1. Continua
3. Plenaries

$R^i$-sets in continua and hyperspaces — Patricia Pellicer-Covarrubias <paty@ciencias.unam.mx>

$R^1$, $R^2$ and $R^3$-continua were defined by S. T. Czuba in 1980, in particular he showed that the existence of any one of these sets in a continuum $X$ implies the noncontractibility of $X$. Also, $R^i$-continua have proved to be useful when studying noncontractibility of hyperspaces. In this talk we recall these concepts and we present some relations between them in continua and hyperspaces.

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1. GeoGT

A Combination Theorem for Relatively Hyperbolic Groups — Darius Alizadeh <daliza2@uic.edu>

Given a group $G$ acting cocompactly on a suitable simply connected cell complex $X$ with relatively hyperbolic cell stabilizers, we show $G$ itself is relatively hyperbolic. Building on work of Dahmani and Martin, the proof constructs a model for the Bowditch boundary by gluing together the boundaries of cell stabilizers. More generally, any cocompact action on a cell complex $X$ induces an algebraic \emph{complex of groups} decomposition which generalizes Bass--Serre theory in the case where $X$ is 1--dimensional. This model connects this algebraic decomposition with a topological decomposition of the boundary which we hope will be useful for answering other questions.

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1. Applied & Data

A Medial-Axis-Based Measure of District Compactness — Greg Malen <gmalen@skidmore.edu>

An essential question for democracy is how to rigorously determine the likelihood that a congressional map has been gerrymandered. A number of state constitutions require districting plans to be "compact," yet no technical legal definition of compactness exists in this context, leaving us to contend with the oft-cited sentiment that "you know it when you see it." In this talk, I will introduce a novel compactness measure based on a geometric structure known as the medial axis. This skeleton-like structure has been shown to have strong ties to the science of how the human brain perceives and processes complex shapes, thus offering a mathematically rigorous version of “the eye test.” I will explain the construction of this metric in detail, and then compare it to a recent machine-learning-based compactness metric introduced by Kaufman, King, and Komisarchik (2021). Specifically, in this work we examine the performance of our measure and theirs in several case studies, including two states whose districting plans were especially contentious and the entire 2016 congressional district map. This is joint work with Ellen Gasparovic at Union College, and Jason D’Amico and Mushan Zhong, who were undergraduates at Union College at the time of their contributions.

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1. Applied & Data

A Schauder Basis for Multiparameter Persistence and Persistence Variants — Zachariah Ross <thomas.z@ufl.edu>

To help combine statistics and machine learning with multiparameter persistence, we would like to map signed barcodes to a Banach space or Hilbert space. We use iteratively refined triangulations to define a Schauder basis of compactly supported Lipschitz functionals. We prove that evaluation of these functionals embeds signed barcodes into sequence space via a map which is both linear, and Lipschitz with respect to the 1-Wasserstein distance. I will illustrate these results with examples for one-parameter persistence, two-parameter persistence, and the variant called mixup barcodes.

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1. Dynamics

A complex dynamics approach to understanding birational maps of the plane arising from cluster algebra mutations — Krishna Chaitanya Kalidindi <kkalidin@iu.edu>

There is a family of birational self-mappings of the plane arising from the theory of cluster algebra mutations that was studied previously by Machacek-Ovenhouse from the perspective of real dynamics. We study this family of mappings from the perspective of complex dynamics and, in particular, show that is most cases there is no conserved quantity. No background on cluster algebras is expected from the audience. This is the joint-work with Andrei Grigorev, Andres Quintero and Roland Roeder.

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1. Applied & Data

A discrete view of Gromov's filling area conjecture — Chris Wells <chris@mathematicaster.org>

In differential geometry, a metric surface $M$ is said to be an isometric filling of a closed metric curve $C$ if $\partial M=C$ and $d_M(x,y)=d_C(x,y)$ for all $x,y\in C$. Gromov's filling area conjecture from 1983 asserts that among all isometric fillings of the Riemannian circle, the one with the smallest surface area is the hemisphere. Gromov's conjecture has been verified if, say, $M$ is homeomorphic to the disk and in a few other cases, but it is still open in general. Admittedly, I'm not a differential geometer, so we consider instead a particular discrete version of Gromov's conjecture which is likely fairly natural to anyone who studies graph embeddings on arbitrary surfaces. We obtain reasonable asymptotic bounds on this discrete variant by applying standard graph theoretic results, such as Menger's theorem. These bounds can then be translated to the continuous setting to show that any isometric filling of the Riemannian circle of length $2\pi$ has surface-area at least $1.36\pi$ (the hemisphere has area $2\pi$). This appears to be the first quantitative lower-bound on Gromov's problem that applies to an arbitrary isometric fillings. (Based on joint work with Joe Briggs)

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1. General & ST

A glance at function spaces with a dense functionally countable subspace — Vladimir Tkachuk <vova@xanum.uam.mx>

We will present some results on existence of dense functionally countable subspaces in spaces $C_p(X)$. It will be shown, among other things, that there is a consistent example of a scattered Lindel\"of $P$-space $X$ for which $C_p(X)$ has no dense functionally countable subspace and that $\mathbb R^{\omega_1}$ has a dense functionally countable subspace of cardinality $\omega_2$ if and only if the Kurepa Hypothesis holds.

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1. Continua

A simple USC bonding function giving $D_m$ as its inverse limit space — Robert Roe <rroe@mst.edu>

We show how the Wa\.zewski universal dendrite of order $m$, for any positive integer $m$ greater than 2, can be obtain as the generalized inverse limit of a single set-valued upper semi-continuous bonding function on $[0,1]$ whose graph consists of exactly $m$ line segments. $D_m$ has been obtained previously as a generalized inverse limit of a single bonding function but in that case the bonding function was extremely complicated consisting of infinitely many line segments. This is joint work with Faruq Mena.

