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Plenary and Semi-Plenary Talks
Events

Submissions closed on 2025-05-15 11:59PM [Central Time (US & Canada)].

SumTopo 2025 will feature plenary and semi-plenary talks from several notable researchers.

Accepted Submissions:

A survey of intrinsically linked and intrinsically knotted graphs, including an outline of an incomplete possible alternate proof of Sachs' linkless embedding conjecture — Joel Foisy

A graph is *intrinsically linked* (resp. *intrinsically knotted*) if it contains, in every spatial embedding, a pair of cycles that form a nonsplit link (resp. a cycle that forms a nontrivial knot). In the early 1980s, Conway-Gordon and Sachs showed that the complete graph on 6 vertices is intrinsically linked, and Conway-Gordon showed that the complete graph on 7 vertices is intrinsically knotted. Sachs' linkless embedding conjecture is that the Petersen Family of graphs (those obtained from K6 by triangle-Y and Y-triangle exchanges) form the complete set of minor-minimal (in some sense, simplest) intrinsically linked graphs. In the early 1990s, Robertson, Seymour and Thomas proved Sachs' linkless embedding conjecture; in a formidable work spanning three journal articles covering 99 pages. In the early 2000s, Flapan encouraged researchers to find another proof, leveraging more topology. Since that time, the speaker has been in and out of the rabbit hole of seeking a new proof. Minor-minimal intrinsically knotted graphs have not yet been fully characterized, and hundreds of such graphs have been found (Foisy, Goldberg-Mattman-Naimi, Kohara-Suzuki Schwartz, etc...). The problem of classifying all such graphs seems elusive at this time. In this talk, we present a survey of minor-minimal intrinsically knotted and intrinsically linked graphs, as well as discuss an outline of an incomplete alternate proof of Sachs' linkless embedding conjecture, in the hopes that someone within earshot of this talk will be inspired to complete Flapan's vision of a new proof.

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Analyzing the Geometric Topology of Artificial (ReLU) Neural Networks — Marissa Masden

This talk is intended as an introduction to and overview of the geometric topology of (some) artificial neural network functions, with aims towards advancing the understanding of deep learning models. First, I will introduce the type of neural network functions under consideration ("ReLU networks," or multilayer perceptrons with ReLU activation) and their relation to contemporary machine learning. I will then discuss some of the perspectives from which geometric and topological measures of ReLU networks may be exploited to understand and analyze the structure of individual such functions as well as the class of all such functions, both theoretically and computationally. I will pay special attention towards the "decision region/boundary" interpretation of classification models, which corresponds to (sub)level set approximation. A concern is that sublevel sets with too high of complexity (as measured via topological invariants) corresponds to "memorization/overfitting" of a machine learning model, but sublevel sets of insufficient complexity will fail to generalize over a large portion of the problem domain. Resultingly, we seek tools to assess the complexity of a given ReLU network. One result in this direction is that, under certain genericity and transversality assumptions on intermediate layers, the (mod-2) Betti numbers of level sets of ReLU networks can be computed exactly by exploiting hyperplane arrangement combinatorics. I will additionally discuss some of the unique challenges faced when extending piecewise linear and discrete Morse theory to this function class, including current progress. This talk is based on, in part, work done jointly with J. Elisenda Grigsby, Kathryn Lindsey, and Robyn Brooks.

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Bounded cohomology of transformation groups of Rn — Francesco Fournier-Facio

Bounded cohomology is a functional analytic analogue of group cohomology, with many applications in rigidity theory, geometric group theory, and geometric topology. A major drawback is the lack of excision, and because of this some basic computations are currently out of reach; in particular the bounded cohomology of some “small” groups, such as the free group, is still mysterious. On the other hand, in the past few years full computations have been carried out for some “big” groups, most notably transformation groups of \mathbb{R}^n, where the ordinary cohomology is not yet completely understood. I will report on this recent progress, which will include joint work with Caterina Campagnolo, Yash Lodha and Marco Moraschini, and joint work with Nicolas Monod, Sam Nariman and Sander Kupers.

