**Week of May 28 - May 24**

**Topic: TBA**

**Nitsan Ben-Gal | 3M**

**Abstract: TBA**

**Week of May 21 - May 17**

**Geometry and Topology**

**Topic: AATRN: Using Algebraic Geometry to detect robustness in Reaction Networks**

**Beatriz Pascual Escudero | Universidad Carlos III de Madrid**

**Abstract: An interesting property of some biological systems is their capacity to preserve certain features against changes in environmental conditions. In particular, we are motivated by Reaction Networks and the evolution in time of the concentrations of the species involved, often modeled by systems of (generalized) polynomial ODEs. For them, the following property is of interest: a system has absolute concentration robustness (ACR) for some species if the concentration of this species does not vary among the different equilibria that the network admits. In particular, this concentration must be independent of the initial conditions. While some classes of networks with ACR have been described, as well as some techniques to check ACR for a given network, finding networks with this property is a difficult task in general.**

**Motivated by this problem, we explore local and global notions of robustness on the set of (real positive) solutions of a system of polynomial equations. The goal is to provide a practical test on necessary conditions for ACR, using algebraic-geometric techniques.**

**This is based on joint work with E. Feliu.**

**Recordings of most of the talks will be posted to the AATRN YouTube Channel.**

**Zoom access: **

**Meeting ID: 924 8568 4682
Zoom meeting starts at 10:00am CT**

**Week of May 14 - May 10**

**Geometry and Topology**

**Topic: AATRN: Data Complexes, Obstructions, Persistent Data Merging**

**Paul Bendich | Florida State**

**Abstract: Data complexes provide a mathematical foundation for semi-automated data-alignment tools that are common in commercial database software. We develop a theory that shows that database JOIN operations are subject to genuine topological obstructions. These obstructions can be detected by an obstruction cocycle and can be resolved by moving through a filtration. Thus, any collection of databases has a persistence level, which measures the difficulty of JOINing these databases.**

**More precisely, collections of measures on compact metric spaces form the model category of "data complexes," with morphisms given by marginalization integrals. The homotopy and homology for this category allow measurement of obstructions to finding measures on larger and larger product spaces.**

**The obstruction theory is compatible with a fibrant filtration built from the Wasserstein distance on measures.**

**This is joint work with Abraham D. Smith and John Harer.**

**https://arxiv.org/pdf/1911.11837.pdf**

**Recordings of most of the talks will be posted to the AATRN YouTube Channel.**

**Zoom access: **

**Meeting ID: 924 8568 4682
Zoom meeting starts at 10:00am CT**

**Dissertation Defense**

**Topic: Stable Harbourne-Huneke Containment and Lower Bounds on Waldschmidt Constant**

**Sankhaneel Bisui | Tulane University**

**Abstract: The following fundamental question was raised by Nagata.**

**Nagata's Question: Given finitely many points {P1,..., Ps} in a complex projective plane, what is the minimum degree of a hyper-surface that passes through the points with multiplicity at least t?**

**This question is very interesting in a very unique way and it attracts a large number of researchers. In general, it is extremely difficult to determine the actual least degree. It is more feasible to study lower bounds of the least degree. There are multiple conjectures involving the lower bound. Nagata himself gave a conjectural lower bound. Chudnovsky also gave a conjectural answer to Nagata's Question in terms of lower bounds on the minimal degree. Demailly generalized the lower bound. We focus on Chudnovsky's and Demailly's conjecture for general points.**

**Zariski-Nagata theorem gives an algebraic platform where those geometric conjectures can be presented in equivalent algebraic formats involving the Waldschmidt constant of the defining ideal of the points. One of the many versatilities of studying containment between the symbolic and the ordinary powers of ideals is that the containment gives lower bounds of the Waldschmidt constant. Harbourne gave a containment conjecture for ideals with big height h. Later on, Harbourne and Huneke strengthen the containment conjectures by adding appropriate powers of the maximal ideals on the right-hand side. The Harbourne-Huneke containment conjectures imply Chudnovsky's as well Demailly's conjecture. We explore those containment conjectures. We prove the stable Harbourne's conjecture for the defining ideal of a general set of points. We also prove stable Harbourne-Huneke containment for the defining ideal of a large number of general points. We use these containment to establish Chudnovsky's and Demailly's conjecture for a large number of general points. We also explore Harbourne - Huneke containment for ideals beyond points. We prove Harbourne-Huneke containment for ideals defining star-configurations of co-dimension h and determinantal ideals of generic matrices. As a consequence, we also prove that these ideals also satisfy Demailly like bounds.**

**In the talk, the motivation of the work will be presented in a very fundamental manner. All the useful concepts and techniques will be introduced with examples. All the known results regarding containment and lower bound conjectures (Chudnsky's and Demailly's) will be presented. I will also present the results from our joint work with Eloísa Grifo, Huy Tài Hà, and Thái Thành Nguyên. Some outlines of our main results will also be demonstrated.**

**Join us:** **Zoom access:** **Time: 10:00AM**

**Week of May 7 - May 3**

**Applied and Computational Mathematics**

**Topic: A general perspective on the Metropolis–Hastings kernel**

**Christophe Andrieu | University of Bristol**

**Abstract: Since its inception the Metropolis–Hastings kernel has been applied in sophisticated ways to address ever more challenging and diverse sampling problems. Its success stems from the flexibility brought by the fact that its verification and sampling implementation rest on a local “detailed balance” condition, as opposed to a global condition in the form of a typically intractable integral equation. While checking the local condition is routine in the simplest scenarios, this proves much more difficult for complicated applications involving auxiliary structures and variables.**

**The aim of the presentation is an attempt to bring together ideas making verification of correctness of complex Markov chain Monte Carlo kernels a purely mechanical or algebraic exercise, while at the same time enabling simpler and unambiguous communication of complex ideas. A motivation behind this work was to bring clarity in the scenario where the proposal distribution involves stopped processes or stopping times, such as NUTS. This will only be alluded to due to time constraints — more in the manuscript!**

**Join us:** **Zoom access:** **Time: 3:30**

**Geometry and Topology**

**Topic: AATRN: Homotopy Types of Vietoris–Rips Complexes of Metric Gluings**

**Bei Wang | Florida State**

**Abstract: We study Vietoris-Rips complexes of metric wedge sums and metric gluings. We show that the Vietoris-Rips complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris–Rips complexes. We also provide generalizations for when two metric spaces are glued together along with a common isometric subset. As our main example, we deduce the homotopy type of the Vietoris-Rips complex of two metric graphs glued together along a sufficiently short path (when compared to lengths of certain loops in the input graphs). As a result, we can describe the persistent homology, in all homological dimensions, of the Vietoris-Rips complexes of a class of metric graphs. We will discuss open research directions concerning metric gluings and metric graphs.**

**This is joint work with Henry Adams, Michal Adamaszek, and our WinComTop working group since 2016 -- Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Yusu Wang, and Lori Ziegelmeier.**

**Recordings of most of the talks will be posted to the AATRN YouTube Channel.**

**Zoom access:
Meeting ID: 882 0916 7343
Zoom meeting starts at 10:00am CT**

**Dissertation Defense**

**Topic: Regularity and resurgence number of homogeneous ideals**

**Abu Chackalamannil Thomas | Tulane University**

**Abstract: Primary objective of this thesis is to study the stability index of algebraic invariants such as $a^*$-invariant and regularity of powers of homogeneous ideals and give bounds to, study the difference between numerical invariants of homogeneous ideals such as resurgence and asymptotic resurgence.**

