Week of November 22 - November 18
Friday, November 22
Applied and Computational &
SCMB-Simons Seminar
Topic: Existence of Stationary Measures for Sdes with Generic, Euler-Type Nonlinearities
Kyle Liss - UDuke University
Abstract: Many physical phenomena involve the nonlinear, conservative transfer of energy from weakly damped degrees of freedom driven by an external force to other modes that are more strongly damped. For example, in hydrodynamic turbulence, energy enters the system primarily at large spatial scales, but at high Reynolds number, dissipative effects are only significant at very high frequencies. In this talk, I will discuss nonlinear energy transfer and the existence of invariant measures for a class of degeneratly forced SDEs on R^d with a bilinear nonlinearity B(x,x) constrained to possess various properties common to finite-dimensional fluid models and a linear damping term -Ax that acts only on a proper subset of the phase space. Existence of an invariant measure is straightforward if kerA = {0}, but when the kerA is nontrivial, an invariant measure can exist only if the nonlinearity transfers enough energy from the undamped modes to the damped modes. We develop a set of sufficient dynamical conditions on B that guarantees the existence of an invariant measure and prove that they hold “generically” within our constraint class of nonlinearities provided that dim(kerA) < 2d/3 and the stochastic forcing acts directly on at least two degrees of freedom.
Location: Gibson Hall 325
Time: 3:00
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Thursday, November 21
SCMB-Simons Seminar
Topic: Parameter estimation for ordinary differential equations with time warping.
John Fricks - Arizona State University
Abstract: Curve registration is a set of techniques to align functional data in the presence of time warping—phase variation in the functional observations. In this talk, we will present a Bayesian framework to estimate the parameters of an ODE model when the observations contain stochastic fluctuations in both amplitude and phase with a Gaussian process defining the time warping model. To facilitate such a framework, a new method for curve registration using Hamiltonian Monte Carlo will be presented along with a hierarchical model that links a basis fit of the data to solutions of an ODE model, allowing for parameter estimation.
Location: Stanley Thomas 316
Time: 11:00 am
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Wednesday, November 20
Probability and Statistics
Topic: Probabilistic Interpretations of the Boltzmann and Enskog Equations
Christian Ennis – Louisiana State University
Abstract: The Boltzmann equation describes the time evolution of the density function in a phase (position-velocity) space for a classical particle (molecule) under the influence of other particles in a diluted (or rarified) gas, evolving in vacuum for a given initial distribution. The Enskog equation introduces a function in the collision operator for the Boltzmann equation, allowing one to take into account the interactions between molecules at a small distance away, rather than solely at the point of collision. In this talk, we discuss modern results on the stochastic treatment of the spatially homogeneous Boltzmann equation, the Enskog equation, and the connection between methods used in each system. Regularity results and the motivation behind this probabilistic treatment will be given.
Location: Gibson 126
Time: 4:00 pm
Wednesday, November 20
Algebra and Combinatorics Seminar
Topic: Rees algebras of linearly presented ideals
Alessandra Costantini - Tulane University
Abstract: Rees algebras represent an essential algebraic tool in the study of singularities of algebraic varieties, as they arise, for instance, as homogeneous coordinate rings of blowups or graphs of rational maps.
In this talk, I will discuss the problem of finding the defining equations of Rees algebras. Although this is wide open in general, the problem becomes treatable in the case of height-two perfect ideals with a linear presentation, where one can use a combination of homological methods and linear algebra, inspired by classical elimination theory.
This is part of joint work with E. Price and M. Weaver (arxiv:2308.16010 and arxiv:2409.14238
Location: Richardson Building - RB-117 (G)
Time: 3:00pm
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Wednesday, November 20
Integrability and Beyond!!!
Topic: Four weeks on the nonlinear Schrodinger equation – week 5!
Ken McLaughlin – Tulane University
Abstract: This will be an informal working group, learning about the long time asymptotic behavior of the defocusing NLS equation, based on a combination of Riemann-Hilbert and d-bar techniques.
Today:
1. A summary of what the past 4 weeks have yielded: a description of the behavior of “all” solutions of the nonlinear Schroedinger equation in the long-time regime.
2. A summary of the fundamental steps in the analysis.
3. If there is any time at the end, some “sick tricks”.
Location: Jones Hall 108
Time: 3:00pm
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Wednesday, November 20
Special Algebraic Geometry
Topic: Lower Bounding the Gromov–Hausdorff Distance on Manifolds and Graphs
Majhi Sushovan - George Washington University DC
Abstract: The Gromov–Hausdorff distance between two abstract metric spaces provides a (dis)-similarity measure quantifying how far the two metric spaces are from being isometric. Although the inception of the distance was due to M. Gromov in the context of hyperbolic groups, it has recently been shown to provide a robust theoretical framework for shape and dataset comparison. Consequently, the computational aspects and various bounds on the Gromov–Hausdorff distance are receiving a lot of attention from both applied and theoretical communities. In this talk, I give an overview of the Gromov–Hausdorff distance, delineating its relation to the well-known Hausdorff distance. The main focus of the talk is to present interesting lower bounds on the former by a constant multiple of the latter on interesting spaces like the circle, closed Riemannian manifolds, and metric graphs.