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1. General & ST

An Adaptation of the Vietoris Topology for Ordered Compact Sets — Jared Holshouser <jholshouser1321@gmail.com>

We discuss a natural topology on powers of a space that is inspired by the Vietoris topology on compact subsets. We then place this topology in context with other product topologies; specifically, we compare this topology with the Tychonoff product, the box product, and Bell's uniform box topology. We identify a variety of topological properties for the specific case when the ground space is discrete. When the ground space is the Euclidean real line, we show that the resulting power is not Lindelöf, and hence, not Menger. This shows that, unlike the the Vietoris topology on unordered compact subsets, covering properties of the ground space need not transfer to the Vietoris power.

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1. GeoGT

An Improved Combination Theorem for A/QI Triples — Olu Olorode <oio3@cornell.edu>

Let $G$ be a group acting by isometries on a hyperbolic space $X$. Given geometrically natural subgroups $H$ and $K$ of $G$, it is natural to ask whether $ \langle H, K \rangle $ inherits the geometric properties of $H$ and $K$, and whether $ \langle H, K \rangle $ admits a nice algebraic structure. In a classical work of Gitik we receive an answer in the case where $G$ is a hyperbolic group, and $H$ and $K$ are quasiconvex. In more recent work of Martínez-Pedroza and Sisto, we receive an answer in the case where $G$ is relatively hyperbolic, and $H$ and $K$ are relatively quasiconvex. In this talk, I will discuss a generalization of a combination theorem of Abbot and Manning that covers a broader class of geometrically natural subgroups of such a group $G$. This is still a work in progress.

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1. GeoGT

An explicit section of the Laudenbach type exact sequence of the big mapping class group of $Map(M_Γ)$ — Jorge Andres Robinson Arrieta <jar064@uark.edu>

Brian Udall proved that there is a short exact sequence of the form: $$1 \xrightarrow{} Twist(M_{\Gamma})\xrightarrow{} Map(M_{\Gamma})\xrightarrow{\Psi} Map(\Gamma) \xrightarrow{}1,$$ where $Twist(M_{\Gamma})$ is the subgroup of $Map(M_{\Gamma})$ generated by sphere twists over sphere systems of $M_{\Gamma}.$ Udall also proved that this short exact sequence splits topologically. The purpose of this talk is to present an explicit formula for a section s of this short exact sequence.

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1. General & ST

An infinite library — KP Hart <k.p.hart@tudelft.nl>

We present an regular space that is not completely regular but only barely so: not only is it first-countable, but in addition all closed sets are $G_\delta$-sets and all points are zero-sets. This answers a question about how the lattice of zero-sets is situated in the lattice of all open sets. Some intermediate results on the Niemytzki plane make excellent homework exercises for a topology course.

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1. GeoGT
3. Plenaries

Arithmeticity in Hyperbolic Geometry — Nick Miller <nicholas.miller@villanova.edu>

Arithmetic manifolds are hyperbolic manifolds constructed from number theoretic data. By their very definition, they exhibit a strong connection between algebraic invariants, such as trace fields, and geometric quantities like lengths of closed geodesics. Despite this, the geometry of these manifolds remains surprisingly mysterious. Nevertheless, a guiding philosophy is that arithmetic manifolds should be the most symmetric hyperbolic manifolds and therefore exhibit geometric phenomena that are rare or absent in generic hyperbolic manifolds. In this talk, I will survey arithmetic hyperbolic manifolds, likely focusing on low dimensions, and discuss several manifestations of this philosophy, both known and conjectural. I will then discuss new work furthering this philosophy by establishing finiteness of closed arithmetic surface bundles, resolving a conjecture of Bowditch, Maclachlan, and Reid.

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1. GeoGT

Asymptotically rigid mapping class groups of infinite graphs — Thomas Hill <thill@math.utah.edu>

We introduce and study asymptotically rigid mapping class groups of certain infinite graphs. We determine their finiteness properties and show that these depend on the number of ends of the underlying graph. In a special case where the graph has finitely many ends, we construct an explicit presentation for the so-called \emph{pure graph Houghton group} and investigate several of its algebraic and geometric properties. Additionally, we show that the graph Houghton groups are not commensurable with other known Houghton-type groups, namely the classical, surface, braided, handlebody, and doubled handlebody Houghton groups, demonstrating that this graph-based construction defines a genuinely new class of groups. This is joint work with Sanghoon Kwak, Brian Udall, and Jeremy West.

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1. Applied & Data

Bayesian Sheaf Neural Networks — Layal Bou Hamdan <lbouhamd@vols.utk.edu>

Equipping graph neural networks with a convolution operation defined in terms of a cellular sheaf offers advantages for learning expressive representations of heterophilic graph data. The most flexible approach to constructing the sheaf is to learn it as part of the network as a function of the node features. However, this leaves the network potentially overly sensitive to the learned sheaf. As a counter-measure, we propose a variational approach to learning cellular sheaves within sheaf neural networks, yielding an architecture we refer to as a Bayesian sheaf neural network. As part of this work, we define a novel family of reparameterizable probability distributions on the rotation group SO(n) using the Cayley transform. We evaluate the Bayesian sheaf neural network on several graph datasets, and show that our Bayesian sheaf models achieve leading performance compared to baseline models and are less sensitive to the choice of hyperparameters under limited training data settings.

