1. GeoGT

Dehn Functions of Coabelian Subgroups — Pratit Goswami <pratit.goswami-1@ou.edu>

The study of Dehn functions has developed into a major area of research in geometric group theory mainly because the growth types of these functions are quasi-isometry invariants of finitely presented groups. The Dehn function of a finitely presented group G is also connected to the complexity of solving the word problem in G namely, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation is recursive. In this talk, we will discuss new methods for computing the precise Dehn functions of coabelian subgroups of direct products of groups, that is, subgroups which arise as kernels of homomorphisms from the direct product onto a free abelian group. This is joint work with Noel Brady and Rob Merrell.

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1. GeoTop

3-Manifolds, Isotopy, and Group Actions — Trent Lucas <trent_lucas@brown.edu>

Suppose a finite group acts on a closed manifold M. Given two equivariant homeomorphisms of M, if we know that they are isotopic, can we conclude that they are equivariantly isotopic? An important theorem of Birman-Hilden and MacLachlan-Harvey says the answer is "yes" if M is a hyperbolic surface; Margalit-Winarski asked whether the same is true when M is a 3-manifold. We answer Margalit-Winarski's question for a wide class of group actions on 3-manifolds; this includes a 3-manifold analog of the hyperelliptic involution, which we can understand particularly well via a connection with geometric group theory.

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1. Continua

A Dendrite Equivalence Relation on Loop Space — Spencer Arnesen <spencer.arnesen@mathematics.byu.edu>

This talk will discuss how to turn a loop space into a group by factoring through dendrites. Inspired by the fact that group homomorphisms between fundamental groups of one-dimensional spaces induce, up to conjugation, a continuous map, and that path homotopies on one-dimensional spaces factor through a dendrite we show that homotopy through a dendrite is an equivalence relation and induces a group structure on a subset of loops. This group is always locally free.

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1. Set-Theoretic

A Forcing Axiom for Preserving a Lindelöf Space — Thomas Gilton <tdgilton@gmail.com>

A topic of continued interest in set-theoretic topology is the question of which topological properties are preserved under which forcings. In recent work, the speaker and Holshouser have shown that strongly proper forcings preserve a wide variety of covering properties (including Lindelöf), generalizing work of Dow, Iwasa, and Kada. In this talk, we will give an overview of yet further work done by the speaker on this topic. Namely, we discuss how to create forcing axioms for proper posets that preserve a given Lindelöf space. This uses Neeman's two-type side conditions machinery in combination with the earlier work of Gilton and Holshouser.

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1. Applied

A discontinuous ham sandwich theorem — Matt Superdock <superdockm@rhodes.edu>

The "ham sandwich" theorem states that any $n$ finite Borel measures on $\mathbb{R}^{n}$ can be simultaneously bisected by a single hyperplane, provided each measure is absolutely continuous with respect to Lebesgue measure. In 1984, Cox & McKelvey showed that even for discontinuous measures, there exists a single hyperplane such that at most half of each measure lies on each side. In this talk, we consider the problem of minimizing the differences of the measures of the two open half-spaces determined by a chosen hyperplane, where the measures may be discontinuous. We show that if the dimension of $\mathbb{R}^{n}$ is much larger than the number of measures, then there exists a hyperplane that divides the measures more fairly than in Cox & McKelvey's result.

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1. GeoTop

A generalization of Cannon's conjecture for cubulated hyperbolic groups — Corey Bregman <corey.bregman@tufts.edu>

We show that cubulated hyperbolic groups with spherical boundary of dimension 3 or at least 5 are virtually fundamental groups of closed, orientable, aspherical manifolds, provided that there are sufficiently many quasi-convex, codimension-1 subgroups whose limit sets are locally flat subspheres. The proof is based on ideas used by Markovic in his work on Cannon's conjecture for cubulated hyperbolic groups with 2-sphere boundary. This is joint work with Merlin Incerti-Medici.

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1. Continua

A notable contractible dendroid — Alejandro Illanes <illanes@matem.unam.mx>

A dendroid is an arcwise connected continuum such that the intersection of any two of its subcontinua is connected. In 1985, Tadeusz Mackiowiak constructed a contractible non-selectible dendroid X. Through the years the originality of the structure of this dendroid has been useful to produce several counterexamples. In this talk we will mention some other important properties of X and some of the examples that have constructed using it, including a new one related to the hyperspace of subcontinua with empty interior of a continuum.

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1. GeoGT

A weak Tits alternative for groups acting on buildings — Chris Karpinski <christopher.karpinski@mail.mcgill.ca>

Buildings are highly symmetrical non-positively curved simplicial complexes introduced by Jacques Tits in the 1950s to study semisimple algebraic groups. Over the years, buildings have garnered interest among geometric group theorists due to their non-positively curved structure and close connections to Coxeter groups. We prove that groups acting properly and cocompactly on buildings satisfy an algebraic dichotomy, commonly encountered among groups with non-positive curvature features, known as the weak Tits alternative: either the group is virtually abelian or it contains a nonabelian free subgroup. This is joint work with Damian Osajda and Piotr Przytycki.

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1. GeoGT

Acylindrical actions on trees and applications to the outer automorphism group of Baumslag-Solitar groups — Bratati Som <bratatis@buffalo.edu>

An acylindrical action generalizes proper and cobounded actions on hyperbolic spaces. Non-elementary acylindrical actions provide acylindrically hyperbolic groups, which includes most mapping class groups of punctured surfaces, 3-manifold groups, and $Out(F_n)$ for $n > 1$. In this talk, we will explore how acylindricity of a group action on a tree can be preserved under quotients by certain subgroups, and discuss the existence of a largest acylindrical action for some groups acting on trees. In addition, we will show when $Out(BS(p,q))$ is acylindrically hyperbolic for non-solvable Baumslag-Solitar groups, despite $BS(p,q)$ itself not being acylindrically hyperbolic, and explore further applications of these acylindricity results. This is a joint work with Daxun Wang.

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1. Set-Theoretic

Adding an uncountable discrete subspace by forcing — Akira Iwasa <akiraiwasa94@gmail.com>

Suppose that a topological space $X$ has no uncountable discrete subspace. We discuss if $X$ can obtain an uncountable discrete subspace in forcing extensions. We prove that for any monotonically normal space $X$ which has no uncountable discrete subspace, $X$ can obtain an uncountable discrete subspace in some forcing extension if and only if $X$ is not separable.

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1. Plenary

All is Rep-Tile — Alexandra Kjuchukova <akjuchuk@nd.edu>

An n-dimensional rep-tile is a PL n-manifold X, embedded in $\mathbb{R}^n$, which can be decomposed as the union of mutually isometric manifolds similar to X which have non-overlapping interiors. For one astonishing example, all knot exteriors are homeomorphic to rep-tiles, by a 2021 result of Blair, Marley and Richards. I will give an isotopy classification of rep-tiles in all dimensions. I will also outline our proof, which is based on a technique called ball swapping. This is joint work with Ryan Blair, Patricia Cahn and Hannah Schwartz.

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1. Continua

An Uncountable Family of Generalized Inverse Limit Spaces Which are Pointwise Self Homeomorphic (Updated). — Faruq Mena <faruq.mena@soran.edu.iq>

In this talk, we will discuss how we found uncountable families of generalized inverse sequences on intervals and also on finite trees such that the inverse limit spaces of these sequences are pointwise self-homeomorphic. We give several examples of pointwise self-homeomorphic continua obtained in this manner including the dendrite $D_3$ and a dendrite containing $D_\omega$.

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1. GeoGT

Anosov representations of cubulated hyperbolic groups — Theodore Weisman <tjwei@umich.edu>

An Anosov representation of a hyperbolic group $\Gamma$ is a representation which quasi-isometrically embeds $\Gamma$ into a semisimple Lie group - say, SL(d, R) - in a way which generalizes and imitates the dynamical behavior of a convex cocompact group acting on a hyperbolic metric space. It is unknown whether every linear hyperbolic group admits an Anosov representation. In this talk, after motivating the theory of Anosov representations from the perspective of geometric group theory, I will discuss joint work with Sami Douba, Balthazar Flechelles, and Feng Zhu which shows that every hyperbolic group acting geometrically on a CAT(0) cube complex admits a 1-Anosov representation into SL(d, R) for some d. The proof exploits the relationship between the combinatorial/CAT(0) geometry of right-angled Coxeter groups and the projective geometry of a convex domain in real projective space on which a Coxeter group acts by reflections.

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1. GeoTop

Arithmeticity and commensurability of links in thickened surfaces — Rose Kaplan-Kelly <rkaplank@gmu.edu>

In this talk, we will consider a generalization of alternating links and their complements in thickened surfaces. In particular, a family of generalized alternating links which each correspond to a Euclidean or hyperbolic tiling and have a right-angled complete hyperbolic structure on their complement. We will determine the arithmeticity of these links and find their pairwise commensurability. This is joint work with David Futer.

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1. GeoGT

Asymptotic dimension of graphs of arcs and curves on infinite-type surfaces — Michael Kopreski <michaelkopreski@gmail.com>

In analogy to the curve complex and its role in the geometry of mapping class groups of finite-type surfaces, a number of authors have defined graphs whose vertices are arcs or curves on a given infinite-type surface S, and on which the mapping class group Map(S) acts by isometries. We show that for a broad class of such graphs, including the grand arc graph, the omnipresent arc graph, and all others defined comparably to Masur-Minsky, the asymptotic dimension is infinite. In particular, if one could construct a graph in this class admitting a Švarc-Milnor-type action of Map(S), then Map(S) would have infinite asymptotic dimension.

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1. GeoTop

Automorphisms of the smooth fine curve graph — Katherine Booth <k.wbooth3@gmail.com>

The smooth fine curve graph of a surface is an analogue of the fine curve graph that only contains smooth curves. It is natural to guess that the automorphism group of the smooth fine curve graph is isomorphic to the diffeomorphism group of the surface. But it has recently been shown that this is not the case. In this talk, I will give several more examples with increasingly wild behavior and give a characterization of this automorphism group for the particular case of continuously differentiable curves.

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1. GeoGT

Automorphisms of the sphere complex of an infinite graph — Thomas Hill <thill@math.utah.edu>

For a locally finite, connected graph $\Gamma$, let $\operatorname{Map}(\Gamma)$ denote the group of proper homotopy equivalences of $\Gamma$ up to proper homotopy. Excluding sporadic cases, we show $\operatorname{Aut}(\mathcal{S}(M_\Gamma)) \cong \operatorname{Map}(\Gamma)$, where $\mathcal{S}(M_\Gamma)$ is the sphere complex of the doubled handlebody $M_\Gamma$ associated to $\Gamma$. We also construct an exhaustion of $\mathcal{S}(M_\Gamma)$ by finite strongly rigid sets when $\Gamma$ has finite rank and finitely many rays, and an appropriate generalization otherwise. This is joint work with Michael Kopreski, Rebecca Rechkin, George Shaji, and Brian Udall.

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1. Applied

Bestvina-Brady discrete Morse theory and Vietoris-Rips complexes — Matthew Zaremsky <mzaremsky@albany.edu>

Bestvina-Brady discrete Morse theory is a topological tool that has historically been most useful in geometric group theory. In this talk I will discuss a version of Bestvina-Brady Morse theory that is particularly conducive to understanding topological properties of Vietoris-Rips complexes of metric spaces, and has applications not only to geometric group theory, but also to applied topology and topological data analysis. In particular I will discuss a recent short proof of a result of Virk, that says the metric space $\mathbb{Z}^n$ with the usual $L^1$ metric has contractible Vietoris-Rips complexes.

