**Algebra and Combinatorics Seminar**

**Topic: **

**Friedrich Wagemann | Universite de Nantes, France**

**Abstract:** Leibniz algebras generalize Lie algebras in the sense that the bracket is not necessarily antisymmetric. We will describe in our talk attempts to integrate Leibniz algebras into Lie racks. Here a rack is an algebraic structure more general than the notion of a group; in fact, one retains from the notion of a group only the conjugation map. M. Kinyon showed in 2007 that the tangent space of a Lie rack carries naturally the structure of a Leibniz algebra. S. Covez showed in 2010 that Leibniz algebras integrate into local Lie racks. Many other integration procedures have been proposed since then. We will focus on the integration procedure of Bordemann-Wagemann (2017), where a general Leibniz algebra is integrated into a Lie rack which is an affine bundle over a Lie group such that in case the Leibniz algebra is a Liealgebra, one obtains the standard integration of Lie algebras. The drawback of this procedure is that it is not functorial.

**Location: **Dinwiddie Hall 102

**Time: **3:00pm

**Applied and Computational Mathematics**

**Topic:** Contact geometry, Hamiltonization and applications

Alessandro Bravetti - IIMAS-UNAM

**Abstract: **The Hamiltonization problem on a smooth manifold is the question of whether a vector field admits a Hamiltonian formulation with respect to either a symplectic (resp. Poisson) or contact (resp. Jacobi) structure. In this talk we briefly introduce some concepts and properties of contact geometry and contact Hamiltonian systems and then we use the latter to address the Hamiltonization problem for different vector fields of interest in the physics and the statistics literatures. In particular, we consider examples from mechanics, thermodynamics, optimization and sampling and for each case we highlight the advantages of the proposed approach.

**Zoom:** TBA

**Time: **3:00 pm

**Joint AG & GT seminar**

**Topic: **

**All Invited - Tulane University**

**Abstract: **We will discuss the format of the seminar and perspectives for next academic year.

Cookies will be provided.

**Location: **Gibson Hall 400D

**Time:** 3:00 pm

**Algebra and Combinatorics Seminar**

**Topic: **

**Mark Skandera - Leigh University**

**Abstract:** For each element z of the symmetric group algebra, we define a symmetric generating function Y(z) = Σλ ελ(z) mλ, where ελ is the induced sign character indexed by λ. Expanding Y(z) in other symmetric function bases, we obtain other trace evaluations as coefficients. We show that all symmetric functions in spanZ{mλ} are Y(z) for some z in Q[Sn]. Using this fact and chromatic symmetric functions, we give new interpretations of permanents of totally nonnegative matrices. For the full paper, see arXiv:2010.00458v2.

**Location: **Dinwiddie Hall 102

**Time: **3:00pm

**Graduate Student Colloquium**

**Topic: **

**Alex Nisbet | Tulane University**

**Abstract: **Most courses and books in point-set topology restrict themselves to the context of Hausdorff spaces, typically motivated as a generalization of metric spaces. However, non-Hausdorff topologies are also very interesting! Often, they can be equivalently described in terms of order theory, and one ends up with a more "algebraic" stylization for topological concepts. Here we look at compactness from this perspective with the aim of developing a better intuition for what it "really means."

**Location: **Stanley Thomas 316

**Time: **4:00 pm

**Modular Forms**

**Topic: **

**Victor H. Moll - Tulane University**

**Abstract: TBA **

**Location:** Gibson Hall 126A

**Zoom access: **https://tulane.zoom.us/j/97255728739?pwd=T09nVENjK1Vnb0pHa1pXLzNwd3F2Zz09

**Time: 2:00**

**Joint AG & GT seminar**

**Topic: **

**Christopher Perez - Loyola University**

**Abstract: **In a remarkable series of papers Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers, and independently Olga Kharlampovich and Alexei Myasnikov did the same using equivalent structures they called regular NTQ groups. It was later proved by Chloé Perin that if H is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group G, then G is a tower over H. We prove a generalization of Perin’s result to toral relatively hyperbolic groups using JSJ and shortening techniques.

**Location: **Gibson Hall 400D

**Time:** 3:00 pm

**Colloquium**

**Topic: **

**Samuli Siltanen - Helsinki (Host: Glatt-Holtz)**

**Abstract: **The structure of vowel sounds in human speech can be divided into two independent components. One of them is the “excitation signal,” which is a kind of buzzing sound created by the vocal folds flapping against each other. The other is the “filtering effect” caused by resonances in the vocal tract, or the confined space formed by the mouth and throat. The Glottal Inverse Filtering (GIF) problem is to divide a microphone recording of a vowel sound into its two components. This “blind deconvolution” task is an ill-posed inverse problem. Good-quality GIF filtering is essential for computer-generated speech needed for example by disabled people (think Stephen Hawking). Also, GIF affects the quality of synthetic speech in automatic information announcements, digital assistants and car navigation systems. Accurate estimation of the voice source from recorded speech is known to be difficult with current glottal inverse filtering (GIF) techniques, especially in the case of high-pitch speech of female or child subjects. In order to tackle this problem, the present study uses two different solution methods for GIF: Bayesian inversion and alternating minimization. The first method takes advantage of the Markov chain Monte Carlo (MCMC) modeling in defining the parameters of the vocal tract inverse filter. The filtering results are found to be superior to those achieved by the standard iterative adaptive inverse filtering (IAIF), but the computation is much slower than IAIF. Alternating minimization cuts down the computation time while retaining most of the quality improvement.

**Zoom ID:** TBA

**Time: **3:30 pm

**Algebra and Combinatorics Seminar**

**Topic: **

**Hop D. Nguyen - Vietnam Academy of Science and Technology**

**Abstract:** Let Inc(N) be the monoid of increasing functions f:N->N on the positive integers. Let Sym(N) be the direct limit of the symmetric groups on n objects, when n tends to infinity. The monoids Inc(N) and Sym(N) act on the infinite polynomial ring k[X]=k[x_1,x_2,...] (where k is a fixed field) via endomorphisms. Ideals of k[X] which are invariants under the Inc(N)- or Sym(N)-action tend to have significant finiteness properties, due to their huge individual orbits. We discuss recent results and problems on the homological invariants of Inc(N)- and Sym(N)-invariant ideals. Joint work with Dinh Van Le.

