Week of April 26 - April 22
Thursday, April 25
Geometry & Topology
Topic: Distortion and Curvature in the Shape Reconstruction Problem
Will Lopez Tran | Tulane University
Abstract: We consider the complete topological reconstruction of a geodesic subspace of Euclidean space from the Vietoris--Rips complexes of a finite, noisy Euclidean sample. Instead of the Euclidean metric, our reconstruction technique uses a path-based metric on the sample to construct the Vietoris--Rips complexes.
We consider the restricted distortion, alpha-distortion, convexity radius, and Alexandrov curvature of a geodesic space as our sampling parameters. With restricted distortion, we guarantee a homotopy-equivalent reconstruction from the sample. With alpha-distortion, we guarantee homology groups equivalence and fundamental group equivlance with reconstruction from the sample. This study provides alternative sampling conditions to the existing and commonly used conditions based on weak feature size and $\mu$--reach.
Location: Newcomb 411 or zoom:https://tulane.zoom.us/s/7138114657
Time: 10:00am
Wednesday, April 24
Algebra and Combinatorics
Topic: Space, Spectra, and Semiring Systems of Equations
William Bernardoni - Case Western Reserve University
Abstract: In this talk we will give two motivations for building theory and methodologies around systems of equations over idempotent semirings. We will show how a theory of equations over idempotent semirings could be used in both real world applications, such as creating a solar system wide internet, as well as to create new mathematical tools in areas such as commutative algebra. We will first briefly discuss how the computational problem of routing in a deep space satellite network can be reduced to solving a matrix equation over specific idempotent semirings and how this model allows one to solve secondary problems such as determining storage requirements in a network. We will then see how idempotent semirings can be used as a tool to study commutative algebra. Through the Giansiracusa's generalized valuation theory one can study the spectrum and structure of commutative rings through valuations into idempotent semirings and the maps between them. We will conclude by examining what it means to "solve a system of equations" and how these problems can be modelled categorically.
Location: Gibson Hall 126A
Time: 3:00 pm
Wednesday, April 24
AMS/AWM Faculty Talk
Topic: A modular framework for Hurwitz class numbers
Olivia Beckwith - Tulane University
Abstract: In this talk I'll give a brief introduction to modular forms and discuss an analytic method of constructing them, focusing on new examples related to class numbers of binary quadratic forms.
Location: Gibson 126A
Time: 4:00 pm
Tuesday, April 23
Graduate Colloquium
Topic: An introduction to Algebraic Coding Theory
Dillon Montero | Tulane University
Abstract: Coding theory has many tools that come from Algebra and Algebraic Geometry. We will explore some of the important families of error-correcting codes that are used today.
Location: Gibson 126
Time: 3:30pm
Monday, April 22
Geometry & Topology
Topic: The wrappingness and trunkenness of volume-preserving flows
Peter Lambert-Cole - University of Georgia
Abstract: Link invariants of long pieces of orbits of a volume-preserving flow can be used to define diffeomorphism invariants of the flow. The wrapping number of a link in the solid torus and the trunk of a link can be generalized and define invariants of links with respect to a fibration on a 3-manifold. Extending work of Dehornoy and Rechtman, we apply this to define diffeomorphism invariants wrappingness and trunkenness of volume-preserving flows on 3-manifolds and interpret these invariants as obstructions to the existence of a global surface of section for the flow. We construct flows and show that wrappingness and trunkenness are not functions of the helicity.
Location: Gibson 308
Time: 2:00pm
Week of April 19 - April 15
Friday, April 19
Applied and Computational Math Seminar
Topic: Pointwise statistics of 2D stochastic heat equations
Cole Graham - Brown University
Abstract: The stochastic heat equation is a fundamental model in statistical physics featuring noise scaled by the solution itself. In this talk, I will discuss the pointwise statistics of a family of nonlinear stochastic heat equations in the critical dimension two. Curiously, these statistics evoke a "forward-backward" SDE and a quasilinear but deterministic heat equation. The well-posedness of the latter is delicate and consequential.
This is joint work with Alexander Dunlap.
Location: Gibson Hall 126
Time: 3:00pm
Thursday, April 18
Colloquium
Topic: Stabilizing phenomenon for incompressible fluids
Jiahong Wu - Notre Dame (Host: Zhao)
Abstract: This talk presents several examples of a remarkable stabilizing phenomenon. The results of T. Elgindi and T. Hou's group show that the 3D incompressible Euler equation can blow up in a finite time. Even small data would not help. But when the 3D Euler is coupled with the non-Newtonian stress tensor in the Oldroyd-B model, small smooth data always lead to global and stable solutions.The 3D incompressible Navier-Stokes equation with dissipation in only one direction is not known to always have global solutions even when the initial data are small. However, when this Navier-Stokes is coupled with the magnetic field in the magneto-hydrodynamic system, solutions near a background magnetic field are shown to be always global in time. The magnetic field stabilizes the fluid. Solutions of the 2D Navier-Stokes in R^2 with dissipation in only one direction are not known to be stable, but the Boussinesq system involving this Navier-Stokes is always stable near the hydrostatic equilibrium. The buoyancy forcing helps stabilize the fluid. In all these examples the systems governing the perturbations can be converted to damped wave equations, which reveal the smoothing and stabilizing effect.