**It is a celebrated result of Cutkosky-Herzog-Trung \cite{CHT}, Kodiyalam \cite{K} that regularity of powers of homogeneous ideals $I^q$ is a linear function for $q\gg 0$ which was later generalized to a standard graded algebra over a Noetherian ring by Trung and Wang in \cite{TW}. We look at the stability index of algebraic invariants of powers of homogeneous ideals such as the Castelnuovo Mumford regularity and the $a^*$-invariants. Our approach will be to introduce an invariant, associated to a coherent sheaf of graded modules over a projective morphism of schemes, which controls when sheaf cohomology can be passed through the given morphism. We then use this invariant to estimate the stability indexes of the regularity and $a^*$-invariant of powers of homogeneous ideals.**

**A celebrated result of Ein-Lazarsfeld-Smith \cite{ELS}, Hochster-Huneke \cite{HoHu}, Ma-Schwede \cite{MS} says that for an ideal $I$ of big height $h$ in a regular ring $R$, $I^{(hn)} \subseteq I^n$, $\forall n \in \mathbb{N}$. One could hope for sharpening the containment by decreasing the power of symbolic powers by a fixed constant. This led to the famous Harbourne containment conjecture in \cite{HaHu}. We also show that for an ideal $I$ defining complement of Steiner configuration of points, stable Harbourne and stable Harbourne-Huneke containment holds. We also investigate the pathological difference between resurgence and asymptotic resurgence numbers which measures the non-containment of ideals of fiber products of projective schemes. Particularly, we show that while the asymptotic resurgence number of the $k$-fold fiber product of a projective scheme remains unchanged, its resurgence number could strictly increase.**

**Join us:** **Zoom access:** **Time: 10:00am**

**Week of April 30 - April 26**

**Applied and Computational Mathematics**

**Topic: New coupling techniques for exponential ergodicity of SPDEs in the hypoelliptic and effectively elliptic settings**

**Oleg Butkovskiy | Weierstrass Institute (WIAS), Berlin**

**Abstract: We will present new coupling techniques for analyzing ergodicity of nonlinear stochastic PDEs with additive forcing. These methods complement the Hairer-Mattingly approach (2006, 2011). In the first part of the talk, we demonstrate how a generalized coupling approach can be used to study ergodicity for a broad class of nonlinear SPDEs, including 2D stochastic Navier-Stokes equations. This extends the results of [N. Glatt-Holtz, J. Mattingly, G. Richards, 2017]. The second part of the talk is devoted to SPDEs that satisfy comparison principle (e.g., stochastic reaction-diffusion equations). Using a new version of the coupling method, we establish exponential ergodicity of such SPDEs in the hypoelliptic setting and show how the corresponding Hairer-Mattingly results can be refined.
(Joint work with Alexey Kulik and Michael Scheutzow)
[1] O. Butkovsky, A.Kulik, M. Scheutzow (2020). Generalized couplings and ergodic rates for SPDEs and other Markov models. Annals of Applied Probability, 30(1), 1–39.
[2] O. Butkovsky, M. Scheutzow (2020). Couplings via comparison principle and exponential ergodicity of SPDEs in the hypoelliptic setting. Communications in Mathematical Physics, 379(3), 1001–1034.**

**Join us:** **Zoom access:** **Time: 10:30am**

**Dissertation Defense**

**Topic: Combinatorial and arithmetical properties of families of sequences**

**Diego Villamizar Rubiano | Tulane University**

**Abstract: This thesis consists of two parts and addresses some combinatorial, statistical, and arithmetical properties of integer sequences. In particular, the sequences on study come from set partitions, permutations, and elementary arithmetic functions.**

**Part I consists of results, problems, and conjectures on the properties of generalizations of set partitions and permutations. Particularly, we study partitions and permutations for which we impose conditions on the size of the blocks and cycles. We study further the properties of $r-$partitions and $r-$permutations that are defined to be partitions and permutations with the condition that certain special elements must be separated. Lastly, we define and study the properties of a statistic dealing with how the elements of a partition or a permutation distribute with respect to their order.**

**Part II focuses on the p-adic valuation of sequences. Particularly, we study the exact p-adic valuation of some arithmetic functions. We give simpler expressions that bound these exact valuations and we discuss the sharpness of this bounds. Connections to Mersenne primes, Fermat primes, Brazilian primes and some diophantine equations are given. We start the study on the arithmetic of coefficients for some polynomials appearing in the context of nonlinear differential equations of Painleve type.**

**Join us:****Zoom access**: **Time: 2:00**

**Dissertation Defense**

**Topic: On the de Silva-Ghrist homological coverage criteria for planar sensor networks (M.Sci. thesis defense)**

**Jack Green | Tulane University**

**Abstract: In their work, Coordinate-free Coverage in Sensor Networks with Controlled Boundaries Via Homology, de Silva and Ghrist propose two criteria for coverage of a planar domain by a "blind” sensor network via homology. The criteria, which we refer to as the weak criterion and the strong criterion, are formulated in terms of the homology of the Rips complex R associated with the communication graph of the sensor network, and the relative homology of the pair (R,F) where F is a 1-dimensional boundary cycle of the domain. In this thesis, we provide background information on the algebraic topology tools used, cover de Silva and Ghrist’s proof of the weak criterion, and as a novel contribution prove the equivalence of the strong and weak criteria in the 2-dimensional case.**

**Join us:****Zoom access**: **Time: 11:00**

**Week of April 23 - April 19**

**Applied and Computational Mathematics**

**Topic: On existence, uniqueness, and parabolic smoothing in scaling-critical spaces for supercritically dissipative hydrodynamic equations**

**Vincent Martinez | Hunter College, CUNY**

**Abstract: In this talk, we review recent results on the parabolic smoothing effect for certain dissipative perturbations of hydrodynamic equations, particularly the family of generalized surface quasi-geostrophic equations, introduced by Constantin, et. al. 2011. These equations are a family of active scalar equations, which include the 2D Euler equations as an endpoint and extrapolate beyond with an increasingly singular relation in the constitutive law between the scalar and its advecting velocity. In the most singular regime of velocities, the equations represent a genuinely quasilinear equation, whose coefficients are of higher-order than the dissipative perturbation, thus serving as a significant structural obstruction to establishing a solution theory due to loss of derivatives. We nevertheless show that this obstruction is only apparent due to the underlying commutator structure of the transport nonlinearity. To exploit this, however, we introduce a new approximation scheme by linear conservation laws that can accommodate these commutators. We note that such an approximation is not needed for the regime of less singular constitutive laws in the family and that this can be tracked precisely to the form of the quasi-linearity of the equation.**

**Join us:****Zoom access**: **Time: **3:30

**Colloquium**

**Topic:** *Examples of MM Algorithms*

**Ken Lange - UCLA (Host: Ji, Xiang)**

**Abstract:**

**Join us:****Zoom access**: **Contact mbrown2@math.tulane.edu****Time**: 3:30

**Geometry and Topology**

**Topic: AATRN: Musing about robotic football**

**Shmuel Weinberger | Chicago**

**Abstract: What would a robot have to do to get past a bunch of blockers to get to the end zone (or score a goal; the football in general)? I will discuss some issues involving modeling, non-fibrations, speed, information, sensing, and things larger than points. We will not solve these problems. The math is related to a paper with D.Cohen and M.Farber.**

