1. Dynamics/CT

$k$-type chaos of $\mathbb{Z}^d$-actions — Anshid Aboobacker <anshidaboobackerk@gmail.com>

In this talk, we define and study the notions of $k-$type proximal pairs, $k-$type asymptotic pairs and $k-$type Li Yorke sensitivity for dynamical systems given by $\mathbb{Z}^d$ actions on compact metric spaces. We prove the Auslander-Yorke dichotomy theorem for $k-$type notions. The preservation of some of these notions under conjugacy is also studied. We also study relations between these notions and their analogous notions in the usual dynamical systems.

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1. Graphs

A Transversal of Planar Graph Faces — Joseph Briggs <joseph.guy.briggs@gmail.com>

Suppose you have a subset $S$ of the vertices of a planar graph which contains at least one vertex from every face. Then $S$ must have at least half of the vertices, and for some planar graphs *every* such $S$ must have at least half of the vertices. We believe this extends to higher dimensions, but don’t really know why, and have found some situational evidence (but also some counter-evidence). This is based on joint work with Michael Dobbins and Seunghun Lee.

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1. General/ST

A measure of Isbell-convexity for a quasi-metric space — Collins Amburo Agyingi <agyinca@unisa.ac.za>

Let $(X,d)$ be a $T_0$-quasi-metric space. Then it has been shown that $X$ has a $q$-hyperconvex hull which is denoted by $Q_X$. It is known that every $q$-hyperconvex $T_0$-quasi-metric space is bicomplete. However, the converse is not true, that is, there exist bicomplete $T_0$-quasi-metric spaces that are not $q$-hyperconvex. In this talk, we shall present a parameter that measures how far a bicomplete $T_0$-quasi-metric space is from being hyperconvex. We will present some characteristics of this new parameter.

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1. Computing

A mechanized characterization of coherent $2$-groups — Perry Hart <hart1262@umn.edu>

In this talk, we will outline an Agda implementation of a higher-dimensional piece of the homotopy hypothesis, which provides a general correspondence between groupoids and homotopy types. The groupoids we look at are monoidal groupoids where every element has the structure of an adjoint equivalence, called coherent $2$-groups. The correspondence for such groupoids takes the form of a biequivalence between the $\left(2,1\right)$-category of coherent $2$-groups and the $\left(2,1\right)$-category of pointed connected (homotopy) $2$-types. We build this biequivalence in homotopy type theory (HoTT) and use Agda to verify the construction (see https://github.com/PHart3/2-groups-agda). Thanks to the univalence axiom, we also obtain a verified identity between the two $\left(2,1\right)$-categories in question. This biequivalence has been suggested at a few places in the literature. Inside HoTT, Buchholtz, van Doorn, and Rijke (2018) propose it as a $2$-dimensional generalization of the equivalence they construct between $\mathbf{Grp}$ and pointed connected $1$-types. It also was suggested in the classical setting by Baez and Lauda (2004). Indeed, the biequivalence we construct generalizes the $1$-dimensional equivalence. It consists of two broad steps. First, we construct the classifying space of a coherent $2$-group $G$ as a higher inductive type by generalizing the first Eilenberg-MacLane space of a group. This defines a function from the type of coherent $2$-groups to the type of pointed connected $2$-types. Second, we equip this function with the structure of a pseudofunctor and prove that it forms a biequivalence with the loop space pseudofunctor, which takes a pointed connected $2$-type to its fundamental $2$-group. Each step is purely algebraic, and all our proofs are constructive. Notably, combined with recent work by Owen Milner (unpublished), our biequivalence computes the Sinh invariant of a coherent $2$-group within a constructive system, whereas the traditional method relies on the axiom of choice. Our formalization, however, involves several huge computations, and we will discuss our experience managing its memory requirements and its arduous type-checking.

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1. Low-Dimensional

A minimality criterion for SL(4) skeins — Anup Poudel <poudel.33@osu.edu>

We provide a 4-valent ribbon model for SL(4) skein category by working with a category with object an oriented marking and morphisms generated by tagged and untagged 4-valent vertices. The category is defined combinatorially in terms of diagrammatic generators and relations. We use linear algebraic and skein theoretic methods to explore topological invariants coming from such a category. As a consequence, we show that a specialization of our parameters provides a 4-valent category that is equivalent to the SL(4) representation category. We further provide a topological evaluation algorithm of closed webs providing a (topological) criterion for reducible webs. We also show that certain HOMFLY relations exist in our category. Our evaluation algorithm works at a very abstract level and doesn’t use any algebraic constraints coming from the representation theory. This is a joint work with Giovanni Ferrer and Jiaqi Lu. 

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1. Plenary

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 <foisyjs@potsdam.edu>

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 $K_6$ 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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1. Graphs

An intrinsically linked simplicial $n$-complex — Ryo Nikkuni <nick@lab.twcu.ac.jp>

We say that a simplicial $n$-complex is intrinsically linked if every embedding of its polyhedron into $(2n+1)$-dimensional Euclidean space contains a pair of disjoint $n$-spheres with a nonzero linking number. Several examples of intrinsically linked $n$-complexes are known. In this talk, we present a new example of such an $n$-complex.

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1. General/ST

An iterable forcing property and universally meager sets — Valentin Haberl <valentin.haberl.math@gmail.com>

By a space we mean a metrizable separable zero-dimensional space. A space $X \subseteq 2^\omega$ is universally meager if for any Polish space $Y$ and any continuous nowhere constant map $f:Y \rightarrow 2^\omega$ the preimage $f^{-1}[X]$ is meager in $Y$. We call a space totally imperfect if it contains no copy of $2^\omega$. We present a forcing property $(\dagger)$, which is a strenthening of properness and implies that no dominating reals are added. It is known that many classical forcing posets like Cohen, Sacks and Miller satisfy this property. We showed that property $(\dagger)$ is preserved by countable support iterations. We then used this preservation result to prove that if we have such an iteration of length $\omega_2$ over a model of CH, where the single forcings have size at most $\omega_1$, all universally meager sets $X \subseteq 2^\omega$ have size at most $\omega_1$ in the forcing extension. \\ This has multiple set-theoretic applications: In the Miller model, we generalized our result of having no concentrated and $\gamma$-sets of size continuum to totally imperfect Hurewicz sets, which are universally meager by a result of Zakrzewski. Moreover, since Bartoszyński showed that all perfectly meager spaces are universally meager in the Miller model, we get that indeed even all perfectly meager spaces have size stricly less than continuum in the Miller model. Miller proved in 2005 that there exists a strong measure zero set of size $\omega_1$ iff there exists a Rothberger space of size $\omega_1$. Goldstern, Judah and Shelah constructed in 1993 a forcing iteration for which there is a strong measure zero set of size $\omega_2$ in the extension. However, this iteration satisfies property $(\dagger)$ and Rothberger spaces are universally meager in this model. Hence our result implies that it is consistent with ZFC to have a strong measure zero set of size $\omega_2$, but no Rothberger spaces of size $\omega_2$. This is joint work with Piotr Szewczak (Cardinal Stefan Wyszyński University in Warsaw) and Lyubomyr Zdomskyy (TU Vienna).

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1. Plenary

Analyzing the Geometric Topology of Artificial (ReLU) Neural Networks — Marissa Masden <mmasden@pugetsound.edu>

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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1. Low-Dimensional

Another proof of functoriality for odd Khovanov homology — Dean Spyropoulos <spyropou@msu.edu>

In 2024, Migdail and Wehrli proved that odd Khovanov homology is functorial with respect to link cobordism (up to sign). Unlike Khovanov's proof that the original theory is functorial, Migdail-Wehrli's is interesting in that it does not depend on any of the recent extensions of odd Khovanov homology to tangles. In recent ongoing work, we adapt Khovanov's original argument to one of these tangle theories to get a proof that Naisse-Putyra's odd tangle invariant is functorial with respect to tangle cobordisms (up to unit). This approach motivates a few novel constructions, including a new generalization of Hochschild (co)homology.

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1. Plenary

Bounded cohomology of transformation groups of $\mathbb{R}^n$ — Francesco Fournier-Facio <ff373@cam.ac.uk>

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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1. Dynamics/CT

Building Continua with non-trivial self covers — Mathew Timm <mtimm@bradley.edu>

We will look at several methods for building continua with non-trivial self covers and discuss their relationships with some problems in topology and group theory.