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1. Applied & Data

Beyond Persistent Homology: Commutative algebra neural network reveals genetic origins of diseases — JunJie Wee <weejunji@msu.edu>

Topological data analysis (TDA) has achieved remarkable success in molecular sciences over the past decade. Integrating TDA with deep learning has led to advances in drug design, materials discovery, protein engineering, and COVID‑19 research. However, many TDA tools rely heavily on persistent homology, which captures only limited aspects of the underlying algebraic and geometric structures. To move beyond these limitations, we develop new mathematical foundations that bridge pure mathematics with modern AI. Recently, we introduced a multiscale commutative algebra embedding that captures intrinsic physical and chemical interactions in molecular systems for the first time. Using Persistent Stanley–Reisner Theory, we extract algebraic invariants—including facet ideals and $f$‑vectors—to construct a Commutative Algebra Neural Network (CANet). Our approach integrates deep learning with rich algebraic information, producing AI models that are mechanistic, interpretable, and highly generalizable. I will present the mathematical framework of CANet and show how these descriptors reveal structural patterns underlying genetic disease–causing mutations, pushing TDA beyond persistent homology.

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1. GeoGT

Boundedness of homeomorphism groups of portable manifolds — Megha Bhatt <mbhat@gradcenter.cuny.edu>

A group is said to be bounded if it has finite diameter with respect to every bi-invariant metric. This is a strong rigidity property for large groups, limiting the large-scale geometry of the group and the types of geometric actions it can admit. Building on ideas of Burago, Ivanov, and Polterovich, Rybicki proved that the identity component of the homeomorphism group of a portable manifold is bounded. In this talk, I will present a simplified proof of this result by constructing a uniform normal generator for the group.

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1. GeoTop

Brackets of disk bundles and configuration space integrals — Robin Koytcheff <robin.koytcheff@louisiana.edu>

In joint work with Xujia Chen and Sander Kupers, we construct a bracket operation on the space of framed disk bundles of fiber dimension at least 4. Kontsevich used integrals over configuration spaces to produce graph homology classes from classes of disk bundles. We prove that our bracket operation is compatible with these Kontsevich characteristic classes via the bracket operation on graph homology. Applying our bracket to Watanabe’s bundles from Borromean surgery on trivalent graphs, we obtain new disk bundles, some of which are nontrivial.

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1. Continua

Bridging Closed Relations and Shift Systems: A Trichotomy of CR-Dynamical Properties — Judy Kennedy <kennedy9905@gmail.com>

Suppose $X$ is a compact metric space and $F$ is a closed relation on $X$. For a classical dynamical property $\mathcal{P}$, we introduce a natural trichotomy of CR-dynamical properties associated with a closed relation $F$: $CR-\mathcal{P}, CR-post\mathcal{P}, CR-pre\mathcal{P}.$ These three notions are designed so that $(X,F)$ satisfies $CR-\mathcal{P}$ exactly when the shift system $(X_F^+,\sigma_F^+)$ satisfies $\mathcal{P}$; whenever the shift system has property $\mathcal{P}$, the relation $F$ has $CR-post\mathcal{P}$; and whenever $F$ has $CR-pre\mathcal{P}$. the shift system has $\mathcal{P}$. We apply this to minimality, dense-orbit transitivity, and transitivity, establishing precise equivalences in each case. Our examples show that, in general, the three CR-versions of a property form a strict hierarchy, with none of the implications reversible without additional assumptions. This is joint work with Iztok Banic, Matevz Crepnjak, Goran Erceg, Ivan Jelic.

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1. General & ST

C-Spaces and Inverse Systems — Leonard Rubin <lrubin@ou.edu>

The concept of a C-space was introduced in 1978 by D. Addis and J. Gresham in order to provide a new class of spaces in dimension theory. We present an internal characterization for an inverse system $\mathbf{X}$ of compact Hausdorff spaces and maps that shows when its limit will be a C-space. This is precisely when $\mathbf{X}$ is a ``C-system,'' whose definition will be given in this presentation. We use this characterization to construct a C-system $\mathbf{Y}$ so that its inverse limit is a weakly infinite-dimensional, strongly countable-dimensional, metrizable compactum that is a C-space. Finally we introduce a new notion into topological game theory called a game-theoretic C-system.

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1. Dynamics

CANCELED -- Dynamics Arising from Group Actions on Primitive Elements — Pratyush Mishra <mishrap@wfu.edu>

There has been some substantial work on studying the structure of a group by analyzing the behavior of primitive elements, sometimes under strong assumptions by work of Platonov, Potapchik, Shpilrain and many others. We formulate and study a conjecture of Platonov and Potapchik for general group actions via analyzing the dynamics of primitive elements for a given action. Such studies led us to further afield to produce results that combines computational, dynamical, geometric, and purely algebraic viewpoints.

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1. Dynamics

Calculating the Malaugh Operations — Forrest Hilton <fmhilton@uab.edu>

Connected Julia sets of a polynomial generally correspond to laminations, sets of chords of the unit disc that reflect the dynamics of the Julia set. If the circle is measured in revolutions and the polynomials studied are of degree $d$, then the dynamics on the lamination are given by the covering map $\sigma_d(t) := td \pmod 1$ where chords are mapped by their end points. Every lamination has at least one laminational invariant set, which is loosely an invariant complementary component of the lamination. That set has a significant impact on the shape of the Julia set. James Malaugh showed how to topologically transform a laminational invariant set in one degree into one in another degree, but he provided no way to execute these operations concretely. In this talk, we show how to calculate those operations.