**Location: **Dinwiddie Hall 102

**Time: **3:00pm

**Probability and Statistics**

**Topic: **

**Giorgio Cipolloni - Princeton Center for Theoretical Science (Didier)**

**Abstract: **We prove that the fluctuations of the eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk. These fluctuations appear on a non-natural scale, due to strong correlations between the eigenvalues. Then, motivated by the long time behaviour of the ODE \dot{u}=Xu, we give a precise estimate on the eigenvalue with largest real part.

**Location:** Gibson 126

**Time: **2:00 pm

**Graduate Student Colloquium**

**Topic: **

**Nestor F. Diaz Morera | Tulane University**

**Abstract: **I will be defining Schubert Varieties through a few cursory examples. To accomplish it, I will be talking about Grassmannian varieties and Flag varieties. It turns out, these objects belong to homogeneous spaces. Along the construction, we will arrive to some combinatorial gadgets.

**Location: **Stanley Thomas 316

**Time: **4:00 pm

**Modular Forms**

**Topic: **

**Olivia Beckwith - Tulane University**

**Abstract: **We'll continue discussing Hecke operators and we'll examine an example from algebraic number theory.** **

**Location:** Gibson Hall 126A

**Zoom access: **https://tulane.zoom.us/j/97255728739?pwd=T09nVENjK1Vnb0pHa1pXLzNwd3F2Zz09

**Time: 2:00**

**Joint AG & GT seminar**

**Topic: **

**G.V. Ravindra | University of Missouri-St. Louis**

**Abstract: **Given a homogeneous polynomial of degree d in n variables, a century old problem due to Dickson asks if some power of this polynomial can be expressed as the determinant of a matrix with smaller degree homogeneous polynomial entries in a non-trivial way. This talk will introduce a precise version of this question and show how this question is intricately related to the geometry of hypersurfaces of degree d in projective space.

**Location: **Gibson Hall 400D

**Time:** 3:00 pm

**Applied and Computational Mathematics**

**Topic:** TBA

Peter Straka - Facebook/Meta

**Abstract: **TBA

**Zoom:** TBA

**Time: **3:00 pm

*****Special Time and Day: Joint AG & GT Seminar*****

**Topic: **

**Daciberg Lima Goncalves - University of Sao Paulo, Brazil**

**Abstract: **

**Location: **Gibson Hall 400 A

**Time:** 1:00 pm

**Topic: **

**Thai Nguyen - Tulane University**

**Abstract: **The symbolic power $I^{(n)}$ of an ideal $I$, which is related to the primary decomposition of the ordinary power $I^n$, has been of interest to many mathematicians, especially in the field of commutative algebra and algebraic geometry for various reasons, one of those is the fact that symbolic powers encode vanishing condition of functions on geometric objects. Monomial ideals and ideals of points are among the most important classes of ideals in polynomial rings with many applications in other fields of study.

While questions regarding polynomials in the symbolic powers of ideals of points are equivalent to questions in polynomial interpolation problem, various questions regarding symbolic powers of monomial ideals and their invariant have interesting connection to questions in graph theory, convex (polyhedral) geometry, and integer programming. In this dissertation, we focus on the problem of comparing the symbolic powers and ordinary powers of an ideal and studying the closely related algebraic invariant. In particular, we prove stable Harbourne-Huneke containment between symbolic powers and ordinary powers of ideals of general points and derive Chudnovsky's and Demailly's Conjecture for those ideals. On the other hand, we use combinatorial data of the Newton-Okounkov body to study the Noetherian property and related algebraic invariant of the Rees algebra of a graded family, which is a generalization of family of symbolic powers, of monomial ideals.

**Zoom ID:** BO 122

**Time: **2:00 pm

**Colloquium**

**Topic: **

Carina Curto - Penn State** **

**Abstract: **Many networks in the nervous system possess an abundance of inhibition, which serves to shape and stabilize neural dynamics. The neurons in such networks exhibit intricate patterns of connectivity whose structure controls the allowed patterns of neural activity. In this work, we examine inhibitory threshold-linear networks whose dynamics are dictated by an underlying directed graph. We develop a set of parameter-independent graph rules that enable us to predict features of the dynamics from properties of the graph. These rules provide a direct link between the structure and function of inhibitory networks, yielding new insights into how connectivity may shape dynamics in real neural circuits. Graph rules also lead us to consider some natural topological structures, such as nerves and sheaves, stemming from various graph covers. We will illustrate the theory with some applications to central pattern generator circuits and other examples of neural computation.

**Zoom ID:** TBA

**Time: **3:30 pm

**Algebra and Combinatorics Seminar**

**Topic: **

**Hop D. Nguyen - Vietnam Academy of Science and Technology**

**Abstract:** There are two different notions for symbolic powers of ideals existing in the literature, one defined in terms of associated primes, the other in terms of minimal primes. Elaborating on an idea known to Eisenbud, Herzog, Hibi, and Trung, we interpret both notions of symbolic powers as suitable saturations of the ordinary powers. We prove a binomial expansion formula for saturated powers of sums of ideals. This gives a uniform treatment to an array of existing and new results on both notions of symbolic powers of such sums: binomial expansion formulas, computations of depth and regularity, and criteria for the equality of ordinary and symbolic powers. Joint work with H.T. Hà, Ạ.V. Jayanthan, and A. Kumar.