Location: Gibson Hall 325
Time: 2:00 pm
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Tuesday, November 19
Graduate Colloquium
Topic: Reed-Solomon Codes in the Uniform Tree Metric
Dillon Montero - Tulane University
Abstract: The classical theory of error-correcting codes primarily uses the Hamming metric to measure distances. Within this framework, Maximum Distance Separable (MDS) codes are highly valued due to their optimal parameters, enabling the correction of the maximum number of errors for a given code rate. In 1997, Rosenbloom and Tsfasman introduced the m-metric codes (now known as NRT metric codes), identifying analogs of Reed-Solomon codes that still possess the MDS property. Later, Skriganov provided an explicit construction of these codes using Hasse derivatives.
In this talk, we will introduce and discuss our analogs of Reed-Solomon codes in a related context, also employing Hasse derivatives for our construction.
This is joint work with Mahir Bilen Can.
Location: F. Edward Hebert Hall 213
Time: 3:30pm
Tuesday, November 19
Special Colloquium
Topic: Topological deep learning on graphs, manifolds, and curves
Dr. Guo-Wei Wei - Affiliation: Michigan State University (Tai Ha)
Abstract: In the past few years, topological deep learning (TDL), a term coined by us in 2017, has become an emerging paradigm in artificial intelligence (AI) and data science. TDL is built on persistent homology (PH), an algebraic topology technique that bridges the gap between complex geometry and abstract topology through multiscale analysis. While TDL has made huge strides in a wide variety of scientific and engineering disciplines, it has many limitations. I will discuss our recent effort in extending TDL from graphs to manifolds and curves, using algebraic topology, geometric topology, and differential geometry. I will also discuss how TDL led to victories in worldwide annual competitions in computer-aided drug design, the discoveries of SARS-CoV-2 evolutionary mechanism, and the accurate forecasting of emerging dominant viral variants.
Special Location: Boggs 243
Special Time: 2:00PM – 3:00PM
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Week of November 15 - November 11
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Friday, November 15
Applied and Computational &
SCMB-Simons Seminar
Topic: Inferring and Interpreting Heterogeneous Models Using Dendrograms
Linh Do - Tulane and Dat Do (Michigan)
Abstract: In the new era of big data, modern datasets (e.g., in genomics) are often large-scale and heterogeneous. To meaningfully model such data, scientists use mathematical models to cluster/segment it into a smaller and interpretable number of subpopulations. Long-standing questions involving those models include: (1) In practice, how to select the number of subpopulations; (2) In theory, what happens if that number is misspecified or the model is incorrect. In this talk, we aim to answer these two questions for two popular models of this class named “mixture models” and “changepoint detection.” We take the multiscaling approach to this problem by first overfitting data with a large number of subpopulations and then sequentially projecting it down to smaller subspaces of models. This results in a binary tree representation (a.k.a., dendrogram) of these nested classes of models, which is useful for visualization and model selection. We then study the convergence rate of the vertices and topology of the inferred dendrogram and show it is statistically optimal. Based on this, we propose a novel consistent model selection method named Dendrogram Information Criteria. Several simulation studies are presented to support our theory. We also illustrate the methodology with applications to single-cell RNA sequence analysis and wind turbines data.
Location: Gibson Hall 325
Time: 3:00
Friday, November 15
Algebraic Geometry
Topic: An Algebraic-Combinatorial Construction of QC-LDPC Codes
Henry Chimal-Dzul - UT San Antonio
Abstract: An Algebraic-Combinatorial Construction of QC-LDPC Codes
Abstract: Quasi-Cyclic LDPC (QC-LDPC) codes constitute one of the most attractive family of linear codes. This is because of their compact representation, existence of efficient encoding and decoding algorithms, rich algebraic structure and their excellent performance when compared to random LDPC codes. These are some of the reasons why QC-LDPC codes now appear in many industry standards, including those developed by the Consultative Committee for Space Data System, NASA deep-space explorations, Digital Video Broadcast, IEEE 802.11a, and the 5G New Radio Mobile communication standard. One of the required properties that a QC-LDPC code must have for all these applications is that their Tanner graph should not have a small girth (often 4 or 6). In this talk we will discuss the problem of designing QC-LDPC codes with Tanner graphs having girth at least 6. To this order, we will present an algebraic representation of QC-LDPC codes from which we will derive combinatorial problems to design them.