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1. General/ST

Cancelled: $Q$-sets, $\Delta$-sets and $L$-spaces — Pourya Memarpanahi <pourya.memarpanahi@utoronto.ca>

The concept of a ∆-set of reals was originally defined by G.M. Reed. An equivalent version was defined by Eric van Douwe and later on was generalized to an arbitrary topology space, (∆-space) . Historically, this notion arose in the study of the normal Moore space conjecture, where Q-sets were used to construct important counterexamples to the conjecture. We prove that Moore's L-space (a hereditarily Lindelöf but not separable space in ZFC) is not a Q-set space and if Aronszajn tree naturally associated with Moore’s L-space is special Moore's L-space will not be a ∆-space.

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1. General/ST

Cancelled: The $F_\sigma$-below relation and some new separation axioms in the category of locales — Mbekezeli Nxumalo <sibahlezwide@gmail.com>

We use $F_\sigma$ sublocales to define a relation on locales called the $F_\sigma$-below relation. This relation is weaker than the rather below relation. Two separation axioms, namely weakly D-completely regularity and $F_\sigma$-regularity, between perfectness and weakly subfitness are introduced using the $F_\sigma$-below relation. We discuss properties of these two locales and find their relationship with other locales such as regularity and subfitness.

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1. General/ST

Cardinal bounds in spaces with a $\pi$-base whose elements have an H-closed closure — Davide Giacopello <dagiacopello@unime.it>

We deal with the class of Hausdorff spaces having a $\pi$-base whose elements have an H-closed closure. Carlson proved that $|X|\leq 2^{wL(X)\psi_c(X)t(X)}$ for every quasiregular space $X$ with a $\pi$-base whose elements have an H-closed closure. We provide an example of a space $X$ having a $\pi$-base whose elements have an H-closed closure which is not quasiregular (neither Urysohn) such that $|X|> 2^{wL(X)\chi(X)}$ (hence, $|X|> 2^{wL(X)\psi_c(X)t(X)}$). Still in the class of spaces with a $\pi$-base whose elements have an H-closed closure, we establish the bound $|X|\leq2^{wL(X)k(X)}$ for Urysohn spaces and we give an example of an Urysohn space $Z$ such that $k(Z)<\chi(Z)$. Lastly, we present some equivalent conditions to the Martin's Axiom involving spaces with a $\pi$-base whose elements have an H-closed closure and, additionally, we prove that if a quasiregular space has a $\pi$-base whose elements have an H-closed closure then such a space is Baire.

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1. Plenary

Cardinal inequalities for non-Hausdorff topological spaces — Ivan Gotchev <gotchevi@ccsu.edu>

For a Hausdorff space $X$, Hajnal and Juhász showed in 1967, that $|X| \le2^{c(X)\chi(X)}$ and $|X| \le 2^{2^{s(X)}}$, where $c(X)$ is the cellularity, $\chi(X)$ is the character and $s(X)$ is the spread of $X$; Arhangel'skii, in 1969, proved that $|X|\le 2^{\chi(X)L(X)}$, where $\chi(X)$ 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 $|X|\le 2^{t(X)\psi(X)L(X)}$, where $t(X)$ is the tightness and $\psi(X)$ is the pseudocharacter of $X$. It has been an open question for a long time if Arhangel'skiĭ's inequality is true for every $T_1$-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 $T_1$-spaces and, in some cases, even for all topological spaces.

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1. Dynamics/CT

Chaos on Peano continua — Klára Karasová <karasova@karlin.mff.cuni.cz>

Among all notions of chaos, there are three widely accepted: Devaney chaos, Li-Yorke chaos and (positive) topological entropy. It is known that exact Devaney chaos, i.e. an exact map with dense set of periodic points, satisfies all these three notions. Various results establish the existence of maps with properties related to chaos (e.g., transitivity) for specific spaces such as the interval, the Cantor set or the Lelek fan, as well as for broader classes, including manifolds and dendrites. Furthermore, chaotic behavior often emerges as a generic phenomenon in the sense of Baire category. Together with Benjamin Vejnar, we prove that every Peano continuum (i.e. a locally connected continuum) admits exact Devaney chaos. Additionally, we generalize some prior results by showing that if a Peano continuum $X$ satisfies the condition that selfmaps locally constant on some dense open subset form a dense subset of all selfmaps, then: • exactly Devaney chaotic maps form a dense subset of chain transitive self- maps of X, • mixing is generic among chain transitive self-maps of X, • shadowing is generic among all self-maps of $X$.

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1. General/ST

Combinatorial covering properties in countable and uncountable contexts — Michał Pawlikowski <michal-pawlikowski4@wp.pl>

Combinatorial covering properties as Rothberger’s, Hurewicz’s and Menger’s are procedures for generating a cover of a given topological space from a sequence of covers of this space. We present the most celebrated such properties together with the most important examples in a classical countable case. We also explore how these notions and examples extend to the uncountable context, where the initial sequence of covers has length $\kappa$ for some uncountable cardinal $\kappa$. In this generalized setting, we replace the classical Baire space $\omega^\omega$ with the generalized Baire space $\kappa^\kappa$. This is joint work with Piotr Szewczak and Lyubomyr Zdomskyy.

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1. General/ST

Combinatorial structures of concentrated sets — Piotr Szewczak <p.szewczak@wp.pl>

Let $X$ be a set of reals and $\kappa$ be an uncountable cardinal number. The set $X$ is $\kappa$-concentrated, if $X$ has size at least $\kappa$ and contains a countable set $D$ such that each closed subset of $X$, disjoint with $D$, has size smaller than $\kappa$. Various forms of concentrated sets play an important role in the study of combinatorial covering properties such as Rothberger’s, Hurewicz’s, and Menger’s properties. We investigate the behavior of such sets in different models of set theory. This is a joint work with Michał Pawlikowski and Lyubomyr Zdomskyy. The research was funded by the Polish National Science Center and Austrian Science Fund; Grant: Weave-UNISONO, Project: Set-theoretic aspects of topological selections 2021/03/Y/ST1/00122.

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1. Computing

Comparison of Precinct and District Voting Data Using Persistent Homology to Identify Gerrymandering in North Carolina — Ananya Shah <ananya.neytri.shah@gmail.com>

We present an extension of Feng & Porter’s 2019 paper on the use of the level-set method for the construction of a filtered simplicial complex from geospatial election data, by applying their method to identify gerrymandering. Using the fact that precincts are regarded to be too small to be gerrymandered, we identify discrepancies between precinct and district level voting data to quantify gerrymandering. Comparing the persistent homologies of democratic voting areas on the precinct and district level shows when areas have been ‘cracked’ or ‘packed’ for partisan gain. This analysis was done for North Carolina House of Representatives elections (2012-2024). NC has been redistricted 4 times in the past 10 years, whereas most states redistrict decennially, allowing us to understand how and when redistricted maps deviate from precinct-level voting data, and when gerrymandering occurs. Comparing persistence barcodes at the precinct and district levels (using the bottleneck distance) shows that precinct-level voting patterns do not significantly fluctuate biannually, while district level patterns do, suggesting that shifts are likely a result of redistricting rather than voter behavior, providing strong evidence of gerrymandering. NC Election data was collected from the public domain. Composite shapefiles were created using QGIS and R, and rasterized using Python. The level-set method was employed to generate filtered similar complexes. Persistence barcodes were produced using GUDHI and PHAT libraries. Additionally, we compare our results with traditional measures such as Polsby-Popper and Reock scores (gerrymandering identification measures). This research presents a novel application of topological data analysis in analyzing gerrymandering.

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1. Computing

Computing path homology chains by inductive construction — Matthew Burfitt <m.burfitt@bimsa.cn>

The path homology introduced by Grigor’yan, Lin, Muranov and Yau plays a central role in digraph topology and the emerging field of digraph homotopy theory more generally. Unfortunately, the computation of the path homology of a digraph is a two-step process, and until now no complete description of even the underlying chain complex has appeared in the literature. In particular, our understanding of the path chains is the primary obstruction to the development of fast path homology algorithms, which in turn would enable the practicality of a wide range of applications to directed networks. I will introduce an inductive method of constructing elements of the path homology chain modules from elements in the proceeding two dimensions. When the coefficient ring has prime characteristic the inductive elements generate the path chains. Moreover, in low dimensions the inductive elements coincide with naturally occurring generating sets up to sign, making them excellent candidates to reduce to a basis. Inductive elements provide a new concrete structure on the path chain complex that can be directly applied to understand path homology, under no restriction on the digraph. During the talk I will demonstrate how inductive elements yield the explicit structure of the dimension 3 path chains and enable the construction of a sequence of digraphs whose path Euler characteristic can differ arbitrarily depending on the choice of coefficients.

