**Week of May 5 - May 1**

**Topic: **

**N/A | Tulane University**

**Abstract: **

We will finalize the speakers for next semester's Graduate Colloquiums and will run elections for AWM, AMS, and SIAM officer positions.

**Location: Gibson 414**

**Time:** 4:00pm

**Week of April 28 - April 24**

**Topic: **

**Iryna Egorova | ILTPE, Kharkiv**

**Abstract: **The long-time asymptotics of steplike solutions of the Korteweg – de Vries equation on constant backgrounds has long been well studied at the physical level of rigor. This talk will present some recent mathematically rigorous results that refine and justify these asymptotics. We will start with an introduction to the classical scattering theory for the Schrodinger operator with fast decaying potential. We then briefly discuss the basics of two most common methods of long-time asymptotic analysis of the integrable systems: the Inverse Scattering Transform and the Nonlinear Steepest Descent, and their applicability to the analysis of the KdV steplike solutions in soliton regions. In addition, we will get acquainted with rigorous asymptotics of the KdV rarefaction and shock waves.

**Location: **Gibson 325

**Time:** 3:00pm

**Topic: **

**Zilin Li | Indiana University, Biostatistics**

**Abstract: **

Large-scale whole-genome sequencing (WGS) studies have enabled the analysis of rare variant associations with complex human diseases and traits. Variant set analysis is a powerful approach to studying rare variant associations. However, existing methods have limited ability to define the variant set in the genome, especially for the noncoding genome. We propose a computationally efficient and robust rare variant association-detection framework, STAARpipeline, to automatically annotate a WGS study and perform flexible rare variant association analysis, including gene-centric analysis and fixed-window and dynamic-window-based non-gene-centric analysis by incorporating variant functional annotations. In gene-centric analysis, STAARpipeline groups coding and noncoding variants based on functional categories of genes and incorporate multiple functional annotations. In non-gene-centric analysis, in addition to fixed-size sliding window analysis, STAARpipeline provides a data-adaptive-size dynamic window analysis. All these variant sets could be automatically defined and selected in STAARpipeline. STAARpipeline also provides analytical follow-up of dissecting association signals independent of known variants via conditional analysis. We applied the STAARpipeline to analyze the total cholesterol in 30,138 samples from the NHLBI Trans-Omics for Precision Medicine (TOPMed) Program. All analyses scale well in computation time and memory. We discover several potentially new significant associations with lipids. In summary, STAARpipeline is a powerful and resource-efficient tool for association analysis of biobank-scale WGS studies.

**Location: Dinwiddie Hall 103**

**Time:** 2:00pm

*Thursday, April 27*

**Topic: A problem on binomial coefficients, and invariable generation of finite simple groups.**

John Shareshian | WUSL (Host: Can)

**Abstract: **I will discuss joint work with Russ Woodroofe (University of Primorska) and, if time permits, further work with Russ and Bob Guralnick (USC).

Given a positive integer n, consider the set NTBC(n) of nontrivial binomial coefficients nCk, 0<k<n. A theorem of Kummer implies that the greatest common divisor of the elements of NTBC(n) is 1 unless n is a prime power. We aim to partition NTBC(n) into as few subsets as possible, so that the gcd of the elements of each subset is larger than 1. Let B(n) be the smallest number of subsets in such a partition. (So, BC(n)=1 if and only if n is a prime power.) We know that BC(n) is at most 2 for all n up to 10^{15} and that the set of n for which BC(n)=2 has asymptotic density 1 in the set of all positive integers. I will explain these facts and discuss how this problem arose when we studied certain topological spaces associated to finite groups. If time permits, I will discuss related problems in group theory.

**Location: **Gibson Hall 414

**Time:** 3:30 pm

**Topic: **

**Dipendranath Mahato | Tulane University**

**Abstract: **

The problem of interpolation is well known. But what happens when we move to a higher dimension space over any arbitrary Field! While investigating such problem, Gregory Volfovich Chudnovsky gave a conjectural lower bound for the minimal degree of a homogeneous polynomial in the Polynomial ring that vanishes on a given finite set of points with a given set of multiplicities. Later Waldschmidth Constant came into the picture, but how this famous geometrical problem boils down to the Containment Problem of Symbolic Powers and Ordinary Powers is the primary objective of my talk. I will also talk on some recent developments in this area.