Location: Gibson Hall 126A
Time: 1:00 pm
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Thursday, November 14
Colloquium
Topic: KPZ limit theorems
Jinho Baik - University: University of Michigan (Host: Gustavo Didier)
Abstract: In probability theory, various models often exhibit similar fluctuation behaviors as the system size or time increases, leading to the formation of universality classes. One such class is the KPZ universality class, which includes randomly growing interfaces, interacting particle systems, and directed polymers. This concept was first introduced by physicists Kardar, Parisi, and Zhang in 1985. We will discuss some key results from the past twenty-five years related to the KPZ universality class, focusing on the last passage percolation models.
Location: Gibson Hall 126A
Time: 3:30 pm
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Wednesday, November 13
AMS / AWM Faculty Talk
Topic: An Introduction to Quantum Error Correction
Mahir Can - Tulane University
Abstract: Quantum information theory sits at the crossroads of physics, mathematics, and computer science, harnessing the immense potential of quantum mechanics for computation. However, this potential is fragile, susceptible to errors from noise and imperfections. To overcome these challenges, quantum error correction (QEC) emerges as a vital tool. In this talk, I will explain the core principles of classical and quantum error correction in elementary terms. I will then explain how a modification of the underlying measurement technique can pave the way for advancements in QEC.
Location: Gibson Hall 310
Time: 4:00pm
Wednesday, November 13
Integrability and Beyond!!!
Topic: Four weeks on the nonlinear Schrodinger equation
John Lopez – Tulane University
Abstract: This will be an informal working group, learning about the long time asymptotic behavior of the defocusing NLS equation, based on a combination of Riemann-Hilbert and d-bar techniques.
Today, John will give us an overview of where we are, and then continue explaining the Parabolic Cylinder parametrix and its use for good and for evil.
Location: Boggs 104
Time: 3:00pm
Wednesday, November 13
Algebra and Combinatorics Seminar
Topic: Slack Matrices of Affine Semigroups
Amy Wiebe - University of BC, Okanagan
Abstract: Slack matrices of polyhedral cones are an important class of nonnegative matrices. They offer canonical representations for cones that can be used for the study of realization spaces of polytopes and are a main ingredient in Yannakakis’ seminal result on lifts of polyhedral cones.
In this talk we generalize the notion of slack matrices to affine semigroups - a discrete analog of polyhedral cones - and present the corresponding result relating lifts of affine semigroups to nonnegative integer factorizations of their slack matrices. We use these generalizations to present new results on nonnegative integer rank of integer matrices.
Location: Richardson Building - RB-117 (G)
Time: 3:00pm
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Tuesday, November 12
Graduate Colloquium
Topic: The unique factorization problem on the rings of integers
Ngoc Trinh Le - Tulane University
Abstract: By the Fundamental Theorem of Arithmetic, we know that the ring of integer numbers ($\mathbb{Z}$) admits the unique factorization into prime integers. However, this property is not retained when we expand our ring $\mathbb{Z}$ (e.g. the ring $\mathbb{Z}[\sqrt{-19}]}$ where $35=5.7=(4+\sqrt{-19})(4-\sqrt{-19}))$. One approach to this problem is generalizing the factorization from the elements into the ideals level. Fortunately, in the cases of ring of integers, this unique factorization property turns out to be true on the latter sense. In this talk, I will give an brief introduction about the unique factorization property on both levels and show how we can use this property to solve some Diophantine equations.
Location: F. Edward Hebert Hall 213
Time: 3:30pm
Tuesday, November 12
SCMB-Simons Seminar
Topic: Modelling Malaria Elimination: Malaria in Zanzibar
Nakul Chitnis - Swiss Tropical and Public Health Institute
Abstract: Malaria cases can be classified as imported, introduced or indigenous cases. The World Health Organization’s definition of malaria elimination requires an area to demonstrate that no new indigenous cases have occurred in the last three years. Here, we present a stochastic metapopulation model of malaria transmission that distinguishes between imported, introduced and indigenous cases, and can be used to test the impact of new interventions in a setting with low transmission and ongoing case importation. We use human movement and malaria prevalence data from Zanzibar, Tanzania, to parameterise the model. We test increasing the coverage of interventions such as reactive case detection; implementing new interventions including reactive drug administration and treatment of infected travellers; and consider the potential impact of a reduction in transmission on Zanzibar and mainland Tanzania. We find that the majority of new cases on both major islands of Zanzibar are indigenous cases, despite high case importation rates. Combinations of interventions that increase the number of infections treated through reactive case detection or reactive drug administration can lead to substantial decreases in malaria incidence, but for elimination within the next 40 years, transmission reduction in both Zanzibar and mainland Tanzania is necessary.