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Cardinal inequalities for non-Hausdorff topological spaces — Ivan Gotchev

For a Hausdorff space X, Hajnal and Juhász showed in 1967, that |X|≤2c(X)χ(X) and |X|≤22s(X), where c(X) is the cellularity, χ(X) is the character and s(X) is the spread of X; Arhangel'skii, in 1969, proved that |X|≤2χ(X)L(X), where χ(X) is the character and L(X) is the Lindelӧf degree of X; and, in 1974, Arhangel'skiĭ and Šapirovskiĭ strengthened Arhangel'skiĭ's inequality by showing that |X|≤2t(X)ψ(X)L(X), where t(X) is the tightness and ψ(X) is the pseudocharacter of X. It has been an open question for a long time if Arhangel'skiĭ's inequality is true for every T1-space X. In this talk we will mention what is known in relation to the above question and how by using other cardinal functions, some of the above inequalities could be extended to be valid for all T1-spaces and, in some cases, even for all topological spaces.

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Formalizing Braid Groups — Hannah Fechtner

I will discuss the formalization, in Lean, of braid groups : from their definition to the ongoing implementation and verification of a polynomial-time algorithm for the braid isotopy problem (Patrick Dehornoy’s subword reversing). Braids, inherently physical objects, were first abstracted to a nascent sort of topology by Vandermonde in the 18th century, and then to a proto-algebraic structure by Gauss in the 19th. More modern authors, from Artin to Markov (Jr.) to Dehornoy, have wrestled with the notion of rigor in this setting, as the associated visual imagery can suggest intuitive leaps. I will discuss one such example, and present a novel, formalized proof, which forms part of the work for the braid isotopy problem.

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Hausdorff vs Gromov-Hausdorff distances — Henry Adams

The goal of this talk is to show how tools from topology can bound or compute quantities arising in metric geometry. I'll begin by introducing the Hausdorff and Gromov-Hausdorff distances, which are ways to measure the "distance" between two metric spaces. Though Hausdorff distances are easy to compute, Gromov-Hausdorff distances are not. Next I will explain the nerve lemma, which says when a cover of a space faithfully encodes the shape of that space. As the main result, I'll show how when X is a sufficiently dense subset of a closed Riemannian manifold M, we can use the nerve lemma to lower bound the Gromov-Hausdorff distance between X and M by 1/2 the Hausdorff distance between them. The constant 1/2 can be improved, and even obtains the optimal value 1 (meaning the Hausdorff and Gromov-Hausdorff distances coincide) when M is the circle.

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Homogeneity degree in local dendrites — Patricia Pellicer-Covarrubias

Let X be a separable space and let D(X) be the set of countable dense subsets of X. Consider an equivalence relation ∼, defined on D(X), as follows: we say that M∼N if and only if there exists a homeomorphism h:X→X such that h[M]=N. Define the countable dense homogeneity degree of X as the cardinality of the set of equivalence classes under the relation ∼. In this talk we discuss the countable dense homogeneity degree for local dendrites.

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Independence, ideal independence and forcing indestructibility — Vera Fischer

Abstract: Two persistent directions in the study of the properties of the, so-called, combinatorial or extremal sets of reals, sets like maximal eventually different families of functions, maximal cofinitary groups or maximal independent families, are the study of their spectra and their projective complexity. In this talk, we will discuss some recent progress in the area, and point out towards interesting remaining open problems.

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Infinite Games — Jocelyn Bell

Nearly a century ago, the first infinite topological game was played on the tabletops of the Scottish Cafe in Poland. Now known as the Banach-Mazur game, it appeared in Problem 43 of the Scottish Book, posed by Banach and answered by Mazur. Since then, many others have been defined. A topological game typically involves two players alternately choosing objects from a space, such as points or open sets, according to a list of rules. They have been used not only to define topological properties but also to prove results seemingly unrelated to games. In this talk, we'll play some of these games and discuss recent results.