**Location: Gibson 414**

**Time:** 4:00pm

**Week of April 21 - April 17**

*Thursday, April 20*

**Topic: Simple ways of encoding roots of polynomials and bounds on number fields**

Robert Lemke Oliver | Tufts (Host: Beckwith)

**Abstract:** From an algebraic perspective, the square root of 2 can be represented quite simply by saying that it's a root of the polynomial x^2-2. This is also essentially the simplest way to represent this number. Similarly, the polynomial x^3-5 is the simplest way to represent the irrational number that's a cube root of 5. But what about the polynomial x^5 - 7810*x^3 - 121055*x^2 + 2116510*x + 18532349? It can't be factored and its roots are all irrational numbers, but is this complicated polynomial really the "simplest" way to encode those roots? It shouldn't be obvious either way! In this talk, I'll tell you how with a little bit of extra information about the polynomial (its Galois group) it's often possible to encode the roots much more efficiently. Pulling back the curtain, this talk is really about studying number fields of bounded discriminant, and is based on joint work with Frank Thorne and others.

**Location: **Gibson Hall 414

**Time:** 3:30 pm

**Topic: **Lie-like structures on parallelizable manifolds

**Sergey Grigorian | The University of Texas Rio Grande Valley**

**Abstract: **In this talk we will explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map $\rho_p:\mathfrak{l} \longrightarrow T_p \mathbb{L}$ for each point $p \in \mathbb{L}$, where $\mathfrak{l}$ is some vector space. This allows us to define a particular class of vector fields, known as fundamental vector fields, that correspond to each element of $\mathfrak{l}$. Furthermore, flows of these vector fields give rise to a product between elements of $\mathfrak{l}$ and $\mathbb{L}$, which in turn induces a local loop structure (i.e. a non-associative analog of a group). Furthermore, we also define a generalization of a Lie algebra structure on $\mathfrak{l}$. We will describe the properties and applications of these constructions.

**Location: **Gibson Hall 126

**Time:** 3:00 pm

**Topic: **

**Oliver Orejola | Tulane University**

**Abstract: **

Spectral clustering is a popular unsupervised learning technique in modern data science. In this talk, I will introduce spectral clustering techniques as well as the relevant spectral graph theory in order to understand the heart of these methods. I will also discuss some applications and results relevant to my own research.

**Location: Gibson 414**

**Time:** 4:00pm

*Monday, April 17*

**Topic: ***Convexity Defect Functions and Reconstruction with Cech complexes (part 2)*

**William Tran | Tulane**

**Abstract: **

We discuss previous results from Attali, Lieutier, Salinas. Given a set of points that sample a shape, can we give conditions -- in terms of convexity of the shape -- that guarantee that a Cech complex built from our sampled points is homotopy equivalent to our shape? We will review the topological results from the previous discussion, then focus on geometric results.

**Location: **DW-103 **(special time and location)**

**Time:** 2:00pm ** (special time and location)**

**Week of April 14 - April 10**

**Topic: **

**Ben Southworth | Los Alamos National Labs**

**Abstract: **Simulating multiphyics phenomena on the computer is a complex task, pulling from many branches of physics and mathematics. Broadly, the goal is to construct robust numerical methods with high physics fidelity, that can be run in parallel environments with thousands of CPUs or GPUs. Some of the challenges include high dimensionality of problems (>>3), large changes in scale of the physical behavior (space and time), and stiff nonlinear coupling between different variables or physics. In this talk I will discuss the numerical solution and evolution of transport equations and the coupling to hydrodynamics, with a particular focus on the time evolution and efficient implicit solution. The main objective is developing computationally feasible ways to approximate the physically stiff behavior. I will review linear algebra theory we have developed as well as physical insight that guides our approximations. We then apply our methods to radiative shocks and a hohlraum problem motivated by inertial confinement fusion.

**Location: **Gibson 325

**Time:** 3:00pm

**Topic: **Symmetric (tropical) rank 2 matrix completion

**May Cai | Georgia Tech**

**Abstract: **The matrix completion problem asks what partially-filled-in matrices can be "completed" to a matrix of certain desired properties. For this talk, we are concerned with completion to a symmetric rank 2 matrix. Equivalently, we may ask for the independent sets of the algebraic matroid arising from the variety of symmetric $$n \times n$$ rank 2 matrices. In this talk, we describe a combinatorial characterisation of these independent sets. We solve the problem by tropicalizing the variety, which is to say solving the analogous problem for symmetric tropical rank 2 matrices. Based on joint work with Cvetelina Hill, Kisun Lee, and Josephine Yu.

