluid demonstrate the creation, interaction, and properties of breathers including their nonlinear dispersion relation. The experiments are shown to be in qualitative agreement with KdV breather solutions and in quantitative agreement with numerical simulations of a strongly nonlinear, long-wavelength model equation.
Short bio: Mark Hoefer is Professor and Chair of the Department of Applied Mathematics at the University of Colorado, Boulder, drawn back to the Rockies in 2014 after obtaining his PhD from the same department in 2006. In between, he held National Research Council and National Science Foundation postdoc positions at NIST and Columbia University before becoming an Assistant Professor in the Mathematics Department of North Carolina State University. Mark's research encompasses the mathematics and physics, including in-house experiments, of multiscale nonlinear waves. He received the National Science Foundation's Career award in 2013 and the T. Brooke Benjamin Prize in Nonlinear Waves from SIAM in 2020. In 2022, he was the principal organizer for a six-month research program on Dispersive Hydrodynamics at the Isaac Newton Institute in Cambridge, UK. He is deputy editor of the recently established Journal of Nonlinear Waves, published by Cambridge University Press.
Location: Gibson Hall 325
Time: 3:00 pm
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Thursday, April 3,
Colloquium
Topic: Matrices, Polynomials, and Linear Recurrences over Finite Fields
Sudhir Ghorpade - IIT Bombay Host: (Can)
Abstract: Let F be a finite field. We will discuss a variety of questions concerning the enumeration of certain classes of matrices with entries F, polynomials with coefficients in F, and homogeneous linear recurrence sequences of elements of F. Here is a sample of the questions, in their simplest form, that we will consider.
1. What is the number of nonsingular matrices of a given size whose order is maximum in the corresponding general linear group?
2 What is the probability that two polynomials of positive degrees are relatively prime?
3. What is the number of Hankel matrices of a given rank?
4. What is the number of primitive linear recurrence sequences of a given order?
We will attempt to outline some results and conjectures concerning these questions and their extensions and generalisations. Connection of these with combinatorics, cryptography, and algebra may also be indicated.
Parts of this talk are a joint work with Sartaj Ul Hasan and Meena Kumari, with Samrith Ram, and with Mario Garcia Armas and Samrith Ram.
Location: Gibson Hall 126
Time: 3:30 pm
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Tuesday, April 1
Graduate Student Colloquium
Topic: Enter Improving Teaching Methods for Effective University Mathematics Education
Speaker: Sang-Eun Lee - Tulane University
Abstract: This talk aims to explore ways in which graduate students can teach mathematics more efficiently at the university level. Mathematics, a discipline that has evolved alongside human history, demands logical rigor and philosophical precision, making it a vast and profound field. However, due to these characteristics, many students perceive mathematics as a challenging subject rather than an engaging one. Upon this importance, modern North American universities require most students to complete at least one mathematics course as part of their graduation requirements, highlighting the need for effective teaching strategies. This talk will discuss the challenges from our experience from the class and potential improvements in teaching methods leading students feel more comfortable.
Location: MA 101
Time: 3:30 PM
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Week of March 28 - March 24
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Thursday, March 27
Colloquium
Topic: INTEGRABLE COMBINATORICS
Philippe Di Francesco - University of Illinois Urbana-Champaign Host: (Ken McLaughlin)
Abstract: Combinatorics has constantly evolved from the mere counting of classes of objects to the study of their underlying algebraic or analytic properties, such as symmetries or deformations. This was fostered by interactions with in particular statistical physics, where the objects in the class form a statistical ensemble, where each element comes with some probability. Integrable systems form a special subclass: that of systems with sufficiently many symmetries to be amenable to exact solutions.
In this talk, we explore various basic combinatorial problems involving discrete surfaces, dimer models of cluster algebra, or two-dimensional vertex models, whose (discrete or continuum) integrability manifests itself in different manners: commuting operators, conservation laws, flat connections, quantum Yang-Baxter equation, etc. All lead to often simple and beautiful exact solutions.
Location: Gibson Hall 126
Time: 3:30 pm
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Thursday, March 27
Defense
Topic: Asymptotics and zeros of a special family of Jacobi polynomials.