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1. Continua

Characterizing local connectedness by non-cut sets in continua — Jorge Vega <vegacevedofc@ciencias.unam.mx>

In this talk we show that for a continuum X, the following conditions are equivalent: (i) the continuum X is locally connected, (ii) each non-cut set of X has arbitrarily small open neighborhoods whose complements are connected, (iii) each non-cut set of X has continuum-wise connected complement, (iv) the continuum X is aposyndetic with respect to each of its non-cut sets, and (v) the continuum X is aposyndetic with respect to each of its nonempty closed sets. Co-authors: Raúl Escobedo and Eduardo García-Muñoz.

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1. GeoTop

Cohomology of Handlebody Torelli Groups — Annie Holden <aholden2@nd.edu>

We begin by introducing the Torelli subgroup of the mapping class group of a surface and outlining known results about its low-dimensional cohomology. We then present recent work extending these results to a Torelli subgroup of the mapping class group of a handlebody.

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1. General & ST

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

Combinatorial covering properties as Rothberger’s, Hurewicz’s and Menger’s are procedures for generating a cover of a given topological space from a sequence of covers of this space. We present the most celebrated such properties together with the most important examples in a classical countable case. We also explore how these notions and examples extend to the uncountable context, where the initial sequence of covers has length $\kappa$ for some uncountable cardinal $\kappa$. In this generalized setting, we replace the Cantor space $2^\omega$ and the classical Baire space $\omega^\omega$ with the $\kappa$-Cantor space $2^\kappa$ and the $\kappa$-Baire space $\kappa^\kappa$, respectively. This is joint work with Piotr Szewczak and Lyubomyr Zdomskyy.

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1. General & ST

Comparative Topology of the Cantor Fan and the Cantor's Teepee — Manuel M. Aguilera <alex.martinez13@upr.edu>

The *Cantor's Fan* is a planar topological space in $\mathbb{R}^2$, constructed from the Cantor set in $[0,1]$ and inspired by the *Cantor's Teepee* introduced in 1921 by Bronisław Knaster and Kazimierz Kuratowski. In this paper, we determine which properties of the Cantor's Teepee persist in the Cantor's Fan; we restate the main properties in contemporary language, provide complete formal proofs, and include illustrative figures.

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1. Dynamics
3. Plenaries

Computable Markov Partitions — Christian Wolf <cwolf@math.msstate.edu>

Computability in dynamical systems is a relatively young field that has attracted significant attention in recent years. One of its central questions is whether dynamically relevant objects can be algorithmically represented by a Turing machine. While this question has been extensively studied in symbolic dynamics, where computability results are known for various thermodynamical quantities such as entropy, pressure, equilibrium states and zero-temperature measures, a corresponding general theory for broader classes of topological and smooth dynamical systems is lacking. In this talk, we present an approach to bridging this gap by introducing the concept of computable Markov partitions. This framework allows us to establish far-reaching computability results for several classes of topological and smooth dynamical systems. The results presented in this talk are joint work with Michael Burr and Tamara Kucherenko.

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1. GeoGT

Conjugator length in finitely presented groups — Francis Wagner <fw294@cornell.edu>

The conjugator length function of a finitely generated group is the function f so that f(n) is the minimal upper bound on the length of a word realizing the conjugacy of two words of length at most n. This function provides a measure for the complexity of a direct approach to the Conjugacy Problem for the finitely generated group. I will discuss what functions can be realized as the conjugator length function of a finitely presented group and the connection of this function with other important invariants of finitely presented groups.

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1. GeoGT

Connectivity of Gromov boundary of the maximized hyperbolic space of right-angled Coxeter groups — Zhihao Mu <zmu@gradcenter.cuny.edu>

The maximized hyperbolic space of a right-angled Coxeter group (RACG) can be obtained from its Davis complex by coning off all standard flats. This space serves as the top-level hyperbolic space in the hierarchically hyperbolic structure of the RACG, analogous to the curve graph for mapping class groups. We provide a necessary and sufficient condition on the defining graph under which the Gromov boundary of the maximized hyperbolic space is connected.

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1. Continua

Coselections on symmetric products — Veronica Martinez-de-la-Vega <vmvm@im.unam.mx>

Given metric continuum X we consider the n-th symmetric product, Fn(X) defined as the hyperspace of nonempty subsets with at most n elements. The continuum X is an Fn-coselection space (n≥2) if for each ε > 0, there exists a mapping gε : X →Fn(X) \F1(X) such that x ∈ gε(x) and diameter(gε(x)) <ε for each x∈X. Answering two questions by Patricia Pellicer-Covarrubias, in this talk we present two significant examples: (a) we prove that a Cook continuum is not an Fn-coselection space for any n ≥2, and (b) there exist two no homeomorphic compactifications of the ray [0,∞) with remainder a simple closed curve which are F2-coselection spaces.

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1. General & ST

Countable dense homogeneity in large products of Polish spaces — Andrea Medini <andrea.medini@tuwien.ac.at>

We will discuss joint work with Juris Steprāns concerning the countable dense homogeneity of products of Polish spaces, with a focus on uncountable products. Our main result states that a product of fewer than $\mathfrak{p}$ Polish spaces is countable dense homogeneous if the following conditions hold: (1) Each factor is strongly locally homogeneous, (2) Each factor is strongly $n$-homogeneous for every $n\in\omega$, (3) Every countable subset of the product can be brought in general position. For example, using the above theorem, one can show that $2^\kappa$, $\omega^\kappa$, $\mathbb{R}^\kappa$ and $[0,1]^\kappa$ are countable dense homogeneous for every infinite $\kappa<\mathfrak{p}$ (these results are due to Steprāns and Zhou, except for the one concerning $\omega^\kappa$). In fact, as a new application, we showed that every product of fewer than $\mathfrak{p}$ connected manifolds with boundary is countable dense homogeneous, provided that none or infinitely many of the boundaries are non-empty. This generalizes a result of Yang.