**Location: **Gibson Hall 126

**Time:** 3:00 pm

*Monday, April 10*

**Topic: ***Graphing, homotopy groups of spheres, and spaces of links and knots*

**Robin Koytcheff | University of Louisiana, Lafayette**

**Abstract: **

We show that the homotopy groups of spaces of 2-component long links, up to knotting, are given by homotopy groups of spheres in a range of degrees that depends on the dimensions of the source manifolds and target manifold. In one degree higher, joining components sends both a parametrized long Borromean rings class and a Hopf fibration to a generator of the first nontrivial homotopy group of the space of long knots. For spaces of two-component long links, we give generators of the homotopy group in this dimension in terms of this class from the Borromean rings and homotopy groups of spheres. A key ingredient in most of our results is a graphing map that increases source and target dimensions by one.

**Location: **Gibson Hall 310; **May be over Zoom**

**Time:** 3:00pm **Time Change for one Seminar**

**Week of March 31 - March 27**

*Thursday, March 30*

**Topic: A survey on mixed multiplicities**

Jonathan Montaño - ASU (Host: Ha)

**Abstract: **

Given a multigraded standard graded algebra, one can define a finite set of numbers called mixed multiplicities. These numbers agree with the multidegrees of multiprojective varieties in the case of algebraically closed fields. If one considers this construction for certain multigraded algebras, one obtains related notions of multiplicity such as mixed multiplicities of ideals. Mixed multiplicities can also be seen in other fields of mathematics and are related to Schubert polynomials, mixed volumes, Milnor numbers, and projective degrees of rational maps. In this talk I will survey the history, properties, and main applications of mixed multiplicities, as well as recent developments in this topic.

**Location: **Gibson Hall 414

**Time:** 3:30 pm

**Topic: **Transfer Systems

**Peter Marcus | Tulane University**

**Abstract: **Transfer systems are combinatorial objects that arise in equivariant homotopy theory. They are defined as a certain type of partial order on the set of subgroups of a fixed finite group. The central question is enumerating all possible transfer systems for a given group. I will discuss this and other related results.

**Location: **Gibson Hall 126

**Time:** 3:00 pm

**Topic: Deleterious CAFI promote more large and more small corals in reefs**

**Louis Nass | Tulane University**

**Abstract: **

We implement a deterministic size-structured population PDE model to simulate the size-density of a coral reef in a space limited environment. We show that this model has a solution that exists, is unique, and conditionally converges to its steady state solution in time. We introduce coral associated fish and invertebrate (CAFI) interactions and implement CAFI dependence on growth and life expectancy of corals. We compare and contrast different levels of CAFI influence and immigration to understand summary features of our coral reefs. We sample from our size-density to visualize sample environments, and utilize our sampling techniques to show that we can estimate the largest maximum sized coral in a given environment. Finally, we show that deleterious CAFI promote more small and more large corals than beneficial CAFI, and try to understand this from a biological standpoint.

**Location: Gibson 414**

**Time:** 4:00pm

*Monday, March 27*

**Topic: ***Vector bundles for data alignment and dimensionality reduction*

**Jose Perea | Northwestern University**

**Abstract: **

Vector bundles have rich structure, and arise naturally when trying to solve synchronization problems in data science. I will show in this talk how the classical machinery (e.g., classifying maps, characteristic classes, etc) can be adapted to the world of algorithms and noisy data, as well as the insights one can gain. In particular, I will describe a class of topology-preserving dimensionality reduction problems, whose solution reduces to embedding the total space of a particular data bundle. Applications to computational chemistry and dynamical systems will also be presented.

**Location: **Over Zoom

**Time:** 4:00pm **Time Change for one Seminar**

**Week of March 24 - March 20**

**Topic: **

**Stuart Humphries | University of Lincoln, UK**

**Abstract: **

The need to understand and predict the effect of micro-scale (<1mm) processes in the oceans is a pressing challenge, requiring the integration of several disciplines and across spatial and temporal scales. Interactions between marine microbes drive nutrient cycling and food webs in our oceans, and ultimately influence biogeochemistry on a global scale. Understanding these microscale processes is essential if we are to understand the dynamics of the oceanic system as a whole. At the microscopic scales at which the life of marine microbes unfolds, the physics is dominated by viscosity. Increasing viscosity slows down both the passive transport of solutes and particles and the swimming of motile microorganisms, and thus directly or indirectly affects all aspects of microbial life. Here I will show how we can reveal spatial heterogeneity of viscosity in planktonic systems by using microrheological techniques that allow measurement of viscosity at length scales relevant to microorganisms. I will show the viscous nature and the spatial extent of the phycosphere, the region surrounding phytoplankton, and discuss the implications of this variation for a number of areas, including how we might consider different diffusive situations in the oceans.