John Lopez - Tulane University
Abstract:
Location: Howard-Tilton Memorial Library, room B11 (basement)
Time: 10:00 am
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Wednesday, March 26
AMS/AWM
Topic: Faculty Talk
Daniel Bernstein - Tulane University
Abstract: TBA
Location: Gibson Hall 310
Time: 4:15 pm
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Wednesday, March 26
Algebra and Combinatorics
Topic: Asymptotic counts of number fields generated by plane curves
Michael Allen - LSU Host: (Olivia Beckwith)
Abstract: Every irreducible polynomial f(x) with integer coefficients corresponds uniquely to a field extension of the rational numbers which consists of Q, a root x of f, and all combinations thereof under the standard arithmetic operations. For example, f(x) = x^2-2 produces the field Q(sqrt{2}) = {a + bsqrt{2} : a, b \in Q}. If f is a polynomial in two or more variables, we can produce infinitely many such fields corresponding to solutions to f=0. For f(x,y) = y^2-x^3-x-1, we have solutions (1, \sqrt{3}), (2, \sqrt{11}), (3, \sqrt{31}) and so on, and so the curve defined by f(x,y)=0 ``generates" the fields Q(\sqrt{3}), Q(\sqrt{11}), and Q(\sqrt{31}).
Recently, Mazur and Rubin suggested using this algebraic information as a means to study the geometric properties of a curve. One can easily ask the reverse question: ``If we know something about a curve C, what can we say about the fields that it generates?" We approach this question through the lens of arithmetic statistics by counting the number of such fields with bounded size---under some appropriate notion of size---for an arbitrary fixed plane curve C. This is joint work in progress with Renee Bell, Robert Lemke Oliver, Allechar Serrano L\'{o}pez, and Tian An Wong.
Location: Gibson Hall 310
Time: 3:00 pm
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Tuesday, March 25
Graduate Student Colloquium
Topic: An Introduction to Riemann-Roch and Serre Duality
Speaker: Naufil Sakran - Tulane University
Abstract: This talk aims to introduce two fundamental theorems in algebraic geometry: the Riemann-Roch theorem and Serre duality. I will develop the necessary background and present these theorems in the setting of Riemann surfaces, following the approach in Algebraic Curves and Riemann Surfaces by Rick Miranda. I will conclude my talk by given few applications of these theorems.
Location: MA 101
Time: 3:30 PM
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Monday, March 24
Probability & Statistics
Topic: Random-Walk Debiased Inference for Contextual Ranking Model with Application in Large Language Model Evaluation
Yichi Zhang - Indiana University Bloomington
Abstract: We propose a debiased inference framework to infer the ranking structure in the contextual Bradley-Terry-Luce (BTL) model. We first adopt a nonparametric maximum likelihood estimation method using ReLU neural networks to estimate unknown preference functions in the model. For the inference of pairwise ranking, we introduce a novel random-walk debiased estimator that efficiently aggregates all accessible estimating scores. In particular, under mild conditions, our debiased estimator yields a tractable distribution, and achieves the semiparametric efficiency bound asymptotically. We further extend our method by incorporating multiplier bootstrap techniques for the uniform inference of ranking structures, and adapting it to accommodate the distributional shift of contextual variables. We provide thorough numerical studies to validate the statistical properties of our method, and showcase its applicability in evaluating large language models based on human preferences under different contexts.
Location: Norman Mayer Building 101
Time: 4:00 pm
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Monday, March 24
Probability & Statistics
Topic: From Myth to Truth: An Introduction to Statisticians’ Role In Drug Development
Cindy Lu and Xinyu Cong - AstraZeneca
Abstract: Our presentation explores the evolving role of statisticians in the pharmaceutical industry, particularly within drug development. It introduces the various stages of the drug development process, from pre-clinical trials through Phase IV post-market, highlighting statisticians critical roles during those processes. The presentation aims to dispel common myths about the statistical profession in pharma, encouraging more talented graduates devote their career to the mission of bringing innovative treatments to patients.