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1. General/ST

Countable dense homogeneity and topological groups — Andrea Medini <andrea.medini@tuwien.ac.at>

All spaces are assumed to be separable and metrizable. A space X is countable dense homogeneous (CDH) if all countable dense subsets of X can be mapped onto each other by homeomorphisms of X. The fundamental theorem of countable dense homogeneity states that every "sufficiently homogeneous" Polish space is CDH. This result motivated a long-standing search for examples of non-Polish CDH spaces. We contribute to this line of research by exhibiting a non-Polish CDH topological group. This is joint work with Claudio Agostini and Lyubomyr Zdomskyy.

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1. Graphs

Decomposing $2$-cycles of graphs — Hein Van der Holst <hvanderholst@gsu.edu>

A $2$-cycle on a graph $G=(V,E)$ is a function $d: E\times E\to \mathbb{Z}$ such that for each edge $e$, both $d(e, \cdot)$ and $d(\cdot, e)$ are circulations on $G$. For an oriented cycle $C$ and an edge $e$ of $G$, define $C(e)=+1$ if $C$ traverses $e$ in forward direction, and $C(e)=-1$ if $C$ traverses $e$ in backward direction. Then examples of $2$-cycles are: take two vertex-disjoint oriented cycles $C$ and $D$ of $G$ and define $d(e,f) = C(e)D(f)$. Also on each $K_{3,3}$- and $K_5$-subdivision are $2$-cycles. In this talk, we show that each $2$-cycle on $G$ can be written as a sum of four types of special $2$-cycles. This is joint work with Serguei Norine and Robin Thomas.

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1. Graphs

Decomposition of Complete Graphs into Arbitrary Trees and Hamiltonian Cycles — Murugan Varadhan <murugan.v@vit.ac.in>

Decomposing the complete graph into arbitrary graph is a challenging and di cult problem in graph theory. In in this paper, we prove that the complete graph K4m+1 can be decomposed into 4m + 1 copies of an arbitrary tree with m edges and m copies of a Hamiltonian cycle whenever 4m+1 is a prime.

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1. Computing

Efficient evader detection in mobile sensor networks — William Ott <william.ott.math@gmail.com>

Suppose one wants to monitor a domain with sensors, each sensing a small ball-shaped region, but the domain is hazardous enough that one cannot control the placement of the sensors. A prohibitively large number of randomly placed sensors could be required to obtain static coverage. Instead, one can use fewer sensors by providing mobile coverage, a generalization of the static setup wherein every possible evader is detected by the moving sensors in a bounded amount of time. Here, we use topology in order to implement algorithms certifying mobile coverage that use only local data to solve the global problem. Our algorithms do not require knowledge of the sensors’ locations, only their connectivity information. We experimentally study the statistics of mobile coverage in two dynamical scenarios. We allow the sensors to move independently (billiard dynamics and Brownian motion), or to locally coordinate their dynamics (collective animal motion models). Our detailed simulations show, for example, that collective motion can enhance performance: The expected time until the mobile sensor network achieves mobile coverage is lower for the D’Orsogna collective motion model than for the billiard motion model. Further, we show that even when the probability of static coverage is low, all possible evaders can nevertheless be detected relatively quickly by the mobile sensor network.

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1. Dynamics/CT

Elliptic sectors and heteroclinic regions in real time holomorphic flows — Nicolas Kainz <nicolas.kainz@uni-ulm.de>

The geometric description of the phase space of holomorphic dynamical systems with real time is a crucial research field. In this context, the globalization of local structures is of particular interest. For example, the local structure of an equilibrium of order $m\in\mathbb{N}\setminus\{1\}$ has already been sufficiently investigated and characterized. Locally, there exist $2m-2$ elliptic sectors, all consisting of homoclinic trajectories tending to the equilibrium in both time directions. Now the question arises how this local structure can be globalized using analytical tools, as is the case, for example, with the basin of attraction of nodes and foci. I present a method to define a global elliptic sector based on so-called “sector-forming orbits” and show some topological properties for it: The global elliptic sector is open, flow-invariant, path-connected, and simply connected. Moreover, all orbits are nested inside each other, consistent with the intuitive notion of an elliptic sector. Furthermore, for the case $m\ge 3$, it coincides with the naively defined global elliptic sector, which merely contains all homoclinic trajectories with adjacent definite directions. This gives us an analytic precise definition of a globalization of a locally defined elliptic sector, together with important and useful topological properties. Moreover, it is possible to investigate the geometrical structure that can occur between two global elliptic sectors with no common boundary near the equilibrium. In this context, the question also arises as to how many so-called “heteroclinic regions” can appear between two such sectors.

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1. Graphs

Embedding Cartesian Products of Graphs on Surfaces — Christian Millichap <christian.millichap@furman.edu>

Determining how to build a minimal genus embedding of a graph is a classical and frequently challenging problem in topological graph theory. Here, we will be interested in Cartesian products of graphs where their fiber structures and symmetries can sometimes be leveraged to efficiently build embeddings and determine the genera of such graphs. More specifically, we will discuss work done towards a classification of all Cartesian products of graphs that embed on the torus where we leverage basic tools from combinatorial topology and determine the genera of certain graphs along the way. This work was part of an undergraduate summer research project with Beppy Badgett, and time permitting, we will briefly discuss ideas for future projects in this area that only requires some background in undergraduate graph theory and surface topology to get started.

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1. Dynamics/CT

Equivalence of equicontinuity and distality for real non-autonomous systems — Sushmita Yadav <yadav.34@iitj.ac.in>

This talk will focus on the topological dynamics of a non-autonomous dynamical system $(X,\mathbb{F})$, where $X$ is a compact metric space and $\mathbb{F}=\{f_1,f_2,\ldots\}$ is a sequence of continuous surjective functions on $X$. In particular we will discuss equicontinuity and distality for non-autonomous systems on the interval. We will discuss the distality of the system using the enveloping cover $E_0(X)=\overline{\{\omega_k:k\in \mathbb{Z} \}}$ (where $\omega_n=f_n \circ f_{n-1}... \circ f_1$). We use analytical tools to establish the equivalence of distality and equicontinuity for non-autonomous systems on the interval.

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1. Computing

Facets in the Vietoris--Rips complexes of hypercubes — Ziqin Feng <zzf0006@auburn.edu>

In this talk, we'll discuss the facets (maximal simplices) of the Vietoris--Rips complex $\mathrm{VR}(Q_n; r)$ where $Q_n$ denotes the $n$-dimensional hypercube. We are particularly interested in those facets which are somehow independent of the dimension $n$. Using Hadamard matrices, we prove that the number of different dimensions of such facets is a super-polynomial function of the scale $r$, assuming that $n$ is sufficiently large. We show also that the $(2r-1)$-th dimensional homology of the complex $\mathrm{VR}(Q_n; r)$ is non-trivial when $n$ is large enough, provided that the Hadamard matrix of order $2r$ exists.

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1. Graphs

Flowers of knots — Kouki Taniyama <taniyama@waseda.jp>

An $(n,k)$-flower $F(n,k)$ is the shadow of the closure of an $n$-braid $(\sigma_{1}\sigma_{2}\cdots\sigma_{n-1})^{k}$.\\ C. Lamm and V. O. Manturov independently showed the following: Let $K$ be a knot and $n\geq \mathrm{braid}(K)$. Then $K$ has $F(n,k)$ as its shadow for some $k$. We show the following: Let $K$ be a knot and $k\geq \mathrm{bridge}(K)$. Then $K$ has $F(n,k)$ as its shadow for some $n$. As a corollary, we show $\mathrm{bridge}(K)=\mathrm{lr}(K)$ where $\mathrm{lr}(K)$ is the left-right number of $K$. This gives us a new definition of the bridge number of a knot.

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1. Graphs

Forbidden complexes for the 3-sphere — Makoto Ozawa <w3c@komazawa-u.ac.jp>

A simplicial complex is said to be critical (or forbidden) for the 3-sphere $S^3$ if it cannot be embedded in $S^3$, but becomes embeddable upon removing the open star of any simplex in its second barycentric subdivision. We classify all critical complexes for $S^3$ that decompose as $(G \times S^1) \cup H$, where $G$ and $H$ are graphs whose intersection $G \cap H$ consists solely of vertices of $H$. This is a joint work with Mario Eudave-Munoz.