**Location: **Gibson 325

**Time:** 3:00pm

**Topic: **Hernán Iriarte | Uinversity of Texas, Austin

**Hernán Iriarte | Uinversity of Texas, Austin**

**Abstract: **We start by giving an overview of what is currently known about tropicalization of algebraic varieties with respect to valuations of rank different from one. In this context, a tropical variety is given by a fibration in which the base and each fiber are tropical varieties as usual. These fibrations admit the structure of a polyhedral complex with coefficients in the ordered ring R[x]/(xk). Moreover, we will show how in the case of hypersurfaces, we can completely understand the combinatorics of this fibration from a layered regular subdivision of its Newton polytope.

**Location: **Gibson Hall 126

**Time:** 3:00 pm

Zachary Bradshaw | Tulane University

**Abstract: **

Explorations in Mathematical Physics Special Function in Quantum Theory and Feynman integrals by the Method of Brackets

**Location: **Navy ROTC Building NA 210

**Time:** 11:00

**Topic: **

**Joshua Agbomola| Tulane University**

**Abstract: **

Ebola virus (EVD) is contingent to African countries and the drastic spread of EVD is devastating over the last decade. The dynamic transmission could be better understood when we consider the various possible transmission routes. Since Ebola virus can only be transmitted via direct pathway; we therefore consider the transmission from human to human, human to fruit bats, fruit bats to fruit bats, human to contaminated environmental surfaces (e.g bedlinen, clothing, doorknobs, needles, or any other medical instrument or other surfaces) and vice versa. A mathematical model was proposed to capture these transmission routes and we check for the necessary conditions to make sure the model is mathematically and biologically meaningful. At the end, the results of the model shown on the graph reveal that the burden of Ebola virus is non-decreased when we incorporate all the transmission pathways. So, considering multiple transmission routes helps in the disease surveillance.

Date & Time: March 21, 4:00 PM (CST)

Location: Gibson 414

**Location: Gibson 414**

**Time:** 4:00pm

*Monday, March 20*

**Topic: ***Polyak-Viro type formulas for high dimensional knots and links*

** Neeti Gauniyal | Kansas State University**

**Abstract: **

I will talk about the problem of finding a high dimensional analogue to Polyak-Viro type formulas given in the classical case of 1-dimensional knots in R^3. We obtained such formulas for invariants of 2- and 3-component links of dimension (2m-1) in R^{3m}. At the end, I will give a conjectural formula for embeddings of R^3 in R^6.

**Location: **Over Zoom

**Time:** 3:00pm

**Week of March 17 - March 13**

**Topic: **

**Angel Pineda | Manhattan College**

**Abstract: **

Magnetic resonance imaging (MRI) is a versatile imaging modality that suffers from slow acquisition times. Accelerating MRI would benefit patients and improve public health both by reducing the time they need to be in the scanner and the cost of healthcare. Under-sampling the acquired data reduces the scan time but creates challenges for creating clinically useful images. Two recent methods for reconstructing images from under-sampled data are compressed sensing with constrained reconstruction and neural networks. Most of the current research focuses on new neural network architectures or training schemes while using mean squared error (MSE) or structural similarity (SSIM) as loss functions. The goal of this project is to optimize the performance of constrained reconstruction and deep learning based on a model for the clinical task of detecting subtle lesions instead of MSE or SSIM. We developed and experimentally validated observer models for estimating ideal and human observer performance. We have found that commonly used metrics like MSE and SSIM over-estimate the benefits of regularization in constrained reconstruction. In neural network reconstructions, we have also seen hallucination artifacts which are captured by MSE and SSIM but do not affect human observer performance in a signal-known-exactly task with varying backgrounds.

**Location: **Gibson 325

**Time:** 3:00pm

*Thursday, March 16*

**Topic: **Number of common zeros of homogeneous polynomials over finite fields

Sudhir Ghorpade - Indian Institute of Technology Bombay (Host: Can)

**Abstract: **

It is elementary and well known that a nonzero polynomial in one variable of degree d with coefficients in a field F has at most d zeros in F. It is meaningful to ask similar questions for systems of several polynomials in several variables of a fixed degree, provided the base field F is finite. These questions become particularly interesting and challenging when one restricts to polynomials that are homogeneous, and considers zeros (other than the origin) that are non-proportional to each other. More precisely, we consider the following question:

Given a system of a fixed number of linearly independent homogeneous polynomial equations of a fixed degree with coefficients in a fixed finite field F, what is the maximum number of common zeros they can have in the corresponding protective space over F?