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1. Continua

Big and large continua — Teja Kac <teja.kac1@um.si>

We generalize the notion of generalized inverse limits of inverse sequences of closed intervals with upper semicontinuous bonding functions to inverse limits of inverse sequences over directed graphs. We show that under certain conditions such inverse limits contain big/large continua.

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1. Plenary

Bounded cohomology and displacement — Francesco Fournier-Facio <ff373@cam.ac.uk>

Bounded cohomology is a functional-analytic analogue of group cohomology that is central to rigidity theory, dynamics, geometric topology, and geometric group theory. A major drawback is the failure of excision, which renders even basic computations currently out of reach. One of the few cases where non-trivial computations are possible is transformation groups with certain displacement properties that are classically used in homology and stable commutator length. I will introduce a new algebraic criterion that captures this, is satisfied in many interesting settings, and implies vanishing in all degrees and with a large class of coefficients. Based on joint work with Caterina Campagnolo, Yash Lodha, and Marco Moraschini

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1. GeoTop

Bounding the Dehn surgery number by 10/8 — Beibei Liu <bbliumath@gmail.com>

In this talk, we provide new examples of 3-manifolds with weight one fundamental group and the same integral homology as the lens space $L(2k,1)$ which are not surgery on any knot in the three sphere. Our argument uses Furuta's 10/8-theorem, and is simple and combinatorial to apply. This is joint work with Piccirillo.

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1. Continua

Building $\mathbb R$-trees — Curtis Kent <curtkent@mathematics.byu.edu>

We discuss a natural way to build actions of the fundamental group of one-dimensional spaces (which might not have universal covers) on $\mathbb R$-trees. We will then discuss how the tools from the study of one-dimensional spaces can be adapted to more general spaces to build actions of locally free groups on $\mathbb R$-trees with prescribed orbit spaces.

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1. Plenary

Building Connections Between Topological Spaces With Games — Jared Holshouser <jholshouser1321@gmail.com>

We will examine three threads of inquiry in topology: convergence/compactness properties, spaces built out of other spaces (i.e. the space of real-valued continuous functions or the hyperspace of closed sets), and topological games. When a space is built out of another space, we can often translate the topological information from the first space to the second. For instance, open covers of the space can produce clustering sequences of real-valued functions. This topological information can be encoded through strategies in certain topological games. Working with Chris Caruvana and Steven Clontz, we have developed techniques for tying all of these threads together and have proven a wide array of connections between spaces and common constructions on those spaces. The general theory will be discussed and specific examples will be displayed.

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1. GeoGT

CAT(0) geometry of complex curve complements and families — Kejia Zhu <kzhumath@gmail.com>

Motivated by the question of whether braid groups are CAT(0), we investigate the CAT(0) behavior of fundamental groups of plane curve complements and certain universal families. If $C$ is the branch locus of a generic projection of a smooth, complete intersection surface to $\mathbb{P}^2$, we show that $\pi_1(\mathbb{P}^2\setminus C)$ is CAT(0). In the other direction, we prove that the fundamental group of the universal family associated with the singularities of type $E_6$, $E_7$, and $E_8$ is not CAT(0). Other examples, both positive and negative, are discussed, with a special emphasis on rational 3-cuspidal curves. This is joint work with C. Bregman and A. Libgober.

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1. Continua

Cantor fences in plane continua — David Lipham <dlipham@ccga.edu>

David Bellamy constructed a surprising example of a smooth dendroid in the plane with a connected set of endpoints. In this talk, I will present the new result that any planable smooth dendroid with $1$-dimensional endpoint set must contain a Cantor fence (a copy of $2^\omega \times [0,1]$) or a Bellamy dendroid (a smooth dendroid whose endpoint set is connected). This is false outside the plane, and it is unknown whether every Bellamy dendroid contains a Cantor fence. More generally, a continuum is said to be non-Suslinian if it contains an uncountable family of pairwise disjoint, non-degenerate subcontinua. I will discuss some open problems about this property in Julia sets and other plane continua with rich dynamical structures. Among these are: If a plane continuum admits a mixing homeomorphism, then is it non-Suslinian? Is the Sierpiński carpet the only locally connected plane continuum that admits a mixing homeomorphism?

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1. GeoGT

Centers of Artin Groups Defined on Cones — MurphyKate Montee <mmontee@carleton.edu>

The Center Conjecture for Artin groups proposes that the center of any infinite type Artin group is trivial. This is known to hold for a wide class of Artin groups, but is not known in general. In this talk we will prove that the Center Conjecture passes to the Artin groups whose defining graphs are cones, if the conjecture holds for the Artin group defined on the set of the cone points. In particular, it holds for every Artin group whose defining graph has exactly one cone point. This is joint work with Kasia Jankiewicz.

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1. Plenary

Chaotic almost minimal actions — Van Cyr <van.cyr@bucknell.edu>

The joint action of $x\mapsto2x$ (mod 1) and $x\mapsto3x$ (mod 1) has a number of remarkable properties. Among them is that ever joint orbit is either finite or dense. Of course any minimal system has that property, but the x2,x3 system is special because it has a dense set of finite orbits that intermingle with dense orbits. In joint work with B. Kra and S. Schmieding, we abstract this property to what we call a chaotic almost minimal (CAM) system. I this talk I will discuss some properties of CAM systems, showing their similarities to and differences from the x2,x3 system.

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1. GeoGT

Characterizations of Stability via Morse Limit Sets — Jacob Garcia <jgarcia46@smith.edu>

An important example of Kleinian groups are the convex cocompact groups: every infinite order element of these groups is a loxodromic, and these groups are exactly the ones which admit Kleinian manifolds. A well known fact of convex cocompact groups is that they can be characterized exactly as the groups whose limit sets, on the visual boundary, are completely conical, or equivalently, completely horospherical. Convex cocompactness has been studied in the context of many non-hyperbolic spaces, such as mapping class groups, and has recently been generalized to the notion of subgroup stability. By using an analog of the visual boundary called the Morse boundary, a quasi-isometry invariant which "sees" hyperbolic directions for non-hyperbolic spaces, we show that subgroup stability is exactly classified by limit set conditions on the Morse Boundary which are analogous to the limit set conditions from the convex cocompact setting.

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1. Set-Theoretic

Characterizing Strong Infinite-Dimension, Weak Infinite-Dimension, and Dimension in Inverse Systems — Leonard Rubin <lrubin@ou.edu>

We present internal characterizations for an inverse system of compact Hausdorff spaces that show when its limit will be strongly infinite-dimensional, weakly infinite-dimensional, or have its dimension $n\in\mathbb{N}_{\geq0}$. Our main tool involves lifting the notion of an essential family into a parallel concept for inverse systems. In our presentation we plan to review the definitions of essential family, strong and weak infinite-dimensionality, finite dimensionality, and inverse systems. After doing that, we will state our main results but will not go into any proofs. The published paper with all details appears in *Rad Hazu. Matematičke Znanosti*, v. 29=564 (2025): 299-318.

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1. Applied

Circle Bundles For Data — Brad Turow <turow.b@northeastern.edu>

We introduce the notion of a discrete approximate circle bundle, as well as theory and algorithms to estimate characteristic classes. We apply these tools to study a benchmark optical flow dataset, where we confirm the toroidal model proposed by Adams et al. and discover larger spaces in other density regimes.

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1. Dynamics

Classification complexity of chaotic systems — Benjamin Vejnar <benvej@gmail.com>

The aim of this talk is first to briefly describe a natural way of measuring simplicity/complexity of classification problems by using Invariant Descriptive Set Theory and then to discuss recent applications in the context of topological dynamics. We mainly deal with the classification of transitive systems on the interval, on the Cantor set and on the Hilbert cube with respect to the topological conjugacy relation. At the end, we provide some attempts to identify the complexity of classification of minimal systems.

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1. GeoTop

Classifying holomorphic maps between spaces of polynomials — Peter Huxford <pjhuxford@uchicago.edu>

Let $\mathrm{Poly}_n\mathbb{C}$ be the space of monic, squarefree, degree $n$ polynomials in one variable over $\mathbb{C}$. Ferrari's solution to the quartic equation gives rise to a holomorphic map $R\colon\mathrm{Poly}_4\mathbb{C}\to\mathrm{Poly}_3\mathbb{C}$. We show that every holomorphic map $\mathrm{Poly}_n\mathbb{C}\to\mathrm{Poly}_m\mathbb{C}$ for $m\leq n$ is equivalent in a certain sense to a constant map, the identity map, or Ferrari's map $R$. This is joint work with Jeroen Schillewaert.

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1. Set-Theoretic

Closed copies of $\mathbb{N}$ in $\mathbb{R}^{\omega_1}$ — KP Hart <k.p.hart@tudelft.nl>

We investigate the existence of closed copies of the discrete space $\mathbb{N}$ of natural numbers in powers of the real line, in particular its $\omega_1$-power, that are not $C^\star$-embedded, or that are $C^\star$-embedded but not $C$-embedded. In the case of non-$C^\star$-embedding we find a whole family of new examples, based on Aronszajn trees and lines, and a combinatorial translation of the existence of such copies. In the case of $C^\star$- but not $C$-embedding we complement an earlier consistency result but showing in consistent with any desired cardinal arithmetic that $\mathbb{R}^{\omega_1}$ contains a closed copy of $\mathbb{N}$ that is $C^\star$- but not $C$-embedded.

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1. Dynamics

Completely invariant sets and Lorenz maps — Piotr Oprocha <piotr.oprocha@osu.cz>

In this talk we will discuss relations between completely invariant sets and renormalizations of expanding Lorenz maps, that is maps $f\colon [0,1]\to [0,1]$ satisfying the following three conditions: 1. there is a critical point $c\in (0,1)$ such that $f$ is continuous and strictly increasing on $[0,c)$ and $(c,1]$; 2. $\lim_{x\to c^{-}}f(x)=1$ and $\lim_{x\to c^{+}}f(x)=0$; 3. $f$ is differentiable for all points not belonging to a finite set $F\subseteq [0,1]$ and $\inf_{x\not\in F} f'(x)>1$; with special emphasis on piecewise linear case. The talk is based on joint works with L. Cholewa.

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1. Applied

Computing the Bottleneck Distance from Every Direction — Elena Wang <wangx249@msu.edu>

One of the most common distances used to compare two persistence diagrams is the bottleneck distance. When the persistence diagrams of a shape in $\mathbb{R}^d$ are computed from every direction in $\mathbb{S}^{d-1}$, we obtain the persistent homology transform (PHT). An efficient way of comparing two PHTs remains unexplored. In this work, we develop a new kinetic data structure to compute the bottleneck distance between two PHTs obtained from shapes in $\mathbb{R}^2$ from every direction. We provide the events and necessary updates to maintain the distance between the diagrams using this structure. Our resulting algorithm runs in $O(n^2\log^2n)$. This is compared to the naive algorithm where $d_B$ is computed at a finite number of smartly chosen directions, which is $O(n^{7/2}\log n)$ complex. It is important to note that our algorithm provides an exact distance in every direction, while the latter is an approximation. Furthermore, we show that this data structure is not limited to the directional transform setting since the techniques apply to more general vineyard structures.