Location: Gibson Hall 126A
Time: 3:30 pm
Wednesday, April 17
Algebra and Combinatorics
Topic: Expanding statistics in phylogenetic tree space
Gill Grindstaff - Oxford
Abstract: For a fixed set of n leaves, the moduli space of weighted phylogenetic trees is a fan in the n-pointed metric cone. As introduced in 2001 by Billera, Holmes, and Vogtmann, the BHV space of phylogenetic trees endows this moduli space with a piecewise Euclidean, CAT(0), geodesic metric. This has be used to define a growing number of statistics on point clouds of phylogenetic trees, including those obtained from different data sets, different gene sequence alignments, or different inference methods. However, the combinatorial complexity of BHV space, which can be most easily represented as a highly singular cube complex, impedes traditional optimization and Euclidean statistics: the number of cubes grows exponentially in the number of leaves. Accordingly, many important geometric objects in this space are also difficult to compute, as they are similarly large and combinatorially complex. In this talk, I’ll discuss specialized regions of tree space and their subspace embeddings, including affine hyperplanes, partial leaf sets, and balls of fixed radius in BHV tree space. Characterizing and computing these spaces can allow us to extend geometric statistics to areas such as supertree contruction, compatibility testing, and phylosymbiosis.
Location: Gibson Hall 126A
Time: 3:00 pm
Wednesday, April 17
Probability and Statistics
Topic: Empowering Business with Statistical Expertise: Working at Eli Lilly and Company
Rong Liu – Senior Director at Eli Lilly
Abstract: This presentation offers an introduction of Eli Lilly and Company, exploring its corporate culture and the sense of belonging experienced by its employees. It delves into the contributions made by individuals to foster a thriving environment at Lilly, including the opportunities presented by internships. Additionally, the presentation shares real-world cases illustrating the pivotal role of statistical expertise in driving business decisions within the pharmaceutical industry. These examples highlight the significance of statistical proficiency in shaping innovative strategies and advancing healthcare initiatives.
Location: Gibson 126
Time: 4:00 pm
Wednesday, April 17
AMS/AWM Faculty Talk
Topic: Rigidity theory and Gaussian graphical models
Daniel I. Bernstein - Tulane University
Abstract: Associated to each graph is something called a Gaussian graphical model. The minimum number of data points required to fit that model (loosely speaking) is called the maximum likelihood threshold of that graph. In this talk, I will show how one can understand the maximum likelihood threshold of a graph by viewing it as a mechanical structure in a high-dimensional space. This talk will be far more mathy and far less statisticsy than the abstract probably makes it seem.
Location: Gibson 126A
Time: 4:00 pm
Tuesday, April 16
Graduate Colloquium
Topic: On the zeros of a special family of Jacobi polynomials with non-classical parameters.
John Jairo Lopez Santander | Tulane University
Abstract: The distribution of zeros of orthogonal polynomials plays a pivotal role in various mathematical analyses. In particular, classical Jacobi polynomials, denoted as p_n(x;a,b), where both a and b are greater than -1, are well-known for having all their zeros confined within the interval (-1, 1) due to orthogonality properties on this interval. However, when either parameter a or b deviates from this classical range, the zeros may migrate into the complex plane, as orthogonality on the interval is no longer guaranteed. In this talk, we will explore a specific family of Jacobi polynomials with varying non-classical parameters and discuss a related Riemann-Hilbert Problem to investigate the distribution of their zeros.
Location: Gibson 126
Time: 3:30pm
Week of April 12 - April 8
Friday, April 12
Algebraic Geometry Seminar
Topic: Maximal Linear Sections of Grassmann and Schubert Varieties and Linear Error Correcting Codes
Sudhir Ghorpade - IIT Bombay
Abstract: Consider the Grassmann variety with its canonical Plucker embedding, or more generally a Schubert variety in a Grassmannian with its nondegenerate embedding in a subspace of the Plucker projective space. We can cut it by linear subspaces of a fixed dimension of the ambient projective space, and ask which of the linear sections are ”maximal”. The term ”maximal” can be interpreted in several ways and we will be particularly interested in maximality with respect to the number of rational points over a given finite field. In general, this is an open problem. This problem is also closely related to questions in the study of linear error correcting codes. We will quickly outline the relevant background, explain the connection with coding theory, and then describe some of the known results and problems.
Location: Gibson Hall 126A
Time: 3:00pm
Week of April 10 - April 8
Wednesday, April 10
Probability and Statistics
Topic: The Proximal Distance Principle for Constrained Estimation
Alfonso Landeros – University of California, Riverside
Abstract: Statistical methods often involve solving an optimization problem, such as in maximum likelihood estimation and regression. The addition of constraints, either to enforce a hard requirement in estimation or to regularize solutions, complicates matters. Fortunately, the rich theory of convex optimization provides ample tools for devising novel methods.