**Location: **Dinwiddie Hall 102

**Time: **3:00pm

**Graduate Student Colloquium**

**Topic: **

**Sinchita Lahiri - Tulane University**

**Abstract: **We come across mostly three different types of geometries in our surroundings; one is Euclidean geometry, and the other two are non-Euclidean geometry, namely Hyperbolic geometry and spherical geometry. This talk will be a description of a few highlights in the early history of non-Euclidean geometry. We will briefly discuss Hyperbolic geometry occurring in nature.

**Location: **Stanley Thomas 316

**Time: **4:30 pm

*****Special Colloquium*****

**Topic: **

**Charles Doran - University of Alberta**

**Abstract: **Calabi-Yau manifolds play a central role in algebraic geometry. We will briefly survey known constructions, working our way up in dimension, and focus on the geometric implications of nesting one Calabi-Yau manifold in another. Mirror symmetry — a phenomenon first suggested by physicists — links (families of) Calabi-Yau manifolds. Mirroring towers of Calabi-Yau manifolds leads us to propose a new conjecture that unifies mirror symmetry for Calabi-Yau manifolds and their Fano manifold cousins. The talk is designed to be broadly accessible to graduate students.

**Zoom ID:** BO 122

**Time: **3:00 pm

**Modular Forms**

**Topic: **

**Olivia Beckwith- Tulane University**

**Abstract: **This week we'll introduce Hecke operators, which act linearly on modular forms. We'll show that one can use Hecke operators to describe bases for spaces of modular forms given by functions with multiplicative coefficients. ** **

**Location:** Gibson Hall 126A

**Zoom access: **https://tulane.zoom.us/j/97255728739?pwd=T09nVENjK1Vnb0pHa1pXLzNwd3F2Zz09

**Time: 2:00**

**Applied and Computational Mathematics**

**Topic:** Abelian and Non-Abelian X-ray transforms. Sharp mapping properties and Bayesian inversion

François Monard - UC Santa Cruz

**Abstract: **

Abelian and Non-Abelian X-ray transforms are examples of integral-geometric transforms with applications to X-ray Computerized Tomography and the imaging of magnetic fields inside of materials (Polarimetric Neutron Tomography). Their study uses tools from classical inverse problems (assessments of injectivity, stability and inversions), and mathematical statistics to deal with cases with noisy data.

After giving a brief introduction to the topic, I plan on covering the following recent results:

(1). We will first discuss a sharp description of the mapping properties of the X-ray transform (and its associated normal operator I*I) on the Euclidean disk, associated with a special L2 topology on its co-domain.

(2). We will then focus on how to use this framework to show that attenuated X-ray transforms (with skew-hermitian attenuation matrix), more specifically their normal operators, satisfy similar mapping properties.

(3). Finally, we will discuss an important application of these results to the Bayesian inversion of the problem of reconstructing an attenuation matrix (or Higgs field) from its scattering data corrupted with additive Gaussian noise. Specifically, I will discuss a Bernstein-VonMises theorem on the 'local asymptotic normality' of the posterior distribution as the number of measurement points tends to infinity, useful for uncertainty quantification purposes. Numerical illustrations will be given throughout.

(2) and (3) are joint work with R. Nickl and G.P.Paternain (Cambridge).

**Zoom:** TBA

**Time: **3:00 pm

**Modular Forms**

**Topic: **

**Naufil Sakran - Tulane University**

**Abstract: **We will give an introduction to the theory of derivatives of modular forms and briefly discuss the proof for Ramanujan congruences using it. We will end by introducing fractional partition functions. ** **

**Location:** Gibson Hall 126A

**Zoom access: **https://tulane.zoom.us/j/97255728739?pwd=T09nVENjK1Vnb0pHa1pXLzNwd3F2Zz09

**Time: 2:00**

**Joint AG & GT seminar**

**Topic: **

**Zvi Rosen - Florida Atlantic University**

**Abstract: **A convex neural code is a combinatorial object arising as the intersection pattern of convex open subsets of Euclidean space. In this talk, we relate the emerging theory of convex neural codes to the established theory of oriented matroids, both categorically and with respect to feasibility and complexity. By way of this connection, we prove that all convex codes are related to some representable oriented matroid, and we show that deciding whether a neural code is convex is NP-hard.

**Location: **Gibson Hall 400D

**Time:** 3:00 pm

**Applied and Computational Mathematics**

**Topic:** Consistent nonparametric Bayesian inference in inverse problems

Matteo Giordano - TBA

**Abstract: **An inverse problem consists in the task of recovering an unknown mathematical object from noisy indirect measurements, a prototypical application being the estimation of a functional parameter governing a PDE from observations of the PDE solution. The Bayesian approach to inverse problems has become an established and popular methodology, after seminal work at the beginning of the last decade; however, only a handful of results are available to provide theoretical guarantees on the statistical consistency of such inversion methodology.

The talk will consider nonparametric Bayesian inference in nonlinear inverse problems, for which MCMC-based sampling methods provide an attractive alternative to optimisation-based estimation techniques and allow to tackle the associated high-dimensional and non-convex inference problems. Focussing on the benchmark example of recovering the unknown conductivity in an elliptic PDE in divergence form, posterior consistency results are provided that show that the posterior distribution arising from certain Gaussian process priors concentrates around the true conductivity as the number of observations increases. As a consequence, a convergence rate (algebraic in sample size) is derived for the reconstruction error of the associated posterior mean estimator. The analysis is based on a posterior contraction rate theory for the general inverse problems with locally Lipschitz forward map, combined with a stability estimate for the solution map of the PDE.

**Zoom:** TBA

**Time: **3:00 pm

**Special Colloquium**

**Topic: **

**Ken McLaughlin - Colorado State University (Host: Victor Moll) **

**Abstract: **

**Abstract: **In this talk, which I hope will be widely accessible, I will explain analytical results in three different areas of mathematics that are connected through the lens of integrability.

