**Week of January 31 - January 27**

Colloquium

**Topic:** *Tropical Algebraic Geometry*

**Kalina Mincheva - Yale University**

**Abstract:**

Tropical geometry provides a new set of purely combinatorial tools to approach classical problems in algebraic geometry. The fundamental objects in tropical geometry are tropical varieties -- combinatorial ``shadows" associated to more traditional geometric objects, algebraic varieties. Until recently, the theory has focused on the geometric aspects of tropical varieties as opposed to the underlying algebra, largely due the lack of tropical analogues to commutative algebra tools. Consequently, there has recently been a lot of effort dedicated to developing such tools using different frameworks -- notably prime congruences, tropical ideals, and tropical schemes. These approaches allow for the exploration of tropical spaces as inherently tropical objects. In this talk, we present a notion of prime congruences and discuss the resulting analogues to classical theorems, such as a Nullstellensatz and aspects of dimension theory. We also demonstrate connections to algebraic geometry via the theory of tropical schemes and ideals.

**Location: **Gibson Hall 126A**Time: 3:30**

Algebra & Combinatorics

**Topic:** *Total nonnegativity and induced sign characters of the Hecke algebra*

**Mark Skandera - Lehigh University**

**Abstract:**

Gantmacher's study of totally nonnegative (TNN) matrices in the 1930's eventually found applications in many areas of mathematics. Descending from his work are problems concerning TNN polynomials, those polynomial functions of n^2 variables which take nonnegative values on TNN matrices. Closely related to TNN polynomials are functions in the Hecke algebra trace space whose evaluations at certain Hecke algebra elements yield polynomials in N[q]. In all cases, it would be desirable to combinatorially interpret the resulting nonnegative numbers. In 2017, Kaliszewski, Lambright, and the presenter found the first cancellation-free combinatorial formula for the evaluation of all elements of a basis of V at all elements of a basis of the Hecke algebra. We will discuss a recent improvement upon this result which also advances our understanding of TNN polynomials. This is joint work with Adam Clearwater.

**Location: **Gibson Hall 127**Time: **3:00

Colloquium

**Topic:** *Numerical methods for ocean models and venous valve simulations*

**Sara Calandrini - Florida State University**

**Abstract:**

**Location: **Stanley Thomas 316**Time: **3:00

Graduate Student Colloquium

**Topic:** *A PDE model for chemotaxis with logistic growth*

**Jiao Xu & Padi Fuster - Tulane University**

**Abstract:**

In this talk, we will derive a PDE model for chemotaxis (the movement of an organism in response to a chemical stimulus) with logistic growth. We will discuss the general derivation of the model and on what phenomena this can be applied to. We will also briefly talk about our results on the existence of solutions for this system of PDE.

**Location: **Stanley Thomas 316**Time: **4:30

Colloquium

**Topic:** *Quantum and symplectic invariants in low-dimensional topology.*

**Nathan Dowlin - Columbia University**

**Abstract:**

Khovanov homology and knot Floer homology are two powerful knot invariants developed around two decades ago. These invariants have been applied to problems all over low-dimensional topology, from detecting exotic smooth structures on 4-manifolds to determining whether a given knot diagram is the unknot. Knot Floer homology is defined using symplectic techniques, while Khovanov homology has its roots in the representation theory of quantum groups. Despite these differences, they seem to have many structural similarities. A well-known conjecture of Rasmussen from 2005 states that for any knot K, there is a spectral sequence from the Khovanov homology of K to the knot Floer homology of K. Using a new family of invariants defined using both quantum and symplectic techniques, I will give a proof of this conjecture and describe some topological applications.

**Location: **Stanley Thomas 316**Time: **2:00

**Week of January 24 - January 20**

Colloquium

**Topic:** *Modeling and simulation of symmetry breaking in cells*

**Calina Copos - New York University**

**Abstract:**

In order to initiate movement, cells need to form a well-defined "front" and "rear" through the process of cellular polarization. Polarization is a crucial process involved in embryonic development and cell motility and it is not yet well understood. Mathematical models that have been developed to study the onset of polarization have explored either biochemical or mechanical pathways, yet few have proposed a combined mechano-chemical mechanism. However, experimental evidence suggests that most motile cells rely on both biochemical and mechanical components to break symmetry. We have identified one of the simplest quantitative frameworks for a possible mechanism for spontaneous symmetry breaking for initiation of cell movement. The framework relies on local, linear coupling between minimal biochemical stochastic and mechanical deterministic systems; this coupling between mechanics and biochemistry has been speculated biologically, yet through our model, we demonstrate it is a necessary and sufficient condition for a cell to achieve a polarized state.

**Location: **Stanley Thomas 316**Time: **3:00

Colloquium

**Topic:** *Hessenberg varieties and the Stanley--Stembridge conjecture*

**Martha Precup - Washington U in St Louis (Host: Mahir Can)**

**Abstract:**

Hessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. In 2015, Brosnan and Chow proved the Shareshian-Wachs conjecture, linking the Stanley-Stembridge conjecture in combinatorics to the geometry of Hessenberg varieties through Tymoczko's permutation group action on the cohomology ring of regular semisimple Hessenberg varieties. This talk will give an overview of that story and present a new set of linear relations satisfied by the multiplicities of certain permutation representations in Tymoczko's representation. As an application of these results, we prove an inductive formula for the multiplicity coefficients corresponding to partitions with a maximal number of parts. This is joint work with M. Harada.