In this talk, I present applications of distance-to-set penalties to statistical learning problems. Specifically, I will focus on proximal distance algorithms, based on the MM principle, tailored to various applications such as regression and discriminant analysis. Special emphasis is given to sparsity set constraints as a compromise between exhaustive combinatorial searches and lasso penalization methods that induce shrinkage.
Location: Gibson 126
Time: 4:00 pm
Tuesday, April 9
Graduate Colloquium
Topic: Rees algebra of graded families of Newton-nondegenerate ideals
Vinh Pham | Tulane University
Abstract: In commutative algebra, if we have an algebra, one of the natural questions is if the algebra is Noetherian. The Noetherian property means that the algebra is finitely generated. It is a significant property because many commutative algebra results require or relate directly to the algebras' finiteness such as Hibert's fourteen problem. Now, given a graded family of ideals, we can consider the Rees algebra of this family. In this talk, we want to introduce the definition and basic properties of a special class of ideals called Newton-nondegenerate ideals and characterize the Noetherian property of the Rees algebra of a graded family of Newton-nondegenerate ideals using the concept of Newton-polyhedron.
Location: Gibson 126A
Time: 3:30pm
Week of April 6 - April 1
Friday and Saturday, April 5 and 6
Friday, April 6
Applied and Computational Math Seminar
Topic: A geometric multigrid method for unstructured grids and point clouds
Grady Wright - Boise state
Abstract: A new geometric multigrid method will be presented for solving linear systems that arise from discretizing elliptic PDEs on unstructured grids and point clouds. The method uses Poisson disk sampling for coarsening the vertices of an unstructured grid or the nodes of a point cloud, and new meshfree restriction/interpolation operators based on radial basis functions for transferring information between the coarsened levels. These components are then combined with standard smoothing and operator coarsening methods in a V-cycle iteration. We demonstrate the applicability of the method both as a solver and preconditioner for several problems based on different discretizations, including finite elements, discontinuous Galerkin, and generalized finite differences, and different geometrically complex domains, including 2D surfaces and graphs. We also perform a side-by-side comparison to algebraic multigrid (AMG) methods for solving the same systems.
Location: Gibson Hall 126
Time: 3:30pm
Friday, April 5
Algebraic Geometry Seminar
Topic: Poisson geometry of cluster algebras and their quantization
Bach Nguyen
Abstract: The relationship between Poisson geometry and cluster algebra was first studied by M. Gekhtman, M. Shapiro, and A. Vainshtein. Following their work, we study the global geometry picture of the affine Poisson varieties associated to a cluster algebra and its quantization, root of unity quantum cluster algebra. In particular, we prove that the spectrum of the upper cluster algebra, endowed with the GSV Poisson structure, has a Zariski open orbit of symplectic leaves and give an explicit description of it. Our result provides a generalization of the Richardson divisor of Schubert cells in flag varieties. Further, we describe the fully Azumaya loci of the root of unity upper quantum cluster algebras, using the theory of Poisson orders. This classifies their irreducible representations of maximal dimension. This is a joint work with Greg Muller, Kurt Trampel and Milen Yakimov.
Location: Gibson Hall 126A
Time: 3:00pm
Thursday, April 4
Colloquium
Topic: Peeling high-dimensional oranges
Anton Dochtermann - Texas State University (Host: Dr. Ha)
Abstract: A `simpicial complex' is a space that one can obtain by gluing together triangles, tetrahedra, and higher dimensional analogues called `simplices'. Simplicial complexes model a wide variety of topological spaces in a way that is accessible to calculation, and also define the Stanley-Reisner rings in commutative algebra. One way to study a simplicial complex X is via a `shelling': an ordering of the top dimensional faces of X that is similar to the way one would (un)peel an orange. The existence of a shelling has important consequences for its topological and algebraic properties of X. A well-known conjecture of Simon posits that one can always extend a given shelling of a shellable complex to the full skeleton of a simplex. We will discuss ideas and new results surrounding Simon's conjecture, including a proof for the special case of `vertex decomposable' complexes, connections to chordal graphs, and certain extremal cases in our search for counterexamples.
Location: Gibson Hall 126A
Time: 3:30 pm
Wednesday, April 3
Probability and Statistics
Topic: Statistical methods used for clinical researc
Hiya Banerjee – Director of Biostatistics at Eli Lilly
Abstract: In the technical presentation, I will showcase an innovative statistical method utilized to address a clinical question in the context of drug marketing. I will provide a comprehensive overview of how statisticians are involved in approaching and solving the problem, shedding light on the formulation of hypotheses and our collective endeavors to reach resolutions.
Besides that I will talk about how our daily responsibilities influence the trajectory of drug development. Furthermore, I will touch upon the essential skills and behaviors that aspiring students can cultivate to successfully embark on a career in the industry. The conversation will be informal, allowing for ample time for interactions and questions, providing insights into potential careers.
Time: 4:00 pm
Location: Gibson Hall 126
Week of March 22 - March 18
Friday, March 22
Applied and Computational Math Seminar
Topic: The restriction of the Laplacian operator on manifolds.