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1. Plenary

Formalizing Braid Groups — Hannah Fechtner <hfechtne@andrew.cmu.edu>

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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1. General/ST

Frechet spaces and M-separable (selectively separable) spaces — Alan Dow <adow@charlotte.edu>

Frechet spaces are selectively separable. Finite products of countable Frechet spaces need not be Frechet but it is independent as to whether they are selectively separable. We will review some recent results on this topic and, at the moment, think we have a new one.

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1. Dynamics/CT

Generalized Proinov-type contractions using simulation functions with applications to fractals — Ramesh Kumar Devaraj <rameshkumard14@gmail.com>

The intention of this article is to introduce a generalization of Proinov-type contraction via simulation functions. We name this generalized contraction map as Proinov-type Z-contraction. This article establishes the existence and uniqueness of fixed points for these contraction mappings in quasi-metric space and also, include explanatory examples with graphical interpretation. As an application, we generate a new iterated function system (IFS) consisting of Proinov-type Z-contractions in quasi-metric spaces. At the end of the paper, we prove the existence of a unique attractor for the IFS consisting of Proinov-type Z-contractions.

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1. Low-Dimensional

Geometric triangulations of the complement of double twist knots — Dionne Ibarra <dionne.ibarra@monash.edu>

In this talk we will present and explain the construction of two different geometric triangulations of the complements of double twist knots of the form $K_{p,q}$ obtained by Dehn filling the crossing circles of the Borromean rings. This is joint work with D. V. Mathews and J. S. Purcell.

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1. Graphs

Graph Linkage on Surfaces — Dong Ye <dong.ye@mtsu.edu>

Let $H$ be a given graph. A graph $G$ is $H$-linked if, for any injective map $\phi: V(H)\to V(G)$, $G$ contains a subdivision of $H$ rooted at the images of $V(H)$. A classic result of Seymour and Thomassen shows that every 4-connected plane triangulation is $K_2$-linked. Ellingham, Plummer and Yu proved that every 4-connected plane triangulation is $K_4^-$-linked. However, not all 4-connected surface triangulation is $K_4^-$-linked. In this talk, we focus on some recent developments on graph linkages on surfaces. This is based on joint work with Moser, Stephens, and Zha.

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1. Plenary

Hausdorff vs Gromov-Hausdorff distances — Henry Adams <henryhughadams@gmail.com>

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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1. General/ST

Holey Vietoris--Rips complex, Batman! — Chris Wells <coc0014@auburn.edu>

Given a metric space and a positive number $d$, the Vietoris--Rips (VR) complex of scale $d$ is the simplicial complex whose faces are all sets of diameter at most $d$. Recently, there's been a push to understand the VR * How many holes (non-trivial homologies) are there? * How big is the largest facet? * How small is the smallest facet? * How many differently-sized facets are there? Based on joint work with Joe Briggs and Ziqin Feng.

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1. Low-Dimensional

Holonomicity from a Heegaard-Floer perspective — Ben Cooper <ben-cooper@uiowa.edu>

I'll discuss $S^r$-colored knot Floer homologies and categorified recurrence relations that they satisfy. The associated Euler characteristic implies $q$-holonomicity of the corresponding sequence of colored Alexander polynomials, inspired by the AJ conjecture for colored Jones polynomials.

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1. Plenary

Homogeneity degree in local dendrites — Patricia Pellicer-Covarrubias <paty@ciencias.unam.mx>

Let $X$ be a separable space and let $D(X)$ be the set of countable dense subsets of $X$. Consider an equivalence relation $\sim$, defined on $D(X)$, as follows: we say that $M \sim N$ if and only if there exists a homeomorphism $h: X \to X$ 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 $\sim$. In this talk we discuss the countable dense homogeneity degree for local dendrites.

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1. Computing

Homotopy connectivity of Cech complexes of spheres. — Sucharita Mallick <sucharitamallick@ufl.edu>

Let $S^n$ be the $n$-sphere with the geodesic metric and of diameter $\pi$. The intrinsic \v{C}ech complex of $S^n$ at scale $r$ is the nerve of all open balls of radius $r$ in $S^n$. In this talk, we will show how to control the homotopy connectivity of \v{C}ech complexes of spheres at each scale between $0$ and $\pi$ in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case $n=1$, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of \v{C}ech complexes of the sufficiently dense, finite subsets of $S^n$. Our bounds imply the new result that for $n\ge 1$, the homotopy type of the \v{C}ech complex of $S^n$ at scale $r$ changes infinitely many times as $r$ varies over $(0,\pi)$. Additionally, we lower bound the homological dimension of \v{C}ech complexes of finite subsets of $S^n$ in terms of their packings. This is joint work with Henry Adams and Ekansh Jauhari.

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1. Low-Dimensional

Ideal Triangulations and Once-Punctured Surface Bundles — Birch Bryant <bbryant3@una.edu>

A well-known result of Walsh states that if $\mathcal{T}^\ast$ is an ideal triangulation of an atoroidal, acylindrical, irreducible, compact 3-manifold with torus boundary components and $\mathcal{T}^\ast$ has essential edges, then every properly embedded, two-sided, incompressible surface $S$ is isotopic to a spun-normal surface in $\mathcal{T}^\ast$ unless $S$ is isotopic to a fiber or virtual fiber. For a given manifold $M$ that fibers over $S^1$, it was previously unknown whether there exists an ideal triangulation in which the fiber appears as a spun-normal surface. We prove that such a triangulation exists and give an algorithm to construct the ideal triangulation provided $M$ has a single boundary component.

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1. Low-Dimensional

In search for homology of Bol-Moufang quasigroups — Jozef Przytycki <przytyck@gmail.com>

0ur goal is to initiate (co)homology theory for quasigroups of Bol-Moufang. Our approach which has its roots in the work of Eilenberg and his coauthors (MacLane, Cartan) is to analyze extensions of a quasigroup $(X, *_X)$ by an affine quasigroup $(A, *_A)$ of the same type. We study these extensions not to classify them but to have the first glimpse at their homology via their second and third boundary operation, $\partial_2(x,y)$ and $\partial_3(x,y,z)$ respectively. We compute the second homology groups for all distinguishing examples of Bol-Moufang quasigroups described by Phillips and Vojtechovsky. We specualte about use of homology of Bol-Moufang quasigroups in Knot Theory. It is a joint work with Anthony Christiana, Ben Clingenpeel, Huizheng Guo, Jinseok Oh, and Anna Zamojska-Dzienio.

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1. General/ST

In the quest for squares in plane continua — Cristina Villanueva-Segovia <cristina@im.unam.mx>

In this talk we will be looking at conditions on a plane continuum $X$ (not necessarily locally connected) that guarantee the existence of four points in $X$ that are the vertices of a Euclidean Square (in which case we say that $X$ admits an inscribed square). In particular we show that ''certain type of square inscription´´ is generic among continua that separate the plane. The motivation of this work comes from the square peg problem: Does every Jordan curve admits an inscribed square?

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1. Plenary

Independence, ideal independence and forcing indestructibility — Vera Fischer <vera.fischer@univie.ac.at>

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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1. Plenary

Infinite Games — Jocelyn Bell <bell@hws.edu>

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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1. Dynamics/CT

Inverse Limits with Smith Functions and Indecomposability — Scott Varagona <svaragona@montevallo.edu>

We say a set-valued u.s.c. function $f$ from $[0,1]$ to $[0,1]$ is a Smith function if $f$ is surjective, the graph of $f$ is connected, and the graph of $f$ is the union of finitely many horizontal and vertical line segments. The author introduced inverse limits with Smith functions in a presentation at the 2021 Spring Topology and Dynamical Systems Conference. Later, in a 2023 paper, the author answered some questions posed by audience members at that 2021 talk, and he raised some new questions as well. This presentation at the 2025 Summer Topology and Its Applications Conference will discuss our further progress on the study of inverse limits with Smith functions, including some new results and conjectures. Our focus will be the case where the inverse limit is a continuum, in which case we wish to determine when such an inverse limit could be indecomposable.

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1. Low-Dimensional

Investigations in Knot Positivity — Lizzie Buchanan <elizabeth-buchanan@uiowa.edu>

A knot is "positive" if it has a diagram in which all crossings are positive. How does having such a diagram force patterns and structure to appear in the Jones polynomial and Khovanov homology? When can these patterns distinguish positive knots from almost-positive knots? In this talk we discuss results from the last few years and ongoing work to understand the Jones polynomial and Khovanov homology of positive knots and links. Particular attention is paid to the class of fibered positive knots, which contains all braid positive knots.

