Spring 2026 Texas Geometry and Topology Conference Texas Christian University Fort Worth, Texas April 10-12, 2026

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Abstracts

Gioacchino Antonelli (University of Notre Dame)

The isoperimetric problem asks a simple question: among all shapes with the same volume, which one has the smallest surface area? In this talk, I explain how ideas coming from the isoperimetric problem can be used to understand the geometry of curved spaces. On the one hand, isoperimetric techniques can be used to prove restrictions on the possible shapes and geometry of a space. On the other hand, the way a space curves influences the structure of optimal shapes and area-minimizing surfaces inside it. A main theme of the talk is that even weak lower bounds on the curvature - based on spectral information rather than point-by-point estimates - are often enough to recover sharp geometric and topological conclusions. I will illustrate how this point of view led to the resolution of the stable Bernstein problem in Euclidean spaces up to dimension six.

Erin Chambers (University of Notre Dame)

In this talk, we will discuss recent work that connects two fundamental but generally distinct areas in computational topology, namely topological data analysis (TDA) and knot theory. Given a function from a topological space to \(\mathbb{R}\), TDA provides tools to simplify and study the importance of topological features: in particular, the \(l^{th}\)-dimensional persistence diagram encodes the topological changes (or \(l\)-homology) in the sublevel set as the function value increases into a set of points in the plane. Given a continuous one-parameter family of such functions, we can combine the persistence diagrams into an object known as a vineyard, which tracks the evolution of points in the persistence diagram as the function changes. If we further restrict that family of functions to be periodic, we identify the two ends of the vineyard, yielding a closed vineyard. This allows the study of knots and braids in vineyards, as well as introducing the concept of monodromy to TDA, which in this context means that following the family of functions for a period permutes the set of points in a non-trivial way. Interestingly, the presence of monodromy in a vineyard also connects in a fundamental way to singularity theory, namely the medial axis and symmetry set of the shape.

After reviewing the basic constructions of persistence diagrams and vineyards, we will show a construction which generates arbitrary knots and monodromy in a vineyard, and (time allowing) will classify why this behavior arises from specific types of singularities in the symmetry set.

Hannah Larson (University of California, Berkeley)

The moduli space \(M_g\) of genus \(g\) curves (or Riemann surfaces) is a central object of study in algebraic geometry. Its cohomology is important in many fields. For example, the cohomology of \(M_g\) is the same as the cohomology of the mapping class group, and is also related to spaces of modular forms. Using its properties as a moduli space, Mumford defined a distinguished subring of the cohomology of \(M_g\) called the tautological ring. The definition of the tautological ring was later extended to the compactification \(\bar M_g\) and the moduli spaces with marked points \(\bar M_{g,n}\). While the full cohomology ring of \(\bar M_{g,n}\) is quite mysterious, the tautological subring is relatively well understood, and conjecturally completely understood. In this talk, I'll discuss several results about the cohomology groups of \(\bar M_{g,n}\), particularly regarding when they are tautological or not. This is joint work with Samir Canning, Sam Payne, and Thomas Willwacher.

Anibal Medina-Mardones (Western University)

A framed polytope is the convex closure of a finite set of points in \(\mathbb{R}^n\) together with an ordered linear basis. An \(n\)-category is a category that is enriched in the category of \((n-1)\)-categories. Although these concepts may initially appear to be distant peaks in the mathematical landscape, there exists a trail connecting them, blazed in the 90's by Kapranov and Voevodsky. We will traverse this path, widening and improving it as we address some of their conjectures along the way.

Maggie Miller (University of Texas at Austin)

A knot is "fibered" if its complement fibers over the circle. Fibered knots in \(S^3\) are well understood. I'll talk about some open problems about the analogue for knotted surfaces in \(S^4\) and \(B^4\), and related interesting constructions.

Efton Park (Texas Christian University, emeritus)

Let \(A\) be a square matrix with complex entries, and suppose \(A\) is normal; i.e., \(AA^* = A^*A\). The Spectral Theorem states that \(A\) is unitarily equivalent to a diagonal matrix.

Now suppose \(A\) has entries that are complex-valued functions on some topological space. Under what conditions does the conclusion of the Spectral Theorem hold? More generally, when are two such normal matrices unitarily equivalent? These questions and related ones have interesting and nontrivial answers, and it turns out that algebraic topology is a useful tool for studying such questions, as I will (I hope!) demonstrate.

This is joint work with Greg Friedman.

Richard Schwartz (Brown University)

To get a paper Moebius band of aspect ratio \(L\) you take a \(1 \times L\) rectangle and twist it in space (smoothly and isometrically) so that the length-\(1\) ends meet, and the space makes a Moebius band. When \(L\) is large these things are easy to make and when \(L\) is small they are impossible to make. In this talk I will prove that a paper Moebius band of aspect ratio L exists if and only if \(L\) is greater than \(\sqrt{3}\). This bound had been conjectured by B. Halpern and C. Weaver in 1978. I will also discuss a few further developments and some of the many open problems related to this.

Xavier Lecture 2: Vertex minimal origami tori

An origami torus is a piecewise affine isometric embedding of a flat torus into \(\mathbb{R}^3\). These things have been known to exist since 1960, and a basic question that then arises is: What is the fewest number of vertices needed to make an origami torus? In this talk I will answer this question, proving that it is impossible to make an origami torus with fewer than 8 vertices and that it is possible to make one with 8 vertices. I will also discuss a very recent sharpening of this result (joint with Peter Doyle) where we show that almost every flat torus is realized by an 8-vertex origami torus.