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1. Applied

Congruence of Planar Curves and the Signature Quiver — Irina Kogan <iakogan@ncsu.edu>

Deciding whether or not two curves are congruent under rotations and translations is a classical, but surprisingly subtle problem. In addition to its theoretical interest, this problem has numerous applications in computer vision and image processing, automated assembly,  signal processing, and more. To address this, as well as more general congruence problems, the signature curve parameterized by differential invariants was introduced by Calabi, Olver, Shakiban, Tannenbaum, and Haker (1998). While congruent curves have identical signatures, the converse is not true, as shown in Muso and Nicolodi (2009).  In a joint work with Eric Geiger (2021), we presented a mechanism for constructing non-congruent, non-degenerate curves with identical signatures. We also introduced a notion of the signature quiver and used it to formulate a congruence criterion for non-degenerate curves with non-simple signatures.

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1. GeoGT

Connected components in Morse boundaries of right-angled Coxeter groups — Annette Karrer <annette.u.karrer@gmail.com>

Every finitely generated group G has an associated topological space, called a Morse boundary, that captures the hyperbolic-like behavior of G at infinity. It was introduced by Cordes generalizing the contracting boundary invented by Charney--Sultan. In this talk, we study subgroups arising from connected components in Morse boundaries of right-angled Coxeter groups and of such that are quasi-isometric to right-angled Coxeter groups. This talk is based on two projects. One is joint work with Bobby Miraftab and Stefanie Zbinden. The other one is joint work in progress with Matthew Cordes and Kim Ruane.

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1. GeoTop

Connectivity in the space of pointed hyperbolic 3-manifolds — Matthew Zevenbergen <zevenber@bc.edu>

I will show that the space of pointed infinite volume hyperbolic 3-manifolds is connected but not path connected. This space is equipped with the geometric topology, in which two pointed manifolds are close if they are almost isometric on large neighborhoods of their basepoints. The proof of connectivity will be an application of the density theorem for Kleinian groups. I will then use a combination of results on representations of Kleinian groups and Chabauty spaces of subgroups to construct an infinite family of path components of this space.

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1. Continua

Coselectibility regarding symmetric products — Patricia Pellicer-Covarrubias <paty@ciencias.unam.mx>

In this talk we consider a concept which is the dual to the concept of a selectible space, namely, a $\Lambda$-coselection space ($\Lambda$ may be any given hyperspace of a space $X$). We consider this concept when $\Lambda$ is the $n$th symmetric product $F_n(X)$. We present sufficient conditions for a continuum to be either an $F_2(X)$-coselection space or an $F_3(X)$-coselection space.

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1. Dynamics

Cubic polynomials and laminations — Nikita Selinger <nikita.selinger@gmail.com>

I will review the notion of laminations as introduced by W. Thurston and explain how laminations can be used to study parameter spaces of polynomials.

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1. Plenary

Data Driven Homological Approaches for Detecting Changes in Dynamical System — Liz Munch <muncheli@msu.edu>

Persistent homology, the flagship method from the field of Topological Data Analysis, is a powerful tool for measuring shape and structure of data. In this talk, we explore methods for using this tool to detect homological changes in the underlying structure of dynamical systems.  As a first step, we can simplify a vineyard of persistence diagrams into a CROCKER plot to provide visual representations of qualitative shifts in the structure of examples such as the Lorenz and Rossler systems. We can also construct a "homological bifurcation plot" to enable the identification of qualitative shifts, namely P-type (phenomenological) bifurcations, within stochastic dynamical systems, defined by structural changes in the probability density functions (PDF) of the state variables. The talk will explore the successful application of this method to stochastic oscillators, showcasing its effectiveness in algorithmically detecting P-bifurcations. This talk is based on joint work with many collaborators, including Firas Khasawneh, İsmail Güzel, Sunia Tanweer, Sarah Tymochko, Audun Myers, and David Muñoz.

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1. GeoTop

Dehn filling in semisimple Lie groups — Theodore Weisman <tjwei@umich.edu>

Thurston's Hyperbolic Dehn Filling Theorem is a seminal result in the theory of 3-manifolds. Given a single noncompact finite-volume hyperbolic 3-manifold M, the theorem provides a construction for a countably infinite family of closed hyperbolic 3-manifolds converging to M in a geometric sense. The theorem is a major source of examples of 3-manifolds admitting hyperbolic structures, and closely connects the topology of a 3-manifold to the analysis of the character variety of its fundamental group in PSL(2, C). In this talk, we discuss some analogs and generalizations of Thurston's theorem in the context of general (arbitrary-rank) semisimple Lie groups. We will explain how our results provide a way to construct new examples of Anosov and relatively Anosov representations into higher-rank Lie groups; time permitting, we will also discuss upcoming joint work with Jeff Danciger, which applies our results to construct exotic new examples of convex cocompact and geometrically finite groups acting on complex hyperbolic 3-space.

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1. GeoTop

Dehn twist and smooth mapping class group of 4-manifolds   — Anubhav Mukherjee <anubhavmaths@princeton.edu>

In this talk, I will present recent advancements in the study of smooth mapping class groups of 4-manifolds. Our work focuses on diffeomorphisms arising from Dehn twists along embedded 3-manifolds and their interaction with Seiberg-Witten theory. These investigations have led to intriguing applications across several areas, including symplectic geometry (related to Torelli symplectomorphisms), algebraic geometry (concerning the monodromy of singularities), and low-dimensional topology (involving exotic diffeomorphisms). This is collaborative work with Hokuto Konno, Jianfeng Lin, and Juan Munoz-Echaniz.

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1. Plenary

Diffeomorphisms of 3-manifolds with boundary — Corey Bregman <corey.bregman@tufts.edu>

Let M be a compact, connected, orientable 3-manifold with non-empty boundary. In this talk, we study the classifying space for the diffeomorphism group of M fixing the boundary pointwise, and show that it has the homotopy type of a finite CW complex. This parallels analogous results of Gramain and Earle-Schatz for surfaces, and confirms a conjecture of Kontsevich for orientable 3-manifolds. The proof will take us on a crash course in 3-manifold topology, and will feature a combination of results on geometrization of 3-manifolds with a topological poset parametrizing embedded spheres in M. This is joint work with Rachael Boyd and Jan Steinebrunner.

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1. Dynamics

Dimension of Lyapunov spectrum for non uniformly hyperbolic settings — Emma Dinowitz <emmad4867@gmail.com>

We study the Hausdorff dimension of the set of points with a fixed lyapunov exponent inside a family of subsets of a 3 dimensional flow with non uniform hyperbolicity properties. Recent work of Sarig, Lima, and others have constructed countable state markov partitions modeling these sets. Using their framework we prove upper bounds analogous to the uniformly hyperbolic situation.

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1. Set-Theoretic

Discrete density number — Alan Dow <adow@charlotte.edu>

A subset $D$ is a discretely dense subset of a space $X$ if every point of $X$ is in the closure of a discrete subset of $D$. The cardinal invariant, $Dd(X)$, was introduced by Juhasz and is the minimum cardinality of a discretely dense subset of $X$. We are reporting on some recent work with Juhasz and van Mill on results that improve upon the, seemingly only, obvious inequalities $d(X)\leq Dd(X)\leq |X|$. We also consider, $Fd(X)$, the free sequence density number.

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1. GeoGT

Distinguishing filling curve types via special metrics — Sayantika Mondal <smondal@gradcenter.cuny.edu>

In this talk, we look at filling curves on hyperbolic surfaces and consider its length infima in the moduli space of the surface as a type invariant. In particular, explore the relations between the length infimum of curves and their self-intersection number. For any given surface, we will construct infinite families of filling curves that cannot be distinguished by self-intersection number but via length infimum. I might also discuss some coarse bounds on the special metrics associated with these infimum lengths.

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1. Plenary

Drilling and Filling in (relatively) hyperbolic groups — Jason Manning <jfmanning@cornell.edu>

Dehn surgery is a classical operation in which one converts one three-manifold to another by first removing a solid torus, and then gluing it back in in a different way. The first operation is called "drilling" and the second "filling". Both of these operations have group-theoretic interpretations in the world of hyperbolic and relatively hyperbolic groups. I will explain those interpretations and applications related to the Cannon conjecture (a special case of Wall's conjecture about $PD(n)$ groups). The most recent work is joint with Groves, Haïssinsky, Osajda, Sisto, and Walsh.

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1. Continua

Dynamics on Hereditarily Decomposable Tree-like Continuum — Christopher Mouron <mouronc@rhodes.edu>

In this talk give an example of a hereditarily decomposable tree-like continuum that admits homeomorphisms that have the following dynamic properties: mixing, the specification property, and continuum-wise turbulence. I will also give results about topological properties (or lack of properties) that prevent hereditarily decomposable tree-like continuum from admitting homeomorphisms with some of the previous properties.

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1. Applied

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. GeoTop

Equivariant Smoothings and the Whitehead Group — Oliver Wang <wang.oliver96@gmail.com>

A closed manifold $M$ of dimension at least $5$ has only finitely many smooth structures. Moreover, the product structure theorem states that the smooth structures on such an $M$ are in bijection with smooth structures on the product $M\times\mathbb{R}$. In this talk, I will describe a construction that gives rise to infinitely many equivariant smooth structures of a closed $G$-manifold $M$ which become isotopic after taking a product with $\mathbb{R}$.

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1. Dynamics

Ergodic Averages along Sequences of Slow Growth — Kaitlyn Loyd <loydka@umd.edu>

Given Birkhoff's pointwise ergodic theorem, it is natural to consider whether convergence still holds along subsequences of the integers. In this talk, we investigate convergence of ergodic averages along the number theoretic sequence $\Omega(n)$, where $\Omega(n)$ denotes the number of prime factors of $n$ counted with multiplicities. In particular, we demonstrate that, although a pointwise ergodic theorem does not hold along $\Omega(n)$, there are multiple instances in which we can recover convergence. We also present a more general criterion for identifying slow-growing sequences possessing a certain divergence property exhibited by $\Omega(n)$. This talk is based on joint work with Sovanlal Mondal (Ohio State).

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1. Dynamics

Ergodic optimization with linear constraints — Kevin McGoff <kmcgoff1@charlotte.edu>

Let $T : X \to X$ be a continuous map on a compact metrizable space, let $f : X \to \mathbb{R}$ be continuous, and let $W \subset C(X)$ be a closed subspace of continuous functions from $X$ to $\mathbb{R}$. We consider the set $M_W(X,T)$ of all $T$-invariant Borel probability measures $\mu$ such that $\int g \, d\mu = 0$ for all $g$ in $W$. Then we consider optimization problems of the form $$ \max \int f \, d\mu + \tau h(\mu),$$ where $\mu$ ranges over $M_W(X,T)$, $h(\mu)$ denotes the entropy of $\mu$ with respect to $T$, and $\tau$ is either $0$ or $1$. Our main results concern the basic properties of such optimization problems, including feasibility, geometry of the solution set, uniqueness of solutions, and realizability. This talk is based on ongoing joint work with Shengwen Guo (UNC Charlotte).

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1. GeoTop

Exotic traces and the shake genus — Kai Nakamura <kainaka@stanford.edu>

The shake genus is the main tool used to detect exotic traces. This is a powerful tool to construct exotic traces, however it has some limitations. We will discuss several desirable properties of exotic traces that are inaccessible using the shake genus. By moving past needing to use the shake genus, we will be able to construct novel examples of exotic traces.

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1. Continua

Fan homogeneity — Rene Gril Rogina <rene.gril1@student.um.si>

We present recent results regarding different types of homogeneity for fans and discuss ongoing research into the topic. We define a larger class of fans with a specific property and use it to prove our results. This is joint work will Will Brian of UNC Charlotte.