(1) The connection between random matrices and the combinatorics of labelled graphs, originating in the quest for a 2-dimensional quantum gravity. The partition function, it turns out, is an uber-generating function - yielding an infinite hierarchy of generating functions which enumerates labelled graphs that can be embedded in Riemann surfaces, according to both the valences of the vertices as well as the genus of the Riemann surfaces.

(2) The emerging rigorous kinetic theory of the interaction of solitons and gasses of solitons. Some nonlinear partial differential equations possess multi-soliton solutions that behave in some way like a collection of particles. The analysis of continuum limits in which these solitons behave collectively like a gas, and the interaction of solitons with that gas, has gained momentum in recent years because of increased interplay between probability and integrability, and because of the development of new methods of analysis of integrable nonlinear partial differential equations.

(3) The behavior of the zeros of Taylor approximants of analytic functions. We will consider this curious observation: the Taylor polynomial of degree N for exp(z) has N zeros, while exp(z) has no zeros in the plane. How do these spurious zeros tend to infinity? What about Taylor approximants of other functions, or other types of approximants?

**Zoom ID:** Dinwiddie 103

**Time: **3:30 pm

**Graduate Student Colloquium**

**Topic: **

**Will Tran - Tulane University**

**Abstract: **Topological Data Analysis (TDA) is a growing field under applied topology. Let's examine some if its basic objects: abstract simplicial complexes, RIPS complexes, Cech complexes. Then, we'll compute a RIPS filtration of a complex by hand. Finally, we'll show how we can use the RIPS filtration and real-world-data from the Bigfoot Field Research Organization to predict Bigfoot sightings in a region.

**Location: **Stanley Thomas 316

**Time: **4:30 pm

**Joint AG & GT seminar**

**Topic: **

**Clayton Shonkwiler - Colorado State University**

**Abstract: **Frames are overcomplete systems of vectors in Hilbert spaces. They were originally introduced in the 1950s in the context of non-harmonic Fourier series, and came to renewed prominence in the 1980s in signal processing applications. More recently, there has been burgeoning interest in frames in finite-dimensional Hilbert spaces, with applications to signal processing, quantum information, and compressed sensing.

In this talk, I will describe some ways in which tools from the differential, Riemannian, and symplectic geometry can be applied to problems in frame theory. For example, frame spaces of interest are often closely related to adjoint or isotropy orbits of compact groups, and hence can be thought of as symplectic manifolds or isoparametric submanifolds. This is joint work with Tom Needham.

**Location: **Gibson Hall 400D

**Time:** 3:00 pm

**Applied and Computational Mathematics**

**Topic:** Understanding movement of flagellated bacteria

Jeungeun Park - State University of New York at New Paltz

**Abstract: ** Bacterial swimming mediated by flagellar rotation is one of the most ubiquitous forms of cellular locomotion, and it plays a major role in many biological processes. A typical swimming path of flagellated bacteria looks like a random walk with no purpose, but the random movement becomes modified as environmental conditions change. The modified random movement is particularly characterized by their motility pattern or a combination of their swimming modes. Further, such individual swimming patterns characterize the collective behavior of a population of bacteria. In this talk, we present several distinct motility patterns exhibited by bacterial species and a mathematical model of an individual bacterium swimming in a fluid. We also discuss how to analyze the collective behavior of bacteria from the individual swimming pattern, particularly, by using an example of E. coli’s swimming behavior in response to chemical signals. This talk is based on two joint works with Zahra Aminzare (Univ. of Iowa) and Sookkyung Lim (Univ. of Cincinnati), Yongsam Kim (Chung-Ang Univ.), and Wanho Lee (National Institute for Mathematical Sciences, South Korea).

**Zoom:** TBA

**Time: **3:00 pm

**Special Colloquium**

**Topic: **

**Philippe G. LeFloch - Sorbonne University **

**Abstract: **I will overview recent advances on nonlinear wave phenomena with singularities. Shock waves are important in compressible fluid dynamics and astrophysics. Propagating interfaces arise in solid materials undergoing phase transitions. The first observation of gravitational waves in our Universe was made in September 2015. Solutions to the Einstein equations may exhibit gravitational singularities. In these areas of research, the problems posed to applied mathematicians are multi-fold and, often, involve several scales and the study of nonlinear wave interactions. Making advances in physical modeling, mathematical analysis, and scientific computation requires to unify techniques from different fields. This has led to exciting mathematical developments on the Euler equations and the Einstein equations. Blog: philippelefloch.org

**Zoom ID:** Dinwiddie 103

**Time: **3:00 pm

**Algebra and Combinatorics Seminar**

**Topic: Weakly Complete Universal Enveloping Algebras of Profinite-Dimensional Lie Algebras**

**Karl H. Hofmann - Tulane University/Darmstadt University**

**Location: **Gibson Hall 400D

**Time: **2:00pm

**Graduate Student Colloquium**

**Topic: **

**Yuwei Bao - Tulane University**

**Abstract: **Under the big umbrella of statistical genetics, we will first do an introduction to the basic constructions of a phylogenetic tree. Then introduce the derivation and calculations of the Felsenstein pruning algorithm for computing the likelihood of a phylogenetic tree from nucleic acid sequence data.

**Location: **Stanley Thomas 316

**Time: **4:00 pm

**Modular Forms**

**Topic: **

**Olivia Beckwith - Tulane University**

**Abstract: **Throughout the seminar, we’ve seen that Fourier coefficients of modular forms often involve arithmetically interesting sequences, such as divisor sum functions, Bernoulli numbers, partition numbers, the number of ways to write an integer as a sum of 4 squares, and the dimensions of the components of the monster Lie algebra. Using Serre’s theory of p-adic modular forms, we’ll examine some of the divisibility properties of such sequences.

We hope to see many of you there!