**Zoom access:
Meeting ID: 924 8568 4682
Zoom meeting starts at 9:30am CT**

**Algebra and Combinatorics Seminar**

**Topic: Steiner Configurations Ideals: Containment and Colorability**

**Abu Thomas | Tulane University**

**Abstract: We show that Stable Harbourne Conjecture and Stable Harbourne--Huneke Conjecture hold for the defining ideal of a Complement of a Steiner configuration of points in $\mathbb{P}^n_k$. We study the relation between a particular notion of colorability of hypergraphs associated to Steiner configurations of points. We also find results on the containment problem for the cover ideal associated to these special hypergraphs. We can also show that Chudnovsky's Conjecture and Demailly's Conjecture are satisfied by the ideal defining Complement of Steiner configuration of points.**

**This is a joint work with E. Ballico, G. Favacchio and E. Guardo. We dedicate the paper to L. Millazzo who passed away in 2019.**

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 3:00pm**

**Dissertation Defense**

**Topic: Qualitative analysis of a PDE model for chemotaxis with logarithmic sensitivity and logistic growth**

**Padi Fuster Aguilera | Tulane University**

**Abstract: This thesis examines the qualitative behavior of solutions to a PDE model for chemotaxis; that is the existence, uniqueness, and asymptotic behavior of solutions. We study initial-boundary value problems for a chemotaxis model with logarithmic sensitivity and logistic growth for the cell population density, and nonlinear growth of the chemical concentration. Extensive work has been done for this particular model without logistic growth on both bounded and unbounded domains. However, the model with logistic growth on a bounded domain has not been studied before. This case is of particular interest given its relevance for modeling tumor angiogenesis. We first establish global well-posedness of strong solutions for large initial data with no-flux boundary conditions and, moreover, establish the qualitative result that both the population density and chemical concentration asymptotically converge to constant states. The population density in particular converges to its carrying capacity. We additionally prove that the vanishing chemical diffusivity limit holds in this regime. Finally, we provide numerical confirmation of the rigorous qualitative results, as well as numerical simulations that demonstrate a separation of scales phenomenon. We then establish global well-posedness of strong solutions for large initial data with dynamic boundary conditions. Moreover, the solutions will asymptotically approach the boundary data under mild and natural assumptions on the boundary functions. We additionally show the formation of a boundary profile in the singular chemical zero diffusive limit. Lastly, we provide numerical simulations that confirm the boundary layer formation, as well as convergence towards certain steady states of the solution when relaxing the assumptions on the boundary data. The main tool developed in these results is a particular Lyapunov functional that helps overcome the mathematical challenges of the non-conservation of the mass due to the logistic growth. These results give a complete study of this particular system on bounded domains with both zero-flux and dynamic moving boundary conditions.**

**Join us:****Zoom access**: **Time: **1:00

**Dissertation Defense**

**Topic: Strong Gelfand Subgroup of Z/p wreath S_n**

**Yiyang She | Tulane University**

**Abstract: The multiplicity-free subgroups (strong Gelfand subgroups) of wreath products are investigated. Various useful reduction arguments are presented. In particular, for any finite group G and its normal subgroup K, if G/K is a cyclic group and its order is a multiplicity-free integer, then (G,K) is a strong Gelfand pair. Furthermore, we classify all multiplicity-free subgroups of Z/p wreath S_n for n>6. Along the way, we derive various decomposition formulas from some special subgroups of Z/p wreath S_n for n>6. **

**Join us:****Zoom access**: **Time: **10:00 am

** **

**Week of April 16 - April 12**

**Applied and Computational Mathematics**

**Topic: Artificial Intelligence in Public Safety and Video Security**

**Chia Ying Lee | Motorola Solutions**

**Abstract: I will begin with an overview of AI/ML research at the core of Motorola Solution's products for Public Safety. Then, focusing on the video security aspect, I will dive deeper into the major computer vision and video analytics problems we face in this arena, specifically tracking and person re-identification, and finally present one of my own projects on self-learning camera adjacency. **

**Join us:****Zoom access**: **Time**: 3:30

**Colloquium**

**Topic: Analysis of free boundary problems in fluid mechanics**

**Ian Tice - Carnegie Mellon University (Host: Glatt-Holtz)**

**Abstract: A free boundary problem in fluid mechanics is one in which the fluid domain is not specified a priori and evolves in time with the fluid. Such problems are ubiquitous in nature and occur at a huge range of scales, from dew drops, to waves on the ocean, to the surface of a star. In this talk I will review the basic features of free boundary problems and how we incorporate interesting interfacial physics effects. I will also survey recent work on various viscous free boundary problems. **

**Join us:****Zoom access**: **Contact mbrown2@math.tulane.edu****Time**: 3:30

**Algebra and Combinatorics Seminar**

**Topic: Nodes on Quintic Spectrahedra**

**Taylor Brysiewicz | MPI Leipzig**

**Abstract: A spectrahedron in R3 is the intersection of a 3-dimensional affine linear subspace of dxd real matrices with the cone of positive-semidefinite matrices. Its algebraic boundary is a surface of degree d in C3 called a symmetroid. Generically, symmetroids have (d^3-d)/6 nodes over C and the real singularities are partitioned into those which lie on the spectrahedron and those which do not. This data serves as a coarse combinatorial description of the spectrahedron. For d=3 and 4, the possible partitions are known. In this talk, I will explain how we determined which partitions are possible for d=5. In particular, I will explain how we used numerical algebraic geometry and an enriched hill-climbing algorithm to find explicit examples of spectrahedra witnessing each partition. **

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 3:00pm**

**Graduate Student Colloquium**

**Topic: A Gentle Introduction to Moduli Spaces**

**Corey Wolfe | Tulane University**

**Abstract: Many spaces of classical interest in algebraic geometry may naturally be regarded as moduli space: flag varieties, Hilbert schemes, moduli spaces of abelian varieties, and so many more! In this talk, we set aside these sometimes intimidating spaces and explore moduli spaces of more familiar objects: lines and triangles. Using concrete examples, we will get acquainted with these moduli spaces and their “natural” geometric structure.**

**Join us:**

**Zoom access: 954 0971 2190
Time: 5:00pm**

**Dissertation Defense**

**Topic: Clans, sects, and symmetric spaces of Hermitian type**

**Aram Bingham | Tulane University**

**Abstract: ****This thesis examines} the geometry and combinatorics of Borel subgroup orbits in classical symmetric spaces $G/L$ where $G$ is complex linear algebraic group and $L$ is a Levi factor of a maximal parabolic subgroup $P$ in $G$. In these cases, known as symmetric spaces of Hermitian type, we show that the canonical projection map $\pi: G/L \to G/P$ has the structure of an affine bundle. This fact yields a cell decomposition of $G/L$ as well as isomorphisms of the cohomology and Chow rings of $G/L$ and $G/P$, and motivates the study of the Borel orbits of $G/L$ in relation to their images under the equivariant map $\pi$. For all of the cases of interest (symmetric spaces of types $AIII$, $CI$, $DIII$ and $BDI$), $G/P$ is a Grassmannian variety, and the Borel orbits of this space are called Schubert cells.**