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1. Dynamics

Flow equivalence and PSL_2(Q)-equivalence — Scott Schmieding <sks7247@psu.edu>

A real number gives rise to a Sturmian system encoding a rotation of the circle, and there are several beautiful connections between these systems and arithmetic properties of the associated parameters. One is a result of Fokkink, which shows that two Sturmian subshifts with parameters \alpha and \beta are flow equivalent if and only if \alpha and \beta lie in the same orbit of the action of PSL_2(Z) on the set of reals via Mobius transformations, a condition which is itself characterized by the tails of their continued fraction expansions. I'll describe some recent work, joint Christopher-Lloyd Simon, describing the action of PSL_2(Q) in terms of a certain relation on systems called isogeny.

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1. GeoGT

Frattini subgroups of hyperbolic-like groups — Ekaterina Rybak <ekaterina.rybak@vanderbilt.edu>

The Frattini subgroup $\Phi(G)$ of a group $G$ is the intersection of all maximal subgroups of $G$; if $G$ has no maximal subgroups, $\Phi(G)=G$ by definition. Frattini subgroups of groups with ``hyperbolic-like" geometry are often small in a suitable sense. Generalizing several known results, we prove that for any countable group $G$ admitting a general type action on a hyperbolic space $S$, the induced action of the Frattini subgroup $\Phi(G)$ on $S$ has bounded orbits, in particular, $\Phi(G)$ has infinite index in $G$. In contrast, we show that the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. The talk is based on a joint work with Gil Goffer and Denis Osin.

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1. Dynamics

Full Groups of Cantor Dynamical Systems: Characters and Invariant Measures — Constantine Medynets <medynets@usna.edu>

Given a Cantor minimal dynamical system $(X, T)$, the topological full group $[[T]]$ consists of all homeomorphisms of $X$ that locally act as powers of $T$. These groups can be viewed as generalized symmetric groups on the continuous orbit equivalence relation of $(X,T)$. A series of works by Giordano–Putnam–Skau, Matui, Medynets, Nekrashevych, and others have demonstrated that the algebraic structure of topological full groups completely determines the orbit structure of the underlying systems. This naturally leads to the question of whether the structure of invariant measures, an invariant of orbit equivalence, is similarly reflected in the full group's algebraic properties. In this talk, we present joint work with Artem Dudko (IMPAN) on the classification of characters of topological full groups of Cantor minimal systems. We establish that every extreme character of the commutator subgroup of $[[T]]$ is of the form $\mu(Fix(g))$, where $\mu$ is an ergodic product measure on $X^n$, thereby confirming Vershik’s conjecture for the class of full groups. As a consequence, we show that prime indecomposable characters are in one-to-one correspondence with ergodic measures.

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1. Continua

Generalization of the specification property to CR-dynamical systems — Ivan Jelić <ivajel@pmfst.hr>

We will recall the definition and basic properties of the notion of the specification property in the case of a standard topological dynamical system (X,f). We will then define a CR-dynamical system (X,F) and introduce different generalizations of the specification property for this type of dynamical system. More precisely, we will introduce and investigate the notions of (strong/weak) specification property and compare them together with their "initial" versions.

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1. Continua

Generalizations of the notion of a hereditarily equivalent continuum — Bryant Rosado Silva <bryantrs99@hotmail.com>

We say that a continuum $X$ is a hereditarily equivalent continuum (HEC) if every non-degenerate subcontinuum of it is homeomorphic to $X$. We can weaken this condition in three different levels: If considered in the hyperspace of continua of $X$, denoted by $\operatorname{Cont}(X)$, being hereditarily equivalent means that $\operatorname{Cont}(X)\setminus \{\{x\} \ | \ x \in X\} = \{ K \in \operatorname{Cont}(X) \ | \ K \simeq X\}.$ This is an open and dense set, hence comeager, thus the first way to weaken it is to ask for the set of homeomorphic copies of $X$ to be a comeager subset of $\operatorname{Cont}(X)$. A continuum with this property we call a generically hereditarily equivalent continuum (GHEC). However, we can go further and consider the hyperspace of maximal order arcs $\operatorname{MOA}(X)$. In the case of an HEC, any maximal order arc is made of an initial unitary set called the root and homeomorphic copies of $X$, hence we can say that - GCHEC holds for a space $X$ if comeager many elements of $\operatorname{MOA}(X)$ have this property of being a chain made of copies of $X$ apart from the root. - GCGHEC holds for $X$ if comeager many elements of $\operatorname{MOA}(X)$ contain comeager many copies of $X$. In this talk, we partially address two natural questions that arise from these definitions: "What kind of spaces satisfy these properties?" and "How are these properties related?"

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1. Continua

Generalized inverse limits with Markov set-valued functions on finite graphs — Hayato Imamura <hayato-imamura@asagi.waseda.jp>

In this talk, we introduce definitions of Markov set-valued functions on finite graphs and the same pattern between two Markov set-valued functions. These functions are defined using the framework of cell complexes. They allow for infinite Markov partitions and have graphs that may contain $2$-cells. We also show that two generalized inverse limits with bonding functions that are Markov set-valued functions following the same pattern are homeomorphic. This is joint work with E. Matsuhashi and Y. Oshima.

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1. Plenary

Geometry of Rips complexes and applications — Florian Frick <frick@cmu.edu>

In geometric group theory, Rips complexes provide a natural construction to give higher structure to a Cayley graph. In topological data analysis, Rips complexes are used to reconstruct a sufficiently nice space from a sample. I will show different but related applications of Rips complexes and similar constructions to Borsuk-Ulam results, understanding Gromov-Hausdorff distances, and roots of zero-mean real-valued maps.

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1. GeoGT

Green metrics on hyperbolic groups and reparameterizations of the geodesic flow — Eduardo Reyes <eduardo.c.reyes@yale.edu>

Teichmüller space is a classical construction that, for a given closed hyperbolic surface, parameterizes the geometric actions of its fundamental group on the hyperbolic plane. I will talk about a generalization of this space, where for an arbitrary hyperbolic group we consider a space parameterizing its geometric actions on Gromov hyperbolic spaces, simultaneously encoding negatively curved Riemannian metrics, Anosov representations, random walks, geometric cubulations, etc. In particular, I will discuss how Green metrics (those encoding admissible random walks on the group) are dense in this space. As an application, for fundamental groups of negatively curved manifolds we produce a dictionary between this space of geometric actions and the space of reparameterizations of the geodesic flow. This is joint work with Stephen Cantrell and Dídac Martínez-Granado.

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1. Applied

Gromov-Hausdorff distance between metric graphs and their subspaces — Nicolò Zava <nicolo.zava@ist.ac.at>

The Gromov-Hausdorff distance, a dissimilarity measure between metric spaces, is used in computational topology and geometry to compare datasets that can be represented as metric spaces. Despite the computational obstructions to its practical use, it still provides a theoretical framework to quantify invariants' stability and information loss. In this talk, we focus on a particular problem regarding the Gromov-Hausdorff distance: Given an object and a sample of it, under what conditions do their Hausdorff and Gromov-Hausdorff distances coincide? As the Gromov-Hausdorff distance describes how far they are from being isometric, and the Hausdorff distance measures the density of the sample, we can less formally restate the question as follows: When is a sample dense enough to describe the original object’s geometry faithfully? In particular, we discuss the case of metric graphs providing both negative and positive results.

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1. Applied

Group Actions on Metric Spaces — Liam Barham <blb0081@auburn.edu>

Given a metric space $X$, the Vietoris-Rips complex VR$(X)$ is a classical simplicial complex obtained from $X$, and a group $G$ acting properly by isometries yields another metric space $X/G$ of the orbits of $X$ under $G$. There is a canonical way in which $G$ can act on VR$(X)$, so instead using the Vietoris-Rips metric thickening VR$^m(X)$ allows a meaningful comparison between VR$^m(X)/G$ and VR$^m(X/G)$ as metric spaces. This talk will survey a variety of properties which a group action on a metric space can have with some examples, and culminate with a discussion of the strong $r$-diameter action, which guarantees that under certain scale parameters VR$^m(X)/G\simeq$ VR$^m(X/G)$. I also discuss a strictly weaker condition and present some open questions concerning the connection between the two. Finally, I will briefly mention some analogous results for the Cech metric thickening.

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1. Continua

Hereditarily Decomposable Continua have Non-Block Points — Daron Anderson <daronanderson@live.ie>

We expand upon our earlier results, to show that every nondegenerate hereditarily decomposable Hausdorff continuum has two or more non-block points, i.e points whose complements contain a continuum-connected dense subset. The celebrated non-cut point existence theorem states that all nondegenerate Hausdorff continua have two or more non-cut points, and the corresponding result for non-block points is known to hold for metrizable continua. It is also known that there are consistent examples of Hausdorff continua with no non-block points, but that non-block point existence holds for Hausdorff continua that are either aposyndetic, irreducible, or separable.

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1. Dynamics

Homomorphisms from aperiodic subshifts to subshifts with the finite extension property — Robert Bland <rbland5@charlotte.edu>

We are inspired by recent efforts to generalize the classical embedding theorem of Krieger for $\mathbb{Z}$ subshifts, which states that if $X$ is an SFT and $Y$ is a mixing SFT, then $X$ embeds into $Y$ if certain necessary conditions on the periodic points and entropy are satisfied. Moving to subshifts over groups $G$ beyond $\mathbb{Z}$, an extra essential hypothesis emerges: that there is a homomorphism (a continuous and shift-commuting map, not necessarily injective) from $X$ to $Y$ at all. This is trivially satisfied if, e.g., $Y$ contains a fixed point, but necessary and sufficient conditions for the existence of a homomorphism are not known in general. In this talk, we present joint work with K. McGoff that constructs a homomorphism $\phi : X \to Y$ in the case that $X$ is aperiodic, $Y$ has the finite extension property, and the underlying group $G$ has the property that every finitely generated subgroup of $G$ has polynomial growth (i.e., $G$ is locally virtually nilpotent by Gromov's theorem). The finite extension property (FEP) can be seen as a very strong mixing-like condition which has been considered before for subshifts over $\mathbb{Z}^d$ [Briceño, McGoff, Pavlov 2016].

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1. Plenary

Hyperbolic actions of Thompson's group $F$ — Sahana H Balasubramanya <hassanba@lafayette.edu>

In this talk, I will present recent results about the poset of hyperbolic structures on Thompson's group $F$. While the global structure of this poset is as simple as one would expect, the local structure turns out to be incredibly rich, in stark contrast with the situation for the $T$ and $V$ counterparts. I will focus on the subposet of quasi-parabolic hyperbolic structures, which contains uncountably many \emph{lamplike} structures, called so as they can be described combinatorially in terms of certain hyperbolic structures on related lamplighter groups. On the other hand, there are also many non-lamplike structures, showing the vastness and complexity of this poset. Lastly, I will talk about how these actions can be extended to more general Thompson's groups $F_n$ for $n \geq 2$. This is joint work with Francesco Fournier-Facio and Matthew C.B.Zaremsky.

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1. Applied

Independence Complexes of Kneser Graphs — Ziqin Feng <zzf0006@auburn.edu>

We will discuss the topological properties of the independence complex of Kneser graphs, Ind(KG$(n, k))$, with $n\geq 3$ and $k\geq 1$. By identifying one kind of maximal simplices through projective planes, we obtain homology generators for the $6$-dimensional homology of the complex Ind(KG$(3, k))$. Using cross-polytopal generators, we provide lower bounds for the rank of $p$-dimensional homology of the complex Ind(KG$(n, k))$ where $p=1/2\cdot {2n+k\choose 2n}$.