Location: Norman Mayer Building 101
Time: 4:00 pm
Week of March 21 - March 17
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Friday, March 21
Applied and Computational Math
Topic: Inverse Problems and In-Context Operator Learning for Mean-Field Games
Siting Liu - UC Riverside Host: (Hongfei Chen)
Abstract: Mean-field game (MFG) systems provide a powerful framework for modeling the collective behavior of multi-agent systems with diverse applications. However, unknown model parameters pose challenges. In the first part of the talk, we address an inverse problem in MFGs, recovering running cost and interaction energy from noisy boundary observations. We formulate it as a constrained optimization problem and solve it efficiently using an operator-splitting algorithm. In the second part, we introduce In-Context Operator Networks (ICON), a neural framework that learns solution operators from prompts to solve MFG problems. ICON demonstrates strong few-shot learning across forward and inverse differential equation tasks, including mean-field control.
Location: Gibson Hall 325
Time: 3:00 pm
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Wednesday, March 19
AMS/AWM
Topic: Faculty Talk
Tai Ha - Tulane University
Abstract: TBA
Location: Gibson Hall 310
Time: 4:15 pm
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Wednesday, March 19
Algebra and Combinatorics
Topic: Geometric vertex decomposition
Thai Nguyen - University of Dayton
Abstract: Geometric vertex decomposition is a useful technique in various algebro-geometric contexts such as liaison theory and Groebner bases theory. It is a degeneration technique that was first used in the work of Knutson-Miller-Yong to study Schubert determinantal ideals. It can also be thought of as an ideal-theoretic generalization of vertex decomposition of simplicial complexes. It was shown in the work of Klein-Rajchgot that geometrically vertex decomposable (gvd) ideals possess various nice algebraic properties as those of the Stanley-Reisner ideal of vertex decomposable simplicial complexes. In this talk, I shall survey some results using this technique. I shall also discuss some homological invariants of gvd ideals, with emphasis on toric ideals of graphs. The talk will include results from my project with Jenna Rajchgot and Adam Van Tuyl.
Location: Gibson Hall, room 310
Time: 3:00 pm
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Tuesday, March 18
Graduate Student Colloquium
Topic: Laurent Series Expansions of $L$ -functions
Tushar Karmakar - Tulane University
Abstract: In this talk, we will be revisiting the well-known Laurent series expansion of Riemann zeta function, Hurwitz zeta function and as a generalization, Dirichlet $L$- function. Additionally, we will briefly discuss modular forms and its $L$- series and then we will see the Laurent series expansion for $L$ - function attached to cusp forms over the full modular group.
Location: Norman Mayer Building - MA-101 (G)
Time: 3:30pm
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Monday, March 17
Integrability and Beyond !!!
Topic: How to compute interesting statistical quantities in random matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics, Part 6.
Ken McLaughlin - Tulane University
Abstract: The plan: investigate the behavior of eigenvalues of random matrices when the size of the matrix grows to infinity, starting with more numerical experiments, and then the asymptotic behavior of the kernel, and the Fredholm determinant. The aim is to understand known asymptotic behavior and identify some interesting but tractable challenging problems.
Location: Gibson Hall 310
Time: 3:00pm
Week of March 14 - March 10
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Scientific Computing Around Louisiana (SCALA): March 14 - 15
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Friday, March 14
Algebra and Combinatorics
Topic: Developments in Interpolation Problem for Projective Spaces
Dipendranath Mahato - Affiliation
Abstract: Classical Interpolation problem of estimating new data from a set of known data is well understood under one variable situation. Here we are more interested in higher dimensional Projective Spaces, where we are trying to find the lowest possible degree of the hyper-surface passing through a given set of points with prescribed multiplicity. There are famous conjectures to tackle such problems: Chudnovsky’s Conjecture, Demailly’s Conjecture. I will be discussing those conjectures and some recent developments in this area.