**Location:** Gibson Hall 126A

**Zoom access: **https://tulane.zoom.us/j/97255728739?pwd=T09nVENjK1Vnb0pHa1pXLzNwd3F2Zz09

**Time: 2:00**

**Joint AG & GT seminar**

**Topic: A duality for sequences and its manifestation for symbolic powers**

**Michael DiPasquale - University of South Alabama**

**Abstract: **

In this talk, we present a duality for sequences of natural numbers. The sequence duality we present takes a sub-additive sequence to a super-additive sequence (and vice-versa) and inverts the coefficients of linear growth. We indicate at least one instance where this duality shows up for symbolic powers of ideals. Concretely, if an ideal defines a projective variety, its nth symbolic power consists of those polynomials which vanish to order n on the variety.

The main example which we will explore is the sequence of initial degrees of symbolic powers, which is a sub-additive sequence giving rise to what is known as the Waldschmidt constant. Via apolarity (or Macaulay-Matlis duality), the sequence of initial degrees is dual (in our sense of sequence duality) to a sequence of regularities of certain ideals generated by powers of linear forms. There will be plenty of examples. This is joint work with Alexandra Seceleanu.

**Location: **Gibson Hall 400D

**Time:** 3:00 pm

**Applied and Computational Mathematics**

**Topic:** From Equation to Innovation: Using Mathematics to Develop Products and Technologies with Impact

Nitsan Ben-Gal - 3M

**Abstract: ** Dr. Ben-Gal will be speaking about her experiences as an Industrial Mathematician, applying mathematical expertise to real-world problems and expanding into new skill domains in Math and beyond. She will discuss a number of technologies and products developed and commercialized over the past 7 years with Applied Mathematics at their core, and the insights and challenges that arise from taking mathematics from the textbook to the lab and beyond. She will also provide advice and insight for those interested in exploring industry as a career choice or source of collaborations.

**Zoom:** TBA

**Time: **3:00 pm

**Colloquium**

**Topic: **

**Cristina Ballantine - Holy Cross **

**Abstract: **Euler’s partitions identity states that the number of partitions of n into odd parts equals the number of partitions of n into distinct parts. In 2017, Beck conjectured and Andrews proved that the difference in the number of parts in all partitions of n into odd parts and the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated) and all other parts odd. Companion identities of Beck type have been established for other partition identities.

I will present such identities accompanying generalizations of Euler’s identity, the Rogers-Ramanujan identities, and identities related to Ramanujan’s mock theta functions. I will also discuss how work on Beck type identities led to a surprising almost partition identity regarding the number of parts in self-conjugate partitions. I will present both analytic and combinatorial approaches.

The talk is based on joint work with George Andrews; Amanda Welch; Hannah Burson, Chi-Yun Hsu, Amanda Folsom, Isabella Negrini, and Boya Wen.

**Zoom ID:** TBA

**Time: **3:30 pm

**Algebra and Combinatorics Seminar**

**Topic: **

**Selvi Kara - University of Utah**

**Abstract:** Abstract: In this talk, we will focus on Rees and multi-Rees algebras of strongly stable ideals. In particular, we will discuss the Koszulness of these algebras through a systematic study of these objects via three parameters: the number of ideals, the number of Borel generators of each ideal, and the degrees of Borel generators. In addition, we utilize combinatorial objects such as fiber graphs to detect Gröbner bases and Koszulness of these algebras.

**Location: **Dinwiddie Hall 102

**Time: **3:00pm

**Probability and Statistics**

**Topic: **

**Cooper Boniece - Utah**

**Abstract: **The squared diffusion coefficient of a Lévy process — the variance of its Gaussian part — can be viewed as an elementary case of a quantity known as the integrated variance, whose estimation in the high-frequency sampling regime has been a topic of active research for the past 2+ decades. However, even in the fundamental case of a univariate Lévy process, there remain significant practical challenges to efficient estimation of this quantity when sample paths exhibit extreme jump behavior.

In this talk, I will discuss some recent work concerning optimal estimation of the diffusion coefficient of a Lévy process in the high-frequency sampling regime under extreme jump activity. This is joint with Yuchen Han and José E. Figueroa-López (Wash. Univ. in St. Louis).

**Location:** Gibson 308

**Time: **2:00 pm

**Graduate Student Colloquium**

**Topic: **

**Louis Nass - Tulane University**

**Abstract: **In this talk, we will discuss various aspects of population models. We will start by introducing basic ordinary differential equation models, modify the models to fit our population’s behavior, and analyze the equilibria. We then extend our simple models to incorporate age-structure, and solve the corresponding partial differential equation using the method of characteristics. Finally, we think about the equilibria associated with the age-structured model, and consider modifications of the age-structure to create size-structure models. We finish by posing a boundary condition that considers modeling coral larvae settlement in a benthic environment.

**Location: **Stanley Thomas 316

**Time: **4:00 pm

**Modular Forms**

**Topic: **

**Vaishavi Sharma - Tulane University**

**Abstract: **Throughout the seminar, we’ve seen that Fourier coefficients of modular forms often involve arithmetically interesting sequences, such as divisor sum functions, Bernoulli numbers, partition numbers, the number of ways to write an integer as a sum of 4 squares, and the dimensions of the components of the monster Lie algebra. Using Serre’s theory of p-adic modular forms, we’ll examine some of the divisibility properties of such sequences.

We hope to see many of you there!

**Location:** Gibson Hall 126A

**Zoom access: **https://tulane.zoom.us/j/97255728739?pwd=T09nVENjK1Vnb0pHa1pXLzNwd3F2Zz09

**Time: 2:00**

**Location: **Gibson Hall 126A

**Zoom info**: https://tulane.zoom.us/j/97255728739?pwd=T09nVENjK1Vnb0pHa1pXLzNwd3F2Zz09

Passcode: The q coefficient of the j-invariant (196884).