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1. GeoGT

Independence of Dehn, conjugator length, and annular Dehn functions of finitely presented groups — Conan Gillis <cg527@cornell.edu>

Brick and Corson introduced annular Dehn functions in 1998 to quantify the conjugacy problem for finitely generated groups and gave the fundamental relationships between it, the Dehn function, and the conjugator length function. I will discuss the key ideas behind these invariants, as well as joint work with T. Riley where we prove that these three invariants are independent—in general, no two of the three functions determine the other.

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1. Continua

Knaster continua in the plane — Ana Anusic <ana.anusic@fer.unizg.hr>

We show that for every Knaster continuum X, and every countable set C of composants of X, there exists a planar embedding of X in which the whole set C is accessible. I will also show that some of these embeddings can be done in dynamically significant way by using a generalization of Barge-Martin construction. This is a joint work with Logan Hoehn.

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1. GeoGT

Latent symmetry of graphs and stretch factors in Out(Fn) — Paige Hillen <paigehillen@ucsb.edu>

Given an irreducible element of Out($F_n$), there is a graph and an irreducible "train track map" on this graph, which induces the outer automorphism on the fundamental group. The stretch factor of an outer automorphism measures the asymptotic growth rate of words in $F_n$ under applications of the automorphism, and appears as the leading eigenvalue of the transition matrix of such a train track representative. I'll present work showing a lower bound for the stretch factor in terms of the number of edges in the graph and the number of folds in the fold decomposition of the train track map. Moreover, in certain cases, a notion of the latent symmetry of a graph G gives a lower bound on the number of folds required for any irreducible train track map on G. I'll use this to classify all single fold train track maps.

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1. GeoTop

Lefschetz fibrations with infinitely many sections — Seraphina Eun Bi Lee <seraphinalee@uchicago.edu>

A Lefschetz fibration $M^4 \to S^2$ is a generalization of a surface bundle which also allows finitely many nodal singular fibers. The Arakelov--Parshin rigidity theorem implies that holomorphic Lefschetz fibrations of genus $g \geq 2$ admit only finitely many holomorphic sections. In this talk, we will show that no such finiteness result holds for smooth or symplectic sections by giving examples of genus-$g$ ($g \geq 2$) Lefschetz fibrations with infinitely many homologically distinct sections. This is joint work with Carlos A. Serván.

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1. GeoGT

Left-invariant Riemannian distances on higher-rank Sol-type groups. — Daniel Levitin <dlevitin@wisc.edu>

Describing the coarse geometry of solvable groups is one of the major projects of geometric group theory. One solvable group whose geometry is well-understood is Sol, a rank-1 group foliated by two families of hyperbolic planes. More generally, Le Donne, Pallier, and Xie recently described the geodesics in Sol-type groups, which are the rank-1 solvable groups foliated by a pair of negatively-curved spaces. Leveraging this description, they show that all left-invariant Riemannian distances on a Sol-type group are roughly similar. In this talk, I will describe the coarse geometry of the broader class of higher-rank Sol-type groups, and discuss my generalization of Le Donne-Pallier-Xie's result to certain distances on these groups.

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1. Applied

Letter Insertion Homology and the Complexity of Word Sets: A Topological Approach to DNA Mutation and Repair — Francisco Martinez Figueroa <fmartinezfigueroa@usf.edu>

When studying mechanisms of DNA repair, short mutations often arise at the repair site, frequently manifesting as the insertion of short nucleotide sequences from the alphabet {A, C, G, T}. Each of these insertions occurs across millions of DNA molecules, generating a set of short words with varying frequencies. Our goal is to identify a suitable mathematical object to analyze these word sets and distinguish patterns across different experimental conditions. In this talk, we introduce the Insertion Chain Complex, a higher-dimensional generalization of insertion graphs, where homology serves as a measure of the complexity of a set of words. We present its construction, fundamental properties, and applications to biological data. In our case study, we analyze data from human cells in which DNA breaks were induced and the repaired sequences were sequenced. Our findings demonstrate that counting the highest-dimensional cells in these insertion complexes effectively distinguishes between different break locations.

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1. Plenary

Mahavier products and Mahavier dynamical systems — Iztok Banic <iztok.banic@um.si>

During the pandemic, Judy Kennedy and I, later joined by Goran Erceg, began investigating the dynamics of closed relations in dynamical systems, which we termed CR-dynamical systems. With travel restrictions in place, we established regular online meetings to collaborate on this research. Our initial focus was on fixed-point problems from the perspective of closed relations, which led us to explore broader dynamical properties and ultimately to introduce Mahavier dynamical systems. Despite being spread across different time zones--Judy in the US, Goran in Croatia, and I in Slovenia--we managed to coordinate meetings in the early evening for Goran and me, and at 1:30 PM for Judy. Since then, we have published numerous papers on Mahavier dynamical systems, with several more in progress. Once travel resumed, Goran and I visited Judy in the US twice, while Judy visited Slovenia and Croatia on many more ocations, allowing us to collaborate in person. While online meetings and screen-sharing have been invaluable, we recognize that nothing fully replaces in-person discussions. Our research has continued to gain momentum, and we are committed to furthering this long-term project. We believe our work is both fundamental and significant, and we remain excited about its potential. Along the way, we named our group the Topology Nerds, and later, Van Nall, Sina Greenwood, Rene Gril Rogina, Chris Mouron and Ivan Jeli\' c joined our efforts. In this talk, I will present an overview of the most important results achieved by the Topology Nerds group.

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1. Applied

Mapp(er)ing brain states using EEG data — Brittany Story <brittany.m.story.civ@army.mil>

There is a lot to be gained by using topological data analysis (TDA) in conjunction with domain knowledge. As an example, consider the task where one wants to cluster brain states based on the underlying neural activity. Electroencephalograms (EEGs) are a common tool used to investigate neural activity by detecting electrical signals through sensors affixed to the scalp. EEG is relatively easy to use and provides high temporal resolution. However, it has low spatial resolution and prone to contamination with artifacts of movement or signals from external sources. Thus, for tasks like clustering brain states, it is difficult to capture the underlying structure and connectivity of individual states from EEG data. TDA, specifically the Mapper algorithm, has been used successfully in these types of problem spaces to pull important and relevant information from datasets. But, when applied directly to EEG data, Mapper does not reveal any structure or information. Luckily, there is a plethora of research and tools that have been developed to process and examine EEG data. Specifically, researchers have found that looking at the signal in the frequency domain can often provide insight into the neural activity. As such, we use the power spectral density paired with Mapper to create MapperEEG (MEEG). MEEG is neuroscience-infused topological tool that can cluster brain states without any pre-labeling or prior knowledge. In this talk, we will illustrate the importance of using prior domain knowledge within the EEG context, introduce the MEEG algorithm as an example of combining domain knowledge and TDA, and demonstrate its use on clustering brain state during a teaming task.

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1. Dynamics

Maximal Pattern Complexity for General Alphabets — Casey Schlortt <casey.schlortt@du.edu>

Maximal pattern complexity was introduced by Teturo Kamae and Luca Zamboni in 2002 as a way to link word complexity and sequence entropy. In this same paper, they introduced the idea of a pattern Sturmian over two letters, an aperiodic sequence with the lowest possible maximal pattern complexity on a two letter alphabet. In this talk, we will introduce some established results about sequences with low maximal pattern complexity and some new results extending the understanding of sequences of low maximal pattern complexity on larger alphabets.

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1. Dynamics

Measures of maximal entropy on coded shift spaces — Christian Wolf <cwolf@ccny.cuny.edu>

In this talk, we present results about the uniqueness of measures of maximal entropy on coded shift spaces. A coded shift space is defined as the closure of all bi-infinite concatenations of words from a fixed countable generating set. We derive sufficient conditions for the uniqueness of measures of maximal entropy and equilibrium states of Hoelder continuous potentials based on the partition of the coded shift into its concatenation set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). We also discuss flexibility results for the entropy on the concatenation and residual sets. Finally, we present a local structure theorem for intrinsically ergodic coded shift spaces. This shows that our results apply to a larger class of coded shift spaces compared to previous works by Climenhaga, Climenhaga and Thompson, and Pavlov. The results presented in this talk are joint work with Tamara Kucherenko and Martin Schmoll.

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1. Applied

Metric thickenings of Vietoris-Rips complexes — Alexandre Karassev <alexandk@nipissingu.ca>

Vietoris-Rips complexes play an important role in geometric topology, geometric group theory, and topological data analysis. For a given scale parameter $r>0$ and a metric space $X$, a Vietoris-Rips complex, $\mathrm{VR(}X,r)$, is defined as a simplicial complex with the vertex set $X$, and so that the simplices are finite collections of points from $X$ of diameter $< r$. One of the main difficulties in working with Vietoris-Rips complexes is that $\mathrm{VR}(X,r)$ is not metrizable unless $X$ is discrete. Moreover, the space $X$, in general, cannot be viewed as naturally embedded in $\mathrm{VR} (X,r)$. To remedy these problems, one can consider so-called metric thickening $\mathrm{VR}^m(X,r)$ of $\mathrm{VR}(X,r).$ To this end, we can view $\mathrm{VR}(X,r)$ as a set of all finitely supported measures with diameter of support $ < r$, and endow it with the Wasserstein metric. The main focus of this talk will be on the relation between the homotopy types of $\mathrm{VR} (X,r)$ and $\mathrm{VR}^m(X,r).$ A recent result by Gillespie implies that $\mathrm{VR}(x,r)$ and $\mathrm{VR}^m(X,r)$ are weekly homotopy equivalent. Therefore, to conclude that they are homotopy equivalent it is sufficient to show that $\mathrm{VR}^m(X,r)$ is an ANR. It has been previously demonstrated by Adams, Frick, and Virk that $\mathrm{VR}^m(X,r)$ is locally contractible. Using different method, we prove that $\mathrm{VR}^m(X,r)$ is strongly locally contractible for a compact metric space $X.$ We also show that if such X is finite-dimensional then $\mathrm{VR}^m(X,r)$ is an ANR. (Note: this is a joint work with Henry Adams and Ziga Virk).

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1. Continua

More on the hyperspace of non-cut subcontinua of a continuum — Jorge E. Vega <vegacevedofcfm@fcfm.buap.mx>

We give conditions under which the Vietoris hyperspace of non-cut subcontinua is the same as the hyperspace of all subcontinua. Also, we give in the class of finite graph conditions under which the hyperspace of non-cut subscontinua is connected. This is joint work with A. Illanes and V. Martínez-de-la-Vega.

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1. Applied

Multiparameter landscapes for latent representations — Evgeniya Lagoda <evgeniya.lagoda@gmail.com>

Recent work by Wayland, Coupette, and Rieck (2024) proposes a method to characterize and compare the latent embedding spaces arising from machine learning models. Their method is based on persistent homology and allows variability and sensitivity analysis of various hyperparameter choices for these models. Inspired by this idea but focusing on the case of classification problems, we would like to develop tools for a similar analysis. In this talk, we define a variant of multiparameter persistence landscapes, which can be seen as a generalization of the definition in the recent work by Vipond (2020). For practical applications, we are interested in the landscapes that are defined over a poset that is a product of $\mathbb R$ and a subposet of an inclusion poset. We discuss the properties of this definition, the theoretical challenges, and future directions.