Location: Gibson Hall 310
Time: 3:00pm
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Thursday, March 13
Colloquium
Topic: The Feynman-Lagerstrom criterion for boundary layers
Theodore Drivas - SUNY Stony Brook Host: (Sam)
Abstract: We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (1904) and Batchelor (1956), any Euler solution arising in this limit and consisting of a single ``eddy" must have constant vorticity. Feynman and Lagerstrom (1956) gave a procedure to select the value of this vorticity by demanding a necessary condition for the existence of a periodic Prandtl boundary layer description. We will show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. This is joint work with S. Iyer and T. Nguyen.
Location: Gibson Hall 126
Time: 3:30 pm
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Wednesday, March 12
Algebra and Combinatorics
Topic: Algebraic Properties of Invariant ideals.
Souvik Dey - Charles University, Prague Host: (Dipendranath Mahato, Tai Ha)
Abstract: Let R be a polynomial ring with mn many indeterminate over complex numbers. We can think of the indeterminates as a matrix, X of size m x n.
Consider the group G = Gl(m) x Gl(n). Then G acts on R via the group action (A,B)X =AXB^{-1}. In 1980, DeConcini, Eisenbud, and Procesi introduced the ideals that are invariant under this group action.
In the same paper, they described various properties of those ideals, e.g., associated primes, primary decomposition, and integral closures. In recent work with Sudipta Das, Tài Huy Hà, and Jonathan Montaño, we described their rational powers and proved that they satisfy the binomial summation formula. In an ongoing work, Alexandra Seceleanu and I are formulating symbolic properties of these ideals. In this talk, I will describe these ideals and the properties we are interested in. I will also showcase some results from my collaborations.
Location: Gibson Hall, room 310
Time: 3:00 pm
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Monday, March 10
Integrability and Beyond !!!
Topic: How to compute interesting statistical quantities in random matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics, Part V.
Ken McLaughlin - Tulane University
Abstract: The plan: (1) Executive summary of the connection between eigenvalues of random matrices and Fredholm determinants. (2) T random matrix theory laboratory – testing the theory. (3) Behavior of eigenvalues when the size of the matrices grows to $\infty$.
Location: Gibson Hall 310
Time: 3:00pm
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Week of February 28 - February 24
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Wednesday, February 26
Algebra and Combinatorics
Topic: A combinatorial method for the reduction number of an ideal
Alessandra Costantini - Tulane University
Abstract: In the study of commutative rings, several algebraic properties are captured by numerical invariants which are defined in terms of ideals and their powers. Among these, of particular relevance are the reduction number and analytic spread of an ideal, which control the growth of the powers of the given ideal for large exponents. Unfortunately, these invariants are usually difficult to calculate for arbitrary ideals, and different methods might be required depending on the specific features of the class of ideals under examination.
In this talk, I will discuss a combinatorial method to calculate the reduction number of an ideal, based on a homological characterization in terms of the regularity of a graded algebra. This is part of ongoing joint work with Louiza Fouli, Kriti Goel, Haydee Lindo, Kuei-Nuan Lin, Whitney Liske, Maral Mostafazadehfard and Gabriel Sosa.
Location: Gibson Hall, room 310
Time: 3:00 pm
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Tuesday, February 25
Graduate Student Colloquium
Topic: Estimation of degradation rate in biological cells
Lan Trinh - Tulane University
Abstract: In our interested biological cells, the particles were born and diffused as in Brownian motions, which could then exit, degrade or stay alive at a specific time. From the figures recording locations of alive particles, we investigate the estimation of the degradation rate and its statistical properties. In this talk, I will discuss the toy model of this problem, based on which will be developed into more sophisticated setup of real cells.
Location: Norman Mayer Building - MA-101 (G)
Time: 3:30pm
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Tuesday, February 25
Colloquium
Topic: Dynamical Systems Insights into Map Enumeration
Joceline Lega - University of Arizona - Host: (Ken McLaughlin)
Abstract: This talk will highlight techniques from discrete dynamical systems theory that have led to significant advances in map enumeration. We will start with a brief overview of generating functions for map counts and their relation to solutions of the discrete Painlevé I equation. We will then present computational and analytical results that lead to specific generating functions and, in the case of 4-valent maps, derive explicit expressions for map counts as functions of the number of vertices and the genus of the surface on which the map is embedded. We will conclude with open as well as recently solved questions associated with this research program. This is joint work with Nick Ercolani and Brandon Tippings..