The case of a single homogeneous polynomial (or in geometric terms, a projective hypersurface) corresponds to a classical inequality proved by Serre in 1989. For the general case, an elaborate conjecture was made by Tsfasman and Boguslavsky, which was open for almost two decades. Recently significant progress in this direction has been made, and it is shown that while the Tsfasman-Boguslavsky Conjecture is true in certain cases, it can be false in general. Some new conjectures have also been proposed. We will give a motivated outline of these developments.

If there is time and interest, connections to coding theory or to the problem of counting points of sections of Veronese varieties by linear subvarieties of a fixed dimension will also be outlined.

This talk is mainly based on joint works with Mrinmoy Datta and with Peter Beelen and Mrinmoy Datta.

**Location: **Gibson Hall 414

**Time:** 3:30 pm

**Topic: **L-functions for Harmonic Maass Forms.

**Larry Rolen | Vanderbilt University**

**Abstract: **The theory of harmonic Maass forms and mock modular forms has seen an explosion of activity in the past 20 years, with applications to physics, partitions, enumerative geometry, and many other topics. Along the way, much has been developed in the theory of harmonic Maass forms. However, until recently, harmonic Maass form theory lacked analogues of key structures that exist for classical holomorphic modular forms and Maass waveforms, such as the theory of L-functions. Recent work with Diamantis, Lee and Raji will be described which gives the first general such theory. In particular, I will describe how we obtain new Weil-type Converse Theorems and a Voronoi-type summation formula in these settings. I will also describe connections with the construction of differential operators on these spaces and a more thorough explanation of a previous formula for a central L-value of the j-invariant, which had been discovered heuristically by Zagier and proven in that case by Bruinier, Funke, and Imamoglu.

**Location: **Gibson Hall 126

**Time:** 3:00 pm

*Monday, February 13*

**Topic: ***Persistent Homology, Merge Trees and Reeb Graphs*

** Tom Needham | Florida State University**

**Abstract: **

Topological Data Analysis is an approach to data science where the main idea is to featurize a dataset via topological methods, such as associating a sequence of homology groups to it. This homological signature is known as the persistent homology of the dataset. In this talk, I will discuss some enriched summaries of persistence - namely, decorated Reeb graphs and decorated merge trees - which capture richer topological information than standard persistent homology. Spaces of such objects admit natural metrics, and I will describe stability properties of these metrics. I will also discuss computational issues and applications to analysis of complex data. This is joint work with Justin Curry, Haibin Hang, Washington Mio, Osman Okutan and Florian Russold.

**Location: **Over Zoom

**Time:** 3:00pm

**Topic: **

**Alexander Hoover | Cleveland State University**

**Abstract: **

Many biomechanical systems are activated by a nervous system that initiates and coordinates muscular contraction. In these systems, there are a number of intrinsic time scales, such as the speed and firing frequency of an action potential or the natural vibrational frequency of an elastic appendage or body. In this talk, we explore the dynamics that neuromuscular activation has in fluid pumping systems and use numerical simulations to describe the interplay between active muscle contraction, passive body elasticity, and fluid forces. This model is then used to explore the interplay between the speed of neuromechanical activation, fluid dynamics, and the material properties of systems, and we use it to describe a phenomenon known as neuromechanical wave resonance. This research is important as a design principle for the actuation of tissue-engineered pumps and soft-bodied robotics.

**Location: **Stanley Thomas 316

**Time:** 3:00pm

**Week of March 10 - March 6**

Scientific Computing Around Louisiana (SCALA) March 10-11

**Topic: **

**Lisa Fauci | Tulane University**

**Abstract: **

The motion of actuated elastic structures in a fluid environment is a common element in many biological and engineered systems. I will present recent work on two very different systems at very different scales, The first is a caricature of the helical flagellar bundle of a bacterium, whose swimming performance improves when confined to a narrow tube. The second model organism I will discuss is the lamprey, the most

primitive vertebrate. Using a closed-loop model that couples neural signaling, muscle mechanics, fluid dynamics and sensory feedback, we examine the hypothesis that amplified proprioceptive feedback could restore effective locomotion in lampreys with spinal injuries.

**Location: *****Hebert Hall, Room 201*** Room Change

**Time:** 3:00pm Time Change

*Thursday, March 9*

**Topic: **Basic Todaism

Carlos Tomei - PUC-Rio (Host: Moll)

**Abstract: **

The Toda lattice is a wonderful mathematical object, its surprises are never ending. In this self-contained lecture, I will present explicit solution formulae and implications to numerical analysis and topology. Emphasis will be given to recent work with Leite, Saldanha and Martinez Torres.

