TAGMaC Fall 2025

The NC State chapter of the AMS will be hosting TAGMaC at SAS Hall (2311 Stinson Dr. Raleigh, NC) on November 15th from 10:00-4:00 pm.

Plenary Talk

We are please to welcome Dr. Jacob Matherne as our plenary speaker this year.

Dr. Jacob Matherne is an Assistant Professor of Mathematics at NC State university. Jacob’s research is focused on the interplay among geometry, combinatorics, and representation theory. Some objects that pop up often are hyperplane arrangements, matroids, cluster algebras, perverse sheaves, quiver representations, algebraic groups, nilpotent cones, and flag varieties.

Title: Chow functions for partially ordered sets

Abstract: Three decades ago, Stanley and Brenti initiated the study of the Kazhdan–Lusztig–Stanley (KLS) theory.  To each kernel in a graded poset, they associate special functions called KLS polynomials.  This unifies and puts on common ground several important theories in combinatorics and representation theory: (i) the classical Kazhdan–Lusztig polynomial of a Bruhat interval in a Coxeter group, (ii) the toric g-polynomial of a polytope, and (iii) the Kazhdan–Lusztig polynomial of a matroid.

Abstracts:

Ala’ Alalabi

Title: Optimal Control in Poroelastic Systems via E-Radiality Theory
Abstract:
Fluid flows through deformable, porous media arise in numerous applications, ranging from geomechanics to biomechanics. These processes are governed by coupled systems of partial differential equations (PDEs) that describe the interaction between fluid transport and mechanical deformation. A widely adopted framework for such systems is Biot’s theory of poroelasticity, which leads, under quasi-static assumptions, to a system of coupled parabolic-elliptic PDEs, falling under the umbrella of implicit, degenerate evolution equations. While the existence and regularity theory for such systems is well established, optimal control problems constrained by poroelastic dynamics, especially relevant in biomedical contexts where internal pressure and displacement must be regulated, have only recently been considered. These control problems are challenging due to the degeneracy of the equations, the implicit nature of the coupling, and the structure of the control-to-state map. In this work, we reinterpret the coupled flow-deformation problem as an implicit dynamical system on Hilbert spaces and analyze it using E-radiality theory – a generalization of the classical Hille-Yosida framework for non-explicit evolution equations. This abstract formulation yields new insights into well-posedness theory and enables a unified treatment of linear control problems constrained by degenerate evolution dynamics.

Alexander Stewart

Title: Combinatorics of Nonsymmetric Macdonald Polynomials
Abstract:
Nonsymmetric Macdonald polynomials are an important family of polynomials related to affine root systems. They were originally defined using as a family of orthogonal polynomials to the Cherednik’s inner product. The original definition is extremely cumbersome so a more general theory using the double affine Hecke algebra of an affine root system was developed to make it easier to study this family of polynomials. Like other Lie-theoretic objects, it turns out that there is a deep combinatorial structure to Macdonald polynomials specifically when they are of Type A. My talk will focus on two different combinatorial models HHL fillings and MLQ whose generating functions turn out to be the Nonsymmetric Macdonald polynomials. The differences between the two models will be explored, and I will share my work to reconcile both models.

Anna Shapiro

Title: The Möbius function of a polymatroid
Abstract:
The Möbius function of a poset assigns an integer to each closed interval of the poset. Polymatroids are a combinatorial object which generalize matroids. We can associate a particular poset to each polymatroid, and we use the Möbius function of this poset to define the Möbius function of a polymatroid. We will not assume any prior knowledge of Möbius functions or polymatroids, so this talk will begin with the relevant background information. We will then define the Möbius function of a polymatroid, and state a nice formula for it.

Arjun Nigam

Title: Geometries
Abstract:

Camryn Thompson

Title: Enumerating Rectangular Domino Tableaux
Abstract:
A domino tableau is a domino tiling of a Young diagram in which the n dominoes are labeled by the distinct integers [n] such that the domino labels across each row and down each column are strictly increasing. In this talk, we enumerate the set of domino tableaux of shape 2xn and the set of rectangular domino tableaux, using two different bijective proofs. We will also define and briefly discuss the cyclic sieving phenomenon (CSP) and how the enumeration results have helped us find new instances of the CSP.