Location: TBA
Time: 4:30 pm
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Monday, February 24
Integrability and Beyond !!!
Topic: A Vision of Integrability: McKean’s Unimodularity Conjecture.
Nick Ercolani - University of Arizona (Host: Ken McLaughlin)
Abstract: In a striking series of papers, titled Geometry of KdV(1) - Geometry of KdV(5), Henry Mckean formulated a precise notion of what should be the function space foliation by invariant sets for the Korteweg-deVries evolution. This is meant to pertain to initial data that is smooth but otherwise only required to be bounded below. This foliation should generalize the picture of (typically infinite dimensional) Arnold-Liouville torii familiar from the particular case of periodic initial data. The proposed answer is phrased in terms Kodaira’s elegant extension of the classical Weyl-Titchmarsh theory for spectral weights of Schrodinger operators.
The goal of the talk will be to first present an overview of McKean’s conjecture and then to describe some recent work, joint with Dylan Murphy, on analogous investigations for the Toda lattice and Jacobi operators.
Location: Gibson Hall 310
Time: 3:30pm
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Week of February 21 - February 17
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Friday, February 7
Applied and Computational Math
Topic: The unreasonable utility of symmetric three-term recurrences
Tom Trogdon - University of Washington Host: (Aikaterini Gkogkou)
Abstract: Symmetric three-term recurrences (STRs) naturally arise in the study of orthogonal polynomials, iterative methods for symmetric matrices and numerical complex analysis. While deceptively simple, STRs allow for many extremely effective numerical methods. This talk will review some classical methods and uses and connect to more recent developments related to the computation of Cauchy integrals, computing matrix functions and spectral density estimation for random matrices.
Location: Gibson Hall 325
Time: 3:00pm
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Friday, February 21
Algebra and Combinatorics
Topic: Ear decompositions of graphs: an unexpected tool in Combinatorial Commutative Algebra
Ngo Viet Trung - Institute of Mathematics, Vietnam Academy of Science and Technology Host: (Tai Ha)
Abstract: Ear decomposition is a classical notion in Graph Theory. It has been shown in [1, 2] that this notion can be used to solve difficult problems on homological properties of edge ideals in Combinatorial Commutative Algebra. This talk presents the main combinatorial ideas behind these results.
[1] H.M. Lam and N.V. Trung, Associated primes of powers of edge ideals and ear decompositions of graphs, Trans. AMS 372 (2019)
[2] H.M. Lam, N.V. Trung, and T.N. Trung, A general formula for the index of depth stability of edge ideals, Trans. AMS, to appear.
Location: Gibson Hall 310
Time: 3:00 pm
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Tuesday, February 18
Graduate Student Colloquium
Topic: Stochastic Differential Equations, Epidemic Models, and a brief overview on Chagas' Disease.
Joshua Agbomola - Tulane University
Abstract: The presentation provides an introduction to Chagas disease, a parasitic infection caused by Trypanosoma cruzi, primarily transmitted by Triatomine bugs. It outlines key transmission pathways, including vector-borne, congenital, and less common mechanisms. The discussion then shifts to the application of a stochastic SIS (Susceptible-Infected-Susceptible) model, illustrating how randomness can enhance epidemic modeling by accounting for variability and uncertainty in disease dynamics.
Location: BO 242
Time: 3:00pm
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Monday, February 17
Integrability and Beyond !!!
Topic: Random Matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics.
TBD - Tulane University
Abstract: We will return to the numerical experiments, to carefully develop intuition. And explore some more precise open problems. Then we will return to complete the proof of the fundamental relation between eigenvalues and Fredholm determinants.