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1. Plenary

Khovanov skein lasagna modules and exotica — Mike Willis <msw188@tamu.edu>

Low dimensional topologists are very interested in "exotic behavior", that is, the difference between the smooth and the merely continuous. One would expect that detecting such subtle differences would require complicated analysis. In this talk I will describe joint work with Qiuyu Ren in which we show that exotica can be detected with a purely combinatorial theory (Khovanov homology and skein lasagna modules). No prior experience with exotica or Khovanov homology will be assumed.

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1. Dynamics/CT

Lelek-like fans — Ivan Jelić <ivajel@pmfst.hr>

The Lelek fan is the only smooth fan that has a dense set of end-points. In this talk, we study non-smooth fans with this property and construct an uncountable family of pairwise non-homeomorphic such fans.

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1. Low-Dimensional

Lie superalgebras and the minimal genus of virtual links — Micah Chrisman <chrisman.76@osu.edu>

For links $L \subset \Sigma \times [0,1]$, where $\Sigma$ is a closed orientable surface, we define a $U_q(\mathfrak{gl}(1,1))$ Reshetikhin-Turaev invariant with coefficients in $\mathbb{Z}[H_1(\Sigma)]$. This is well-defined up to multiples of the quantum supergroup variable $q$. This invariant turns out to be equivalent to an infinite cyclic version of the Carter-Silver-Williams (CSW) polynomial. The importance of the CSW polynomial is that half its symplectic rank gives strong lower bounds on the virtual genus. Recall that given a virtual link type $L$, the virtual genus of $L$ is the smallest genus of all closed orientable surfaces $\Sigma$ on which $L$ can be represented by a diagram $D$ on $\Sigma$. The main objective of this paper is to extend the CSW bound on the virtual genus to all Lie superalgebras $U_q(\mathfrak{gl}(m,n))$ with $n>0$. For links in thickened once-punctured surfaces $\Sigma$, we define a $U_q(\mathfrak{gl}(m,n))$ Reshetikhin-Turaev invariant with coefficients in $\mathbb{Z}[H_1(\Sigma)]$. We show that half its symplectic rank is also a lower bound on the virtual genus. Changing the value of the pair $(m,n)$ can give lower bounds better than those available from other known methods. We compare the $U_q(\mathfrak{gl}(m,n))$ lower bounds to those coming from the CSW polynomial, the surface bracket, the arrow polynomial, hyperbolicity, and the Gordon-Litherland determinant test. As an application, we show that the Seifert genus of homologically trivial knots in thickened surfaces is not additive under the connected sum operation of virtual knots. This is joint work with Killian Davis and Anup Poudel.

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1. General/ST

Metrization theorem in forcing extensions — Akira Iwasa <iwakira@yahoo.com>

We study ``metrizaion theorem in forcing extensions.'' That is, for a non-metrizable space $X$, we study what topological property $X$ has to have to become metrizable in forcing extensions. We provide such property for a class of spaces with weight $\leq\kappa$ and each point has a neighborhood of density $<\kappa$, where $\kappa$ is a regular uncountable cardinal.

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1. Low-Dimensional

Multiple Virtual Knot Theory — Lou Kauffman <loukau@gmail.com>

This talk will discuss a generalization of virtual knot theory (stabilized embeddings of knots and links in thickened surfaces) that uses many types of virtual crossings. The theory is motivated by graph coloring problems and their analogs as bracket polynomials for multiple virtual knots. We discuss a number of invariants of virtuals, conjectures and open problems.

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1. Dynamics/CT

Neighborhood N-Shadowing — Ellie Stephens <ellie_stephens2@baylor.edu>

We discuss a variation of the shadowing property, called neighborhood N-shadowing, and various dynamical systems with this property. Specifically, we consider neighborhood 2-shadowing with a focus on shift spaces. We discuss progress on characterizing neighborhood 2-shadowing in shift spaces in terms of the language of the shifts, drawing parallels to the known result that shifts of finite type are exactly those shift spaces with the shadowing property.

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1. Graphs

Obstructions to knotless embedding — Hyoungjun Kim <kimhjun@knu.ac.kr>

The Graph Minor Theorem of Robertson and Seymour implies a finite set of obstructions for any minor closed graph property. We show that there are only three obstructions to knotless embedding of size 23, which is far fewer than the 92 of size 22 and the hundreds known to exist at larger sizes. We describe several other topological properties whose obstruction set demonstrates a similar dip at small size. For order ten graphs, we classify the 35 obstructions to knotless embedding and the 49 maximal knotless graphs. This work is collaborated with Thomas Mattman.

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1. Graphs

On Distance-Scaling Transformations and Isomorphisms of Euclidean Distance Graphs on the Rational Points — Matt Noble <matthew.noble@mga.edu>

For any d > 0, define $G(\mathbb{Q}^n, d)$ to be the graph whose vertices are points of the rational space $\mathbb{Q}^n$ with any two vertices being adjacent if and only if they are a Euclidean distance $d$ apart. Such a graph is only of interest if $d$ is a distance actually realized between points of $\mathbb{Q}^n$, so we might as well assume that is the case. In this talk, we will ask for which $n$ and distances $d_1, d_2$ the graphs $G(\mathbb{Q}^n, d_1)$ and $G(\mathbb{Q}^n, d_2)$ are isomorphic. A resolution will be given for $n \leq 4$, and we will then present, by way of drawing a bunch of pictures, a method that, perhaps with some ingenuity, could be extended to answer this question for general $n$.

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1. Low-Dimensional

On algebraic invariants of multi-virtual knots — Sujoy Mukherjee <sujoymukherjee.math@gmail.com>

Multi-virtual knot theory is a generalization of virtual knot theory that associates labels to the virtual crossings of a virtual knot. After discussing basic ideas in multi-virtual knot theory, I will talk about algebraic invariants of multi-virtual knots constructed using operator quandles. The talk is based on joint work with Louis H. Kauffman and Petr Vojtechovsky.

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1. General/ST

On the modular metric topology — Olivier Olela-Otafudu <olivier.olela-otafudu@ul.ac.za>

Following earlier authors in this subject, the topology induced by a modular metric is herein called a modular topology. We show that such a topology is metrizable. More precisely, we show that the uniform topology induced by the uniformity on the modular set of a modular pseudometric is metrizable. In addition, we observe that such a topology is coarser than the underlying topology of the uniformity induced by the corresponding pseudometric. Other related immediate observations are also presented. \begin{references}{99} \bibitem{Chistyakov2} V.V. Chistyakov, \emph{Modular metric spaces, I}: Basic concepts, Nonlinear Anal. 72 (1)(2010), 1-14. \bibitem{Chistyakov3} V.V. Chistyakov, \emph{Modular metric spaces, II}. Application to superposition operators, Nonlinear Anal. 72 (1)(2010), 15-30. \bibitem{Chistyakov-book} V.V. Chistyakov, Metric modular spaces: Theory and applications, SpringerBriefs in Mathematics, Springer, Switzerland, 2015. \bibitem{Olela-Otafudu} Z. Mushaandja and O. Olela-Otafudu, On the modular metric topology, Topology Appl. (in press). \end{references}

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1. Low-Dimensional

On the neighborhood of knots in the space of open curves — Eleni Panagiotou <eleni.panagiotou@asu.edu>

In this talk we will discuss a new framework for classifying knots by exploring the neighborhood of knot embeddings in the space of (collections of) simple open curves in 3-space with no constraints at their endpoints. The latter gives rise to a knotoid (or linkoid) spectrum of a knot that consists of a knot-type knotoid and pure knotoids. We will examine to what extent the pure knotoids of the knotoid spectrum determine the knot type. For example, we will prove that the pure knotoids in the knotoid spectra of a knot, which are individually agnostic of the knot type, can distinguish knots of Gordian distance greater than one. We will also prove that the open curve neighborhood of, at least some, embeddings of the unknot can be distinguished from any embedding of any non-trivial knot that satisfies the cosmetic crossing conjecture. Topological invariants of knots can be extended to their open curve neighborhood to define continuous functions in the neighborhood of knots. We will discuss their properties and prove that invariants in the neighborhood of knots may be able to distinguish more knots than their application to the knots themselves. For example, we will prove that an invariant of knots that fails to distinguish mutant knots (and mutant knotoids), can distinguish them by their neighborhoods, unless it also fails to distinguish non-mutant pure knotoids in their spectra. Studying the neighborhood of knots opens the possibility of answering questions, such as if an invariant can detect the unknot, via examining possibly easier questions, such as whether it can distinguish height one knotoids from the trivial knotoid.