**Time: **2:00 pm

**Joint AG & GT seminar**

**Topic: Multiparameter Persistence and Discrete Morse Theory**

**Robyn Brooks - Boston College**

**Abstract: **Persistent Homology is a tool of Computation Topology which is used to determine the topological features of a space from a sample of data points. In this talk, I will introduce the (multi-)persistence pipeline, as well as some basic tools from Discrete Morse Theory which can be used to better understand the multi-parameter persistence module of a filtration. In particular, the addition of a discrete gradient vector field consistent with a multi-filtration allows one to exploit the information contained in the critical cells of that vector field as a means of enhancing geometrical understanding of multi-parameter persistence. I will present results from joint work with Claudia Landi, Asilata Bapat, Barbara Mahler, and Celia Hacker, in which we are able to show that the rank invariant for nD persistence modules can be computed by selecting a small number of values in the parameter space determined by the critical cells of the discrete gradient vector field. These values may be used to reconstruct the rank invariant for all other possible values in the parameter space.

**Location: **Gibson Hall 400D

**Time:** 3:00 pm

**Applied and Computational Mathematics**

**Topic:** Identifying improbable plates at a commercial COVID diagnostic testing laboratory

Shishi Luo - Helix

**Abstract: ** A story of how a probabilist in the right place, at the right time, was able to make a difference at a laboratory processing tens of thousands of COVID tests a day. The probability in this talk should be accessible to an undergraduate mathematics audience.

**Zoom:** TBA

**Time: **3:00 pm

**Colloquium**

**Topic: **

**Vladimir Sverak - U Minnesota (Host: TBA)**

**Abstract: **We discuss topics in PDE analysis of the Navier-Stokes equations, including open problems and recent advances, mostly surrounding questions around uniqueness.

**Zoom ID:** TBA

**Time: **3:30 pm

**Algebra and Combinatorics Seminar**

**Topic: Kronecker coefficients, polytopes, and complexity**

**Aram Bingham - UNAM**

**Abstract:** The Kronecker coefficients problem is one of the last major open questions in the classical representation theory of symmetric groups. It asks for a combinatorial rule describing the decomposition of tensor products of irreducible symmetric group representations, which is unknown outside of certain special cases. Kronecker coefficients have also been the subject of much recent research motivated by the geometric complexity theory (GCT) program, which hypothesizes efficient computation of these numbers as part of a strategy to separate the computational complexity classes P and NP. We will give an overview of this strategy, explain how the Kronecker problem arises in the context of GCT, and report some progress on computing these coefficients as discrete volumes of polytopes, joint with Ernesto Vallejo.

**Location: **Dinwiddie Hall 102

**Time: **3:00pm

**Colloquium**

**Topic: **

**Samuli Siltanen- University of Helsinki **

**Abstract: **The structure of vowel sounds in human speech can be divided into two independent components. One of them is the “excitation signal,” which is a kind of buzzing sound created by the vocal folds flapping against each other. The other is the “filtering effect” caused by resonances in the vocal tract, or the confined space formed by the mouth and throat. The Glottal Inverse Filtering (GIF) problem is to divide a microphone recording of a vowel sound into its two components. This “blind deconvolution” task is an ill-posed inverse problem. Good-quality GIF filtering is essential for computer-generated speech needed for example by disabled people (think Stephen Hawking). Also, GIF affects the quality of synthetic speech in automatic information announcements, digital assistants and car navigation systems. Accurate estimation of the voice source from recorded speech is known to be difficult with current glottal inverse filtering (GIF) techniques, especially in the case of high-pitch speech of female or child subjects. In order to tackle this problem, the present study uses two different solution methods for GIF: Bayesian inversion and alternating minimization. The first method takes advantage of the Markov chain Monte Carlo (MCMC) modeling in defining the parameters of the vocal tract inverse filter. The filtering results are found to be superior to those achieved by the standard iterative adaptive inverse filtering (IAIF), but the computation is much slower than IAIF. Alternating minimization cuts down the computation time while retaining most of the quality improvement.

**Zoom ID:** TBA

**Time: **3:30 pm

**Algebra and Combinatorics Seminar**

**Topic: Critical values of complex polynomials**

**Tewodros Amdeberhan - Tulane University**

**Abstract: **

Given a list of complex numbers, say $w_1,\dots,w_n$ does there exist a complex polynomial $P(z)$ and another list $z_1,\dots,z_n$ so that $P’(z_k)=0$ and $P(z_k)=w_k$ for $k=1,2\dots,n$? We will consider how some basic Analysis and basic Linear Algebra come to the rescue, working in tandem.

**Location: **Dinwiddie Hall 102

**Time: **3:00pm

**Graduate Student Colloquium**

**Topic: **

**Vinh A. Phạm - Tulane University**

**Abstract: **Secant varieties was one of the topics studied by Italian schools in the 19th century. Recently the interest of mathematicians for the secant varieties has increased dramatically because of its application in studying some well-known problems such as the Waring problem for polynomials. In this talk, we want to introduce the concept of secant varieties of projective varieties including some basic definitions, examples, and properties. Then, we will consider the secant varieties of Veronese varieties. The main result of the talk is going to be the Alexander-Hirschowitz Theorem, which classifies completely the secant varieties of Veronese varieties that don't have the expected dimension.

**Location: **Stanley Thomas 316

**Time: **4:00 pm

**Applied and Computational Mathematics**

**Topic:** Frameworks for non-reversibility: GENERIC and hypocoercivity

Michela Ottobre - Heriot Watt University

**Abstract: **The study of non-reversible processes has attracted the attention of various communities in the last decade (and for a much longer if we consider the conceptual problems they pose within statistical mechanics). For example, in biology, non-reversible processes constitute important and popular modelling tool to describe animal navigation; in statistical sampling the hope that such processes could be used as a framework to accelerate convergence has been a catalyst for important theoretical progress in the field.

We consider two approaches to study non-reversible Markov processes, namely the Hypocoercivity Theory (HT) and GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling); the premise and purpose of these two theories is quite different, but the basic idea behind both of them is substantially the same, namely to split the process into a reversible component and a non-reversible one, and then quantify the way in which they interact. We compare such theories and provide explicit formulas to pass from one formulation to the other; as a bi-product we give a simple proof of the link between reversibility of the dynamics and gradient flow structure of the associated Fokker-Planck equation. We will discuss connection with large deviation principles as well. Based on joint work with H. Duong.