**Borel orbits of most of these symmetric spaces are parametrized by certain combinatorial objects called clans. This thesis provides enumerative formulae for the orbits in type $CI$, $DIII$ and $BDI$, and gives bijections between sets of clans and other families of well-studied objects such as (fixed-point free) partial involutions, rook placements, and set partitions. Clans also come with a poset structure given by the closure containment relation of the corresponding Borel orbits and we supply rank generating functions for these posets in types $CI$ and $DIII$. We further provide a combinatorial description of the closure order relations in types $AIII$, $CI$, and $DIII$. This description allows us to resolve part of a conjecture of Wyser on the restriction of the closure order from type $AIII$ to other types.**

**In the course of this description, we also identify the preimages of Schubert cells under the map $\pi$ as collections of clans called ``sects.'' Our combinatorial description of the sects identifies Borel orbits whose closures generate the Chow ring of $G/L$ and reveals additional structure in the closure poset of clans. In particular, the preimage of the largest Schubert cell coincides variously with well-known posets of matrix Schubert varieties and congruence Borel orbit closures. Furthermore, we also show that the closure order restricted to a given sect can still be described combinatorially in terms of ``rank tableaux.'' **

**Join us:**

**Zoom access:
Time: 10:00**

**Geometry and Topology**

**Topic: AATRN: High-dimensional data, level-set geometry, and Voronoi analysis of spatial point sets**

**Menachem Lazar (Bar-Ilan University)**

**Abstract: Physical systems are regularly studied as spatial point sets, and so understanding the structure of such sets is a very natural problem. However, aside from special cases, describing the manner in which a set of points is arranged in space can be quite challenging. In the first part of this talk, I will show how consideration of the configuration space of local arrangements of neighbors can shed light on essential challenges of this problem, and in the classification of high-dimensional data more generally. In the second part of the talk I will introduce some ideas from Voronoi cell topology and show how they can be used to define crystals, defects, and order more generally in a somewhat precise manner. **

**Recordings of most of the talks will be posted to the AATRN YouTube Channel.**

Zoom access:

Meeting ID: 924 8568 4682

Zoom meeting starts at 10:00 CT

** **

**Week of April 9 - April 5**

**Applied and Computational Mathematics**

**Topic: Transition from academia to industry**

**Camelia Pop | TBA**

**Abstract: I plan to speak about my transition from an academic career to one in the financial industry. I will talk about my background, how I prepared for this change, and the similarities and differences between the two career paths, from my experience. **

**Join us:
Zoom access:
Time**: 3:30

**Colloquium**

**Topic: Mixing, transport, and enhanced dissipation**

**Anna Mazzucato | Penn State (Host: Glatt-Holtz)**

**Abstract: I will discuss transport of passive scalars by incompressible flows and measures of optimal mixing and stirring. I will present two examples of opposite effects of mixing: one leads to irregular transport and a dramatic, instantaneous loss of regularity for transport equation, the other is enhanced dissipation, which can lead to global existence in non-linear, dissipative systems. In particular, I will show how mixing leads to global existence for the 2D Kuramoto-Sivashisky equation, a model for flame propagation. **

**Join us:****Zoom access: Contact mbrown2@math.tulane.edu
Time: 3:30pm**

**Geometry and Topology**

**Topic: AATRN: How optimal transport can help us to determine the curvature of complex networks?**

**Marzieh Eidi | Max Planck**

**Abstract: Ollivier Ricci curvature is a notion that originated from Riemannian Geometry and is suitable for applying on different settings from smooth manifolds to discrete structures such as (directed) hypergraphs. In the past few years, alongside Forman Ricci curvature, this curvature as an edge-based measure has become a popular and powerful tool for network analysis. This notion is defined based on optimal transport problem (Wasserstein distance) between sets of probability measures supported on data points and can nicely detect some important features such as clustering and sparsity in their structures. After introducing this notion for (directed) hypergraphs and mentioning some of its properties, as one of the main recent applications, I will present the result of the implementation of this tool for the analysis of chemical reaction networks. **

**Recordings of most of the talks will be posted to the AATRN YouTube Channel.**

Zoom access:

Meeting ID: 924 8568 4682

Zoom meeting starts at 10:00 CT

** **

**Week of April 2 - March 29**

**Applied and Computational Mathematics**

**Topic: Learning Temporal Evolution of Spatial Dependence**

**Shiwei Lan | Arizona State University School of Mathematical and Statistical Sciences**

**Abstract: We are living in an era of data explosion usually featured with `big data' or `big dimension'. However, there is another big challenge in data science that we cannot ignore - complex relationship. Spatiotemporal data are ubiquitous in our life and have been a trending topic in the scientific community, e.g. the dynamic brain connectivity study in neuroscience. There is usually complicated dependence among spatial locations and such relationship does not necessarily stay static over time. The temporal evolution of spatial dependence (TESD) is often of scientific interest in understanding the underlying mechanism behind natural phenomena such as cognition and disease progression. **

**In this talk, I will introduce two novel statistical methods to learn TESD in various applications. The first is a semi-parametric method modeling TESD as dynamic covariance matrices [1]. A spherical product representation of covariance matrix is introduced to ensure its positive-definiteness along the process. An efficient MCMC algorithm based on the representation is implemented for Bayesian inference. The second is a fully nonparametric generalization of the first model based on spatiotemporal Gaussian process (STGP) [2]. It further enables scientists to extend the learned TESD to new territory where there are no data. While classic STGP with a covariance kernel separated in space and time fails in this task, I propose a novel generalization to introduce the time-dependence to the spatial kernel that can effectively and efficiently characterize TESD. The utility and advantage of the proposed methods will be demonstrated by a number of simulations, a study of dynamic brain connectivity and a longitudinal neuroimaging analysis of Alzheimer's patients.**

**Join us:****Zoom access**: **Time**: 3:30

**Algebra and Combinatorics Seminar**

**Topic: Uniform Asymptotic Growth of Symbolic Powers of Ideals**

**Robert Walker | University of Wisconsin-Madison**

**Abstract: Algebraic geometry (AG) is a major generalization of linear algebra which is fairly influential in mathematics. Since the 1980's with the development of computer algebra systems like Mathematica, AG has been leveraged in areas of STEM as diverse as statistics, robotic kinematics, computer science/geometric modeling, and mirror symmetry. Part one of my talk will be a brief introduction to AG, to two notions of taking powers of ideals (regular vs symbolic) in Noetherian commutative rings, and to the ideal containment problem that I study in my thesis. Part two of my talk will focus on stating the main results of my thesis in a user-ready form, giving a "comical" example or two of how to use them. At the risk of sounding like Paul Rudd in Ant-Man, I hope this talk will be awesome.**