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1. General & ST

Covers, Stars, and Points, Oh My! — Jocelyn Bell <bell@hws.edu>

The proximal game, introduced in 2014, is a two-player infinite game played in a uniform space. It relies on the uniform structure in an inherent way: the first player chooses elements of the uniformity while the other selects points. A winning strategy for the first player implies the space has certain additional topological properties, which as such are independent of the particular uniform structure with which the game was played. So, is the uniform structure really necessary? I will discuss some recent progress in divorcing the proximal game from its reliance on a uniform structure, resulting in the creation of purely topological "point-star" games.

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1. GeoGT

Criterion for Finiteness of BMS Measure — Rou (Vicky) Wen <rwen5@wisc.edu>

In many useful settings, having a finite Bowen-Margulis-Sullivan (BMS) measure on a flow space allows people to normalize the BMS measure into a probability measure and facilitates powerful ergodic theoretic tools. This often leads to asymptotic estimates for counting orbital points and establishing equidistribution results. Hence, it is important to know when a dynamical system admits a finite BMS measure. In this talk, I will first introduce what is a BMS measure, and then state a criterion that detects the finiteness of BMS measure on a flow space associated to a discrete subgroup of higher rank semi-simple Lie group.

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1. Dynamics

Critical Orbit Relation Curves and Degenerations — Jan Kiwi <jkiwi@uc.cl>

Critical orbit relation curves of rational maps acting on the Riemann sphere are dynamically natural complex one-dimensional slices of moduli space. The aim of the talk is to review some known and new results (work in progress with Caroline Davis and Alex Kapiamba) about degenerations along these curves.

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1. Applied & Data

Cubical Persistent Homology of Hyperspectral Retinal Images — Desiree Paczay <dapaczay629@my.nipissingu.ca>

Topological Data Analysis (TDA) has emerged as a powerful framework for extracting meaningful structure from complex, high-dimensional data. In particular, persistent homology is widely used for its ability to quantify multiscale topological features while exhibiting robustness to noise. In this work, we apply persistent homology to hyperspectral images of retinal tissue in order to further investigate Spaceflight Associated Neuro-ocular Syndrome. Hyperspectral imaging captures vast spectral information, but its high dimensionality poses challenges for analysis and interpretation. For each spectral band, we treat pixel intensity as a scalar function and construct a sublevel set filtration of cubical complexes, which provide a natural cell-complex structure for image data. From the resulting persistence diagrams, we derive summary statistics including total persistence and feature counts in dimension 0. Preliminary results indicate that these persistence-based summaries distinguish between pigmented and albino retinal tissue. Ongoing work focuses on further interpretation of the detected topological structure and the implementation of additional persistence-based methods.

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1. Continua

Degree of homogeneity on some spaces — Alicia Santiago Santos <alicia@mixteco.utm.mx>

Given a positive integer n, a non-empty topological space is said to be 1/n-homogeneous provided there are exactly n orbits for the action of the group of homeomorphisms of the space onto itself, in which case we say that the degree of homogeneity of X, is n. In this talk, I will present our recent contributions to this lines of research.

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1. GeoGT

Dense Conjugacy Classes in the Mapping Class Group of Graphs — Rocky Klein <klein@brandeis.edu>

The mapping class group of a locally finite graph Maps$(X)$ is the set of proper homotopy equivalences of $X$ up to proper homotopy. It is meant to be the analogue of the mapping class group of an infinite-type surface one dimension lower, but it also generalizes Out$(F_n)$ to a much larger class of possibly infinitely generated groups, establishing a "Big Out$(F_n)$." In this talk, I plan to define the mapping class group for a locally finite graph, discuss its topology, and give motivation. I will then discuss which locally finite graphs $X$ are such that Maps$(X)$ contains a dense conjugacy class. Along the way, we will discuss end spaces and their structures.

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1. General & ST

Do monotonically semi-neighborhood refining spaces have well-ordered neighborhood (F)? — Ted Porter <jporter@murraystate.edu>

In 1996, Stares introduced monotonically semi-neighborhood refining (MSNR) spaces and showed that well-ordered neighborhood (F) spaces are MSNR spaces. In this talk, the question of whether MSNR spaces have well-ordered neighborhood (F) is explored. We show that MSNR spaces have well-ordered (F) and hence are monotonically normal and hereditarily paracompact. MSNR spaces are also shown to be lob-spaces. The relationships between MSNR spaces with other monotone covering properties will also be explored.

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1. Dynamics

Dynamical approximation of post-singularly finite entire functions — Malavika Mukundan <mmukunda@bu.edu>

An entire map is said to be post-singularly finite if the forward orbit of its set of singular values is finite. Such maps play a crucial role in understanding natural families of entire maps. Motivated by previous work of Devaney, Goldberg and Hubbard, we ask the following question: _Given a post-singularly finite entire function f, can f be realized as the limit of a sequence of post-singularly finite polynomials?_ In joint work with Nikolai Prochorov and Bernhard Reinke, using techniques from Teichmüller theory, we show how we may answer this question in the affirmative.