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1. Dynamics/CT

On the set function $\wp$ — Sergio Macias <macias@unam.mx>

Inspired on the work that Professor Janusz R. Prajs did on homogeneous metric continua in his paper *Mutually Aposyndetic Decomposition of Homogeneous Continua*, [Canad. J. Math., 62 (2010), 182-201] and the version of his work for Hausdorff continua with the uniform property of Effros done by this author, we introduce a new set function, $\wp$, and present properties of it.

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1. Plenary

On tracing properties, invariant measures, and entropy. — Piotr Oprocha <piotr.oprocha@osu.cz>

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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1. Dynamics/CT

On various forms of independence and minimality for general triangular systems — Deepanshu Dhawan <dhawan.1@iitj.ac.in>

In this talk, we will discuss various notions of independence for general non-autonomous systems. Further, we use the notions to investigate dynamics of a general triangular system. In particular, we investigate the dynamics of a minimal triangular system and relate it to the dynamics of its component systems.

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1. Graphs

Optimization of the lattice stick number in handcuff graphs — Sungjong No <sungjongno@kyonggi.ac.kr>

A handcuff graph is a graph consisting of disjoint two loops and connected by an edge. The lattice stick number of a handcuff graph is the minimum number of sticks required to embed the graph in a lattice space. Previous studies have shown that the lattice stick numbers of the trivial handcuff graph and the Hopf-linked handcuff graph are 9 and 11, respectively, and these are the only graphs whose lattice stick number is 13 or less. In this talk, we utilize a squeezing method to identify all models with a lattice stick number of at most 14.

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1. General/ST

Partial metric spaces - topological properties and applications — Dariusz Bugajewski <ddbb@amu.edu.pl>

The notion of a partial metric space was introduced by Matthews in 1994 who showed, roughly speaking, how metric--like tools can be extended to non--Hausdorff topologies. He also indicated some applications of this class of spaces in the study of denotational semantics of a programming language. In this talk we are going to present some necessary and sufficient conditions under which the topology generated by a partial metric is equivalent to the topology generated by a suitably defined metric. Next, we are going to focus on two basic topological properties of partial metric spaces, namely completeness and compactness. In particular, it appears that in these spaces compactness is equivalent to sequential compactness. Finally, we will focus on a very general fixed point theorem for mappings acting in partial metric spaces. In that theorem we impose some conditions on behavior of considered mappings on orbits and a condition relating orbits of points of small size.

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1. Computing

Persistence-Augmented Neural Networks — Elena Wang <wangx249@msu.edu>

Topological Data Analysis (TDA) provides tools to describe the shape of data, but integrating topological features into deep learning pipelines remains challenging, especially when preserving local geometric structure rather than summarizing it globally. We propose a persistence-based data augmentation framework that encodes local gradient flow regions and their hierarchical evolution using the Morse–Smale complex. This representation, compatible with both convolutional and graph neural networks, retains spatially localized topological information across multiple scales. Importantly, the augmentation procedure itself is efficient, with computational complexity $O(n \log n)$, making it practical for large datasets. We evaluate our method on histopathology image classification and 3D porous material regression, where it consistently outperforms baselines and global TDA descriptors such as persistence images and landscapes. We also show that pruning the base level of the hierarchy reduces memory usage while maintaining competitive performance. These results highlight the potential of local, structured topological augmentation for scalable and interpretable learning across data modalities.

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1. Dynamics/CT

Plane embeddings of continua, and accessible points — Logan Hoehn <loganh@nipissingu.ca>

A point $p$ in a plane continuum $X \subset \mathbb{R}^2$ is accessible if there exists an arc $A \subset \mathbb{R}^2$ such that $A \cap X = \\{ p \\}$. I will describe our recent results about plane embeddings of continua and their accessible points. Specifically, I will discuss arc-like continua (the Nadler-Quinn problem), Knaster continua, and Ingram's atriodic triod-like continuum. This is joint work with Andrea Ammerlaan and Ana Anušić.

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1. Graphs

Problem Session — Elena Pavelescu <elenapavelescu@southalabama.edu>

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1. Plenary

Projections of spatial graphs — Erica Flapan <elf04747@pomona.edu>

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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1. Graphs

Properties of the Penrose Polynomial — Dan Silver <silver@southalabama.edu>

In joint work with Louis Kauffman and Susan Williams, we give an elementary introduction to the Penrose polynomial for cubic graphs and explore the combinatorial significance of its coefficients. No special background is assumed.

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1. General/ST

Pseudo-$\aleph_1$-compactness in $\mathbb R$-factorizable groups — Olga Sipacheva <ovsipa@gmail.com>

This is a joint work with Evgenii Reznichenko. A topological group is said to be $\mathbb R$-factorizable if, given any continuous function $f\colon G\to \mathbb R$, there exists a continuous homomorphism $h\colon G \to H$ to a second-countable topological group $H$ and a continuous function $g\colon H\to \mathbb R$ such that $f = g \circ h$. The main unsolved problems of the theory of $\mathbb R$-factorizable groups are as follows: 1. Is the property of being an $\mathbb R$-factorizable group topological? In other words, is any topological group homeomorphic to an $\mathbb R$-factorizable one $\mathbb R$-factorizable? 2. Is the square of an $\mathbb R$-factorizable group $\mathbb R$-factorizable? 3. Is any $\mathbb R$-factorizable group pseudo-$\aleph_1$-compact, that is, contains no uncountable locally finite family of open sets? 4. Is the image of an $\mathbb R$-factorizable group under a continuous homomorphism $\mathbb R$-factorizable? We show that if the answer to question 2 is positive, then so is the answer to question 1. Also, if the answer to question 4 is positive, then so is the answer to question 3, and if the answer to question 3 is negative, then so are the answers to questions 1 and 2. Note that there are examples of $\mathbb R$-factorizable groups $G$ and $H$ such that $G\times H$ is not $\mathbb R$-factorizable. Our main concern is the pseudo-$\aleph_1$-compactness of $\mathbb R$-factorizable groups. We prove that an $\mathbb R$-factorizable group $G$ is pseudo-$\aleph_1$-compact if it satisfies any of the following conditions: (i) the weight of $G$ is at most $\omega_1$; (ii) the pseudocharacter of $G$ equals $\omega_1$; (iii) $G^2$ is $\mathbb R$-factorizable; (iv) $G$ contains a nonmetrizable compact subspace; (v) $G$ contains a Lindel\"of subspace of uncountable pseudocharacter.

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1. General/ST

Reflecting topological properties in closures of countable sets. — Vladimir Tkachuk <vova@xanum.uam.mx>

This talk's purpose is to present some results on whether a topological space $X$ has a property $\mathcal P$ given that $\overline A$ has $\mathcal P$ for any countable set $A\subset X$. The respective line of research was outlined in a recent paper of A. Dow and the author. We prove, among other things, that there is a consistent example of a metric space $X$ such that $\overline A$ is \v Cech-complete for any countable $A\subset X$ but $X$ has no dense \v Cech-complete subspace. This talk will also feature some related results on general locally convex spaces.

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1. Computing

Representations of Micrograph Geometry for Machine Learning — Benjamin Schweinhart <bschwei@gmu.edu>

Two-dimensional micrographs are a common data format in several important machine learning applications. One example is histopathological classification: the detection of disease-related abnormalities in 2D scans of biological tissue. Another, from materials science, is the classification of polycrystalline materials and the prediction of their physical properties based on images of their microstructures. In both applications, the local topology and geometry --- namely the shape and arrangement of cells/grains --- are thought to be essential. However, information about these features may be lost in traditional machine learning pipelines such as those involving convolutional neural nets (CNNs). In this talk, I will discuss two methods to represent the geometry of micrographs in formats amenable to machine learning. The first augments images of biological tissue with additional fields representing the persistent homology of local windows. The second represents the grain structure of a polycrystal as a metric measure space of local configurations.

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1. Dynamics/CT

Retract or Not: A Tale of Two Fans — Iztok Banic <iztok.banic@um.si>

In this talk, we present structural and dynamical aspects of certain arcwise connected continua known as fans. First, we present conditions under which embeddings of the Lelek fan admit retractions, focusing on how features such as wedges and cuts influence retraction properties. Second, we address a classical open question about characterizing fans as unions of arcs intersecting in a single point. This is joint work with Goran Erceg, Sina Greenwood, Ivan Jelic, Judy Kennedy, and Van Nall.