Corbin Balitactac

Title: A higher-gradient incompressible viscous fluid model
Abstract:
In this talk we will introduce a higher-gradient incompressible fluid model and how it compares to the Navier-Stokes equations. We will discuss an explicit solution to the model in a well-studied flow with pressure-dependent viscosity (joint with C. Rodriguez). We then will discuss current work in establishing a weak solution theory for the second-gradient model with constant viscosity.

Diego Cornejo

Title: Approximating Proximity Operators of Composite Convex Functions.
Abstract:
“Proximity operators lie at the heart of many modern optimization algorithms. Yet, for composite functions—where a convex function is composed with a linear transformation—computing this operator explicitly is often impossible.
These structures appear in areas like signal and image processing, inverse problems, and machine learning.
This talk will explore iterative approaches for constructing and approximating these operators, offering both theoretical insight and algorithmic tools.”

Ezra Aylaian

Title: C. T. C. Wall’s Amazing Theorem on Non-Smoothable 8-Manifolds
Abstract:
In dimensions 1, 2, and 3, every topological manifold admits a unique triangulation and a unique smooth structure. In dimension 4 this breaks down, as there are 4-manifolds that are homeomorphic but not diffeomorphic. However, it is still true for dimensions 4, 5, and 6 that a triangulable manifold admits a unique smooth structure. In dimension 7, even this begins to erode—a triangulable manifold still admits a smooth structure, but it doesn’t need to be unique. For example, Kervaire and Milnor proved that the 7-sphere admits 28 distinct smooth structures. In dimension 8, everything falls apart completely: there are triangulable 8-manifolds that are non-smoothable. It is this last situation that is the subject of an amazing theorem of C. T. C. Wall with a surprisingly short proof: the classes of orientable triangulable 8-manifolds modulo smoothability with connected sum form a group isomorphic to Z/28. We will prove this amazing theorem. If time permits, we will also discuss a result of Nicholaas H. Kuiper that every class of triangulable 8-manifold modulo smoothability can be represented by a complex algebraic hypersurface in CP^5, and explicit polynomial equations for non-smoothable triangulable 8-manifolds can be written down.

Fang-Rong Zhan

Title: Khovanov homology and exotic surfaces
Abstract:
We will discuzz Khovanov homology and exotic surfaces in 4-manifolds, and an example of exotic surfaces from [HS21] which can be distinguished by Khovanov homology.

Gleb Gribovskii

Title: Data-driven machine learning approaches to modeling pertussis vaccine scare behavior
Abstract:
Vaccine scares erode public trust in vaccination, posing significant challenges to disease control, as exemplified by the 1970s pertussis vaccine scare in England and Wales. While evolutionary game theory provides a powerful framework for modeling vaccination behavior, calibrating these models is hindered by the need to estimate unobserved parameters. To address this challenge, we apply physics-informed neural networks and Kolmogorov-Arnold networks to the game-theoretic model of the pertussis vaccine scare by Bauch and Bhattacharyya (2012). This methodology embeds the model’s governing differential equations into the network’s loss function, ensuring consistency with theoretical dynamics while simultaneously estimating system parameters and states from observational data. Using historical vaccination and incidence data, we accurately recovered key behavioral parameters and reconstructed the system’s dynamics. Our findings show that bridging game theory with physics-informed machine learning provides a robust tool for uncovering the drivers of vaccination behavior from epidemiological data. This approach can improve our understanding of disease dynamics and ultimately inform the design of more effective public health interventions.

Isaac Weiss

Title: Representations of Categories of Finite G-Sets
Abstract:
In 2010, Church and Farb introduced the idea of representation stability by looking at representations of categories of finite sets and injections. Representation stability allowed algebraic topologists to consider sequences of topological spaces under the lens of sequences of S_n representations. Now, we look at a natural extension of this idea by considering the representations of categories of finite G-sets, which are really just sequences of wreath product representations. What information can we gather about sequences of topological spaces in this way?