Location: Stanley Thomas 316
Time: 11:00
Week of November 8 - November 4
Friday, November 8
Applied and Computational Math Seminar
Topic: Multi-Physics and Multi-Model Integration with the Suite of Nonlinear and Differential/Algebraic Equation Solvers (SUNDIALS)
Steven Roberts - Affiliation: LLNL
Abstract: Operator splitting is a simple but powerful technique to evolve coupled systems of differential equations forward in time. In the context of multi-physics simulations, where they are ubiquitous, operator splitting methods allow each process to be solved independently, possibly using a different integrator and time step tailored to the unique characteristics of that process. In this talk, I will discuss a new implementation of operator splitting methods in the SUNDIALS library and two applications. The first is a unique approach to leverage an approximate surrogate model to accelerate the integration of an expensive, full system of differential equations. Second, I will cover the benefits of high order integrators, including operator splitting, multirate, and implicit-explicit methods, for a cloud microphysics model based on a subset of the Predicted Particle Properties (P3) scheme used in the Energy Exascale Earth System Model (E3SM).
Location: Gibson Hall 325
Time: 3:00pm
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Thursday, November 7
Colloquium
Topic: The partition function and modular forms
Scott Ahlgren - University of Illinois at Urbana-Champaign (Host: Olivia Beckwith)
Abstract: The partition function p(n), which counts the number of ways to break a positive integer into parts, is a basic function in additive number theory and combinatorics.
Modular forms are hyper-symmetric complex functions which play a central role in number theory.
The fact that the generating function for partitions is a modular form opens the door to study its properties using the theory of modular forms. There are two branches to this study; the analytic side involves Maass forms and spectral theory and the arithmetic side involves holomorphic modular forms and Galois representations. In all cases the study can be viewed as a "testing ground” for more general theorems about modular forms.
I will discuss (in a non-technical way) the history of this subject as well as a number of results which have been proved with various collaborators in the last few years.
Location: Gibson Hall 126A
Time: 3:30 pm
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Wednesday, October 30
Algebra and Combinatorics Seminar
Topic: Tropicalization in Combinatorics
Greg Blekherman - Georgia Tech
Abstract: I will survey some recent applications of tropicalization in combinatorics. Tropicalization captures possible orders of growth of counted quantities (such as number of certain subgraphs of a graph, or the number matroids of certain rank). This provides a coarse picture of the combined behavior of several quantities, while exact counting results in combinatorics are usually capable of capturing the joint behavior of only two quantities.
Location: Richardson Building - RB-117 (G)
Time: 3:00pm
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Tuesday, November 5
Graduate Colloquium
Topic: Interpolation Problem and some Recent Developments
Dipendranath Mahato - Tulane University
Abstract: The classical Interpolation problem is a numerical analysis problem that deals on estimating some new data, from a known set of data. There have been different approaches to provide more refined numerical results on Interpolation, like Polynomial Interpolation, Spline Interpolation, Mimetic Interpolation etc. But in higher dimensions (specifically in $\mathbb{P}^N$), the main problem is to find the minimal degree homogeneous polynomial that vanishes on a finite set of points with given set of multiplicities. To deal such problem G.V. Chudnovsky and J.P. Demailly provided some conjectural bounds to the minimal degree, which I am going to discuss in my talk. I will also talk on some recent developments on this topic.
Location: F. Edward Hebert Hall 213
Time: 3:30pm
Tuesday, November 5
SCMB-Simons Seminar
Topic: Quantifying approximate symmetries in biological systems
Adriana Dawes - Ohio State
Abstract: Symmetry is a fundamental characteristic of natural systems, and is often linked to survival, reproductive success, and evolvability. While symmetry is ubiquitous and often intuitively obvious, biological symmetry is rarely perfect, making it challenging to apply mathematical definitions of idealized symmetry. To address this challenge, we developed a flexible, entropy-based method for quantifying symmetry that requires very little user input. I will highlight some novel insights arising from applications of this measure, including evidence for convergent evolution in flowering plants, classification of biopolymer networks, and visualization of the emergence and loss of symmetries in pattern formation systems.