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1. GeoTop

Multiplane diagrams of surfaces in 4-space — Roman Aranda <jarandacuevas2@unl.edu>

Surfaces in 4-space can be described using tuples of b-string tangles called multiplane diagrams. In this talk, we will discuss local modifications for multiplane diagrams that affect the embedded surface in a controlled way. This talk will explore such operations in the context of bridge multisections. We show a uniqueness result for multiplane diagrams representing isotopic surfaces. If time permits, we will show that any n-valent graph with an n-edge coloring is the spine of a bridge multisection of an unknotted surface.

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1. Dynamics

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

We define neighborhood $N$-shadowing property and discuss the relationship of this property to mixing sofic shifts. Specifically, we show all mixing sofic shifts over a finite alphabet have neighborhood 2-shadowing.

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1. Plenary

On a generalization of the Ingram conjecture — Matevž Črepnjak <matevz.crepnjak@um.si>

After Ingram's conjecture was proven, new questions arose concerning tent functions. One of them is to identify all skew tent maps with their top vertices in the unit square whose (generalized) inverse limits are homeomorphic. In particular, it is interesting to identify the regions of top vertices in the unit square for which inverse limits are homeomorphic. In this talk, we revisit the skew tent maps problem and give some partial results.

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1. Set-Theoretic

On cardinal inequalities for topological spaces — Ivan Gotchev <gotchevi@ccsu.edu>

In this talk some recent results about cardinal inequalities for topological spaces will be presented and some open questions will be discussed.

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1. Set-Theoretic

On discrete and disjoint shrinking properties — Vladimir Tkachuk <vvtmdf@gmail.com>

A space $X$ has the disjoint (discrete) shrinking property if for any family $\{U_n: n\in\omega\}$ of non-empty open subsets of $X$ there exists a disjoint (discrete) family $\{V_n: n\in\omega\}$ of non-empty open sets such that $V_n \subset U_n$ for every $n\in\omega$. We present a topological equivalent of the disjoint shrinking property in general spaces and apply it to characterize the disjoint shrinking property in topological groups and locally convex spaces.

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1. Set-Theoretic

On expansive homeomorphisms on a quasi-uniform space — Olivier Olela-Otafudu <olivier.olela-otafudu@ul.ac.za>

In this talk, we present the concepts of expansive homeomorphisms in the context of quasi-uniform spaces. We continue with our analysis on expansive homeomorphisms by extending the results from quasi-metric spaces to quasi-uniform spaces. It turns out that an expansive homeomorphism on a quasi-uniform space is also an expansive homeomorphism on its induced quasi-uniformity but the converse does not hold in general. We show that if a homeomorphism on a quasi-uniform space is expansive then the quasi-uniform space is always a Kolmogorov space. Moreover, we generalize the concept of expansive measures in the sense of Morales and Sirvent to quasi-uniform spaces point of views.

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1. Continua

On increasing and persistent Whitney properties — Hugo Villanueva <hugo.villanueva@udlap.mx>

In 2009, increasing Whitney properties were defined by F. Orozco and give results and examples of topological increasing Whitney properties. In this talk we define Whitney persistent and locally Whitney persistent properties. We present results and examples of continua and topological properties to establish relations between these concepts and those of Whitney and increasing Whitney properties. This is a joint work with José Gerardo Ahuatzi-Reyes and Norberto Ordoñez-Ramírez.

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1. Set-Theoretic

On some Selection Principles and Games involving Countable Networks — Davide Giacopello <dagiacopello@unime.it>

We introduce new selection principles involving networks, namely, M-nw-selective, R-nw-selective, and H-nw-selective. These spaces represent a strengthening of both M-separability, R-separability, and H-separability, as well as the Menger, Rothberger, and Hurewicz properties. We also define and investigate two new games: the R-nw-selective game and the M-nw-selective game, which arise naturally from their corresponding selection principles. We give consistent results, and we define trivial R-, H-, and M-nw-selective spaces the ones with countable netweight and cardinality and weight strictly less than $\text{cov}(\mathcal{M})$, $\mathfrak{b}$, and $\mathfrak{d}$, respectively. We establish that spaces with cardinalities greater than $\text{cov}(\mathcal{M})$, $\mathfrak{b}$, and $\mathfrak{d}$ fail to possess the R-, H-, and M-nw-selective properties, respectively. Non-trivial examples, therefore, should eventually have weight greater than or equal to these small cardinals. Using forcing methods, we construct consistent countable non-trivial examples of R-nw-selective and H-nw-selective spaces. Finally, we study relations between nw-selective properties and a strong version of the HFD property.

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1. Dynamics

On the Equivalence of Equilibrium and Freezing States — Evans Hedges <evans.hedges@du.edu>

This talk is concerned with freezing phase transitions in general dynamical systems. A freezing phase transition is one in which, for a given potential $\phi$, there exists some inverse temperature $\beta_0 > 0$ such that for all $\alpha, \beta > \beta_0$, the collection of equilibrium states for $\alpha \phi$ and $\beta \phi$ coincide. In this sense, below the temperature $1 / \beta_0$, the system "freezes" on a fixed collection of equilibrium states. We will provide an overview of this direction of study, and conclude with some novel results related to the obtainability of a given measure as a freezing state, as well as the fact that the collection of potentials that freeze is dense in $C(X)$ under certain conditions on the dynamical system.

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1. Dynamics

On the existence of freezing phase transitions for lattice systems — Tamara Kucherenko <tkucherenko@ccny.cuny.edu>

We establish the existence of freezing phase transitions in the settings of multi-dimensional shift spaces. Precisely, given an arbitrary proper subshift $X$ of a d-dimensional shift space we explicitly construct a continuous potential $\phi$ such that for all $\beta$ above some critical value $\beta_c$ the equilibrium states of $\beta\phi$ are the measures of maximal entropy of $X$, whereas for $\beta$ below $\beta_c$ no equilibrium state of $\beta\phi$ is supported on $X$. This phenomenon is referred to as a freezing phase transition for potential $\phi$ with the motivation stemming from quasicrystal models in statistical physics. To contrast this result we establish sufficient conditions on the potential which guaranty that the system never freezes. This is a joint work with J.-R. Chazottes and A. Quas.

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1. Plenary

Order-reversing maps on $\mathbb N^\ast$ and $\mathbb H^\ast$ — Will Brian <wbrian.math@gmail.com>

I will discuss two related questions concerning the two spaces in the title: the Čech-Stone remainder of the natural numbers $\mathbb N$, and the Čech-Stone remainder of the half-line $\mathbb H = [0,\infty)$. Both $\mathbb N$ and $\mathbb H$ are naturally ordered from left to right. These orders on $\mathbb N$ and $\mathbb H$ are reflected in their Čech-Stone remainders, in certain dynamical systems on $\mathbb N^\ast$ and in certain subcontinua of $\mathbb H^\ast$. Are these left-to-right aspects of $\mathbb N^\ast$ and $\mathbb H^\ast$ truly topological, or can either of the spaces be "reversed" via some self-homeomorphism?

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1. GeoTop

Parametrized Legendrian Surgery — Eduardo Fernández <eduardofernandez@uga.edu>

Given a parametrized Legendrian $\Lambda$ in a contact manifold $(M, \xi)$, there is a well-defined operation called Legendrian surgery, which produces a new contact manifold $(M(\Lambda), \xi(\Lambda))$. The contactomorphism type of the surgered manifold depends only on the Legendrian isotopy class of the initial Legendrian. Given a loop of Legendrians $\Lambda_t$, it is also possible to realize a 1-parameter family of Legendrian surgeries. From this, we naturally obtain a bundle over the circle with fiber $(M(\Lambda), \xi(\Lambda))$. The non-triviality of the bundle depends on the contact isotopy class of a gluing contactomorphism, which we call the “Legendrian surgery contactomorphism.” Its contact isotopy class depends only on the homotopy class of the given loop of Legendrians within the space of parametrized Legendrians. The obvious realization problem is: which contactomorphisms of a given contact manifold are Legendrian surgery contactomorphisms? In this talk, I will address this question by showing that every formally trivial contactomorphism arises as a Legendrian surgery contactomorphism associated with a certain loop of Legendrians in some overtwisted contact manifold with controlled topology. As a consequence, in 3-dimensional contact topology, we will deduce the existence of formally contractible but non-contractible loops of loose Legendrians in every overtwisted contact 3-manifold. This is a joint work in progress with Fabio Gironella.

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1. GeoGT

Pattern preserving quasi-isometries in lamplighter groups — Beibei Liu <bbliumath@gmail.com>

In this talk, I will explore the interplay between aspects of the geometry and algebra of three families of groups of the form $B\rtimes Z$, namely Lamplighter groups, solvable Baumslag-Solitar groups and lattices in $SOL$. In particular we examine what kind of maps are induced on $B$ by quasi-isometries that coarsely permute cosets of the $Z$ subgroup. By the results of Schwartz(1996) and Taback(2000) in the lattice in $SOL$ and solvable Baumslag-Solitar cases respectively such quasi-isometries induce parallelogram preserving maps of $B$. We show that this is no longer true in the lamplighter case but the induced maps do share some features with parallelogram preserving maps. This is joint work with Dymarz, Macura and Morris-Wright.

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1. Dynamics

Percolation Theory and the Diversity of Cellular Automata on Groups — Felipe García-Ramos <felipegra@yahoo.com>

We will explain a connection between the diversity of cellular automata observable on a given countable group and the percolation threshold associated with the Cayley graphs of such groups. As a consequence, we show that Gilman's dichotomy holds for the endomorphism semigroup of a countable group if and only if the group is locally virtually cyclic.

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1. Applied

Persistent cohomology operations and Gromov-Hausdorff estimates — Ling Zhou <zhouling0903@gmail.com>

We establish the foundations of the theory of persistent cohomology operations, derive decomposition formulas for wedge sums and products, and prove their Gromov–Hausdorff stability. We use these results to construct pairs of Riemannian pseudomanifolds for which the Gromov-Hausdorff estimates derived from persistent cohomology operations are strictly sharper than those obtained using persistent homology. This work is joint with Anibal M. Medina-Mardones.

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1. Dynamics

Phase transitions in the Potts model on Cayley tree. — Diyath Pannipitiya <dinepann@iu.edu>

The Ising model is one of the most important theoretical models in statistical physics, which was originally developed to describe ferromagnetism. A system of magnetic particles, for example, can be modeled as a linear chain in one dimension or a lattice in two dimensions, with one particle at each lattice point. Then each particle is assigned a spin $\sigma_i\in \{\pm 1\}$. The $q$-state Potts model is a generalization of the Ising model where each spin $\sigma_i$ may take on $q\geq 3$ number of states $\{0,\cdots, q-1\}$. Both models have temperature $T$ and an externally applied magnetic field $h$ as parameters. Many statistical and physical properties of the $q$-state Potts model can be derived by studying its partition function. This includes phase transitions as $T$ and/or $h$ are varied. The celebrated Lee-Yang Theorem characterizes such phase transitions of the $2$-state Potts model (the Ising model). This theorem does not hold for $q>2$. Thus, phase transitions for the Potts model as $h$ is varied are more complicated and mysterious. We give some results that characterize the phase transitions of the $3$-state Potts model as $h$ is varied for constant $T$ on the binary rooted Cayley tree. Similarly to the Ising model, we show that for fixed $T>0$ the $3$-state Potts model for the ferromagnetic case exhibits a phase transition at one critical value of $h$ or not at all, depending on $T$. However, an interesting new phenomenon occurs for the $3$-state Potts model because the critical value of $h$ can be non-zero for some range of temperatures. The $3$-state Potts model for the antiferromagnetic case exhibits phase transition at up to two critical values of $h$. The recursive constructions of the $(n+1)^{st}$ level Cayley tree from two copies of the $n^{th}$ level Cayley tree allow one to write a relatively simple rational function relating the Lee-Yang zeros at one level to the next. This allows us to use techniques from dynamical systems.