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1. Dynamics

Dynamical systems as enriched functors — Suddhasattwa Das <iamsuddhasattwa@gmail.com>

A new advancement is presented in a broad ongoing effort to develop Dynamical systems theory in the language of Category theory. A new idea will be presented to describe a general dynamical systems as an enriched functor, and change of variables as enriched natural transformations. This framework is essential to establish the equivalence of three descriptions of dynamics -- a semigroup action on the domain; a parameterized family of endomorphisms; and a transformation of time-space into the collection of endomorphisms. A collection of categorical axioms are presented that provides a complete categorical language to develop dynamical systems theory. True to the philosophy of dynamical systems, none of these assumptions are rooted in specific contexts such as topology and measure spaces. The equivalence of the three descriptions is further used to construct other related notions,such as transfer operators, orbits and sub-shifts. All of these objects are defined by their structural role and universal properties, instead of their usual pointwise definitions. Source : https://arxiv.org/pdf/2509.05900

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1. Continua

Dynamics of homeomorphisms of the Lelek and Cantor fans — Christopher Mouron <mouronc@rhodes.edu>

This is a continuation of Van Nall's talk {\it Specification on the Lelek Fan}. I will be discussing examples and non-examples of homeomorphisms of the Lelek and Cantor fans with the following properties: transitivity, mixing, shadowing, the specification property and maybe a few more.

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1. Continua

Dynamics on Fences — Jernej Cinc <jernej.cinc@um.si>

We call a fence any compact metric space whose connected components are either points or arcs. In this talk I will present a very general method for raising maps of the Cantor space to various fences with dense set of endpoints, such as the Lelek Fence (in Complex Dynamics known also as the Hairy Cantor set) and Fraïssé Fence, while preserving the dynamics of the base homeomorphism of the Cantor space. As simple corollaries we obtain that Lelek Fan admits homeomorphisms with various dynamical properties.

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1. Applied & Data

Early Functional Brain Network Alterations and Longitudinal Progression of Asymptomatic Alzheimer’s Disease — Altansuren Tumurbaatar <altaamgl@gmail.com>

As Alzheimer’s disease pathology begins decades before clinical symptoms, early functional brain changes in asymptomatic Alzheimer’s disease (AsymAD) remain poorly characterized, particularly from a longitudinal perspective. Although AsymAD individuals are biomarker-positive for Alzheimer’s pathology, they remain cognitively unimpaired, representing a preclinical stage that is clinically silent yet biologically active. At this stage, conventional MRI markers and cross-sectional analysis often lack sensitivity to detect the subtle functional alterations that precede symptom onset. Consequently, compared to symptomatic Alzheimer’s disease, AsymAD is substantially more difficult to identify using imaging markers alone, necessitating more sensitive, network-level, and longitudinal approaches. Resting-state functional MRI provides a noninvasive framework for probing intrinsic functional brain networks and detecting early network-level disruptions. Since brain networks exhibit a small-world topology, defined by high local clustering (segregation) and short average path lengths (integration), graph-theoretical metrics sensitive to subtle perturbations related to these properties may provide early network-level signatures of AsymAD. In this study, we examine longitudinal changes in functional connectivity in AsymAD compared with cognitively normal controls using complementary connectivity approaches, including region-to-region connectivity (RRC), graph-theoretical metrics, and seed-based connectivity (SBC). General linear models are used to assess between-subject and repeated-measure effects, with primary emphasis on group-by-session interactions.

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1. GeoGT
3. Plenaries

End-periodic homeomorphisms and volumes of mapping tori — Elizabeth Field <ecfield@uw.edu>

In this talk, we will introduce the notion of an end-periodic homeomorphism of an infinite-type surface. We will explore how the geometry of the associated mapping torus is related to certain topological and dynamical features of the end-periodic gluing map. In particular, we will see how the hyperbolic volume of the 3-manifold can be bounded both above and below in terms of a certain dynamical feature of the homeomorphism. This talk represents joint work with Autumn Kent, Heejoung Kim, Christopher Leininger, and Marissa Loving (in various configurations)

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1. GeoTop
3. Plenaries

Exotic aspherical 4-manifolds — Kyle Hayden <kyle.hayden@rutgers.edu>

The Borel conjecture predicts that closed, aspherical manifolds (i.e., those with contractible universal cover) are topologically rigid: they are determined up to homeomorphism by their fundamental group. I will discuss the smooth version of this conjecture (concerning manifolds up to diffeomorphism), which is true in dimensions ≤ 3 but long known to be false in all dimensions ≥ 5. I will explain joint work with Davis, Huang, Ruberman, and Sunukjian that resolves the remaining 4-dimensional case by detecting exotic smooth structures on certain closed aspherical 4-manifolds.

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1. Applied & Data
3. Plenaries

Fiber bundles of toric arrangements — Christin Bibby <bibby@math.lsu.edu>

We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups and discuss topological consequences. These bundles are akin to those of Fadell-Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. Of particular focus is the case of a toric arrangement: a finite collection of codimension-one subtori in a complex torus. If the intersection pattern of the subtori satisfies the combinatorial condition of supersolvability, the complement of the toric arrangement sits atop a tower of fiber bundles. This structure provides insight into topological invariants of these toric arrangement complements, including the homotopy groups, cohomology, and topological complexity. Based on joint work with Daniel C. Cohen and Emanuele Delucchi.

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1. GeoGT

Filling Links and Essential Systole — Yandi Wu <yw220@rice.edu>

The systole of a hyperbolic 3-manifold is the length of the shortest closed geodesic. Given a closed 3-manifold M and link L such that M\L is hyperbolic, the essential systole of M\L is the length of the shortest closed geodesic which is not nullhomotopic in M. In this talk, we will discuss and motivate the study of essential systoles of hyperbolic link complements, including their application towards answering a question of Freedman and Krushkal about the existence of "filling links" in closed 3-manifolds.

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1. Dynamics
3. Plenaries

Finite time evolution and finite time predictions for dynamical and random systems. — Leonid Bunimovich <leonid.bunimovich@math.gatech.edu>

TBA

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1. Continua

Fraïssé fence "with" pseudo-arcs? — Bryant Rosado Silva <bryantrs99@hotmail.com>

Using as inspiration known and new results about the Fraïssé fence and the flow of its homeomorphism group on the space of chains of compacta, we suggest a new space by using pseudo-arcs instead of arcs and briefly discuss some of our questions and motivations. This is an ongoing project with Benjamin Vejnar (Charles University).