Padi Fuster Aguilera - University of Colorado Boulder
Abstract: On a Riemannian manifold, there is no canonical Laplace operator for vectors fields or forms, and it is not clear what is the “correct” Laplacian to use when formulating fluid dynamics equations. In this talk, we will walk through different approaches for obtaining a viscosity operator when considering a Riemannian submanifold in the Euclidean space, as well as present some concrete examples.
Location: Gibson Hall 126
Time: 3:00pm
Thursday, March 21
Colloquium
Topic: Mathematics of magic angles for twisted bilayer graphene
Maciej Zworski - UC Berkley (Host: Punshon-Smith)
Abstract: Magic angles refer to a remarkable theoretical (Bistritzer--MacDonald, 2011) and experimental (Jarillo-Herrero et al 2018) discovery, that two sheets of graphene twisted by a certain (magic) angle display unusual electronic properties such as superconductivity.
Mathematically, this is related to having flat bands of nontrivial topology for the corresponding periodic Hamiltonian and their existence be shown for the chiral model of twisted bilayer graphene (Tarnopolsky-Kruchkov-Vishwanath, 2019). A spectral characterization of magic angles (Becker--Embree--Wittsten--Z, 2021, Galkowski--Z, 2023) also produces complex values and the distribution of their reciprocals looks remarkably like a distribution of scattering resonances for a two-dimensional problem, with the real magic angles corresponding to anti-bound states. I will review various results on that distribution as well as on the properties of the associated eigenstates.
The talk is based on joint works with S Becker, M Embree, J Galkowski, M Hitrik, T Humbert and J Wittsten.
Location: Gibson Hall 126A
Time: 3:30 pm
Wednesday, March 20
AMS/AWM
Topic: Universality, random matrices, and data science through the lens of high-dimensional scale invariance
Gustavo Didier | Tulane University
Abstract: In this talk, we show how the topic of scale invariance (fractality) in high dimensions naturally brings together some major topics of modern mathematical research such as universality, random matrix theory, high-dimensional probability, data science and machine learning. No prior knowledge of these topics will be assumed.
Location: Gibson Hall 126A
Time: 4:00 PM
Wednesday, March 20
Algebra and Combinatorics
Topic: Invariants of SDP Exactness in Quadratic Programming
Julia Lindberg - UT Austin
Abstract: In this talk I will consider the Shor relaxation of quadratic programs by fixing a feasible set and considering the space of objective functions for which the Shor relaxation is exact. I first give conditions under which this region is invariant under the choice of generators defining the feasible set. I then will describe this region when the feasible set is invariant under the action of a subgroup of the general linear group. If time permits, I will conclude by applying these results to quadratic binary programs by giving an explicit description of objective functions where the Shor relaxation is exact and discuss algorithmic implications of this insight.
Location: Gibson Hall 126A
Time: 3:00pm
Tuesday, March 19
Graduate Colloquium
Topic: Geometric Realization: How to add shape to otherwise shapeless data sets
Will Tran - Tulane University
Abstract: We will learn how to add shape to data sets, even when those sets are not necessarily plottable or graphable in n-dimensional real space. Then, we’ll learn what the shape of our data sets could tell us about our data
Location: Gibson 126
Time: 3:30pm
Week of March 15 - March 11
Friday, March 15
Applied and Computational Math Seminar
Topic: Response theory for dissipative SPDEs.
Giulia Carigi
Abstract: A framework suitable to establish response theory for a class of nonlinear stochastic partial differential equations is presented. With response theory we mean in this context the following: one considers a dynamical system whose dynamical law depends on a parameter (here given by an SPDE where the parameter is in the forcing) and we say that one has a response theory if one can show a regularity in the dependence of the invariant measure on the parameter (here differentiability or Hölder continuity in weak topology). The results are applied to the 2D stochastic Navier-Stokes equation and the stochastic two-layer quasi-geostrophic model, an intermediate complexity model popular in the geosciences to study atmosphere and ocean dynamics. This work is jointly with Jochen Bröcker (University of Reading) and Tobias Kuna (University of L’Aquila).
Location: Gibson Hall 126
Time: 3:00pm
Thursday, March 14
Colloquium
Topic: The fractional Yamabe equation on homogeneous groups
Dimiter Vassilev - University of New Mexico (Host: Dr. Can)
Abstract: The general themes of the talk are Dirichlet forms, fractional (non-local) operators and associated Sobolev type spaces on groups of homogeneous type. I will recall some general motivating examples for considering non-local operators and particular equations before focusing on the respective questions in the setting of homogenous groups. The considered groups are not assumed to be Carnot groups or to satisfy a Hörmander type condition. I will describe a result on sharp asymptotic decay of solutions to non-linear equations modeled on the fractional Yamabe equation.