Jacob Folks

Title: Some approaches to Roth’s theorem on arithmetic progressions
Abstract:
Here, we cover “hard” and “soft” analytic approaches to Roth’s theorem, which concerns monochromatic arithmetic progressions in finite colorings of the natural numbers.

Jeremy Wall

Title: Hessian Bounds and Doubling Inequalities
Abstract:
In this talk we will introduce elliptic PDE’s with some examples before moving on to the motivation for finding Hessian estimates. From there we will delve into the method of doubling inequalities and all of the tools involved. We will end with applications to a range of elliptic PDE.

Matt Broussard

Title: Analysis of Multiscale Interface Couplings of Poroelasticity and Lumped Hydraulic Systems
Abstract:
In biomechanics, local phenomena, such as tissue/organ perfusion, are strictly related to the global features of the surrounding blood circulation. We consider a heterogeneous model where a local, accurate, 3D description of tissue perfusion by means of poroelastic equations is coupled with a systemic 0D lumped model of the remainder of the circulation. This represents a multiscale strategy, which couples an initial boundary value problem to be used in a specific tissue region with an initial value problem in the rest of the circulatory system. We present new results related to well-posedness analysis and solution methods for the multiscale model under consideration.

Matthew Elpers

Title: Characterising slopes for more 3 manifolds
Abstract:
For any homotopy class h in any compact orientable 3-manifold M which is closed or has exclusively torus boundary components, we produce infinitely many pairs of distinct knots representing h with orientation-preserving homeomorphic 0-surgeries.

Nico Fontova

Title: A Survey of Rasmussen s-Invariants
Abstract:
Rasmussen’s s-invariant is a knot invariant arising from Khovanov homology, which gives a lower bound on the slice genus. We explore the relationship between the different types of s-invariants, as well as their relationships with the knot signature. We also discuss how the s-invariant behaves under satellite operations.

Patrick Martin II

Title: Hypergraph Magnitude Homology
Abstract:
In 2017 Hepworth and Willerton categorified the magnitude of a graph, introducing the theory of the magnitude homology of a graph. Since then, magnitude homology has been generalized to metric spaces and enriched categories. Bi, Li, and Wu recently applied this theory to hypergraphs. Here I discuss the consequences of Bi, Li, and Wu’s work — introducing a Mayer-Vietoris sequence and a Künneth formula for the magnitude homology of hypergraphs.

Paul Spears

Title: Numerical Simulation and Identification of Electromagnetic Vortex Knots under Uncertainties
Abstract:
Topological vortex knots propagated by electromagnetic beams have been modeled theoretically using the paraxial wave equation. They have also been observed experimentally, but their identification in the received field required human intervention. This project uses a combination of analytical and computational methods to fully automate vortex knot identification, accompanied by numerical simulations to assess its robustness. The results can be used for estimating the capacity of communication channels where the transmitted and received knots serve as alphabet symbols in information theory.

Perry Beamer

Title: Rings of Rings: Persistent Homology for Nested Topological Structure
Abstract:
Topological data analysis (TDA) is a mathematical framework for describing the shape and organization of complex data across multiple scales and dimensions. Persistent homology (PH) a central method in TDA, tracks how topological structures such as connected components, loops, and higher-dimensional cavities appear and disappear with varying geometric scales, quantifying their significance by how long each feature persists across scales. But the topological structure of these homological features has not been studied: how might we identify loops of loops, clusters of cavities, or other nested topological structures? By extracting cycle-representatives from persistent homology computations, we can geometrically represent components, loops, and cavities from point cloud data in the original data space. We can then build filtrations on the set of cycle generators induced by the Wasserstein distances between generators. We can then repeat the persistent homology computation on these generator filtrations, detecting higher-order topological structures like nested loops. Furthermore, we propose natural applications in material science and biochemistry.