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1. Graphs

Ribbonlength along three-page presentation of links — Hyungkee Yoo <hyungkee@scnu.ac.kr>

Since Kauffman first introduced the concept of ribbonlength for knots and links, many researchers, including Denne, have studied ribbonlength using folded ribbons. In several studies, upper bounds for ribbonlength were obtained by folding the ribbon in the shape of a right isosceles triangle. In this presentation, we introduce a new method of folding the ribbon into an equilateral triangle shape instead of the right isosceles triangle. We will then relate this method to three-page presentations, which are variations of the arc presentation. In this process, we present new upper bounds for the ribbonlength of the Hopf link and the trefoil knot.

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1. General/ST

Sequential and countable compactness associated to Ramsey-like properties — Cesar Corral <cicorral@ciencias.unam.mx>

In this talk, we will examine variants of sequential compactness and countable compactness that are associated with Ramsey-like properties. These notions have arisen naturally in various topological and combinatorial contexts. We will present several results obtained by treating these compactness properties as central objects of study. The talk will conclude with some connections to classical problems and open questions.

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1. Computing

Sequential topological complexity of symmetric products of surfaces — Ekansh Jauhari <ekanshjauhari@ufl.edu>

Sequential topological complexities (denoted $\text{TC}_m$ for each $m\ge 2$) are numerical homotopy invariants of topological spaces motivated by the motion planning problem in robotics. Given a space $X$, $\text{TC}_m(X)$ measures the discontinuity of planning a motion between any given sequence of $m$ points in $X$ for any robot whose configuration space is $X$. Usually, cohomological data of a space helps estimate its $\text{TC}_m$ values. In this talk, we focus on the case when our space $X$ is a symmetric product of a closed orientable surface. Using Macdonald's description of the cohomology ring of these spaces, we completely determine all sequential topological complexities of all symmetric products of closed orientable surfaces. Our methods involve explicit computations of their Lusternik--Schnirelmann category and rational zero-divisor cup-lengths. Using our computations, we also verify the “TC-rationality conjecture" of Farber and Oprea for these spaces.

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1. Plenary

Shadowing in $\mathcal C(X)$ — Jonathan Meddaugh <jonathan_meddaugh@baylor.edu>

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 $\mathcal T(X)$ of maps with shadowing as a subset of $\mathcal C(X)$, the space of continuous self-maps on a compact metric space $X$. We will discuss the structure of $\mathcal T(X)$ for certain classes of spaces, with a special focus on the question of whether $\mathcal T(X)$ is a generic set in $\mathcal C(X)$.

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1. Computing

Sheaf Cohomology and the Algebraic Path Problem — Kaelyn Willingham <will4247@umn.edu>

Routing problems in computer science often involve computing efficient routes for moving entities between different points in some defined space. Given a semi-ring $R$ and a graph $G$, the Algebraic Path Problem provides a unifying framework for analyzing various routing problems mathematically by abstracting the notion of combining weighted paths on $G$ under the additive operation defined on $R$. Routing problems defined on planar graphs are fairly understood, but these same problems remain elusive when defined on more-complex topological structures. In this talk, I will discuss current work that utilizes sheaf cohomology to understand the nature of routing problems defined on cellular complexes. In so doing, we will find a nice generalization of the Algebraic Path Problem. This is joint work with Russell Funk and Thomas Gebhart.

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1. Low-Dimensional

Skein Modules for 3-Manifolds and Their Structure — Rhea Bakshi <rheapalak@ucsb.edu>

Skein modules were introduced by Przytycki and independently by Turaev as generalizations of the polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. Among these, the Kauffman bracket skein module (KBSM) has been studied most extensively. Recently, Gunningham, Jordan, and Safronov demonstrated that for any closed 3-manifold, the KBSM is finite-dimensional over $\mathbb Q(A)$; however, this finiteness does not extend to the KBSM over $\mathbb Z[A^{\pm 1}]$. Moreover, computing the KBSM of a 3-manifold remains a notoriously challenging problem, especially over this ring. In this talk, we will survey these developments and explore several open questions concerning the structure of the KBSM over $\mathbb Z[A^{\pm 1}]$.

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1. Low-Dimensional

Skein identities at roots of unity — Vijay Higgins <higginsv@math.ucla.edu>

The skein algebra of an oriented surface is spanned by framed links in the thickened surface subject to the Kauffman bracket relations. Multiplication of links is given by stacking in the direction of the thickening. We will discuss special skein identities which hold when the quantum parameter $q$ is specialized to a root of unity. The identities involve Jones-Wenzl projectors and are certain incarnations of special cases of Steinberg tensor product identities from the representation theory of $U_q(sl_2).$ We will discuss how the easiest such identity can be used to recover the Chebyshev-Frobenius homomorphism of Bonahon-Wong. This is joint work with Indraneel Tambe.

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1. Dynamics/CT

Specification in Mahavier Systems via Closed Relations — Goran Erceg <gorerc@pmfst.hr>

We study two types of specification properties - standard and initial - and extend them to CR-dynamical systems, where the dynamics are given by closed relations instead of continuous functions. Although these properties are often equivalent in classical settings, we show they can behave differently in this broader context. We define new specification-type properties for Mahavier dynamical systems and present several examples that highlight their differences. Each new property matches the classical specification property when applied to continuous functions. This is joint work with Iztok Banič, Ivan Jelić and Judy Kennedy

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1. Dynamics/CT

Speedups of Toeplitz Flows — Lori Alvin <lori.alvin@furman.edu>

Given a minimal Cantor system $(X,T)$, a topological speedup of $(X,T)$ is a dynamical system $(X,S)$ where $S$ is a homeomorphism such that $S(x) =T^{p(x)}(x)$ for some function $p:X\to \mathbb{N}$. We assume the function $p$ is continuous (and thus bounded) and the resulting system $(X,S)$ is minimal. One can ask what properties of the underlying initial system $(X,T)$ are preserved under minimal bounded speedups. We investigate the class of Toeplitz flows, which are minimal symbolic almost one-to-one extensions of odometers. Although the minimal bounded speedup of an odometer is always a conjugate odometer, we demonstrate that the minimal bounded speedup of a Toeplitz flow need not be Toeplitz. We then provide sufficient conditions to guarantee that the minimal bounded speedup will be a Toeplitz flow; in this case, it is never conjugate to the original Toeplitz flow but has the same underlying odometer.

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1. Computing

Squares inscribed in ray compactifications — Ulises Morales-Fuentes <ulises.morales@uaem.mx>

In this talk we will prove that ray compactifications in the plane admit inscribed squares provided that their remaider is a piecewise linear simple triod. Also we will show visualizations of sections of Vaughan's function implemented in python and visualized with Ipyvolume. We will discuss how this visualizations have proven to be very useful in finding squares inscribed in plane sets.

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1. General/ST

Stone-Cech extensions of pseudocompact convex spaces. — Evgenii Reznichenko <erezn64@gmail.com>

All spaces are assumed to be Tychonoff spaces. Let $X$ be a convex pseudocompact subspace of some locally convex space (LCS). Question 1. Is it true that the Stone-Cech extension $\beta X$ has the structure of a convex compact set? Is it true that $\beta X$ is homeomorphic to a convex compact subset of some LCS? The answer to this question is positive if $X=P(Y)$, where $P(Y)$ is the space of probability Radon measures on $X$ in the weak topology [1]. In this case, $Y$ is a pseudocompact space and $\beta P(Y)=P(\beta Y)$. There is a convex compact set $K$ and its dense convex pseudocompact subset $C$ such that $\beta C\neq K$. Proposition 1. $\beta X$ is path-connected. This fact is related to the fact that the structure of a convex set with $X$ extends to $\beta X$. A space $S$ with a (separately) continuous operation $p: [0,1]\times X\times X\to X$ is called a (semi)topological convex set if there is an embedding of $S$ into a linear space (without topology) such that $p(\lambda,x,y)=\lambda x+(1-\lambda)y$. Theorem 1. If $S$ is a pseudocompact topological convex set, then $\beta S$ is a semitopological convex set. Clearly, a convex subset of some LCS is a topological convex set. Theorem 1 implies Proposition 1. Theorem 2. If $S$ is a topological convex set and $S^2$ is pseudocompact, then $\beta S$ is a topological convex set. Theorem 3. If $S$ is a countable compact semitopological convex set, then $\beta S$ is a semitopological convex set. The theorems imply that the convex set structure from $S$ extends to $\beta S$. A (semi)topological convex set $S$ is a universal (semi)topological algebra with continuum operations $p_\lambda: S\times S\to S$, $p_\lambda(x,y)=p(\lambda,x,y)$, where $\lambda\in [0,1]$. The signature of $S$ is continuous, is a segment of $[0,1]$. The theorems are proved using results on the extension of operations in universal algebras obtained in [2]. [1] Reznichenko, E., "Stone-Cech extensions of probability measure spaces." arXiv preprint arXiv:2412.11838 (2024). [2] Reznichenko, E., "Extensions and factorizations of topological and semitopological universal algebras." Topology and its Applications (2025): 109256.