**Location: **Gibson Hall 414

**Time:** 3:30 pm

**Topic: **

**Gene Kopp | LSU**

**Abstract: **

Class field theory abstractly specifies the abelian Galois extensions of a number field in terms of data intrinsic to the base field. The classical formulation involves ray class groups and associated ray class fields. Not every abelian extension is a ray class field, but every abelian extension is contained in some ray class field. There are also ring class groups associated to arbitrary orders in the base field, with associated ring class fields, this time not containing or generating arbitrary abelian extensions, but arising naturally, for example, in the theory of complex multiplication. We define a "ray class group of an order" and a "ray class field of an order," common generalizations of the ray and ring class concepts. We explain how these objects fit in to class field theory and its applications. Along the way, we encounter some of the pitfalls of working with non-maximal orders. This is joint work with Jeffrey Lagarias.

**Location: **Gibson Hall 126

**Time:** 3:00 pm

**Topic: **

**Peter Marcus | Tulane University**

**Abstract: **

Transfer systems are combinatorial objects that arise in equivariant homotopy theory. They are defined as a certain type of partial order on the set of subgroups of a fixed finite group. The central question is enumerating all possible transfer systems for a given group. I will discuss this and other related results.

**Location: Gibson 414**

**Time:** 4:00pm

**Week of March 3 - February 27**

**Topic: **

**David Herzog | Iowa State University**

**Abstract: **

We discuss the method of stochastic characteristics to help motivate laws of the iterated logarithm. This method is often used to solve second-order linear boundary-valued PDEs, such as the Dirichlet or Poisson Problem in a bounded domain in R^n. If the operator L defining the equation is uniformly elliptic in a neighborhood of the domain and the boundary is smooth, then we can always use this method with ease to find an expression for the unique classical solution, provided the data is smooth enough. However, if we drop the ellipticity assumption, say L is only hypoelliptic or "weakly" hypoelliptic (i.e. the randomness is degenerate), then the candidate expression typically makes sense, but it may no longer satisfy the equation. The problem moving from the elliptic to hypoelliptic settings is that the natural elliptic approximation of L may not approximate the defining stochastic characteristics near the boundary, even though the processes defined by these operators are close in a very strong sense. In order to determine when it does approximate L near the boundary, we establish laws of the iterated logarithm for the diffusion at time zero.

**Location: **Gibson Hall 325

**Time:** 3:00pm Time Change

**Topic: **

**Sang-Eun Lee | Tulane University**

**Abstract: **

In this talk, we discuss some immersed structures in the various fluid controlling the scale of Reynolds number from 0 to infinity. The immersed structure in the low Reynolds number flow is developed by the resistive force theory (RFT1), and the analogy on the high Reynolds number flow is called by the reactive force theory (RFT2). We mainly focus on the difficulties of each theory to develop them in computational and mathematical perspectives. For instance, the slender body theory is modified the non-local discrepancy of RFT1. If time permits, the slender body theory on the high Reynolds number flow can be discussed.

**Location: Gibson 414**

**Time:** 4:00pm

*Monday, February 13*

**Topic: **

**Thomas Brazelton | UPenn**

**Abstract: **

Classical enumerative geometry asks geometric questions of the form "how many?" and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of "equivariant enumerative geometry", which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the sum of regular representations of the orbits of solutions to an equivariant enumerative problem are conserved. We leverage this to compute the S4 orbits of the 27 lines on any symmetric cubic surface.

**Location: **DW-103

**Time:** 3:00pm

**Week of February 24 - February 20**

Spring 2023 Math For All February 24-25, 2023

**Topic: **

**Matias Delgadino | University Texas Austin**

**Abstract: **

In this talk, we will study the mean field limit of weakly interacting diffusions for confining and interaction potentials that are not necessarily convex. We explore the relationship between the large N limit of the constant in the logarithmic Sobolev inequality (LSI) for the N-particle system, and the presence or absence of phase transitions for the mean field limit. The non-degeneracy of the LSI constant will be shown to have far reaching consequences, especially in the context of uniform-in-time propagation of chaos and the behaviour of equilibrium fluctuations. This will be done by employing techniques from the theory of gradient flows in the 2-Wasserstein distance, specifically the Riemannian calculus on the space of probability measures.

**Location: **Gibson Hall 325

**Time:** 3:30pm Time Change

*Thursday, February 23*

**Topic: **Universality limits for orthogonal polynomials

Milivoje Lukic - Rice (Host: Moll)

**Abstract: **

It is often expected that the local statistical behavior of eigenvalues of some system depends only on its local properties; for instance, the local distribution of zeros of orthogonal polynomials should depend only on the local properties of the measure of orthogonality. This phenomenon is studied using an object called the Christoffel-Darboux kernel. The most commonly studied case is known as bulk universality,

where the rescaled limit of Christoffel-Darboux kernels converges to the sine kernel.