**Zoom:** TBA

**Time: **3:00 pm

**Colloquium**

**Topic: **

**Greg Blekherman - Georgia Tech **

**Abstract: **A real polynomial is called nonnegative if it takes only nonnegative values. A sum of squares or real polynomials is clearly nonnegative. The relationship between nonnegative polynomials and sums of squares is one of the central questions in real algebraic geometry. It is also important in applications, such as optimization, as sums of squares are a computationally tractable set of nonnegative polynomials. A modern approach is to look at nonnegative polynomials and sums of squares on a real variety, where unexpected links to complex algebraic geometry and commutative algebra appear. I will describe some of these connections and applications of sums of squares.

Zoom ID: TBA

**Time: **3:30 pm

**Algebra and Combinatorics Seminar**

**Topic: **

**Zach Walsh - LSU**

**Abstract: **

Given a graph G with edges labeled by a finite group, a construction of Zaslavsky gives a rank-1 lift of the graphic matroid M(G) which respects the group-labeling.

For which groups can we construct a rank-t lift of M(G) with t > 1 which respects the group-labeling?

We show that this is possible only if the group has a non-trivial partition, and further conjecture that it is possible only if the group is the additive subgroup of a non-prime finite field.

**Location: **Dinwiddie Hall 102

**Time: **3:00pm

**Graduate Student Colloquium**

**Topic: **

**Kendall Gibson - Tulane University**

**Abstract: **While images may not appear to be a collection of finite points, they can be represented by a finite array of pixels. Each pixel in this array gives an intensity value to determine the color of that pixel. When we interpret an image in this way, we can apply known matrix operations to these arrays to achieve some new image. In this talk, we will explore the underlying mathematics of these manipulations and use MATLAB to visualize the results.

**Location: **Stanley Thomas 316

**Time: **4:00 pm

**Joint AG & GT seminar**

**Topic: The trunkenness of a volume-preserving vector field**

**Pierre Dehornoy - Université Grenoble Alpes**

**Abstract: **This work (joint with Ana Rechtman) deals with the construction of invariants for volume-preserving vector fields in the 3-sphere, up to homeo- (or diffeo- )morphism of the underlying manifold. Not that many such invariants exist, the most famous one being the helicity. It turns out that it can be recovered in many ways, and that it is the only smooth invariant. Here we construct another invariant, inspired by and connected to the trunk for knots, which has the interest of being independent of helicity.

**Location: **TBA

**Time:** 3:00 pm

**Applied and Computational Mathematics**

**Topic:** Dirichlet fractional Gaussian fields on the Sierpinski gasket and their discrete graph approximations

Li Chen - LSU

**Abstract:** In this talk, we discuss the Dirichlet fractional Gaussian fields on the Sierpinski gasket. We show that they are limits of fractional discrete Gaussian fields defined on the sequence of canonical approximating graphs. This is a joint work with Fabrice Baudoin (UConn).

**Zoom:** TBA

**Time: **2:00 pm

**Colloquium**

**Topic: Dissipation Enhancement, Mixing and Blow-up Suppression**

**Gautam Iyer - CMU (Host: TBA) **

**Abstract: **Diffusion and mixing are two fundamental phenomena that arise in a wide variety of applications. In this talk we quantitatively study the interaction between diffusion and mixing in the context of problems arising in fluid dynamics. The first question we address is how fast the energy can decay i the advection diffusion equation. Even though this is a simple linear equation, the energy decay rate is intrinsically related to the mixing properties of the advecting velocity field, and there are many unresolved open questions. I will present a few recent results involving both upper and lower bounds, and then consider applications to studying the long time dynamics of a few model

non-linear equations arising in chemotaxis and phase separation.

**Zoom ID: **TBA

**Time: **3:30 pm

**Algebra and Combinatorics Seminar**

**Topic: The nilpotent variety of an asymptotic semigroup**

**Mahir Can - Tulane University**

**Abstract: **

The asymptotic semigroup of a semisimple group is the semigroup that is obtained by the procedure called "contraction". In this talk, we will discuss the geometry and combinatorics of the nilpotent variety of an asymptotic semigroup. In particular, we will show that the top dimensional homology of this variety affords a permutation representation of the Weyl group.

**Zoom Only: TBA
Time: **3:00pm

**Graduate Student Colloquium**

**Topic: Chevalley Groups**

**Victor Bankston | Tulane University**

**Abstract: **We will discuss how Lie theory is modified to apply in the finite field setting, and thus identify structural motifs in a large class of finite simple groups.

**Location: **Stanley Thomas 316

**Time: **4:30 pm

**Colloquium**

**Topic: Fokker-Planck Equations and Machine Learning**

**Yuhua Zhu - Stanford University Stanford, California (Host: Lisa Fauci)**

**Abstract: **

As the continuous limit of many discretized algorithms, PDEs can provide a qualitative description of algorithm's behavior and give principled theoretical insight into many mysteries in machine learning. In this talk, I will give a theoretical interpretation of several machine learning algorithms using Fokker-Planck (FP) equations. In the first one, we provide a mathematically rigorous explanation of why resampling outperforms reweighting in correcting biased data when stochastic gradient-type algorithms are used in training. In the second one, we propose a new method to alleviate the double sampling problem in model-free reinforcement learning, where the FP equation is used to do error analysis for the algorithm. In the last one, inspired by an interactive particle system whose mean-field limit is a non-linear FP equation, we develop an efficient gradient-free method that finds the global minimum exponentially fast.