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1. Applied & Data
3. Plenaries

From Descriptors to Interfaces: Visual Analytics for Topological Data Analysis — Federico Luricich <fiurici@clemson.edu>

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1. Continua

Gehman Dendrite G3 as a Generalized Inverse Limit Space — Faruq Mena <faruq.mena@soran.edu.iq>

We show that the family of functions in the paper by Sherzad and Mena that give $G_3$ as the inverse limit space, can be expanded to include upper semi-continuous functions whose graphs have a finite number (or even one) of ``short'' line segments of the form $[x_1,\alpha]\times \{a_i\}$ and $[\alpha,x_2] \times \{b_i\}$ where $0 < x_1 < \alpha < x_2 <1$. This is joint work with Sarezh R. Rasul.

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1. Continua
3. Plenaries

General Topology in Dynamical Systems — Hisao Kato <kato.hisao.fw@u.tsukuba.ac.jp>

Research in general topology is important for the study of dynamical systems. The complexity of dynamical systems suggests the existence of complex topological structures in their base spaces. This lecture will discuss the following two topics: (Part 1) Extended Takens-type reconstruction theorems for one-sided dynamical systems, and (Part 2) The existence of indecomposable continua in chaotic dynamical systems.

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1. General & ST

General and Set-Theoretic Topology Problem Session — Will Brian <wbrian.math@gmail.com>

Session participants are invited to join an interactive problem session discussing open questions in our discpline.

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1. Continua

Generalized Inverse Limits and a Property of Kelley — Mardan A. Pirdawood <mardan.ameen@koyauniversity.org>

Ingram, \cite[Problem 6.56, p.81]{ingram2012introduction}, asked what can be said about the Property of Kelley in the generalized inverse limit space $\varprojlim\{X_i,f_i\}$ where $\{f_i\}$ is a sequence of upper semi-continuous bonding functions. In this work, we give conditions on the projection maps from the graph of the functions $f_i$ to the domain and co-domain such that if the first factor space, in the case of Theorem 2.2, or all factor spaces, in the case of Theorem 2.4, have the Property of Kelley then the generalized inverse limit space $\varprojlim\{X_i,f_i\}$ has the Property of Kelley. Furthermore, we present examples demonstrating that if any condition is dropped then the inverse limit space may not have the Property of Kelley. These results also answers several questions by Charatonik, Mena and Roe \cite{Charatonik2020}. This is joint work with Faruq Mena and Robert Roe.

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1. Continua

Generalized Inverse Limits on Circles — Scott Varagona <svaragona@montevallo.edu>

It has now been twenty years since the publication of W. T. Ingram and W. S. Mahavier’s landmark paper, “Inverse limits of upper semi-continuous set valued functions” (Houston Journal of Mathematics, 2006, vol. 32, no. 1, p. 119-130). For all these years, generalized inverse limits whose factor spaces are arcs have been studied intensively by researchers around the world. However, generalized inverse limits whose factor spaces are circles have been far less thoroughly studied, and could offer researchers a whole new frontier to explore. We state some questions about these spaces and provide various examples, including an example of a generalized inverse limit on circles (with a single, continuum-valued bonding function) that gives rise to an indecomposable continuum.

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1. Dynamics

Generic continuous Lebesgue measure preserving interval maps are nowhere monotone but invertible a.e. — Jozef Bobok <jozef.bobok@cvut.cz>

We consider all continuous maps of the interval preserving the Lebesgue measure $\lambda$ equipped with the uniform topology. Except for the identity map or $1 - id$ all such maps have topological entropy at least $\log2/2$ and generically they have infinite topological entropy. In this talk we discuss two generic properties: (i) invertibility $\lambda$-a.e. implied by the zero measure-theoretic entropy with respect to $\lambda$, and (ii) complicated structure of level sets. We also recall that there are Besicovitch maps (having no finite or infinite unilateral derivative at any point) preserving $\lambda$ and show that each such map has positive measure-theoretic entropy with respect to $\lambda$.

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1. Continua

Genericity of Shadowing — Jonathan Meddaugh <jonathan_meddaugh@baylor.edu>

A dynamical system is said to have the shadowing property provided that approximate orbits are well-approximated by true orbits. It has previously been established that for a continuum belonging to certain classes of continua, shadowing is a common, i.e. generic, property in its space of continuous self-maps. In particular, this is known for manifolds and for locally connected one-dimensional continua. We demonstrate that shadowing is a generic property in the space of continuous self-maps for any continuum which admits retractions onto graphs.

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1. Applied & Data

Graph polynomial encoding for RNA structure data analytics — Pengyu Liu <pengyu.liu@uri.edu>

Advancements in sequencing technologies have produced a wealth of genomic data. In parallel, the development of artificial intelligence has enabled novel folding models that predict molecular structures from sequences. These advancements have resulted in a myriad of biomolecular structure data. Analytics of structure data offers more accurate approaches to genotype-to-phenotype analyses, as biomolecular structures are more evolutionarily conserved than sequences and more directly linked to biological functions. A major challenge of structure data analytics is the lack of efficient and accurate structure encodings. In this talk, we introduce encodings of RNA secondary structures using polynomial invariants of graphs. We show that the graph polynomial encodings enable efficient, accurate and interpretable RNA secondary structure analyses using modern data analytics tools. 