In this talk, we will survey this subject, prior results, and a recent result which gives for the first time a completely local sufficient condition for bulk universality. The new approach is based on a matrix version of the Christoffel-Darboux kernel and the de Branges theory of canonical systems, and it applies to other self-adjoint systems with 2x2 transfer matrices such as continuum Schrodinger and Dirac operators.

The talk is based on joint work with Benjamin Eichinger (Technical University Wien) and Brian Simanek (Baylor University).

**Location: **Gibson Hall 414

**Time:** 3:30 pm

**Week of February 17 - February 13**

**Topic: **

**Naufil Sakran | Tulane University**

**Abstract: **

On the wonderful occasion of Valentine's Day, I would like to discuss the stable marriage problem which relates to finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element. We then discuss its application in modern technology in the hope that the talk adds flavor to your wonderful day.

**Location: Gibson 414**

**Time:** 4:00pm

*Monday, February 13*

**Topic: **

**William Tran | Tulane University**

**Abstract: **

We discuss previous results from Attali, Lieutier, and Salinas. Given a set of points that sample a shape, can we give conditions -- in terms of convexity of the shape -- that guarantee that a Cech complex built from our sampled points is homotopy equivalent to our shape?

**Location: **DW-103 (special time and location)

**Time:** 2:00pm

**Week of February 10 - February 6**

**Topic: **

**Ken McLaughlin | Tulane University**

**Abstract: **

~~I will provide more information, including background information, about solitons and their interactions, some sort of a definition of a gas of solitons, and the interaction between a single soliton and a gas of solitons. I will finish with an explanation of an initial connection to random matrix theory, to introduce randomness into a large collection of solitons. This is joint work with Manuela Girotti, Tamara Grava, Robert Jenkins, and Alexander Minakov.~~

**Location: **Gibson Hall 325

**Time:** 3:00pm

*Thursday, February 9*

**Topic: **The mathematics of taffy pulling

Jean-Luc Thiffeault ** - **University of Wisconsin, Madison** (Host: **

Punshon-Smith**)**

**Abstract: **

Taffy is a type of candy made by repeated 'pulling' (stretching and

folding) a mass of heated sugar. The purpose of pulling is to get air bubbles into the taffy, which gives it a nicer texture. Until the late 19th century, taffy was pulled by hand -- an arduous task. The early 20th century saw an avalanche of new devices to mechanize the process. These devices have fascinating connections to the topological dynamics of surfaces, in particular with pseudo-Anosov maps. The motion of the pins of the taffy puller cab be related to orbits of singularities on closed surfaces of genus one and higher. Special algebraic integers such as the Golden ratio and the lesser-known Silver ratio make an appearance, as well as more exotic numbers. We examine different designs from a mathematical perspective, and discuss their efficiency. This will be a "colloquium style" talk that should be accessible to graduate students.

**Location: **Gibson Hall 414

**Time:** 3:30 pm

**Topic: **

**Haoxi Hu | Tulane University**

**Abstract: **

We have seen a lot of concepts from computability theory, like "computable", Turing Machine, NP and P etc. However, most people don't really have a chance to take a close look at some compatibility theory, so for this talk, I will introduce some basic definitions, theorems, and history from compatibility theory.

**Location: Gibson 414**

**Time:** 4:00pm

**Week of February 3 - January 30**

**Topic: **

**Ken McLaughlin | Tulane University**

**Abstract: **

I will describe the interaction between a single soliton and a gas of solitons, providing for the first time a mathematical justification for the kinetic theory as posited by Zakharov in the 1970s. Then, if time permits, I will explain an initial connection to random matrix theory, in order to introduce randomness into a large collection of solitons. This is joint work with Manuela Girotti, Tamara Grava, Robert Jenkins, and Alexander Minakov.

**Location: **Gibson Hall 325

**Time:** 3:00pm

**Topic: **

**Kalina Mincheva | Tulane University**

**Abstract: **

Working towards endowing tropical varieties with extra structure, we study the algebra of convergent tropical power series and the topological spaces (of prime congruences) it corresponds to. We characterize the (nice) prime congruences of this algebra and we show that the dimension behaves as expected.