Reese Lance

Title: Integrals of stable envelopes on cotangent bundles to Grassmannians
Abstract:
We consider cohomological stable envelopes for a natural torus action on the cotangent bundle to the Grassmanian, T^*Gr(k,n), introduced by Maulik-Okounkov. We define a notion of “equivariant integral” of the stable envelope using equivariant localization. The integral of such a class is an integer times a power of $\hbar$, and the main result is a combinatorial formula for these integers. In 3d mirror symmetry, these non-equivariant limits are expected to reflect some curve counting phenomena on the 3d mirror dual. When $k=1$, we obtain the binomial coefficients, and for higher $k$ these integers have very interesting combinatorial properties which are not fully understood yet.

Shannon Jenkins

Title: Uncovering the Cultural Topology of Paris, France
Abstract:
“Topological Data Analysis (TDA) leverages principles of algebraic topology to analyze large and complex data sets by describing the shape of data. TDA tools are qualitative, so they are robust to noise and can be used to analyze shapes in high-dimensional datasets that are indistinguishable for standard statistical tools [DW22]. BallMapper (BM) is one computationally efficient TDA tool that provides topology-based graph visualizations, enabling exploratory analysis [D lo19].
Inspired by Rudkin et al’s 2023 study, “Economic Topology of the Brexit Vote” [al24], we apply the BallMapper algorithm to explore a different problem in the social sciences: the classification of subcultures. Our data consist of 19,133 words from 4,301 headlines of the Marie webpages of the 18 municipalities corresponding to 20 arrondissements in Paris, France. These arrondissements are often said to be cities within a city due to their distinct cultures. Similar to computational text analysis approaches, we analyze the relative frequency of words in our dataset as an expression of the cultural artifact of language [al19]. BM is used as a dimension reduction and visualization tool, providing a BM graph that we analyze through the BM coloring feature. We discuss the frequency- and topic-induced topologies and what they imply about the presence of subcultures in Paris, France. Using the linguistic Distribution Hypothesis [Gri24], we conclude that the arrondissements personalize French by assignment different meanings to the same corpus and that the northwestern arrondissements have a more topic-diverse discourse, indicating distinct subcultures. Furthermore, we compare this approach to text analysis approaches such as structural topic modeling.
Keywords: topological data analysis (TDA), BallMapper (BM), Paris, culture studies, subcultures, linguistics, French, text analysis, structural topic modeling (STM)

References
[al19] Margaret E. Roberts et al. “stm: An R Package for Structural Topic Models”. In: Journal of Statistical Software 91 (2 2019). doi: https://doi.org/10.18637/jss.v091.i02.
[al24] Simon Rudkin et al. “An economic topology of the Brexit vote”. In: Regional Studies 58.3 (2024), pp. 601–618. doi: https://doi.org/10.1080/00343404.2023.2204123.
[D lo19] Pawel D lotko. “BallMapper: a shape summary for topological data analysis”. In: arXivLabs (2019). doi: https : / / doi . org / 10.48550/arXiv.1901.07410.
[DW22] Tamal Krishna Dey and Yusu Wang. Computational Topology for Data Analysis. Cambridge UP, 2022. doi: https://doi.org/10.1017/9781009099950.014.
[Gri24] Stefan T. Gries. Frequency, Dispersion, Association, and Keyness: Revising and tupelizing corpus-linguistic measures. Studies in Corpus Linguistics, 2024. doi: https://doi.org/10.1075/scl.115.

(NOTE: I will give background on simplicial complexes, so this talk is accessible to undergraduates)

Zachary Parker

Title: Cohomology of $\mathbb{Q}[i]$ with Level
Abstract:
Automorphic forms are intimately linked to the cohomology of arithmetic groups. In this talk, we give an overview of how to explicitly compute such forms. We motivate the objects of interest, then visit the classical case where we illustrate the general approach. We conclude with some collected data, highlighting some exciting surprises, and describe future directions for this work.

Parking and Transportation

Parking is free on NC State’s campus on the weekend. There is a parking lot behind SAS hall which connects to the first floor lobby. More travel information can be found at this link.