Location: Stanley Thomas 316
Time: 11:00
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Monday, November 4
SCMB-Simons Seminar
Topic: Two results in Optimal Transport with applications to biomedical data
Natalia Kravtsova - Ohio State
Abstract: This talk presents two results in applied Optimal Transport. The first part of the talk is based on the joint work by N. Kravtsova, R. L. McGee II and A. T. Dawes (https://link.springer.com/article/10.1007/s11538-023-01175-y) and work by N. Kravtsova (https://arxiv.org/abs/2408.06525) on applications of the Gromov-Wasserstein distance defined by F. Memoli in 2011. We modify the NP-hard to compute Gromov-Wasserstein distance to construct a distance between time series that is computable in polynomial time. Our distance retains excellent performance of the Gromov-Wasserstein distance in machine learning tasks, including the ability compare objects in metric spaces with different dimensions. The second part of the talk is based on the work by N. Kravtsova, Asymptotic inference for Multimarginal Optimal Transport cost (submitted). Here we derive the asymptotic distribution of the empirical estimator for the Multimarginal Optimal Transport cost. We use the results to construct statistical inference procedures to compare probability measures. We illustrate the utility of the proposed approach on various datasets, including publicly available real data on cancers in US in 2004 – 2020.
Location: Gibson 126A
Time: 4:00
Week of November 1 - October 28
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Friday, November 1
Algebraic Geometry
Topic: Relative Ideals & Unipotent Numerical Monoids
Naufil Sakran - Tulane University
Abstract: In earlier work, we introduced unipotent numerical monoids as complement finite submonoids within a finitely generated submonoids in the unipotent linear algebraic group $(U(n,\mathbb{N})$. This talk further develops the theory by defining relative ideals and their associated invariants. We will introduce irreducible relative ideals and classify their structure with respect to symmetric and pseudo-symmetric ideals. Additionally, we will introduce the notion of reduction number, blowup, and the Arf closure of an ideal, and study the structure of a unipotent numerical monoid with respect to them.
In this talk, we will introduce and discuss our analogs of Reed-Solomon codes in a related context, also employing Hasse derivatives for our construction.
This is joint work with Mahir Bilen Can.
Location: Gibson Hall 126A
Time: 1:00 pm
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Thursday, October 31
Colloquium
Topic: Subsets of Groups in Public-Key Cryptography
Antonio Malheiro - Universidade Nova de Lisboa, Portugal (Host: Mahir Can)
Abstract: This presentation introduces group-based cryptography, focusing on a novel method that employs algebraic subsets instead of subgroups in public-key cryptography. The initial part reviews the essential concepts of public-key cryptography and the motivation for using groups, including a brief introduction to formal languages and algebraic subsets.
The second part presents an adaptation of well-known protocols, such as those by Shpilrain and Ushakov, where finitely generated subgroups are replaced by algebraic subsets. Examples are provided to illustrate how these subsets offer greater resistance to length- and distance-based attacks. The practical challenges associated with implementing this approach are also discussed. The presentation concludes by proposing new group-theoretic problems arising from this technique and exploring potential applications in other cryptographic systems.
This is joint work with André Carvalho (University of Porto, Portugal)
Location: Gibson Hall 126A
Time: 3:30 pm
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Wednesday, October 30
Algebra and Combinatorics Seminar
Topic: Asymptotic regularity and depth of invariant chains of edge ideals
Hop D. Nguyen - Institute of Mathematics, Vietnam Academy of Science and Technology
Abstract: We consider the asymptotic behavior of chains of monomial ideals that are stable under the action of the monoid Inc of increasing functions N → N. It is conjectured that for such chains, the regularity and projective dimension are eventually linear functions. We confirm the conjecture and provide complete description of the regularity and projective dimension (equivalently, the depth) in the case of chains of edge ideals. Remarkably, if the ideals in the chain are non-zero, then the regularity function is eventually constant with only two possible limiting values, and the same thing happens for the depth. Our results and their proofs also reveal many interesting combinatorial and topological properties of Inc-invariant chains of graphs and their independence complexes. Joint work with Tran Quang Hoa, Do Trong Hoang, Dinh Van Le, and Thái Thành Nguyễn.
Location: Richardson Building - RB-117 (G)
Time: 3:00pm
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Tuesday, October 29
Graduate Colloquium
Topic: The Darker Side of Mathematics
Nathaniel Vaduthala - Tulane University
Abstract: In this talk, we will discuss the mathematical contributions of history’s most notorious mathematicians, accompanied by a brief biographical overview of their lives.
Location: F. Edward Hebert Hall 213
Time: 3:30pm
Tuesday, October 29
SCMB-Simons Seminar
Topic: Questionable Cooperation Between Swimming Cells or Molecular Motors
Peter Kramer - Rensselaer Polytechnic Institute
Abstract: We examine the effective dynamics of two model systems consisting of stochastically active biological agents coupled together in a manner reflective of natural settings. The first concerns colonies of swimming flagellated cells such as choanoflagellates. We study how the swimming behavior of the colony could be derived from those of the constituent cells, including the effects of taxis and kinesis. Secondly, molecular motors are proteins in biological cells which perform various sorts of biophysical work. For two dissimilar types of kinesin transporting a common cargo, we provide approximate analytical characterizations for how the motors cooperate in carrying the cargo, with attention to incorporating slack in the tether connecting the motor with the cargo. The methodology combine multiscale asymptotic analysis, renewal theory, and first passage time calculations.