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1. Continua

Planarity of compactifications of R with arc-like remainder — Andrea Ammerlaan <ajammerlaan879@my.nipissingu.ca>

In 1972, Nadler and Quinn asked if for any arc-like continuum $X$, and point $x \in X$, there exists a plane embedding of $X$ in which $x$ is accessible. A continuum $X$ is arc-like if it can be expressed as an inverse limit on arcs and, if $X$ is in the plane $\mathbb{R}^2$, a point $x \in X$ is called accessible if there exists an arc $A \subset \mathbb{R}^2$ such that $A \cap X =$ {$x$}. The question was recently answered in the positive (AA, Anušić, Hoehn 2024). This talk will discuss some consequences of the result: if $X$ is an arc-like continuum, then any continuum which is the disjoint union of $X$ and a ray $R$, with cl$(R) \setminus R \subseteq X$, is embeddedable in the plane, as is any compactification of a line having remainder $X$. Joint work with Logan Hoehn.

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1. GeoTop

Prym Representations and Twisted Cohomology of the Mapping Class Group with Level Structures — Xiyan Zhong <xzhong4@nd.edu>

The Prym representations of the mapping class group are an important family of representations that come from abelian covers of a surface. They are defined on the level-$\ell$ mapping class group, which is a fundamental finite-index subgroup of the mapping class group. One consequence of our work is that the Prym representations are infinitesimally rigid, i.e. they can not be deformed. We prove this infinitesimal rigidity by calculating the twisted cohomology of the level-$\ell$ mapping class group with coefficients in the Prym representation, and more generally in the $r$-tensor powers of the Prym representation. Our results also show that when $r\ge 2$, this twisted cohomology does not satisfy cohomological stability, i.e. it depends on the genus $g$.

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1. GeoGT

Random branched covers of 2-complexes — Jean-Francois Lafont <lafont.1@osu.edu>

I'll introduce a random model for branched covers of 2-complexes. I'll explain why asymptotically almost surely, a random branched cover has Gromov hyperbolic fundamental group. This is joint work with Hyeran Cho (OSU) and Rachel Skipper (Univ. Utah).

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1. Applied

Random complexes with free involution — Andrew Newman <anewman@andrew.cmu.edu>

In this talk I will discuss a new model for random simplicial complexes. Unlike other models, which are generically simply-connected when sufficiently dense, the complexes in this model generically have fundamental group $\mathbb{Z}/2\mathbb{Z}$. I'll describe results on the asymptotic behavior of the homology and homotopy groups in these complexes as well as how these results imply a "random Borsuk--Ulam theorem". This talk is based on joint work with Florian Frick.

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1. GeoTop

Rational points on quartic del Pezzo surfaces via homological stability — Philip Tosteson <philip.tosteson@gmail.com>

A quartic del Pezzo surface $X$ is an intersection of two degree $2$ hypersurfaces in $$\mathbb P^4$$. So rational points on $X$ correspond to solutions of a pair of homogeneous quadratic equations in $5$ variables. I will discuss joint work with R. Das, B. Lehmann, and S. Tanimoto, using topological methods to determine statistics of rational points on $X$ (over the function field $\mathbb F_q(t)$)

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1. Applied

Restricted Distortions of Embedded Length Spaces — Atish Mitra <atish.mitra@gmail.com>

In geometric and topological reconstruction of compact length spaces embedded in some metric space, one needs an appropriate notion of distortion of the embedding. We consider variants of the classical notion of distortion, by controlling the coarseness of the distance scale of the ambient space and the discreteness of the coarse paths used to generate the length structure. In addition to discussing the stability and convergence of these notions of distortion, we compare them with existing notions of sampling parameters used in shape reconstruction and show some applications of our approach. This is based on joint work with Rafal Komendarczyk and Sushovan Majhi.

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1. GeoTop

Rigidity Phenomena for Surface Amalgams — Yandi Wu <yandi.wu@rice.edu>

Geometric rigidity theory aims to determine a geometric object with the smallest amount of data possible. For instance, one could ask whether the volume or length set of a manifold determines its metric. In this talk, I will motivate and present some results related to length spectrum and volume rigidity for negatively curved surface amalgams, natural generalizations of negatively curved surfaces.

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1. Plenary

Semi-Kelley fans have contractible hyperspaces — David Maya <dmayae@outlook.com>

Semi-Kelley continua were introduced by J. J. Charatonik and W.J. Charatonik in 1998, who proved that every Kelley continuum is semi-Kelley. Since then, this class of continua has been studied by several authors. The most important problem in this area is determining whether the hyperspaces of a semi-Kelley continua are contractible. In this talk, we present a positive partial answer to this question in the case of fans.

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1. Dynamics

Sets of pointwise recurrence and answers to some questions of Host, Kra, and Maass — Anh Le <anh.n.le@du.edu>

A subset of the positive integers is dynamically central syndetic if it contains the times of return of a point to a neighborhood of itself in a minimal dynamical system. This class of syndetic sets forms an important bridge between dynamics and combinatorics. We show that a set is dynamically central syndetic if and only if it is a member of a syndetic, idempotent filter. We elaborate on the consequences of this characterization for the dual family: sets of pointwise recurrence. For example, we provide several combinatorial characterizations of sets of pointwise recurrence, show that these sets do not have the Ramsey property, and they are sets of multiple recurrence. These results answer several questions asked by Host, Kra, and Maass. This talk is based on an joint work with Daniel Glasscock (University of Massachusetts Lowell).

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1. Set-Theoretic

Some recent results on $\Delta$-spaces — Paul Szeptycki <szeptyck@yorku.ca>

A $\Delta$-space is a Tychonoff space with the property that every partition of the space (into arbitrary sets) has a point finite open expansion. M. Reed defined a set of reals with this property to be a $\Delta$-set and was motivated by the characterization of a $\Delta$-set as those sets of reals $X$ for which the Moore plane over $X$ is countably paracompact. Recently, Leiderman and Kąkol characterized $\Delta$-spaces as those $X$ for which the locally convex space $C_p(X)$ is distinguished. I will survey some recent results concerning $\Delta$-spaces and mention a number of open problems.

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1. Dynamics

Specification and $\omega$-chaos in non-compact systems — Jonathan Meddaugh <jonathan_meddaugh@baylor.edu>

We demonstrate conditions under which a dynamical system on a Lindelöf space exhibits $\omega$-chaos. In particular, we show that a system which satisfies a generalized version of the specification property and which contains at least three mutually separated orbit closures exhibits dense $\omega$-chaos.

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1. Plenary

Steps on the Way to and from Persistent Homology — Herbert Edelsbrunner <edels@ist.ac.at>

The formation of topological data analysis (TDA) as a research area with dedicated meetings and funding happened during the years around the beginning of this millenium. A crucial step in this development was the introduction of persistent homology. The root system of this idea goes back to the dependent and independent work of a number of mathematicians, including Marston Morse. This talk recalls a few of the steps on my personal journey leading to this concept, and steps that expand the basic notion toward other branches of mathematics and applications outside of mathematics.

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1. Plenary

Suspending the pigeonhole principle: amenability, dynamics, and C*-algebras — David Kerr <kerrd@uni-muenster.de>

The Rokhlin lemma is a finite approximation property that underpins a great many constructions in classical ergodic theory, including most spectacularly those at the basis of the Ornstein isomorphism theory for Bernoulli shifts. In the 1970s Ornstein and Weiss showed amenability to be the natural setting for finite approximation in dynamics by establishing a general form of the Rokhlin lemma in this setting, and this led, among other things, to a much broader recasting of the Ornstein isomorphism theory. Over the last couple of decades a growing interest in the interplay between dynamics and the geometric and analytic structure of groups has set the stage for a resurgence of applications of the Ornstein-Weiss Rokhlin lemma, not only in its original measure-theoretic incarnation but also as a versatile tiling principle that has turned out be intimately connected, on the topological side, to the remarkable recent successes in the Elliott classification program for separable nuclear C*-algebras. I will sketch a picture of these various developments at the interface of measure, topology, dynamics, geometric group theory, and operator algebras.

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1. GeoTop

Symmetry and arithmetic structures — Yanlong Hao <ylhao@umich.edu>

Arithmetic manifolds admit many 'Hidden' symmetries. In this talk, we want to discuss the inverse problem: if a object admit many symmetries, is it arithmetic? We will invest the question from variety of aspects: algebra, differential geometry and coarse geometry and answer the question for non-compact negatively curved manifolds.

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1. Plenary

The Borel Conjecture for compact aspherical 4-manifolds with boundary — James Davis <jfdavis@iu.edu>

The Borel Conjecture for closed manifolds implies that two closed aspherical manifolds with isomorphic fundamental group are homeomorphic. The Borel conjecture for compact aspherical manifolds with boundary states that a homotopy equivalence which is homeomorphism on the boundary is homotopic to a homeomorphism. Jonathan Hillman and I classify and prove the Borel Conjecture for all compact aspherical four manifolds with boundary with good (= elementary amenable) fundamental group. We classify all possible fundamental groups and all possible 3-manifold boundaries.

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1. Plenary

The Mandelbrot set and its Satellite copies — Luna Lomonaco <luna@impa.br>

For a polynomial on the Riemann sphere, infinity is a (super) attracting fixed point, and the filled Julia set is the set of points with bounded orbit. Consider the quadratic family $P_c(z)=z^2+c$. The Mandelbrot set $M$ is the set of parameters $c$ such that the filled Julia set of $P_c$ is connected. Computer experiments quickly reveal the existence of small homeomorphic copies of $M$ inside itself; the existence of such copies was proved by Douady and Hubbard. Each little copy is either primitive (with a cusp on the boundary of its main cardioid region) or a satellite (without a cusp). Lyubich proved that the primitive copies of $M$ satisfy a stronger regularity condition: they are quasiconformally homeomorphic to M. The satellite copies are not quasiconformally homeomorphic to $M$ (as we cannot straighten a cusp quasiconformally), but are they mutually quasiconformally homeomorphic? In joint work with C. Petersen we prove that the answer is negative in general, but positive in the case the satellite copies have rotation number with same denominator.

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1. Plenary

The Milnor-Wood Inequality: Geometry, Topology, and Flat Bundles — Sam Nariman <snariman@purdue.edu>

The Milnor-Wood inequality, introduced in two landmark papers by John Milnor (1958) and John W. Wood (1971), is a striking result at the intersection of geometry, topology, and dynamics. It establishes sharp bounds on the Euler number of flat $\mathbb{S}^1$-bundles over surfaces, revealing deep connections between geometric curvature and topological invariants. Milnor’s original inequality highlights the boundedness of Euler invariants for flat bundles with "linear" structures, which Gromov later generalized using bounded cohomology. Wood extended Milnor's result to "non-linear" flat circle bundles, offering a perspective rooted in 1-dimensional dynamics. In the 1980s, Étienne Ghys posed the intriguing question of whether Wood’s inequality could be extended to flat-oriented $\mathbb{S}^3$-bundles. In this talk, we will also discuss the surprising ways in which inequality fails in higher-dimensional non-linear cases, showcasing the new calculations in the bounded cohomology of diffeomorphism groups.