**Zoom Only: TBA
Time: **3:00pm

**Week of February 4 - January 31**

**Colloquium**

**Topic: The failure of injectivity in number theory**

**Jeremy Rouse - TBA (Host: Beckwith) **

**Abstract: **To classify mathematical objects, mathematicians create invariants: functions f defined on the objects that one seeks to classify, so that if A and B are isomorphic objects, then f(A) = f(B). I will give examples of several situations where these invariants fail to classify number theoretic objects, as well as give a discussion of two reasons why these invariants do not suffice. Most of the talk will consist of examples, including non-isometric lattices with the same theta function, non-isomorphic number fields with the same Dedekind zeta function, non-equivalent trinomials defining the same number field, and further examples involving elliptic curves and modular curves.

**Zoom ID: **TBA

**Time: **3:30 pm

**Graduate Student Colloquium**

**Topic: Real Root Counting**

**Naufil Sakran - Tulane University**

**Abstract: **Finding roots of polynomials have always been a centric object of study in Mathematics. Much of Modern Mathematics owes its motivation to questions arising in this field. Observing roots of polynomials in real field (ordered field) give rise to the subject of real algebraic geometry. In this realm, one can find interesting relations between the roots of the polynomial and the list of the coefficients of the polynomial. We will discuss Descarte's Law of signs, Budan Fourier Theorem, and Sturm's Theorem which completely establishes the relationship between real roots and the list of coefficients of the polynomial.

**Location: **Stanley Thomas 316

**Time: **4:30 pm

**Colloquium**

**Topic: Controlling the spread of invasive biological species**

**Maria Chiri - Penn State (Host: Glatt-Holtz )**

**Abstract: **

We consider a controlled reaction-diffusion equation, modeling the spreading of an invasive population. Our goal is to derive a simpler model, describing the controlled evolution of a contaminated set. We first analyze the optimal control of 1-dimensional traveling wave profiles. Using Stokes’ formula, explicit solutions are obtained, which in some cases require measure- valued optimal controls. Then we introduce a family of optimization problems for a moving set and show how these can be derived from the original parabolic problems, by taking a sharp interface limit. In connection with moving sets, we show some results on controllability, existence of optimal strategies, and necessary conditions.

This is a joint work with Prof. Alberto Bressan and Dr. Najmeh Salehi (Penn State University).

References

[1] A. Bressan, M. T. Chiri, and N. Salehi, On the optimal control of propagation fronts, to appear on Math. Models & Methods in the Applied Sciences.

[2] A. Bressan, M. T. Chiri, and N. Salehi, Optimal control of moving sets. Submitted.

**Location:
Time: **3:30 pm

**Joint AG & GT seminar**

**Topic: On Anosovity, divergence and bi-contact surgery**

**Surena Hozoori - Georgia Tech**

**Abstract: **I will revisit the relation between Anosov 3-flows and invariant volume forms, from a contact geometric point of view. Consequently, I will give a contact geometric characterization of when a flow with dominated splitting is Anosov based on its divergence, as well as a Reeb dynamical interpretation of when such flows are volume-preserving. Moreover, I will discuss the implications of this study on the surgery theory of Anosov 3-flows. In particular, I will conclude that the Goodman-Fried surgery of Anosov flows can be reconstructed, using a bi-contact surgery of Salmoiraghi.

**Location: **TBA

**Time:** 3:30 pm

**Week of January 28 - January 24**

**Colloquium**

**Topic: Novel Time Integration Methods for Solving Stiff Systems**

**Tommaso Buvoli - University of California at Merced (Host: ) **

**Abstract: **Many scientific and engineering disciplines depend on efficient numerical methods to model complex physical systems. Since the underlying dynamics frequently involve a range of temporal and spatial scales, specialized methods that can handle numerical challenges such as stiffness and high dimensionality are required. In this talk I will present several novel approaches for constructing time integration techniques that achieve improved efficiency on specific application problems and computational hardware. In particular, I will introduce a framework based on interpolating polynomials that can be used to construct time integrators with desirable properties such as parallelism, high-order of accuracy, and varying degrees of implicitness. In addition I will also discuss the construction of parallel-in-time integrators for non-diffusive systems. Specifically, I will show how convergence and stability analysis can be used to select stable Parareal configurations that allow for massively parallel simulations of hyperbolic and dispersive equations.

**Location: **Stanley Thomas 316

**Time: **3:00 pm

**Applied and Computational Mathematics**

**Topic: Manifold MCMC methods for Bayesian Inference in Diffusion Models**

**Alexandros Beskos - University College London (UCL) **

**Abstract: **

Bayesian inference for nonlinear diffusions, observed at discrete times, is a challenging task that has prompted the development of a number of algorithms, mainly within the computational statistics community. We propose a new direction, and accompanying methodology - borrowing ideas from statistical physics and computational chemistry - for inferring the posterior distribution of latent diffusion paths and model parameters, given observations of the process. Joint configurations of the underlying process noise and of parameters, mapping onto diffusion paths consistent with observations, form an implicitly defined manifold. Then, by making use of a constrained Hamiltonian Monte Carlo algorithm on the embedded manifold, we are able to perform computationally efficient inference for a class of discretely observed diffusion models. Critically, in contrast with other approaches proposed in the literature, our methodology is highly automated, requiring minimal user intervention and applying alike in a range of settings, including: elliptic or hypo-elliptic systems; observations with or without noise; linear or non-linear observation operators. Exploiting Markovianity, we propose a variant of the method with complexity that scales linearly in the resolution of path discretisation and the number of observation times. Python code reproducing the results is available at

doi.org/10.5281/zenodo.5796148.

The talk is based on manuscript https://arxiv.org/pdf/1912.02982.pdf accepted for publication at JRSSB.

**Location: TBA
Time:** 10:00 pm

**Colloquium**

**Topic: Linear PDE with Constant Coefficients**

**Bernd Sturmfels - MPI Leipzig and UC Berkeley (Host: Bernstein ) **

**Abstract: **We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the Fundamental Principle of Ehrenpreis–Palamodov from the 1960s. We develop this further using recent advances in computational commutative algebra.

**Location: **TBA

**Time: **3:30 pm