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 3:00pm**

**Geometry and Topology**

**Topic: AATRN: Graph representation learning and its applications to biomedicine**

**Marinka Zitnik | Harvard University**

**Abstract: he success of machine learning depends heavily on the choice of representations used for prediction tasks. Graph representation learning has emerged as a predominant choice for learning representations of networks. In this talk, I describe our efforts to expand the scope and ease the applicability of graph representation learning. First, I outline SubGNN, a subgraph neural network for learning disentangled subgraph embeddings. SubGNN generates embeddings that capture complex subgraph topology, including structure, neighborhood, and position of subgraphs in a graph. Second, I will discuss applications in biology and medicine. The new methods predicted disease treatments that were experimentally verified in the wet laboratory. Further, the methods helped to discover dozens of combinations of drugs safe for patients with considerably fewer unwanted side effects than today's treatments. Lastly, I describe our efforts in learning actionable representations that allow users of our models to receive predictions that can be interpreted meaningfully.**

**Recordings of most of the talks will be posted to the AATRN YouTube Channel.**

**Zoom access:
Meeting ID: 924 8568 4682
Zoom meeting starts at 10:00 CT**

**Graduate Student Colloquium**

**Topic: The Representation Theory of Angular Momentum in Quantum Mechanics**

**Zachary Bradshaw | Tulane University**

**Abstract: Often, the theory of Lie groups, Lie algebras, and their representations make an appearance in physics. The underlying symmetries responsible for this appearance are exploited by physicists to reduce the complexity of a given problem. For example, the rotational symmetry in the Hydrogen atom is used to split the 3-dimensional problem of finding the eigenvalues of the associated Hamiltonian into two parts: finding a solution for the angular part (a 2-dimensional problem) and finding a solution for the radial part (1-dimensional). These solutions are then pieced back together. In this talk, I will discuss the theory of angular momentum in quantum mechanics from the perspective of the representation theory of Lie groups and Lie algebras.**

**Join us:**

Zoom access: 954 0971 2190

Time: 5:00pm

** **

**Week of March 26 - March 22**

**Applied and Computational Mathematics**

**Topic: Diffuse Interface modeling for two-phase flows: the journey from the model H to the AGG model**

**Andrea Giorgini | Indiana University**

**Abstract: In the last decades, the Diffuse Interface theory (also known as Phase Field theory) has made significant progresses in the description of multi-phase flows from modeling to numerical simulations. A particularly active research topic has been the development of thermodynamically consistent extensions of the well-known Model H in the case of unmatched fluid densities. In this talk, I will focus on the AGG model proposed by H. Abels, H. Garcke and G. Grün in 2012. The model consists of a Navier-Stokes-Cahn-Hilliard system characterized by a concentration-dependent density and an additional flux term due to interface diffusion. Using the method of matched asymptotic expansions, it was shown that the sharp interface limit of the AGG model corresponds to the two-phase Navier-Stokes equations. In the literature, the analysis of the AGG system has only been focused on the existence of weak solutions. During the seminar, I will present the first results concerning the existence, uniqueness and stability of strong solutions for the AGG model in two dimensions.**

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 3:30pm**

**Geometry and Topology**

**Topic: AATRN: Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius**

**Facundo Mémoli | Ohio State**

**Abstract: The persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. We consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space X into a larger ambient metric space E and then considering neighborhoods of the original space X inside E.**

**We then prove that the persistent homology of the Vietoris-Rips filtration is isomorphic to our geometric persistent homology provided that the ambient metric space E satisfies a property called injectivity.**

**As an application of this isomorphism result we are able to precisely characterize the type of intervals that appear in the persistence barcodes of the Vietoris-Rips filtration of any compact metric space and also to give succinct proofs of the characterization of the persistent homology of products and metric gluings of metric spaces.**

**Our results also permit proving several bounds on the length of intervals in the Vietoris-Rips barcode by other metric invariants.**

**As another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov and show some consequences related to (1) the homotopy type of the Vietoris-Rips complexes of spheres which follow from work of M. Katz and (2) characterization (rigidity) results for spheres in terms of their Vietoris-Rips persistence barcodes which follow from work of F. Wilhelm.**

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 10:00 CT**

**Algebra and Combinatorics Seminar**

**Topic: Quantum Groups, R-Matrices, Quantum Yang-Baxter Equation and Solvable Lattice Models**

**Mike Joyce | Tulane University**

**Abstract: In this talk, we will briefly define quantum groups and focus on one specific example, quantum sl_2. Quantum group modules yield interesting R-matrices that arise from the almost cocommutativity of the quantum group. These R-matrices satisfy the Quantum Yang-Baxter Equation (QYBE). We'll connect this to an area of intense current research, solvable lattice models, which have applications in algebraic combinatorics, number theory, and probability. We'll see how quantum sl_2 explains properties of the six vertex model, the simplest and most well-understood solvable lattice model.**

**Join us:****Zoom access**: **Time**: 3:00

**Graduate Student Colloquium**

**Topic: Convex sets associated to algebraic objects**

**Thai Nguyen | Tulane University**

**Abstract: I will talk about a way to associate convex sets to polynomials and ideals in polynomial rings. This idea can be traced back to an idea of Issac Newton (~1676) that was used to prove Newton-Puiseux theorem. In 1996, Okounkov introduced a way to associate convex sets to algebraic varieties in order to show the log-concavity of the degrees of those algebraic varieties. Later on, this construction was studied systematically by the works of Kaveh-Khovanskii and Lazarsfeld-Mustata (~2009). It has become a very active research area recently and is now known as the theory of Newton-Okounkov bodies. Among numerous applications of this theory, I will present a beautiful one, which is to count the number of solutions of a system of (general) polynomials (so-called Kouchnirenko-Bernstein theorem that was proved many years ago). I will also discuss how this relates to my current project, involving Newton polyhedra and symbolic polyhedra.**

**Join us:
Zoom access: 954 0971 2190
Time: 5:00pm**

** **

**Week of March 19 - March 15**

**Applied and Computational Mathematics**

**Topic: Monte Carlo methods for the Hermitian eigenvalue problem**

**Robert Webber | NYU-Courant**

**Abstract: In quantum mechanics and the analysis of Markov processes, Monte Carlo methods are needed to identify low-lying eigenfunctions of dynamical generators. The standard Monte Carlo approaches for identifying eigenfunctions, however, can be inaccurate or slow to converge. What limits the efficiency of the currently available spectral estimation methods and what is needed to build more efficient methods for the future? Through numerical analysis and computational examples, we begin to answer these questions. We present the first-ever convergence proof and error bounds for the variational approach to conformational dynamics (VAC), the dominant method for estimation eigenfunctions used in biochemistry. Additionally, we analyze and optimize variational Monte Carlo, which combines Monte Carlo with neural networks to accurately identify low-lying eigenstates of quantum systems.**

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 3:30pm**

**Algebra and Combinatorics Seminar**

**Topic: On combinatorics of Arthur's trace formula, convex polytopes, and toric varieties**

**Kiumars Kaveh | University of Pittsburgh**

**Abstract: I start by discussing two beautiful well-known theorems about decomposing a convex polytope into an signed sum of cones, namely the classical Brianchon-Gram theorem and Lawrence-Varchenko theorem. I will then explain a generalization of the Brianchon-Gram which can be summerized as "truncating a function on the Euclidean space with respect to a polytope". This is an extraction of the combinatorial ingredients of Arthur's ''convergence'' and ''polynomiality'' results in his famous trace formula. Arthur's trace formula concerns the trace of left action of a reductive group G on the space L^2(G/Γ) where Γ is a discrete (arithmetic) subgroup. The combinatorics involved is closely related to compactifications of ''locally summetric spaces'' (which btw are hyperbolic manifolds). Our ''combinatorial truncation'' can be thought of as an analogue of Arthur's truncation over a toric variety (in place of a compactification of a locally symmetric space). This is joint work in progress with Mahdi Asgari (Oklahoma State).**