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1. GeoGT

The RAAG Recognition Problem for Bestvina--Brady Groups — Yu-Chan Chang <yuchanchang74321@gmail.com>

Right-angled Artin groups (RAAGs) are an important class of objects of study in geometric group theory. It is interesting to know which groups are isomorphic to a RAAG. In this talk, we will explore how to recognize a Bestvina–Brady group as a RAAG. In particular, I will focus on Bestvina–Brady groups defined on 2-dimensional flag complexes. This is joint work with Lorenzo Ruffoni.

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1. Dynamics

The Variational Principle for Entropy of Countable State Shift Spaces With Specification — Alexander Paschal <ampasch@unc.edu>

We define and discuss specification properties for countable state shift spaces, which are special cases of definitions from an upcoming paper by Climenhaga, Thompson, and Wang and generalize the well-studied compact specification property to non-compact shift spaces. We present an infinite class of examples of such shift spaces and prove the variational principle for these spaces.

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1. Applied

The connectivity of Vietoris-Rips complexes of spheres — Johnathan Bush <bush3je@jmu.edu>

Although Vietoris--Rips complexes are frequently used in topological data analysis to approximate the “shape” of a dataset, their theoretical properties are not fully understood. In the case of the circle, these complexes exhibit a surprising progression of homotopy types (from $S^1$ to $S^3$ to $S^5$, etc.) as the scale increases. However, much less is known about the Vietoris--Rips complexes of higher-dimensional spheres. I will present work that explores Vietoris--Rips complexes of the $n$-sphere $S^n$ and shows how the appearance of nontrivial homotopy groups of $\mathrm{VR}(S^n; t)$ can be controlled by covering properties of $S^n$ and real projective space $\mathbb{R}P^n$. Specifically, if the first nontrivial homotopy group of $\mathrm{VR}(S^n; \pi-t)$ occurs in dimension $k$, then $S^n$ can be covered by $2k+2$ balls of radius $t$, but there is no covering of $\mathbb{R}P^n$ by $k$ balls of radius $t/2$. This is joint work with Henry Adams and Žiga Virk.

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1. Dynamics

The stabilized automorphism group of minimal systems — Jennifer N. Jones-Baro <jenniferjones2024@u.northwestern.edu>

The stabilized automorphism group of a dynamical system (X,T) is the group of all self-homeomorphisms of X that commute with some power of T. In this talk, we will describe the stabilized automorphism group of minimal systems. The main result we will prove is that if two minimal systems have isomorphic stabilized automorphism groups and each has at least one non-trivial rational eigenvalue, then the systems have the same rational eigenvalues.

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1. Continua

The weak Extension Principle — Alessandro Vignati <ale.vignati@gmail.com>

We study the weak Extension Principle $\mathrm{wEP}$ allowing us to completely understand maps between \v{C}ech-Stone remainders of locally compact noncompact second countable spaces, generalising work of Farah in the 2000s. In short, the $\mathrm{wEP}$ asserts that all maps between such remainders come from maps between the underlying spaces. We show that once assuming fairly mild axioms (namely the Open Colouring Axiom and Martin's Axiom) the $\mathrm{wEP}$ holds, while this is not the case if the Continuum Hypothesis holds. This is joint work with D. Yilmaz.

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1. Applied

Topological Analysis of U.S. City Demographics — Thomas Weighill <t_weighill@uncg.edu>

Topological data analysis is naturally suited to “data with shape”. In this talk, I will use a recent joint project with Jakini Auset Kauba as a demonstration of how TDA can uncover shape in geospatial data. In our project, we looked at persistence diagrams given by the demographics of 100 U.S. cities, and used them to perform various investigations and comparisons. Towards the end of the talk, I will highlight some of the pitfalls of using persistent homology on this kind of data, and pitch some challenges for those interested in TDA and geospatial data.

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1. Applied

Topological Stability and Latschev-type Reconstruction Theorems for $\boldsymbol{\operatorname{CAT}(\kappa)}$ Spaces (part 2) — Rafal Komendarczyk <rako@tulane.edu>

We address the problem of homotopy-type reconstruction of compact shapes $X\subset\mathbb{R}^N$ that are $\operatorname{CAT}(\kappa)$ in the intrinsic length metric. The reconstructed spaces take the form of Vietoris–Rips complexes, computed from a compact sample $S$ that is Hausdorff-close to the unknown shape $X$. Instead of employing the Euclidean metric on the sample, our reconstruction technique utilizes a path-based metric to compute these complexes. Naturally emerging in the reconstruction framework, we also explore the Gromov–Hausdorff topological stability and the finiteness problem for general compact $\operatorname{CAT}(\kappa)$ spaces. Our techniques offer novel sampling conditions as alternatives to the existing and commonly used methods based on the weak feature size and $\mu$-reach.

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1. Applied

Topological Stability and Latschev-type Reconstruction Theorems for CAT(k) Spaces (part 1) — Sushovan Majhi <s.majhi@gwu.edu>

We discuss the problem of homotopy-type reconstruction of compact shapes $X\subset\mathbb{R}^N$ that are $\mathrm{CAT}(\kappa)$ in the intrinsic length metric. The reconstructed spaces are Vietoris–Rips complexes computed from a compact sample $S$, Hausdorff–close to the unknown shape $X$. Instead of the Euclidean metric on the sample, our reconstruction technique leverages a path-based metric to compute these complexes. As naturally emerging in the reconstruction framework, we also study the Gromov–Hausdorff topological stability and finiteness problem for general compact $\mathrm{CAT}(\kappa)$ spaces. Our techniques provide novel sampling conditions as an alternative to the existing and commonly used techniques using weak feature size and $\mu$–reach. In particular, we introduce a new parameter, called the restricted distortion, which is a generalization of the well-known global distortion of embedding. We show examples of Euclidean subspaces, for which the known parameters such as the reach, $\mu$–reach and weak features size vanish, whereas the restricted distortion is finite, making our reconstruction results applicable for such spaces.

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1. Continua

Unary Topological Algebras on Continua, and Some Associated Function Algebras — Matt Insall <insall.at.mst@gmail.com>

Let $D^2$ denote the unit disk in the plane, and let $C$ denote the set of (continuous) self-maps of $D^2$. Using $\circ$, as is common, to denote the binary operation on $C$ that takes a pair of continuous functions to another continuous function, we study some properties of the following algebras and their subalgebras: $$ \mathbb{A}=\langle D^2; C\rangle, $$ and $$ \mathbb{F}=\langle C; \circ\rangle. $$ The algebra $\mathbb{A}$ can be naturally endowed with a topology, and we will suppress any notation the choice of topology on it, because we are interested in only the usual topology, so we think of $\mathbb{A}$ as a topological algebra; it is a {\bf multi-unary topological algebra} on the continuum $D^2$. As is well-known, there are various reasonable topologies that can be given to the algebra $\mathbb{F}$, but we will treat it only as an algebra for now. Note that the algebra $\mathbb{F}$ is a semigroup, and it is a subalgebra of a function algebra (an algebra of functions over a set that is closed under composition and contains the projection functions) over $D^2$. Recall that in a semigroup, a left translation, $\lambda_a$ is a self-map of the semigroup defined using a parameter $a$, an element of the given semigroup, using the formula $\lambda_a(f)=af$. In our case, the parameters are continuous functions on $D^2$ and the semigroup operation is composition, so instead of juxtaposition of symbols, we will write $\lambda_a(f)=a\circ f$. Similarly, a right translation is defined by the other order of the ``multiplication'': $\rho_a(f)=f\circ a$. Given an element $a\in C,$ we call the set $$\Lambda_a=\left\{(f,g)\in C^2\vert \lambda_a(f)=\lambda_a(g)\right\}=\left\{(f,g)\in C^2\vert a\circ f=a\circ g\right\}$$ the {\bf kernel} of $\lambda_a$. kernels of right translations are defined similarly: $${\rm P}_a=\left\{(f,g)\in C^2\vert \rho_a(f)=\rho_a(g)\right\}=\left\{(f,g)\in C^2\vert f\circ a=g\circ a\right\}.$$ These are {\bf congruences} of the algebra (semigroup) $\mathbb{F}$; i.e. they are equivalence relations $\theta$ on the set $C$ that are compatible with the semigroup operation (composition). The compatibility property can be described via the containment $\{(b\circ f,c\circ g)\vert (b,c),(f,g)\in\theta\}\subseteq\theta$. On a set $X$, two special equivalence relations are congruences for any structure on $X$, namely the {\bf identity relation}, $\Delta_X=\{(p,p)\vert p\in X\}$, and the {\bf all relation}, $\nabla_X=\{(p,q)\vert p, q\in X\}$. It is clear that all kernels of left translations on a semigroup are congruences on that semigroup, and similarly, kernels of rigjt translations on a semigroup are congruences on that semigroup. We will sketch a proof of the following: Theorem. The algebra $\mathbb{F}$ has only three kinds of congruences, namely the identity relation, the all relation, and kernels of left translations by members of $C$. Our proof of the above result will employ nonstandard methods and results from the theory of function algebras on finite sets, and interestingly, the above immediately entails the below consequence Corollary. In the semigroup $\mathbb{F}$, every right translation equalizer is a left translation equalizer, and vice versa. This is joint work with Malgorzata Marciniak (mmarciniak@lagcc.cuny.edu)

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1. Continua

Uncountable family of Lelek-like fans — Judy Kennedy <kennedy9905@gmail.com>

Defining an appropriate equivalence relation on a Lelek fan L we construct an uncountable family of pairwise non-homeomorphic Lelek-like fans. In this talk plan is to explain the construction of that family. This is joint work with Iztok Banič, Goran Erceg, and Ivan Jelić.

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1. Continua

Uniqueness of cones for some not locally connected continua — Daria Michalik <dmichalik@mimuw.edu.pl>

A continuum $X$ has unique cone provided that the following property holds: if $Y$ is a continuum and ${\rm Cone}(X)$ is homeomorphic to ${\rm Cone}(Y)$, then $X$ is homeomorphic to $Y$. In this talk we consider the problem of the uniqueness of cones for some not locally connected continua, e.g. the indecomposable continua and the compactifications of the ray.

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1. Applied

Vaughan's function orthogonal sections — Ulises Morales-Fuentes <ulises.morales@uaem.mx>

A plane set X admits an inscribed polygon P, if every vertex of a polygon similar to P lies in X. It is still not known whether every Jordan curve admits an inscribed square. In 1977 H.Vaughan proved that every homeomorphic copy of $S^1$ in $ \mathbb{R}^2$ admits at least one inscribed rectangle. In this talk, we present an algorithm implemented in Python that helps us visualize Vaughan's function, and we classify locally connected plane continua that inscribe rectangles.

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1. Applied

Čech Complexes of Certain Finite Metric Spaces — Naga Chandra Padmini Nukala <padmini.nukala@mines.edu>

Topological Data Analysis (TDA) is an emerging field that aims to extract the shape and structure of the data. The key idea here is to build a higher-dimensional graph by connecting more than two nearby data points- resulting in simplicial complexes. There are different ways to build a simplicial complex on a metric space including Vietoris-Rips Complexes, Čech complexes. In this talk, we specifically examine Čech complexes constructed from the finite union of finite metric spaces at scales 2 and 3, using the symmetric difference metric. Our primary focus is on determining the homotopy types of these Čech complexes. Using these homotopy types, we also derived a precise formula for the homotpy types of Čech complex of a hypercube graph. This is a joint work by me and my PhD advisor Dr. Ziqin Feng.

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