Location: Stanley Thomas 316
Time: 11:00
Week of October 25 - October 21
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Friday, October 25
Algebraic Geometry
Topic: Reed-Solomon Codes in the Uniform Tree Metric
Dillon Montero - Tulane University
Abstract: The classical theory of error-correcting codes primarily uses the Hamming metric to measure distances. Within this framework, Maximum Distance Separable (MDS) codes are highly valued due to their optimal parameters, enabling the correction of the maximum number of errors for a given code rate. In 1997, Rosenbloom and Tsfasman introduced the m-metric codes (now known as NRT metric codes), identifying analogs of Reed-Solomon codes that still possess the MDS property. Later, Skriganov provided an explicit construction of these codes using Hasse derivatives.
In this talk, we will introduce and discuss our analogs of Reed-Solomon codes in a related context, also employing Hasse derivatives for our construction.
This is joint work with Mahir Bilen Can.
Location: Gibson Hall 126A
Time: 1:00 pm
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Thursday, October 24
Colloquium
Topic: Extreme first passage times for populations of identical rare events
Jay Newby - University of Alberta (Host: McKinley)
Abstract: A collection of identical and independent rare event first passage times is considered. The problem of finding the fastest out of $N$ such events to occur is called an extreme first passage time. The rare event times are singular and limit to infinity as a positive parameter scaling the noise magnitude is reduced to zero. In contrast, previous work has shown that the mean of the fastest event time goes to zero in the limit of an infinite number of walkers. The combined limit is studied. In particular, the mean time and the most likely path taken by the fastest random walker are investigated. Using techniques from large deviation theory, it is shown that there is a distinguished limit where the mean time for the fastest walker can take any positive value, depending on a single proportionality constant. Furthermore, it is shown that the mean time and most likely path can be approximated using the solution to a variational problem related to the single-walker rare event.
Location: Gibson Hall 126A
Time: 3:30 pm
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Wednesday, October 23
Algebra and Combinatorics Seminar
Topic: Newton non-degenerate ideals in regular local domains
Vinh Nguyen - University of Arkansas
Abstract: The concept of Newton non-degenerate (NND) ideals in rings of holomorphic germs was introduced by M. J. Saia in 1996 to understand geometric invariants of complex-valued functions with an isolated singularity. We extend this notion to regular local domains and investigate algebraic invariants and properties of graded families of NND ideals in terms of associated convex bodies. This is joint work with Tai Huy Ha and Thai Thanh Nguyen.
Location: Richardson Building - RB-117 (G)
Time: 3:00pm
Week of October 18 - October 14
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Friday, October 18
Applied and Computational Math Seminar
Topic: Passive tracers advected by 2D Navier–Stokes equations with degenerate stochastic forcing
Keefer Rowan - NYU Courant
Abstract: I provide a high-level discussion of recent work with William Cooperman in which we prove the presence of various passive tracer phenomena in the physical model of a fluid with large-scale stirring given by the 2D Navier--Stokes equations with a degenerate stochastic forcing. This model was considered in the groundbreaking work of Hairer and Mattingly '06. The passive tracer phenomena were proved for the case of non-degenerate forcing by Bedrossian, Blumenthal, and Punshon-Smith '21, '22, '22. Our work can be viewed as a union of these frameworks.
Location: Gibson Hall 325
Time: 3:00pm
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Wednesday, October 16
Probability and Statistics
Topic: MCMC Importance Sampling via Moreau-Yosida Envelopes
Eric Chi – Rice University
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: Gibson 126
Time: 4:00 pm
Wednesday, October 16
Algebra and Combinatorics Seminar
Topic: Symbolic Powers of Matroids
Vinh Nguyen - University of Arkansas
Abstract: In general, it is quite hard to explicitly describe the minimal generators of the symbolic powers of any class of ideals, even in the case of square-free monomial ideals. In recent work with Paolo Mantero, we provide a structure result on the minimal generators of symbolic powers of a class of square-free monomial ideals that come from matroids. Matroids are combinatorial structures which abstract the structure of linear independence of vectors. Their Stanley-Reisner ideals have nice properties. For instance, every symbolic power is Cohen-Macaulay. In fact, they are the only square-free monomial ideals for which every symbolic power is Cohen-Macaulay.
In this talk I will introduce symbolic powers, matroids and their related ideals, and discuss our structure result along with various applications. If time permits, I would also like to talk about the minimal resolution of the symbolic powers of matroids. It turns out that their Betti numbers are supported on their symbolic powers. In fact, this is yet another characterization of matroids; they are the only square-free monomial ideals where this is true.