**Join us:****Zoom access**: **Time**: 3:30

**Graduate Student Colloquium**

**Topic: Ulam´s problem**

**John Lopez | Tulane University**

**Abstract: In this talk we will take a look to Ulam´s problem, which is related to the length (l_n) of the longest increasing subsequence in a random permutation of {1, 2, 3,...,n}. This question has been studied by a variety of increasingly technically sophisticated methods over the last 30 years. We will study the asymptotic behavior of the expected value of l_n, (E(l_n)), by analyzing the limit of E(l_n)/sqrt(N). Specifically, we will derive a lower bound for this limit using some combinatorial arguments and will also introduce some probabilistic tools which are used to determine its existence.**

**Join us:****Zoom access**: 954 0971 2190**Time**: 5:00pm

** **

**Week of March 12 - March 8**

**Applied and Computational Mathematics**

**Topic: Is MCMC Really Slower Than Variational Inference?**

**Matt Hoffman | Google Research**

**Abstract: Variational inference (VI) and Markov chain Monte Carlo (MCMC) are approximate posterior inference algorithms that are often said to have complementary strengths, with VI being fast but biased and MCMC being slower but asymptotically unbiased. We analyze gradient-based MCMC and VI procedures and find theoretical and empirical evidence that these procedures are not as different as one might think. In particular, a close examination of the Fokker- Planck equation that governs the Langevin dynamics (LD) MCMC procedure reveals that LD implicitly follows a gradient flow that corresponds to a VI procedure based on optimizing a nonparametric normalizing flow. This result suggests that the transient bias of LD (due to the Markov chain not having burned in) may track that of VI (due to the optimizer not having converged), up to differences due to VI’s asymptotic bias and parameterization. Empirically, we find that the transient biases of these algorithms (and their momentum-accelerated counterparts) do evolve similarly. This suggests that practitioners with a limited time budget may get more accurate results by running an MCMC procedure (even if it doesn't quite converge) than a VI procedure, as long as the variance of the MCMC estimator can be dealt with (e.g., by running many parallel chains on a GPU). I will also briefly discuss ChEES-HMC, an adaptive Hamiltonian Monte Carlo method that is better suited to GPU parallelization than the widely used NUTS algorithm**

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 3:30pm**

** **

**Geometry and Topology**

**Topic: Vietoris-Rips thickenings: Problems for birds and frogs**

**Henry Adams | Colorado State**

**Abstract: An artificial distinction is to describe some mathematicians as birds, who from their high vantage point connect disparate areas of mathematics through broad theories, and other mathematicians as frogs, who dig deep into particular problems to solve them one at a time. Neither type of mathematician thrives without the help of the other! In this talk, I will survey open problems related to Vietoris-Rips complexes that are attractive to both birds and frogs. Though Vietoris-Rips complexes are frequently used to approximate the shape of a dataset, many questions remain about their mathematical properties. Frogs may delight in open problems such as the homotopy types of Vietoris-Rips complexes of spheres, ellipsoids, tori, graphs, Cayley graphs of groups, geodesic spaces, subsets of the plane, and even the integer lattice Z^n with the taxicab metric for n >= 4. Birds may enjoy emerging connections between Vietoris-Rips complexes and a variety of areas in pure mathematics, including metric geometry (Gromov-Hausdorff distances), quantitative topology (Gromov's filling radius), measure theory (optimal transport), topological combinatorics (Borsuk-Ulam theorems), geometric group theory (finiteness properties of groups), and geometric topology (thick-thin decompositions).**

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 10:00 CT **

**Algebra and Combinatorics Seminar**

**Topic: When is a (projectivized) toric vector bundle a Mori dream space?**

**Christopher Manon | University of Kentucky**

**Abstract: **Like toric varieties, toric vector bundles are a rich class of varieties which admit a combinatorial description. Following the classification due to Klyachko, a toric vector bundle is captured by a subspace arrangement decorated by toric data. This makes toric vector bundles an accessible test-bed for concepts from algebraic geometry. Along these lines, Hering, Payne, and Mustata asked if the projectivization of a toric vector bundle is always a Mori dream space. Su\ss and Hausen, and Gonzales showed that the answer is "yes" for tangent bundles of smooth, projective toric varieties, and rank 2 vector bundles, respectively. Then Hering, Payne, Gonzales, and Su\ss showed the answer in general must be "no" by constructing an elegant relationship between toric vector bundles and various blow-ups of projective space, in particular the blow-ups of general arrangements of points studied by Castravet, Tevelev and Mukai. In this talk I'll review some of these results, and then show a new description of toric vector bundles by tropical information. This description allows us to characterize the Mori dream space property in terms of tropical and algebraic data. I'll describe new examples and non-examples, and pose some questions. This is joint work with Kiumars Kaveh.

**Join us:****Zoom access**: Contact mbrown2@math.tulane.edu**Time**: 3:30pm

**Week of March 5 - March 1**

**Applied and Computational Mathematics**

**Topic: TBA**

**Nitsan Ben-Gal | 3M**

**Abstract: ****TBA**

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 3:30pm**

**Colloquium**

**Topic: TBA**

**Mark Girolami | Cambridge (Host: Glatt-Holtz)**

**Abstract: **TBA

**Join us:****Zoom access**: Contact mbrown2@math.tulane.edu**Time**: 3:30pm

**Algebra and Combinatorics Seminar**

**Topic: A Survey of Hopf Algebras and Quantum Groups**

**Mike Joyce | Tulane University**

**Abstract: **We will introduce Hopf algebras and quantum groups through some of their key properties and some of the simplest examples. We will discuss R-matrices and the Yang-Baxter equation and then survey some of their manifestations in other areas of mathematics.

**Join us:****Zoom access**:**Time**: 3:30

**Graduate Student Colloquium**

**Topic: An Intro to Knot Invariants**

**Will Tran | Tulane University**

**Abstract: **The word "knot" and phrase "knot theory" have been intimidating students since the early 1900s. We may see the word "knot" and immediately shut down, thinking "I don't know enough topology to study knots." It turns out that understanding knot invariants -- a primary focus in knot theory -- requires little to no topology at all! In fact, with some undergraduate linear algebra and an open mind, you too can get started on understanding knot theory. In this talk, we'll compute elementary knot invariants such as p-colorability, the knot determinant, and the Alexander Polynomial. If time permits, we'll see how other knot invariants like the Gauss Linking Integral, Linking Number, and Moebius Energy can connect to number theory, algebra, analysis, biology, and physics.