Location: Gibson Hall 126A
Time: 3:30 pm
Wednesday, March 13
Probability and Statistics
Topic: Overview of Master Protocol Trials and Statistical Considerations
Xiaoyun(Nicole) Li – Senior Director at BeiGene
Abstract: Master protocol is a trial structure that evaluates multiple diseases or multiple drugs (or drug combinations) within the same trial. There are three main types of master protocols, i.e., basket trials, umbrella trials and platform trials. Basket trials evaluate the same drug/drug combination in different diseases within the same trial, with the assumption that similar drug activities may seen and data may be borrowed. Various basket trial designs have been proposed over the years and I will give a flavor of it. Umberlla trials evaluate multiple drugs/drug combinations in the same disease within a trial. There is usually a shared control arm for all the different experimental arms to increase the efficiency. I will talk about the statistical consideration in terms of type I error control and other statistical errors in terms of umbrella trials. Platform trials refer to umbrella trials in a perpetual manner. Statistical considerations arise as to whether we could use the non-contemporaneous (non-concurrent) control and if so, how to use it. I will also talk about a phase 3 umbrella trial design as an illustration.
Location: 4:00pm
Time: Zoom with meeting ID: 932 4354 5612
Wednesday, March 13
Algebra and Combinatorics
Topic: Differential operators: simplicity and combinatorial properties of affine semigroup rings
Janet Vassilev - University of New Mexico
Abstract: We will discuss the ring of differential operators of an affine semigroup ring $R$ and how combinatorial properties of the affine semigroup translate into both the simplicity of the ring of differential operators,$ D(R)$, and the simplicity of the ring as a $D(R)$-module. This is joint work with Berkesch, Chan, Matusevich, Page and Traves.
Location: Gibson Hall 126A
Time: 3:00pm
Tuesday, March 12
Graduate Colloquium
Topic: A few non-classical time stepping methods to study fluid flow problems at low Reynolds number
Moslem Uddin - Tulane University
Abstract: Very often the dynamics of a system mimicking real-world phenomena seem to be well modeled by a system of ordinary differential equations. In practice, it's nearly impossible to solve such systems analytically, and this is why lots of efforts have been made to approximate those numerically. Usually, explicit methods(those that require knowledge from previous steps only) are very popular due to less computational effort required. However, those methods tend to return unstable solutions(the solution becomes unbounded in finite time). In this talk, I'll try to review a few non-explicit time integrators with the intention reduce the level of this type of shortcoming considering an example emerging from fluid flow governed by Stokes' equation.
Location: Gibson 126
Time: 3:30pm
Tuesday, March 12
Integrability and beyond!!!
Topic: Umbral calculus, a method for symbolic computation
Christophe Vignat - Tulane University
Abstract:
Umbral calculus is a computation method that represents a sequence of numbers or functions as a sequence of moments. It allows a significant simplification in the computation of some sequences, such as those associated with orthogonal polynomials.
This talk will introduce umbral calculus through some examples such as Hermite or Gegenbauer polynomials, and will show some applications.
Location: Dinwiddie 102
Time: 2:00pm
Week of March 4 - March 8
Friday, March 8
Applied and Computational Math Seminar
Topic: A tractable algorithm, based on optimal transport, for computing adversarial training lower bounds.
Nicolas Garcia Trillos - University of Wisconsin Madison
Abstract: Despite the success of deep learning-based algorithms, it is widely known that neural networks may fail to be robust to adversarial perturbations of data. In response to this, a popular paradigm that has been developed to enforce robustness of learning models is adversarial training (AT), but this paradigm introduces many computational and theoretical difficulties. Recent works have developed a connection between AT in the multiclass classification setting and multimarginal optimal transport (MOT), unlocking a new set of tools to study this problem. In this talk, I will leverage the MOT connection to discuss new computationally tractable numerical algorithms for computing universal lower bounds on the optimal adversarial risk. The key insight in the AT setting is that one can harmlessly truncate high order interactions between classes, preventing the combinatorial run times typically encountered in MOT problems. I’ll present a rigorous complexity analysis of the proposed algorithm and validate our theoretical results experimentally on the MNIST and CIFAR-10 datasets, demonstrating the tractability of our approach. This is joint work with Matt Jacobs (UCSB), Jakwang Kim (UBC), and Matt Werenski (Tufts).
Location: Gibson Hall 126
Time: 3:00pm
Wednesday, March 6
Algebra and Combinatorics
Topic: Pick's formula and Castelnuovo polytopes
Takayuki HIbi - Osaka University
Abstract: Pick’s formula and Castelnuovo polytopes Let P ⊂ Rd be a lattice polytope of dimension d. Let b(P) denote the number of lattice points belonging to the boundary of P and c(P) that to the interior of P. It follows from the lower bound theorem of Ehrhart polynomials that, when c(P) > 0, vol(P) ≥ (d · c((1) P) + (d − 1) · b(P) − d2 + 2)/d!, where vol(P) is the (Lebesgue) volume of P. Pick’s formula guarantees that, when d = 2, the inequality (1) is an equality. One calls P Castelnuovo if c(P) > 0 and if the equal sign holds in (1). A quick introduction to Ehrhart theory of lattice polytopes will be presented. Furthermore, a historical background on polarized toric varieties to explain the reason why one calls Castelnuovo will be briefly reviewed.