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1. General/ST

The Borel hierarchies of function spaces consisting of metrics — Katsuhisa Koshino <ft160229no@kanagawa-u.ac.jp>

Function spaces have been studied in the theory of infinite-dimensional topology, and their Borel hierarchies play important roles in recognizing topologies on them. In this talk, we shall investigate the Borel hierarchies and the complete metrizability of function spaces consisting of metrics on metrizable spaces, and as an application, we will decide their topological types.

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1. General/ST

The Shape of Generating Families — Paul Gartside <paulmgartside@gmail.com>

In his survey article in the Handbook of Set-theoretic Topology on cardinal characteristics of the continuum and small cardinals arising in topology, van Douwen introduced three such invariants of a separable metrizable space, M, namely cof(K(M)), kc(M) and k(M). Each invariant asks for the minimum size of a family of compact subsets of M with certain properties.The third invariant, k(M), requires that the compact subsets witness the k-space property of M. In this talk we aim to understand not just the size, but the "shape" of compact families witnessing the k-space property (k-structures), and the "shape" of families of convergent sequences witnessing sequentiality (sequential structures), of a separable metrizable space. Our primary tool will be an extension, due to Vojtas, of the Tukey order on directed sets to general relations. A natural question arising from this work will have as its answer, `the omega_1 st fixed point of the aleph function'.

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1. Plenary

The Shape of Relations: From Knot Invariants to Cancer Genomics — Radmila Sazdanovic <rsazdan@ncsu.edu>

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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1. Low-Dimensional

Thompson's Groups, Annular Links, and Tangles — Louisa Liles <lml2tb@virginia.edu>

Vaughan Jones showed how to associate links in the $3$-sphere to elements of Thompson’s group $F$ and proved that $F$ gives rise to all link types. This talk will introduce Jones’s construction and discuss two recent extensions– the first is a method of building annular links from Thompson’s group $T$, which contains $F$ as a subgroup, and the second is a method of building $(n,n)$-tangles, which give rise to an action of $F$ on Khovanov's chain complexes. This talk includes joint work with Slava Kruskhal and Yangxiao Luo.

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1. Graphs

Topological Characterization of Carbon Allotrope Graphs Using Multifractal Analysis — D. Easwaramoorthy <easandk@gmail.com>

The intrinsic properties of carbon nanosheets derived from their basic molecular structure through a self-similar pattern have attracted much interest among researchers. Graph-theoretical methods are used to identify certain molecular descriptors known as topological indices, which are highly useful in connecting molecules to their physical attributes. Several chemical characteristics have been correlated with degree and neighborhood degree sum-based topological indices, which have been investigated extensively. Our current research establishes the use of topological indices in studying the newly synthesized carbon allotropes $\delta$-graphene, $\delta$-graphyne and $\delta$-graphdiyne. Furthermore, the complexity and information of carbon allotropes can be discussed in terms of Generalized Fractal Dimensions $(\mathrm{GFD})$, which are newly constructed based on Renyi entropy using some types of topological indices. The study of GFD indices of graphs is gaining importance as a measure of the complexity of basic coupling and as a tool for characterizing structural properties. We have estimated various topological indices, including graph-based GFD values of these structures obtained using the GFD method based on Renyi entropy.

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1. Computing

Topological Feature Selection for Time Series Data — Johnathan Bush <bush3je@jmu.edu>

I will describe how tools from applied topology may be used to identify components of time series most responsible for cyclic dynamics observed in orbits of an underlying dynamical system. In this setting, I will show that derivatives of the persistent homology may be computed explicitly and describe a simple algorithm for gradient descent. As an example, we will consider neuronal data from the model organism C. elegans and identify subsets of neurons driving global cyclic brain dynamics in the spirit of dimensionality reduction.

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1. General/ST

Topological spaces after forcing — Pedro Marun <marun@math.cas.cz>

If $(X,\tau)$ is a topological space and $P$ is a poset, then $\tau$ may cease to be a topology after forcing with $P$, for example if new subsets of $X$ are added. Nevertheless, in the generic extension, $\tau$ is a basis for a topology, call it $\tau^P$, which is finer than $\tau$. One can then ask which properties of $\tau$ are inherited by $\tau^P$. In this talk, we will look at what happens to the Lindelöf property under different classes of forcing notions.

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1. Low-Dimensional

Triple point numbers of 2-twist spun knots — Scott Carter <carter@southalabama.edu>

The minimal triple point number of the $2$-knot that is the 2-twist spin of the knot $5_2$ is bounded between 8 and 12. The movie suggested by Fox's example 15 has triple point number 12. To improve this bound we follow Shin Satoh's work on triple point number and construct virtual surfaces that have small triple point number and for which Mochizuki's 3-cocycle vanishes. This is joint work with Seonmi Choi, Hongdae Kim, Sangsu Lee, and Seong Yeop Yang.

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1. Dynamics/CT

Turbulent closed relations — Judy Kennedy <kennedy9905@gmail.com>

In classical dynamical systems, turbulence has played a pivotal role in understanding chaotic behavior, particularly for interval maps. This talk extends the notion of turbulence from continuous functions to closed relations on compact metric spaces, utilizing Mahavier products and associated shift maps. We define and explore CR-turbulence (Closed Relation Turbulence) and its variants, establishing connections between turbulence and topological entropy in the setting of closed relations. This is joint work with Chris Mouron and Van Nall.

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1. Graphs

Unavoidable Induced Subgraphs of Large Graphs — Sarah Allred <sarahallred@southalabama.edu>

In 1930, Ramsey proved that for every positive integer $r$, every sufficiently large graph contains as an induced subgraph either $K_r$ or an independent set of size $r$. In this talk, I will give analogous characterizations for increasing levels of connectivity. This presentation combines work from two projects: the first with Guoli Ding and Bogdan Oporowski, and the second with Mark Ellingham.

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1. Plenary

Unknotting numbers of spatial graphs, knots and DNA — Danielle O'Donnol <dodonnol@marymount.edu>

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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1. Computing

Unmapped Territory for Topological Data Analysis: Mathematics Education — Devin Hensley <dkh0009@auburn.edu>

Topological data analysis has been used to analyze social systems, disease spread, and polling locations, but it is not typically used in mathematics education research. In this study, university calculus students created concept maps–visual representations of connections–which were analyzed using homology groups. We argue that homology is an innovative and useful tool for analyzing concept maps, complementing previous analyses conducted via qualitative techniques or scoring systems. We further argue that topological data analysis can be a valuable tool for mathematics education research.

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1. Computing

Unveiling Topological Structures in Text & Speech — Adaku Uchendu <adaku.uchendu@ll.mit.edu>

The surge of data available on the internet has led to the adoption of various computational methods to analyze and extract valuable insights from this wealth of information. Among these, the field of Machine Learning (ML) has thrived by leveraging data to extract meaningful insights. However, ML techniques face notable challenges when dealing with real-world data, often due to issues of imbalance, noise, insufficient labeling, and high dimensionality. To address these limitations, some researchers advocate for the adoption of Topological Data Analysis (TDA), a statistical approach that discerningly captures the intrinsic shape of data despite noise. Despite its potential, TDA has not gained as much traction within the Natural Language Processing (NLP) domain compared to structurally distinct areas like computer vision. Nevertheless, a dedicated community of researchers has been exploring the application of TDA in NLP, yielding 93 papers we comprehensively survey in this paper. Our findings categorize these efforts into theoretical and non-theoretical approaches. Theoretical approaches aim to explain linguistic phenomena from a topological viewpoint, while non-theoretical approaches merge TDA with ML features, utilizing diverse numerical representation techniques. We conclude by exploring the challenges and unresolved questions that persist in this niche field.

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