Location: Richardson Building - RB-117 (G)
Time: 3:00pm
Wednesday, October 16
Integrability and Beyond!!!
Topic: Four weeks on the nonlinear Schrodinger equation
Katerina Gkogkou and Ken McLaughlin – Tulane University
Abstract: This will be an informal working group, learning about the long time asymptotic behavior of the defocusing NLS equation, based on a combination of Riemann-Hilbert and d-bar techniques.
Location: Boggs 104
Time: 3:00pm
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Tuesday, October 15
Graduate Colloquium
Topic: Regularized Non-uniform Segments
Zheng Wang - Tulane University
Abstract: Regularized Stokeslet segment is a method used in fluid dynamics to model the motion of slender, flexible filaments (such as biological flagella, cilia, or fibers) immersed in a viscous fluid with low Reynolds numbers. The regularization parameter is usually an approximation of the filament cross-section radius. In this study, we present a modified model where the regularization parameter is not a constant but a continuous function. This method could be used to model filaments who has non-uniform radius.
Location: Stanley Thomas 316
Time: 3:30pm
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Week of October 11 - October 7
Thursday, October 10
Colloquium
Topic: Invariant Embeddings
Shlomo Gortler - Harvard (Host: Bernstein)
Abstract: Fix a dimension d and graph H, with n vertices and m edges. Let p be a configuration of n points in R^d. Then we can measure the configuration, mod the Euclidean group, by recording the squared length between each point pair associated with an edge of H. When H is generically globally rigid in d-dimensions, then this measurement map is an almost everywhere injective map from R^{nd}/E(d) to R^m. In this talk, I will discuss the general question of how one can create fully injective maps from R^{nd}/G to R^m where G is some group and m is roughly 2nd.
Location: Dinwiddie Hall 108
Time: 3:30 pm
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Wednesday, October 9
Probability and Statistics
Topic: SOFARI: High-Dimensional Manifold-Based Inference
Jinchi Lv – University of Southern California
Abstract: Multi-task learning is a widely used technique for harnessing information from various tasks. Recently, the sparse orthogonal factor regression (SOFAR) framework, based on the sparse singular value decomposition (SVD) within the coefficient matrix, was introduced for interpretable multi-task learning, enabling the discovery of meaningful latent feature-response association networks across different layers. However, conducting precise inference on the latent factor matrices has remained challenging due to the orthogonality constraints inherited from the sparse SVD constraints. In this paper, we suggest a novel approach called the high-dimensional manifold-based SOFAR inference (SOFARI), drawing on the Neyman near-orthogonality inference while incorporating the Stiefel manifold structure imposed by the SVD constraints. By leveraging the underlying Stiefel manifold structure that is crucial to enabling inference, SOFARI provides easy-to-use bias-corrected estimators for both latent left factor vectors and singular values, for which we show to enjoy the asymptotic mean-zero normal distributions with estimable variances. We introduce two SOFARI variants to handle strongly and weakly orthogonal latent factors, where the latter covers a broader range of applications. We illustrate the effectiveness of SOFARI and justify our theoretical results through simulation examples and a real data application in economic forecasting. This is a joint work with Yingying Fan, Zemin Zheng and Xin Zhou.
Location: Gibson 126
Time: 4:00 pm
Wednesday, October 9
Algebra and Combinatorics Seminar
Topic: Reconstructing configurations and graphs from unlabeled distance measurements
Shlomo Gortler - Harvard
Abstract: Place a configuration of n points (vertices) generically in R^d. Measure the Euclidean lengths of m point-pairs (edges). When is the underlying graph determined by these $m$ numbers (up to isomorphism)? When is the point configuration determined by these $m$ numbers (up to congruence). This question is motivated by a number of inverse problem applications. In this talk, I will talk about what is known about this question.
Location: Richardson Building - RB-117 (G)
Time: 3:00pm
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Week of October 4 - September 30
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Wednesday, October 2
Algebra and Combinatorics Seminar
Topic: Explicit Hypergeometric Modularity and Applications
Brian Grove - LSU
Abstract: The existence of hypergeometric motives predicts that hypergeometric Galois representations are modular. More precisely, explicit identities between special values of hypergeometric character sums and coefficients of certain newforms on appropriate arithmetic progressions of primes are expected. I will discuss a general method to prove these hypergeometric modularity results in dimensions two and three. Then I will use this method to explore new connections between hypergeometric functions and modular forms in the complex and p-adic settings. This is joint work with Michael Allen, Ling Long, and Fang-Ting Tu.