Location: Gibson Hall 126A
Time: 3:00 pm
Tuesday, March 5
Graduate Colloquium
Topic: Intro to Continued Fractions
Peter Marcus - Tulane University
Abstract: Continued fractions are representations of real numbers that use infinitely nested fractions, in contrast to decimal representations which use infinite sums. They provide excellent rational approximations and don't require choosing a base beforehand, which are benefits over decimal representations. However, there are issues of convergence and uniqueness which need to be addressed. I will discuss this as well as more examples and properties of continued fractions.
Location: Gibson 126
Time: 3:30pm
Monday, March 4
Joint AG & GT seminar
Topic: Knot Invariants, Categorification, and Representation Theory
Arik Wilbert - University of South Alabama
Abstract: I will provide a survey highlighting connections between representation theory, low-dimensional topology, and algebraic geometry central to my research. I will recall basic facts about the representation theory of the Lie algebra sl2 and discuss how these relate to the construction of knot invariants such as the well-known Jones polynomial. I will then introduce certain algebraic varieties called Springer fibers and explain how they can be used to geometrically construct and classify irreducible representations of the symmetric group. These two topics turn out to be intimately related. More precisely, I will demonstrate how one can study the topology of certain Springer fibers using the combinatorics underlying the representation theory of sl2. On the other hand, I will show how Springer fibers can be used to categorify certain representations of sl2. As an application, one can upgrade the Jones polynomial to a homological invariant which distinguishes more knots than the polynomial invariant. Time permitting, I will discuss how this picture might generalize to other Lie types beyond sl2.
Location: Gibson 308
Time: 2:00pm
Week of March 1 - February 26
Wednesday, March 1
Algebra and Combinatorics
Topic: Algebraic Matroids, Monodromy, and the Heron Variety
Barbara Prinari - University at Buffalo
Abstract: We present the inverse scattering transform to solve the Maxwell-Bloch system of equations that describes two-level systems with inhomogeneous broadening, in the case of optical pulses that do not vanish at infinity in the future. The inverse problem is formulated in terms of a suitable matrix Riemann-Hilbert problem, and the formulation of the direct scattering problem combines features of the methods with decaying as well as non-decaying fields. We also discuss the asymptotic state of the medium and of the optical pulse.
Location: Gibson Hall 126
Time: 3:00 pm
Thursday, February 29
Colloquium
Topic: A broad conjectural framework for the parity of eta-quotients
Fabrizio Zanello - Michigan Tech (Host: Dr. Ha)
Abstract:
One of the classical and most fascinating problems at the intersection between combinatorics
and number theory is the study of the parity of the partition function. Even though p(n) is widely
believed to be equidistributed modulo 2, progress in this area has always proven exceptionally hard. The
best results we have today, obtained incrementally over several decades by Serre, Soundararajan, Ono
and many others, do not even guarantee that, asymptotically, p(n) is odd for √x values of n ≤ x.
In this colloquium talk, we present a new, general conjectural framework that naturally places the
parity of p(n) into the much broader, number-theoretic context of eta-quotients. We discuss the history of
this problem as well as recent progress on our “master conjecture,” which includes novel results on multiand
regular partitions. We then show how seemingly unrelated classes of eta-quotients carry surprising
(and surprisingly deep) connections modulo 2. One instance is the following striking result: If any tmultipartition
function, with t ̸≡ 0 (mod 3), is odd with positive density, then so is p(n). (Note that
proving either fact unconditionally seems entirely out of reach with current methods.)
Throughout our talk, we will also try to give a sense of the many interesting mathematical techniques
that come into play in this area. They include a variety of algebraic and combinatorial ideas, as well as
tools from modular forms and number theory.
Much of this work is in collaboration with my former Ph.D. student S. Judge or with W.J. Keith (see
my papers in the J. Number Theory, 2015, 2018, 2021, 2022, and 2023; Annals of Comb., 2018; Int. J.
Number Theory, 2021 and 2023).
Location: Gibson Hall 126A
Time: 3:30 pm
Tuesday, February 27
Graduate Colloquium
Topic: Inverse PDE Problem: Comparison Between Numerical Method and Physics-informed Neural Network
Lan Trinh | Tulane University
Abstract: In our problem, we’re interested in the number of particles inside biological cells, which is governed by a Poisson spatial process with the intensity measure u(x). This u(x) is shown to satisfy a PDE with two unknown parameters z (source location) and lambda (nondimensional quantity) constructed from the diffusivity constant, emerging rate, and size of the cell. We also let u(x) equal 0 on the domain's boundary U, assuming that the particles are absorbed once hitting it. In this talk, I will discuss a simple version of this model in the 1D case using two methods: finite difference technique and Physics-informed Neural Network, then discuss the advantages, disadvantages as well as a potential combination of these methods for the full model in the 2D case.
Location: Gibson 126A
Time: 3:30pm
Week of February 25 - February 19
Clifford Lectures, February 22-25
Information: Here
Registration: Here
Wednesday, February 21
AMS/AWM
Topic: Mathematical Crossroads: some connections between very different areas of mathematics
Ken McLaughlin - Tulane University
Abstract: I will try to create a snapshot of the research interests of our small group by taking examples from combinatorics, complex analysis, probability theory, and other areas. There will be pictures and there will be mad, mad limits.