Location: Gibson Hall 310
Time: 3:30pm
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Week of February 14 - February 10
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Clifford Lectures: February 11 - 14
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Wednesday, February 12
Prob & Stats seminar
Topic: Multidimensional Empirical Wavelet Transform
Charles-Gérard Lucas- San Diego State University
Abstract: The empirical wavelet transform, inspired by empirical mode decomposition, is an adaptive time-frequency representation that extracts the different modes of a signal or image by designing a bank of adaptive wavelet filters. The data robustness of this transform has made it the subject of intense development and a growing number of applications over the past decade. However, to date, it has mainly been studied theoretically for signals, and its extension to images is limited to a specific wavelet kernel. This presentation will focus on a multidimensional extension of this transform formulated from a wavelet kernel. Theoretical and numerical properties of this formulation will be particularly detailed. Its interest for texture segmentation will also be highlighted.
Location: 104 Norman Mayer
Time: 1:00pm
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Tuesday, February 11
Graduate Student Colloquium
Topic: Partial Betti splittings for binomial edge ideals
Aniketh Sivakumar - Tulane University
Abstract: The Free resolution of a module is an object which contains important information about the structure of the module. These resolutions are used to define several invariants associated to a module, including their Betti numbers. In this talk, we will introduce the notion of a partial Betti splitting of a homogeneous ideal, generalizing the notion of a Betti splitting first given by Francisco, Hà, and Van Tuyl. We will also define an ideal associated to a graph known as its binomial edge ideal and describe an explicit partial Betti splitting for this class of ideals.
Location: BO 242
Time: 3:30pm
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Monday, February 10
Algebra and Combinatorics
Topic: Versions of the circle method
Edna Jones - Tulane University
Abstract: The circle method is a useful tool in analytic number theory and combinatorics. The term "circle method" can refer to one of a variety of techniques for using the analytic properties of the generating function of a sequence to obtain an asymptotic formula for the sequence. We will discuss different versions of the circle method and some results that can be obtained by using the circle method.
Location: Gibson Hall, room 310
Time: 3:00 pm
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Monday, February 10
Integrability and Beyond !!!
Topic: Random Matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics.
Ken McLaughlin - Tulane University
Abstract: We will continue exploring the connection between random matrices and Hermite polynomials, and start the proof of the relation between eigenvalue probabilities and Fredholm determinants. Time permitting, we will return to the numerical experiments, to carefully develop intuition.
Location: Gibson Hall 310
Time: 3:30pm
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Monday, February 10
Algebra and Combinatorics
Topic: From Interpolation problems to matroids
Paolo Mantero - University of Arkansas - Host: (Alessandra Costantini)
Abstract: Interpolation problems are long-standing problems at the intersection of Algebraic Geometry, Commutative Algebra, Linear Algebra and Numerical Analysis, aiming at understanding the set of all polynomial equations passing through a given finite set X of points with given multiplicities.
In this talk we discuss the problem for matroidal configurations, i.e. sets of points arising from the strong combinatorial structure of a matroid. Starting from the special case of uniform matroids, we will discover how an interplay of commutative algebra and combinatorics allows us to solve the interpolation problem for any matroidal configuration. It is the widest class of points for which the interpolation problem is solved. Along the way, we will touch on several open problems and conjectures.
The talk is based on joint projects with Vinh Nguyen (U. Arkansas).
Location: Dinwiddie Hall, room 103 (note unusual location and day)
Time: 3:00 pm
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Week of February 7 - February 3
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Friday, February 7
Applied and Computational Math
Topic: Admissible behavior of Krylov subspace iterative methods for non-symmetric systems
Kirk Soodhalter - Trinity College Dublin (Lisa Fauci)
Abstract: In this talk, we introduce the audience Krylov subspace iterative methods, the work-horse matrix-free methods for solving linear systems and eigenvalue problems in the case that matrix is large and sparse or otherwise not available to be solved by accessing all entries of the matrix. These methods are built on the core assumption that we only have access to a procedure that multiplies the matrix times vectors at relatively low computational cost. After discussing some basic convergence theory related to polynomial interpolation, we demonstrate the complicated nature of estimating rate of convergence for such methods based on quantities such as eigenvalues when the matrix is non-normal. In particular, we discuss and contextualize a constructively proven theorem that in pathological cases, the eigenvalues need not be at all descriptive with regard to convergence behavior [Greenbaum, Pták, Strakoš 1996]. We build the language needed to describe the mechanics of how to construct such cases. We then discuss our currently running project on developing more robust theory of convergence for these methods applied to non-symmetric Toeplitz systems, the type of which arise in a variety of problems from the computational sciences. Toeplitz matrices are constant along each diagonal; thus the linear system is governed by a matrix with fewer degrees of freedom than in general. We demonstrate that tools developed in the aformentioned constructive proof can be repurposed to develop refine previous convergence theory for Toeplitz systems. This is ongoing work; thus we finish by discussing what remains to be proven and how we intend to extend this theory to generalizations of Toeplitz matrices encompassing a larger class of matrix structures often arising in the computational sciences.