Location: Richardson Building - RB-117 (G)
Time: 3:00pm
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Tuesday, October 1
Graduate Colloquium
Topic: Tevelev Degrees of $\mathbb{P}^1$
Naufil Sakran - Tulane University
Abstract: In this talk, I will introduce the field of enumerative geometry and discuss recent developments in this area. Let $C$ be a general curve with $n$ marked points, and consider a degree $d$ map $\pi: C \to \mathbb{P}^1$, subject to specific incidence conditions.
Tevelev degree is defined as the number of such maps $\pi: C \to \mathbb{P}^1$. Interestingly, the computation of Tevelev degree presents intriguing connections with combinatorics, particularly through Dyck path counting and Schubert calculus. I will explore these topics, discussing their role in computing Tevelev degrees and presenting the latest formulas, as featured in recent work by R. Pandharipande, A. Cela, C. Lian, and others.
Location: MA(Norman Mayer)-G106
Time: 3:30pm
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Week of September 27 - September 23
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Tuesday, September 24
Graduate Colloquium
Topic: Modular Forms and Orders of Vanishing
Peter Marcus | Tulane University
Abstract: Modular forms are holomorphic functions with specific symmetry properties. Their Fourier expansions are generating functions for various sequences of interest, such as partition numbers and divisor sums. These functions live in finite-dimensional vector spaces, so by studying these spaces we can learn about these number-theoretic sequences. The dimensions of these spaces have well-known formulas, but there is no known formula for the maximal order of vanishing. In other words, if you write a row-reduced basis of Fourier expansions, when will the first nonzero Fourier coefficient of the last function occur? I will give an overview of this subject and progress on this problem.
Location: MA(Norman Mayer)-G106
Time: 3:30pm
Week of September 20 - September 16
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Wednesday, September 18
Integrability and Beyond!!!
Topic: Numerical method for the solution of Riemann-Hilbert problems
Katerina Gkogkou - Tulane University
Abstract: This week Katerina Gkogkou will explain elements of her project on a numerical method for the solution of Riemann-Hilbert problems.
Location: Gibson Hall 400A
Time: 1:00 pm
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Tuesday, September 17
Graduate Colloquium
Topic: A Kirchhoff Rod Model to Study Dynamics of Flexible Fibers and Rotating Helical Flagella in Stokes Flow
Rubaiyat Islam | Tulane University
Abstract: At the microscale, fluid motion is governed by Stokes equations. To study how microfibers behave in an ambient flow or how bacteria propel themselves with their rotating helical flagellum, we need a model to compute forces and torques along the fiber body and a way to include fluid interactions. Our computational framework is a Kirchhoff rod model coupled to regularized Stokeslet segments. This model takes advantage of the slenderness of fibers or flagella and uses a set of orthonormal triads to compute forces and torques. Passive filaments show rich shape deformations depending on their length and stiffness when subject to a background flow. Using a system of images for Stokeslet segments, we also show flagellated bacteria swimming in circles near a rigid wall.
Location: MA(Norman Mayer)-G106
Time: 3:30pm
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Week of September 2 - September 6
Friday, September 6
Applied and Computational Math Seminar
Topic: Interpretable AI: data driven and mechanistic modeling for chemical toxicity and drug safety evaluations.
Hao Zhu - Tulane University
Abstract: Abstract: Addressing the safety aspects of new chemicals has historically been undertaken through animal testing studies, which are expensive and time-consuming. Computational toxicology is a promising alternative approach that utilizes machine learning (ML) and deep learning (DL) techniques to predict toxicity potentials of chemicals. Although the applications of ML and DL based computational models in chemicals toxicity predictions are attractive, many toxicity models are “black box” in nature and difficult to interpret by toxicologists, which hampers the chemical risk assessments using these models. The recent progress of interpretable ML (IML) in the computer science field meets this urgent need to unveil the underlying toxicity mechanisms and elucidate domain knowledge of toxicity models. In this new modeling framework, the toxicity feature data, model interpretation methods, and the use of toxicity knowledgebase in IML development advance the applications of computational models in chemical risk assessments. The challenges and future directions of IML modeling in toxicology are strongly driven by heterogenous big data and newly revealed toxicity mechanisms. The big data mining, analysis, and mechanistic modeling using IML methods will advance artificial intelligence in the big data era to pave the road to future computational chemical toxicology and will have a significant impact on the risk assessment procedure and drug safety.
This is joint work with Alexander Dunlap.
Location: Gibson Hall 414 The Location is different than normal.
Time: 3:00pm
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Week of September 13 - September 9
Thursday, September 12
Colloquium
Topic: On the flow of zeros of derivatives of polynomials
Andrei Martinez-Finkelshtein - Baylor University (Host: Ken McLaughlin)
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.
I also wanted to mention that between the two return options I sent, the latest one is unreasonably expensive, so I am happy to fly in the early morning.
Location: Dinwiddie Hall 108
Time: 3:30 pm