Location: Gibson Hall 126A
Time: 4:00 PM
Wednesday, February 21
Algebra and Combinatorics
Topic: Algebraic Matroids, Monodromy, and the Heron Variety
Taylor Brysiewicz - University of Western Ontario
Abstract: Heron's formula gives the area of a triangle in terms of the lengths of its sides. More generally, the volume of any simplex is determined by its edge-lengths via a Cayley-Menger determinant. In this talk, I will discuss which sets of volumes of faces of an n-simplex determine other volumes. The answer to this question is encoded in the algebraic matroid of the Heron variety. Whether this determination is in terms of a formula in terms of radicals is controlled by the monodromy groups of certain branched covers. We answer these questions for n<5 by combining techniques in computational group theory, computer algebra, field theory, and numerical algebraic geometry. Of particular focus is recovering the 10 edge lengths of a 4-simplex from its 10 triangular face areas, a problem motivated by applications in theoretical physics.
Location: Gibson Hall 126A
Time: 3:00 pm
Tuesday, February 20
Graduate Colloquium
Topic: Statistical Phylogenetic Approach to Characterize the Evolutionary Impact of Interlocus Gene Conversion (IGC)
Yufei Zou | Tulane University
Abstract: The Interlocus Gene Conversion (IGC) is a type of mutation that homogenizes repeated DNA sequences. Although substantial progress has been made with regard to inferring nucleotide substitutions that result from point mutations, IGC has typically been ignored when the genomes of related species are studied. This can potentially lead to misleading inferences about evolutionary history and process. Here we apply a composite likelihood approach to IGC inference. By applying this approach to data sets from segmentally-duplicated regions of primates, our results show that evolutionary changes from IGC occur at substantially different rates in different segmentally-duplicated regions.
Location: Gibson 126A
Time: 3:30pm
Week of February 16 - February 12
Friday, February 16
Applied and Computational Math Seminar
Topic: Self-Similar Blow up Profiles for Fluids via Physics-Informed Neural Network
Javier Gomez Serrano - Brown University
Abstract: In this talk I will explain a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution (or asymptotically self-similar solution) for different equations in fluid dynamics, such as Euler or Boussinesq. The new numerical framework is shown to be both robust and readily adaptable to several situations. Joint work with Tristan Buckmaster, Gonzalo Cao-Labora, Ching-Yao Lai and Yongji Wang.
Location: Gibson Hall 126
Time: 3:00pm
Thursday, February 15
Colloquium
Topic: DeLTA: Changing Teaching Evaluation through Departmental Action
Paula P Lemons - University of Georgia (Host: TBA)
Abstract: The University of Georgia DeLTA project works toward new core commitments in undergraduate STEM education: collaboration about teaching, basing educational decisions on evidence, and continuously improving our teaching. Modernizing our teaching evaluation is a primary way to achieve these commitments. In the DeLTA project we have achieved change in teaching evaluation by working at the departmental level. A leadership action team of department chairs convenes several times per year to learn about national models for effective teaching evaluation and to exchange ideas about the teaching evaluation practices in their units. Department chairs recruit faculty members who collaborate with faculty from other units to understand, revise, and implement new teaching evaluation practices, such as peer observation by trained peers and instructor self-reflection. The change we have achieved at the department level has been facilitated by changes in policy at the university level. DeLTA research shows that change takes place in departments at different rates and suggests factors that may contribute to departmental outcomes. This seminar will present the UGA DeLTA model, including principles and details about implementation, and will explain our research findings.
Location: Gibson Hall 126A
Time: 3:30 pm
Week of February 9 - February 5
Wednesday, February 7
AMS/AWM
Topic: Tropical Geometry
Kalina Mincheva - Tulane University
Abstract: In this talk I will give a brief overview of tropical geometry and the philosophy behind it. I will introduce algebraic varieties and their tropical counter parts. I will give some applications and open problems associated to them related to toric degenerations and dual curves.
Location: Gibson Hall 126A
Time: 4:00 PM
Wednesday, February 7
Algebra and Combinatorics
Topic: On partial trace ideals of one-dimensional local rings
Souvik Dey - Charles University, Czech Republic
Abstract: In this talk, based on joint work with S. Kumashiro, we define and study a slight generalization of the notion of partial trace ideals and h-invariant of S. Maitra. We show that for one-dimensional local rings, h-invariant of a module is finite if and only if the co-length of its trace is so. For ideals in nice enough local domains of dimension one, we give an explicit tangible formula for the h-invariant. We also discuss some characterizations of rings, including three-generated numerical semigroup rings, whose canonical ideal have low h-invariant, and how the h-invariant of the canonical module changes with respect to forming fiber products and gluing of numerical semigroup rings.