Location: Gibson Hall 325
Time: 3:00pm
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Tuesday, February 4
Graduate Student Colloquium
Topic: Symbolic powers via extension
Haoxi Hu - Tulane University Host: (Moslem Uddin)
Abstract: Symbolic Powers of ideals are well-studied objects, in this talk, We investigates under which conditions the symbolic powers of the extension of an ideal is the same as the extension of the symbolic powers. Our result generalizes the known scenarios. As an application, we prove formulas for the resurgence of sum of two homogeneous ideals in finitely generated k-algebra domains, where k is algebraically closed. Initially, these were known for ideals in polynomial rings.
Location: Norman Mayer Building - MA-101 (G)
Time: 3:30pm
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Monday, February 3
Integrability and Beyond !!!
Topic: Random Matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics.
Ken McLaughlin - Tulane University
Abstract: We will start calculating some basic number statistics for finite sized random matrices using Hermite polynomials, and compare to numerical experiments. In the second half, I will explain the connection between random matrices and Hermite polynomials.
Location: Gibson Hall 310
Time: 3:30pm
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Week of January 31 - January 27
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Friday, January 31
Colloquium
Topic: Partitions Detect Primes
Ken Ono - University of Virginia Host: ( )
Abstract: This talk presents “partition theoretic” analogs of the classical work of Matiyasevich that resolved Hilbert’s Tenth Problem in the negative. The Diophantine equations we consider involve equations of MacMahon’s partition functions and their natural generalizations. Here we explicitly construct infinitely many Diophantine equations in partition functions whose solutions are precisely the prime numbers. To this end, we produce explicit additive bases of all graded weights of quasimodular forms, which is of independent interest with many further applications. This is joint work with Will Craig and Jan-Willem van Ittersum.
Location: Gibson Hall 126 Subject to change
Time: 4:00 pm
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Friday, January 31
Applied and Computational Math
Topic: Towards efficient deep operator learning for forward and inverse PDEs: theory and algorithms
Ke Chen - University of Delaware Host: (Hongfei Chen)
Abstract: Deep neural networks (DNNs) have been a successful model across diverse machine learning tasks, increasingly capturing the interest for their potential in scientific computing. This talk delves into efficient training for PDE operator learning in both the forward and inverse PDE settings. Firstly, we address the curse of dimensionality in PDE operator learning, demonstrating that certain PDE structures require fewer training samples through an analysis of learning error estimates. Secondly, we introduce an innovative DNN, the pseudo-differential auto-encoder integral network (pd-IAE net), and compare its numerical performance with baseline models on several inverse problems, including optical tomography and inverse scattering.
Location: Gibson Hall 325 (Room Change)
Time: 3:00pm
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Tuesday, January 28
Graduate Student Colloquium
Topic: Unearthing Latent Time Series in Repeated Survey Data
John V Argentino - Tulane University Host: (Moslem Uddin)
Abstract: In today’s world of readily available data, repeated survey data is commonplace in a variety of settings. These data are collected to model the impacts of particular variables of interest, which are often estimated using methods that assume temporal independence in the observed noise. This premise is shaky given how easily conceivable it is that variables not accounted for have some degree of continuity over time. This talk will present a method that seeks to reconcile mixed effect models and time series analysis while employing classic results from linear algebra to identify it’s potential pitfalls.
Location: DW 102
Time: 3:30pm
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Monday, January 27
Integrability and Beyond !!!
Topic: Random Matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics.
Ken McLaughlin - Tulane University
Abstract: How to compute interesting statistical quantities in random matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics.