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Khovanov skein lasagna modules and exotica — Mike Willis

The Khovanov skein lasagna module S(X;L) is a smooth invariant of a 4-manifold X with link L in its boundary. In this talk I will outline the construction of Khovanov skein lasagna modules, as well as new computations and applications including the detection of some exotic 4-manifolds. This work is joint with Qiuyu Ren.

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On tracing properties, invariant measures, and entropy. — Piotr Oprocha

In 1970s Bowen related hyperbolic dynamics with specification property and used this to show existence of a unique measure of maximal entropy. Almost the same time Sigmund used specification property as a tool in characterization of simplex of invariant measures. These results have several consequences. First, it became clear that (broadly understood) tracing of well well-chosen trajectories can provide good insight into the simplex of invariant measures. Second, tracing of trajectories can lead to emerging structures and properties in dynamics (e.g. uniform spread of some trajectories necessary for measure of maximal entropy; forming of some patterns; irregular motions, etc.). Finally, well defined tracing may be stable under perturbations, leading to better understanding of features of typical dynamics. Over the years, these results were inspiration for numerous mathematicians in various studies of dynamical systems. In this talk we will present selected questions and recent results fitting into the above framework of research.

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Projections of spatial graphs — Erica Flapan

One would not expect to be able to conclude much about a link or a spatial graph by looking at a single projection. Yet in 1984, Menasco proved that if G is a reduced, alternating, connected projection of a link L, then there is a sphere meeting L in two points splitting the link into two non-trivial pieces if and only if there is a circle meeting G in two points splitting the projection into two non-trivial pieces. Then in 1987, Kauffman, Murasugi, and Thistlethwaite proved Tait's Conjecture of more than a century that any reduced, alternating projection of a link has a minimal number of crossings. Since the 80's, these two important results have been generalized to other classes of links, tangles, and spatial graphs. In this talk we review known results, present counterexamples to some prior results about spatial graphs, and present new results for spatial graphs.

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Shadowing in C(X) — Jonathan Meddaugh

Informally, a continuous self-map f on a compact metric space X has the shadowing property provided that behaviors witnessed by the pseudo-orbits of a system (i.e. orbits with some allowed amount of error) are representative of true behaviors of the system in the sense that every pseudo-orbit has an orbit which approximates it. Surprisingly, despite being quite a strong property and having connections to many other dynamical properties, shadowing has been shown to be a generic property of continuous self-maps for certain classes of spaces. Motivated by this, in this talk we examine the set T(X) of maps with shadowing as a subset of C(X), the space of continuous self-maps on a compact metric space X. We will discuss the structure of T(X) for certain classes of spaces, with a special focus on the question of whether T(X) is a generic set in C(X).

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The Shape of Relations: From Knot Invariants to Cancer Genomics — Radmila Sazdanovic

Topological Data Analysis (TDA) provides a powerful framework for extracting structure from complex data by studying its shape. This talk presents recent work on visualizing maps between high-dimensional spaces to detect correlations between datasets, alongside new adaptations of TDA to settings where representative sampling is impossible. This includes the integration of TDA with machine learning methodologies, particularly in contexts where traditional sampling is impractical, to analyze infinite datasets effectively. A central theme is the application of these methods to knot theory, where the exponential growth in knot complexity places the space of knots and their invariants firmly in the realm of big data. Additional examples from cancer genomics and game theory highlight the broad applicability of these techniques across mathematics and the sciences.

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Unknotting numbers of spatial graphs, knots and DNA — Danielle O'Donnol

The unknotting number of a knot K, denoted u(K), is the minimum number of times the knot must pass through itself to result in the unknot. Determining unknotting numbers is a widely studied and subtle problem. Unknotting number can be extended to (abstractly planar) spatial graphs in a natural way. In this talk we will explain what known about unknotting numbers for spatial graphs, and how this relates to what is known for knots. We will also look at the connections with knotting in DNA, and unknotting numbers of knotoids.

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