Location: Gibson Hall 126A
Time: 3:00 pm
Tuesday, February 6
Graduate Colloquium
Topic: Rigid Microspheres in a Stokes Fluid: Motion Due to White Noise
Irene Erazo Estrada | Tulane University
Abstract: This talk will center around the dynamic behavior of small spherical particles subjected to externally applied random forces while immersed in a viscous fluid. In contrast to the stochastic immersed boundary method which averages fluctuating random forces within the particle location, here, these forces are in the surrounding fluid, external to the particle surfaces.
Location: Gibson 126A
Time: 3:30pm
Week of February 2 - January 29
Friday, February 2
Applied and Computational Math Seminar
Topic: The planar Coulomb gas on a Jordan curve
Klara Courteaut - NYU Courant
Abstract: The eigenvalues of a uniformly distributed unitary matrix (CUE) have the physical interpretation of a system of particles subject to a logarithmic pair interaction, restricted to lie on the unit circle and at inverse temperature 2. In this talk, I will present a more general model in which the unit circle is replaced by a sufficiently regular Jordan curve, at any positive temperature. In a paper with Johansson, we obtained the asymptotic partition function and the Laplace transform of linear statistics at any positive temperature. These can be expressed using either the exterior conformal mapping of the curve or its associated Grunsky operator.
Location: Gibson Hall 126
Time: 3:00pm
Friday, February 2
Applied Math / Probability and Statistics
Topic: Stochastics in medicine: Delaying menopause and missing drug doses
Sean Lawley - University of Utah
Abstract: Stochastic modeling and analysis can help answer pressing medical questions. In this talk, I will attempt to justify this claim by describing recent work on two problems in medicine. The first problem concerns ovarian tissue cryopreservation, which is a proven tool to preserve ovarian follicles prior to gonadotoxic treatments. Can this procedure be applied to healthy women to delay or eliminate menopause? How can it be optimized? The second problem concerns medication nonadherence. What should you do if you miss a dose of medication? How can physicians design dosing regimens that are robust to missed/late doses? I will describe (a) how stochastics theory offers insights into these questions and (b) the mathematical questions that emerge from this investigation. The first problem is based on joint work with Joshua Johnson (University of Colorado School of Medicine), John Emerson (Yale University), and Kutluk Oktay (Yale School of Medicine).
Location: 12:00 pm
Time: Stanley Thomas 316
Thursday, February 1
Colloquium
Topic: Exponential generating functions and their congruences in enumerative combinatorics
Ira Gessel - Brandeis University (Host: Amdeberhan)
Abstract: Enumerative combinatorialists study sequences of integers that count things, and some of us like to find congruences for these integers. I will talk about sequences that have nice exponential generating functions, which are power series in which the numbers of interest are the coefficients of x^n/n!. An important example is the exponential generating function exp(exp(x) -1) for the Bell numbers, which count partitions of a set. I will first discuss how exponential generating functions are used in enumeration. Then I will discuss three methods for finding congruences for coefficients of exponential generating functions. The first method (which does not actually use the generating function) is the combinatorial method: Suppose that we have a finite group acting on a set S. If every element of S is in an orbit of size divisible by m, then the size of S is divisible by m. The second method, the umbral method, works with recurrences that are not so easily derived directly from generating functions. The third method uses the algebra of exponential generating functions modulo a prime, and differential operators on this algebra.
Location: Gibson Hall 126A
Time: 3:30 pm
Tuesday, January 30
Graduate Colloquium
Topic: Student Activities in Mathematics at Tulane
Sang-Eun Lee - Tulane University
Abstract: We will wrap up the activity we did last semester and propose this semester's events.
Location: TBA
Time: 3:30pm
Week of January 19 - January 15
Thursday, January 18
Math Club
Maggie Lai, Tulane Math Club President
Topic: Floer homology and algebraic geometry
Nikolai Saveliev - University of Miami (Host: Komendarczyk)
Abstract: Machine learning is quickly becoming embedded in everyday applications. It’s becoming essential for
students and educators to adopt this technology to solve complex real-world problems. MATLAB and
Simulink provide a flexible and powerful platform to develop and automate data analysis, deep learning,
AI, and simulation workflows in a wide range of domains and industries. In this workshop we will
introduce machine learning with MATLAB. We will utilize a previously trained network and modify it,
using the MATLAB Deep Network Designer. The Deep Network Designer allows you to interactively
build, visualize, and train neural networks. Individuals can generate the code for the neural network and
fine-tune parameters. Users can use popular pre-trained networks or construct their own. We will also
look at the MATLAB Classification Learner to run several models on a single data set. These visual
approaches create a more efficient workflow.
Jon Loftin is a Customer Success Engineer at MathWorks. Jon’s background is in mathematics. More
specifically, implementing mathematics in a computer. He holds degrees in mathematics: a BS from
Southern Arkansas University, a MS from the University of Arkansas, and a Ph.D. from Texas Tech
University. He has had years of teaching experience, from teaching at the Naval Nuclear Power School to
teaching as an Assistant Professor. Jon’s research focus is building efficient integration techniques in
finite element methods.
Location: Newcomb Institute 300, Diboll Gallery (3rd floor of Commons)
Time: 5:00-6:30PM