With the assistance of the whole group, this should be very introductory.
Speakers: to be determined, starting with Ken McLaughlin on January 27.
Location: Gibson Hall 310
Time: 3:30pm
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Week of January 24 - January 20
Friday, January 24
Applied and Computational Math
Topic: Manuela Girotti - Emory University Host: (Aikaterini Gkogkou)
Di Fang - Duke University
Abstract: N. Zabusky coined the word "soliton" in 1965 to describe a curious feature he and M. Kruskal observed in their numerical simulations of the initial-value problem for a simple nonlinear PDE. The first part of the talk will be a broad introduction to the theory of solitons/solitary waves and integrable PDEs (the KdV and modified KdV equation in particular), describing classical results in the field. The second (and main) part of the talk will focus on some new developments and growing interest into a special case of solutions defined as "soliton gas".
I will describe a collection of works done in collaborations with K. McLaughlin (Tulane U.), T. Grava (SISSA/Bristol), R. Jenkins (UCF) and A. Minakov (U. Karlova).
We analyze the case of a regular, dense KdV soliton gas and its large time behaviour with the presence of a single trial soliton travelling through it.
We are able to derive a series of physical quantities that precisely describe the dynamics, such as the local phase shift of the gas after the passage of the soliton, and the velocity of the soliton peak, which is highly oscillatory and it satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El (at leading order).
I will finally present some ongoing work where we establish that the soliton gas is the universal limit for a large class of N-solutions with random initial data.
Location: Gibson Hall 325
Time: 3:00pm
Week of January 17 - January 13
Friday, January 17
Applied and Computational Math Seminar
Topic: _______
Di Fang - Duke University
Abstract: Unbounded Hamiltonian Simulation: Quantum Algorithm and Superconvergence
Location: Simulation of quantum dynamics, emerging as the original motivation for quantum computers, is widely viewed as one of the most important applications of a quantum computer. Quantum algorithms for Hamiltonian simulation with unbounded operators Recent years have witnessed tremendous progress in developing and analyzing quantum algorithms for Hamiltonian simulation of bounded operators. However, many scientific and engineering problems require the efficient treatment of unbounded operators, which may frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure, quantum differential equations solver and quantum optimization. We will introduce some recent progresses in quantum algorithms for efficient unbounded Hamiltonian simulation, including Trotter type splitting and Magnus expansion based algorithms in the interaction picture. (The talk does not assume a priori knowledge on quantum computing.)
Location: Gibson Hall 126
Time: 3:00pm
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Thursday, January 16
Colloquium
Topic: On the flow of zeros of derivatives of polynomials
Andrei Martinez-Finkelshtein - Baylor University (Host: Ken)
Abstract: Assume we have a sequence of polynomials whose asymptotic zero distribution is known. What can be said about the zeros of their derivatives? Especially if we differentiate each polynomial several times, proportional to its degree? This simple-to-formulate problem has recently attracted the attention of researchers. Both the problem and the methods of its solution have exciting connections with free probability, random matrices, and approximation theory on the complex plane. In this talk, I will explain some known results in this direction and our approach to the problem, which uses only some elementary complex analysis. This is a joint work with E. Rakhmanov from the University of South Florida.
Location: Gibson Hall 126
Time: 3:30 pm
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Wednesday, January 15
Algebra and Combinatorics
Topic: Rings with extremal cohomology annihilator
Souvik Dey - Charles University, Prague Host: (Dipendranath Mahato, Tai Ha)
Abstract: The cohomology annihilator of Noetherian algebras was defined by Iyengar and Takahashi in their work on strong generation in the module category. For a commutative Noetherian local ring, it can be observed that the cohomology annihilator ideal is the entire ring if and only if the ring is regular. Motivated by this, I will consider the question: When is the cohomology annihilator ideal of a local ring equal to the maximal ideal? I will discuss various ring-theoretic and category-theoretic conditions towards understanding this question and describe applications for understanding when the test ideal of the module closure operation on cyclic surface quotient singularities is the maximal ideal.
Location: Gibson Hall, room 310
Time: 3:00 pm
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