**Join us:**

**Meeting ID: 954 0971 2190 Time: 5:00pm**

**Week of February 26 - February 22**

**Applied and Computational Mathematics**

**Topic: An examination of school reopening strategies during the SARS-CoV-2 pandemic**

**Alfonso Landeros | UCLA**

**Abstract: ****The SARS-CoV-2 pandemic led to closure of nearly all K-12 schools in the United States of America in March 2020. Although reopening K-12 schools for in-person schooling is desirable for many reasons, officials understand that risk reduction strategies and detection of cases are imperative in creating a safe return to school. Furthermore, consequences of reclosing recently opened schools are substantial and impact teachers, parents, and ultimately educational experiences in children.**

**In this talk, I will present a compartmental model developed to explore scenarios under which reopening schools may be deemed safe and to evaluate mitigation strategies. Specifically, the question of differences in transmissibility will be discussed alongside a multiple cohort approach.**

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 3:30pm**

**Algebra and Combinatorics Seminar**

**Topic: Invariants and properties of symbolic powers of edge and cover ideals**

**Joseph Skelton | Tulane University**

**Abstract: **In this talk I will address several questions about symbolic powers of edge and cover ideals. The containment between ordinary and symbolic powers of edge ideals has been an active area of research for decades. As a result the resurgence number and Waldschmidt constant are of particular interest. The regularity of symbolic powers of edge ideals has been motivated by a conjecture of N.C. Minh which states that $\reg I(G)^{(s)} = \reg I(G)^s$ for any $s\in \NN$.

**For cover ideals we are motivated by the results of Villlarreal showing that whiskering a graph results in a Cohen-Macaulay graph which, in turn, implies the cover ideal of the whiskered graph has linear resolution. Necessary and sufficient conditions on $S\subset V(G)$ cover ideal of the graph whiskered at $S$, $J(G\cup W(S))$ is Cohen-Macaulay. While symbolic powers of the cover ideal do not necessarily have linear resolution I will show necessary conditions on $S$ such that symbolic powers of $J(G\cup W(S))$ have componentwise linearity.**

**Join us:****Zoom access**: **Time**: 3:00pm

**Graduate Student Colloquium**

**Topic: A tutorial on Peirce's Graphical Logic**

**Alex Nisbet | Tulane University**

**Abstract: **Charles S. Peirce was in important figure in the development of the ``algebra of logic'' in the second half of the 19th and early 20th century. In particular, he is known for his contributions in developing Boole's system, the logic of relatives, and quantification. Indeed, the algebraic notation he used is more-or-less what we use today, after some symbolic changes by Peano. Less well-known is Peirce's system of graphical logic, which he preferred to the algebraic notation. Here I will give a brief tutorial on the alpha and beta parts of this system. The alpha part corresponds to propositional logic and the beta to first order logic with equality.

**Join us:****Time**: 5:00pm**Zoom ID**: **954 0971 2190**

**Week of February 19 - February 15**

**Applied and Computational Mathematics**

**Topic: Convergence of unadjusted Hamiltonian Monte Carlo for mean-field models**

**Katharina Schuh | Hausdorff Center for Mathematics, University of Bonn**

**Abstract: In the talk, we consider the unadjusted Hamiltonian Monte Carlo algorithm applied to highdimensional probability distributions of mean-field type. We evolve dimension-free convergence and discretization error bounds. These bounds require the discretization step to be sufficiently small, but do not require strong convexity of either the unary or pairwise potential terms present in the mean-field model. To handle high dimensionality, we use a particlewise coupling that is contractive in a complementary particlewise metric. This talk is based on joint work with Nawaf Bou-Rabee.**

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 3:30pm**

**Algebra and Combinatorics Seminar**

**Topic: A Survey of Classical Representation Theory**

**Mike Joyce | Tulane University**

**Abstract: **Representation theory is a vast field which has applications in many other areas of mathematics, including algebra and combinatorics. This talk will review some of the classical theory of representations of groups and Lie algebras, with an emphasis thats lead to more modern aspects of representation theory, which will be addressed in a second talk.

**Join us:****Zoom access**: **Time**: 3:30pm

**Week of February 12 - February 8**

**Applied and Computational Mathematics**

**Topic: Unravelling A Geometric Conspiracy**

**Michael Betancourt | Symplectomorphic, LLC**

**Abstract: The Hamiltonian Monte Carlo method has proven a powerful approach to efficiently exploring complex probability distributions. That power, however, is something of a geometric conspiracy: a sequence of delicate mathematical machinations that ensure a means to explore distributions not just in theory but also in practice. In this talk I will discuss the coincident geometrical properties that ensure the scalable performance of Hamiltonian Monte Carlo and present recent work developing new geometric theories that generalize each of these properties individually, providing a foundation for generalizing the method without compromising its performance."**

The geometrical concepts get nontrivial towards the end but hopefully it will be sufficiently engaging for many!

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 3:30pm**

**Colloquium**

**Topic: Class numbers of quadratic fields**

**Olivia Beckwith | University of Illinois Urbana-Champaign (Host: Lisa Fauci)**

**Abstract: **Gauss was the first to count classes of binary quadratic forms with a fixed discriminant up to matrix equivalence. The number of equivalence classes, the class number, measures the obstruction to unique factorization into primes for quadratic number fields. Information about class numbers percolates into many branches of number theory, including the theory of L-functions via Dirichlet's class number formula, and elliptic curves in view of the work and conjecture of Birch and Swinnerton-Dyer. This talk will begin with a brief introduction to algebraic number theory and class numbers, as well as some of the important results in the history of their study. Then I will discuss some of my work in this area, which is about the divisibility properties of class numbers.

**Join us:****Zoom access**: Contact mbrown2@math.tulane.edu**Time**: 3:30pm

**Graduate Student Colloquium**

**Topic: Linearity Testing**

**Victor Bankston | Tulane University**

**Abstract: **We will discuss techniques for deciding if a Boolean function is linear.

**Join us:**>

Zoom ID: 954 0971 2190

Time: 5:00pm

**Colloquium**

**Topic: Brownian Dynamics with Constraints**

**Brennan Sprinkle | New York University**

**Abstract: **At the scale of a few micrometers, objects suspended in a fluid are subject to random kicks from collisions with the solvent molecules. This leads to a random motion of the suspended objects which must be reconciled with any geometric or mechanical constraints, like rigidity or inextensibility. After discussing numerical methods to simulate the Brownian dynamics of rigid bodies, I will primarily focus on simulation methods for inextensible filaments. Filaments at the cellular scale can take the form of beating flagella that propel sperm cells and bacteria; or they can tangle into the vast, interconnected networks that make up the cellular cytoskeleton. I’ll introduce a method where fibers are treated as a chain of beads and use it to interrogate experimental observations on magnetic filaments which can be made to swim using an applied field. Motivated by this study, I’ll present ongoing work concerning a method more suited to fiber networks in which inextensible fiber motions are parametrized as curves on the unit sphere.

**Join us:****Zoom access**: Contact mbrown2@math.tulane.edu**Time**: 3:30pm

**Special Colloquium**

**Topic: Edge Ideals of Random Graphs**

**Arindam Banerjee | (at Ramakrishna Mission Vivekananda Educational and Research Institute)**

**Abstract: **The theory of edge ideal studies finite simple graphs from an algebraic perspective. It attaches an ideal with every finite simple graph and tries to interpret the combinatorics of the graph in terms of various algebraic invariants of that ideal. Philosophically speaking, one may ask what happens to those invariants on an average when one considers all possible graphs. The study of the edge ideals of Erdos-Renyi random graphs gives a nice mathematical framework to properly pose that philosophical question. In this talk we shall develop this framework and use that to discuss average behaviours of some important algebraic invariants. In particular we shall discuss a new result which shows that algebraic invariants Krull Dimension and Castelnuovo-Mumford regularity (a measure of size and a measure of complexity respectively) satisfy some law of large number when number of vertices of the underlying Erdos-Renyi random graph goes to infinity.