**Location: **Gibson Hall 126

**Time:** 3:00 pm

**Topic: **

**Lan Trinh | Tulane University**

**Abstract: **

Spatial Point processes have been applied in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications,… In this talk, I will give an overview about some typical spatial point processes (Poisson, Cox, and Markov point processes) and explain their first- and second-order moments, the useful summary descriptions to explore the spatial point processes at the first glance.

**Location: **Stanley Thomas 316

**Time:** 4:00pm

**Week of January 27 - January 24**

**Topic: **

**Rudi Shuech | Tulane**

**Abstract: **

Flagella are crucial to the interactions of many microorganisms with their surrounding fluid environment. The single-celled dinoflagellates have a unique but remarkably conserved flagellation morphology: a trailing longitudinal flagellum and an exquisitely complex transverse flagellum that encircles the cell. What are the selective advantages offered by this arrangement? We investigated the dinoflagellate design in silico using a high-performance regularized Stokeslet boundary element method, comparing to µPIV observations of swimming cells and quantifying how the morphology affects swimming performance. We found that the helical transverse flagellum provides most forward thrust and, despite its near-cell position, is more hydrodynamically efficient than the trailing flagellum; however, the latter is nonetheless required to enable steering. Flagellar hairs and the sheet-like structure of the transverse flagellum allow dinoflagellates to exert strong propulsive forces and maintain high clearance rates without extending a long conventional flagellum far into the surroundings. This unique morphology has thus been essential to the evolution of the generally large, fast-swimming dinoflagellates.

**Location: **Stanley Thomas 316

**Time:** 3:00pm

*Monday, January 24*

**Topic: **

**Desmond Coles | University of Texas, Austin**

**Abstract: **

Tropicalization is the process by which algebraic varieties are assigned a "combinatorial shadow". I will review the notion of the tropicalization of a toric variety and recent work on extending this construction to spherical varieties. I will then present how one can construct a deformation retraction from the Berkovich analytification of a spherical variety to its tropicalization.

**Location: **Temporary location: GI 126

**Time:** 3:00pm

**Week of January 20 - January 16**

**Topic: **

**Melanie Tian | Tulane University**

**Abstract: **

We introduce the field of mathematical linguistics in two flavors. First we introduce mathematical methods in linguistics, where we give examples on verifying equivalences and non-equivalences using tools such as lambda calculus and determiners as relations. Then we look at two problems on deciphering unfamiliar writing systems: numeral system in Nahuatl, and Transcendental Algebra constructed by Jakob Linzbach.

**Location: **Stanley Thomas 316

**Time:** 4:00pm

**Week of January 20 - January 16**

**Topic: **

**Rudi Shuech | Tulane**

**Abstract: **

In this two-part talk, first I will summarize my previous work on the effects of curved-rod bacterial shapes on swimming performance and other ecologically important tasks. We used a regularized Stokeslet boundary element method to compute the motion of curved-rod microswimmers propelled by rotating helical flagella. We then showed that Pareto-optimal tradeoffs between efficient swimming, chemotaxis, and cell construction cost can explain the morphological diversity of extant curved bacterial species.

In the second part, I will transition to thinking about the complex environments that microorganisms swim through, which are often composed of a viscous fluid with suspended microstructures such as elastic polymers and filamentous networks. These microstructures can have similar length scales to the microorganisms, leading to complex swimming dynamics. Some microorganisms are also known to remodel the viscoelastic networks they move through. To gain insight into the coupling between the dynamics of the swimmer and the network, we combined our computational framework for microswimmer motion with a model of a discrete viscoelastic network. The network is represented by a cloud of points with virtual Maxwell element links, whose properties (i.e., stiffness, relaxation time) can have non-obvious effects on the swimmer dynamics. We model enzymatic dissolution of the network by bacteria or microrobots by breaking links based on their distance to the microswimmer. We investigate how swimming performance is affected by properties of the network and swimmer.

If time allows, I will also introduce our new work on microswimmers penetrating thin, membrane-like interfaces.

**Location: **Stanley Thomas 316

**Time:** 3:00pm

**Week of December 9 - December 5**

**Topic: TBA**

**Speaker - University (Host: TBA )**

**Abstract: **

**TBA**

**Location: **TBA

**Time:** 3:30 pm

**Topic: TBA**

**Speaker | University**

**Abstract:**

TBA

**Location: TBA**

**Time:** 4:00 PM

**Topic: TBA**

**Speaker | Tulane University**

**Abstract: **

TBA

**Location: **Stanley Thomas 316

**Time:** 5:00pm

*Monday, February*

**Topic: TBA**

**Speaker | Tulane University**

**Abstract: **

**TBA**

**Location: **TBA

**Time:** 3:00pm