**Location: **Gibson Hall 126A**Time: 3:30**

AWM Coffee discussion

**Topic:** *Coffee Discussion*

**Martha Precup - Washington U in St Louis**

**Abstract:**

The AWM chapter is organizing coffee discussion with this week's colloquium speaker, Martha Precup from Washington University in St.Louis.

This is an informal meeting to have coffee and good conversation with our colloquim speaker. Come say hi! There will be coffee and cookies! Don't forget your reusable mugs.

**Location: **Stanley Thomas 404**Time: 11:00**

Algebra & Combinatorics

**Topic:** *Gorenstein polytopes*

**Takayuki Hibi - Osaka University**

**Abstract:**

A Gorenstein polytope is a lattice polytope one of whose dilated polytopes is a reflexive polytope. In my talk, after reviewing Gorenstein polytopes from a viewpoint of enumeration of lattice points, several conjectures arising from Gorenstein polytopes will be reported. No special knowledge will be required to understand my talk.

**Location: **Gibson Hall 127**Time: **3:00

Graduate Student Colloquium

**Topic:** *Shape Comparison and Gromov-Hausdorff Distance*

**Sushovan Majhi | Tulane University**

**Abstract:**

The Gromov-Hausdorff distance between any two metric spaces was first introduced by M. Gromov in the context of Riemannian manifolds. This distance measure has recently received increasing attention from researchers in the field of topological data analysis. In applications, shapes are modeled as abstract metric spaces, and the Gromov-Hausdorff distance has been shown to provide a robust and natural framework for shape comparison. In this talk, we will introduce the notion and address the difficulties in computing the distance between two Euclidean point-clouds. In the light of our recent findings, we will also describe an O(n log n)-time approximation algorithm for Gromov-Hausdorff distance on the real line with an approximation factor of (1+ 1/4).

**Location: **Stanley Thomas 316**Time: **4:30

Colloquium

**Topic:** *Math in the lab: mass transfer through fluid-structure interactions*

**Jinzi Mac Huang - University of California San Diego**

**Abstract:**

The advancement of mathematics is closely associated with new discoveries from physical experiments. On one hand, mathematical tools like numerical simulation can help explain observations from experiments. On the other hand, experimental discoveries of physical phenomena, such as Brownian motion, can inspire the development of new mathematical approaches. In this talk, we focus on the interplay between applied math and experiments involving fluid-structure interactions -- a fascinating topic with both physical relevance and mathematical complexity. One such problem, inspired by geophysical fluid dynamics, is the experimental and numerical study of the dissolution of solid bodies in a fluid flow. The results of this study allow us to sketch mathematical answers to some long standing questions like the formation of stone forests in China and Madagascar, and, how many licks it takes to get to the center of a lollipop. We will also talk about experimental math problems at the micro-scale, focusing on the mass transport process of diffusio-phoresis, where colloidal particles are advected by a concentration gradient of salt solution. Exploiting this phenomenon, we see that colloids are able to navigate a micro-maze that has a salt concentration gradient across the exit and entry points. We further demonstrate that their ability to solve the maze is closely associated with the properties of a harmonic function – the salt concentration.

**Location: **Stanley Thomas 316**Time: **3:30

**Week of January 17 - January 13**

Algebra and Combinatorics

**Topic:** *Decomposable Specht modules*

**Liron Speyer - Okinawa Institute of Science and Technology**

**Abstract:**

I will give a brief survey of the study of decomposable Specht modules for the symmetric group and its Hecke algebra, which includes results of Murphy, Dodge and Fayers, and myself, before reporting on recent work with Louise Sutton, in which we have studied decomposable Specht modules for the Hecke algebra of type $B$ indexed by `bihooks’. I will present our conjectured classification of decomposable Specht modules indexed by bihooks, which we proved `one half of’, and some ongoing work in explicitly determining the structure of those decomposable Specht modules.

**Location: **Gibson Hall 127**Time: **3:00

Colloquium

**Topic:** *From Zariski-Nagata to local fundamental groups*

**Jack Jeffries - Mathematics Research Center, Mexico**

**Abstract:**

Hilbert's Nullstellensatz gives a dictionary between algebra and geometry; e.g., solution sets to polynomial equations over the complex numbers (varieties) translate to (radical) ideals in polynomial rings. A classical theorem of Zariski-Nagata gives a deeper layer to this correspondence: polynomial functions that vanish to certain order along a variety correspond to a natural algebraic notion called symbolic powers.

In this talk, we will explain this theorem, and then pursue a couple of variations on this theme. First, we will consider how the failure of this theorem over ambient spaces with bends and corners allows us to study the geometry of such spaces; in particular, we will give bounds on size of local fundamental groups. Second, we will consider what happens when we replace the complex numbers by the integers; we will show that "arithmetic differential geometry" (in the sense of Buium) allows us to obtain a Zariski-Nagata theorem in this setting. Only a passing familiarity with polynomials and complex numbers is assumed.

This is based on joint projects with Holger Brenner, Alessandro De Stefani, Eloísa Grifo, Luis Núñez-Betancourt, and Ilya Smirnov.

**Location: **Stanley Thomas 316**Time: **2:00

Research Seminar Name

**Topic:** *Title*

**Speaker - Institution**

**Abstract:**

TBA

**Location: **TBA**Time: **TBA