**Join us:****Zoom access**: Contact mbrown2@math.tulane.edu**Time**: **10:30AM – 11:30AM**

**Week of February 5 - February 1**

**Colloquium**

**Topic: The Batchelor spectrum and mixing in stochastic fluids**

**Samuel Punshon-Smith | Brown University (Scott McKinley)**

**Abstract: **In 1959 George Batchelor predicted that a passively advected quantity in a fluid (like small temperature fluctuations or some chemical concentration), in a regime where the scalar dissipation is much lower than the fluid viscosity, should reach an equilibrium with an L2 spectral density proportional to 1/|k| over an appropriate inertial range, known as "the Batchelor spectrum". This prediction has since been observed experimentally and in various numerical experiments. However, despite strong evidence in its favor, rigorous derivations are only known in very special circumstances.

In this talk, I will consider the problem of a passive scalar undergoing advection-diffusion when the advecting velocity field belongs to a class of stochastic incompressible fluid motions, including models like the 2d incompressible stochastic Navier-Stokes equations (among a host of other stochastic fluid models in both 2d and 3d). I will discuss how a version of Batchelor's prediction is actually a general consequence of uniform-in-diffusivity exponential mixing properties of a fluid. Based on an argument inspired by techniques of Furstenberg for random dynamical systems, I will present a result deducing almost sure chaotic motion of the particle trajectories, known as Lagrangian chaos. I will then discuss how Lagrangian chaos can be used along with spectral theory and quantitative techniques from the ergodic theory of Markov processes to deduce almost sure, uniform-in-diffusivity exponential mixing, a powerful property that is not known to hold in the deterministic setting. This result shows that a version of Batchelor's prediction is indeed fairly robust and holds quite generally for a variety of suitably non-degenerate, stochastic incompressible fluid motions.

**Join us:****Zoom access**: Contact mbrown2@math.tulane.edu**Time**: 3:30pm

**Algebra and Combinatorics Seminar**

**Topic: The Least Generating Degree of Symbolic Powers and Ideal Containment Problem**

**Thai Nguyen | Tulane University**

**Abstract: **What is the smallest degree of a homogeneous polynomial that vanishes to order m on a given finite set of points, or more generally on some algebraic variety in projective space? A classical result of Zariski and Nagata tells us the set of such polynomials is the m-th symbolic power of the defining ideal I of the variety. To bound the generating degree of the symbolic powers of I, we can study containment between symbolic powers and ordinary powers of I. Conversely, knowing bounds for generating degree can help us study containment. My talk will be an introduction to this subject. I will also present some results from our joint work with Sankhaneel Bisui, Eloísa Grifo and Tài Huy Hà.

**Join us:****Zoom access**: **Time**: 3:00pm

**Graduate Student Colloquium**

**Topic: Living your best life with symbolic math software**

**Dana Ferranti | Tulane University**

**Abstract: **I've only recently started using symbolic math software in my research and I wish I started earlier. This talk will focus on a few ways that one can use symbolic math software to make their workflow more efficient, as well as its limitations. For my examples I will be using the open source SymPy, however, the concepts will be applicable to other software.

**Join us:Time: 5:00pm
Zoom ID**: 954 0971 2190

** **

**Colloquium**

**Topic: Rigidity of plane frameworks with forced symmetry**

**Daniel Bernstein | Fields Institute**

**Abstract: **Rigidity theory asks and answers questions about how a given mechanical structure can deform. This area extends back into the nineteenth century with work of Cauchy and Maxwell, and continues to be an active area of research with a wide range of applications. I will begin my talk with a broad overview of this area. Then, I will narrow my focus onto symmetry-forced rigidity of plane frameworks to discuss a recent result. I will discuss the algebraic-geometric ideas involved in the proof, and how these same ideas can be used to address certain problems in matrix completion.

**Join us:****Zoom access**: Contact mbrown2@math.tulane.edu**Time**: 3:30pm

**Week of January 29 - January 25**

**Applied and Computational Mathematics**

**Topic: ***DNA methylation-based aging biomarkers in health and disease *

**Mary Sehl | UCLA**

**Abstract: **DNA methylation-based estimates of age are strongly correlated with chronologic age across many cell types and tissues. Importantly, these biologic aging estimates are accelerated in disease states, and predictive of both lifespan and healthspan. Recent evidence suggests that female breast tissue ages faster than other parts of the body in healthy women, based on the Horvath pan-tissue epigenetic clock. Estrogens are thought to contribute to breast cancer risk through cell cycling and accelerated breast aging. We hypothesize that epigenetic breast aging is driven by lifetime estrogen exposure. In this talk, we will review the development and key features of several epigenetic clocks including Horvath’s pan-tissue clock and the Hannum clock, as well as second generation clocks including the Phenotypic age, Grim age, and Skin and Blood age clocks. We will describe findings from a recent study examining associations between hormonal factors (including earlier age at menarche, and body mass index) and these epigenetic aging measures in healthy women. We will further describe additional applications of peripheral blood methylation age estimates to study biologic age acceleration in HIV-infected men pre- and post-initiation of antiretroviral therapy, and in early stage breast cancer survivors undergoing radiotherapy and chemotherapy.

**Zoom access**:

**Meeting ID:
Zoom meeting starts at 3:30pm**

**Colloquium**

**Topic:** *Mathematically Modeling the Mechanisms Behind Intra-Droplet and Droplet Field Patterning in Phase Separated Systems*

**Dr. Kelsey Gasior | Florida State University**

**Abstract:**

**Join us:****Zoom access**: Contact mbrown2@math.tulane.edu**Time**: 3:30pm

**Algebra and Combinatorics**

**Topic:** *Stable Harbourne-Huneke containment and Chudnovsky's Conjecture*

**Sankhaneel Bisui | Tulane University**

**Abstract:**

**Join us:****Zoom access**:**Time**: 3:30pm

**Colloquium**

**Topic:** *Modeling and analysis of complex systems — with a basis in zebrafish patterns*

**Alexandria Volkening | Northwestern University**

**Abstract:** Many natural and social phenomena involve individual agents coming together to create group dynamics, whether they are cells in a skin pattern, voters in an election, or pedestrians in a crowded room. Here I will focus on the specific example of pattern formation in zebrafish, which are named for the dark and light stripes that appear on their bodies and fins. Mutant zebrafish, on the other hand, feature different skin patterns, including spots and labyrinth curves. All these patterns form as the fish grow due to the interactions of tens of thousands of pigment cells. The longterm motivation for my work is to better link genes, cell behavior, and visible animal characteristics — I seek to identify the specific alterations to cell interactions that lead to mutant patterns. Toward this goal, I develop agent-based models to simulate pattern formation and make experimentally testable predictions. In this talk, I will overview my models and highlight future directions. Because agent-based models are not analytically tractable using traditional techniques, I will also discuss the topological methods that we have developed to quantitatively describe cell-based patterns, as well as the associated nonlocal continuum limits of my models.

**Join us:****Zoom access**: Contact mbrown2@math.tulane.edu**Time**: 3:30pm