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1. Applied & Data
3. Plenaries

Gromov-Wasserstein distances and distributional invariants — Tom Needham <tneedham@fsu.edu>

Gromov-Wasserstein (GW) distances provide a method for comparing probability measures defined on different metric spaces, thereby giving an optimal transport-inspired variant of the well-known Gromov-Hausdorff distance. As GW distances admit computationally tractable approximations, they have become popular in machine learning applications where one wishes to learn trends in a dataset consisting of incomparable spaces, such as ensembles of graphs. In this talk, I will overview recent advances in the theory of GW distances. In particular, I will discuss a certain approximation technique which relies on comparing the distributions of pairwise distances between metric measure spaces. This approach naturally gives rise to fascinating questions about the geometrical and topological features that are encoded in this distributional information, and I will explain some partial answers to these questions.

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1. GeoTop

Gromov’s Conjecture for Graph Product of Groups — Satyanath Howladar <showladar@ufl.edu>

Gromov defined macroscopic dimension of metric spaces to study manifolds admitting a Positive Scaler Curvature (PSC) metric, via their largeness properties. He conjectured universal cover of PSC n-manifolds should have macroscopic dimension at most n-2. This conjecture depends heavily on the fundamental group of the n-manifold. Under the assumption of the Strong Novikov Conjecture, we prove that closed spin manifolds having fundamental group a Graph Product of geometrically finite groups satisfies the conjecture, provided the vertex groups have classifying space which becomes wedge sum of Moore Spaces, after finitely many suspension. This generalizes our previous result when the fundamental group is a RAAG. We developed a crucial property called 1-Step Stabilization Property (1-SSP) for groups to prove the above. We also found an examples of groups not satisfying 1-SSP, inspiring more investigation towards possible counter example related to Gromov’s conjecture.

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1. Applied & Data

Group actions on Vietoris-Rips complexes of hypercube graphs — Federico Galetto <f.galetto@csuohio.edu>

The hyperoctahedral group is the group of symmetries of the hypercube graph. It acts on the Vietoris-Rips complexes of the hypercube graph with the Hamming distance and, therefore, on their homology groups. I will present a method to understand this action and show how it can be used as an alternative approach to compute homology. This is joint work with Jonathan Montaño and Zoe Wellner.

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1. General & ST

Higher Lindelöf trees — Pedro Marun <marun@math.cas.cz>

Given an infinite cardinal $\kappa$ and a $\kappa$-splitting $\kappa^+$-tree $T$, we topologize $T$ as follows: if $x\in T$, then sets of the form $\uparrow x \setminus \uparrow F$, for $F$ a set of immediate successors of $x$ with $|F|<\kappa$, form a basis of neighbourhoods of $x$. We then ask whether $T$ is $\kappa^+$-compact with respect to this topology and characterize this property in purely order-theoretic terms. Such trees are necessarily $\kappa^+$-Aronszajn, so they may (consistently) not exist when $\kappa\ge\aleph_1$. In this talk, discuss how to construct such trees using Proxy Principles, introduced by Brodsky and Rinot. We will also mention a further consitency result on the non-existence of such trees together with the failure of the tree property at $\aleph_2$. This is joint work with Ari Meir Brodksy.

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1. Continua
3. Plenaries

How complex is the arc-connection relation? — Benjamin Vejnar <benvej@gmail.com>

For a continuum, we consider the equivalence relation in which two points are equivalent if they can be joined by an arc. This equivalence relation is analytic in general (i.e. a continuous image of a Polish space). Recently, Debs and Saint Raymond proved that, for planar continua, this equivalence relation is always Borel measurable. We show that for every planar continuum, the arc-connection relation is in fact Borel reducible to the Vitali equivalence relation, where two real numbers are equivalent if their difference is rational. Moreover, the Knaster continuum is an example where this complexity is attained. This is joint work with Michal Hevessy and Yusuf Uyar. We also investigate several related questions concerning continuum-wise connectivity.

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1. GeoTop
3. Plenaries

Hyperbolic manifolds: Past, present, future — Matthew Stover <mstover@temple.edu>

The most basic Riemannian manifolds are those admitting a complete metric of constant curvature. The classification of closed manifolds with metrics of positive and zero curvature is has been relatively well-understood for quite a long time. Constant curvature -1 manifolds, hyperbolic manifolds, remain quite a bit more mysterious, particularly in high dimensions. I will give a (biased) narrative regarding what we know, including a number of exciting recent results with connections to dynamics and geometric group theory, and look forward to some problems I hope to see solved in the coming years.

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1. GeoTop

Immersed incompressible surfaces in hyperbolic manifolds — Zhenghao Rao <zhenghao.rao@rutgers.edu>

The study of surface subgroups in 3-manifolds has drawn sustained attention for decades, motivated both by their intrinsic geometric richness and by their broad consequences in geometric topology, geometric group theory, and dynamics. A landmark result is the Surface Subgroup Theorem of Kahn–Markovic, which states that every cocompact Kleinian group contains a ubiquitous collection of closed surface subgroups. In this talk, we will introduce some key developments in the subject and highlight our recent progress, including joint work with Jeremy Kahn, and with Xiaolong Han and Jia Wan.

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1. Continua

Induced Dynamics on Hyperspaces: Periodic Points and Li–Yorke Chaos — Leonel Rito Rodríguez <leonel_rito@ciencias.unam.mx>

In this talk, we study the dynamical behavior of hyperspace maps induced by continuous functions on dendrites. Our main goal is to show that if $X$ is a dendrite and $f : X \to X$ is a continuous map for which every point of $X$ is periodic, then the induced map \[ 2^f : 2^X \to 2^X \] does not admit Li--Yorke pairs. To establish this result, we analyze two fundamental cases that capture the combinatorial structure of dendrites: closed intervals and trees.

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1. GeoTop

Infinitely many Lefschetz pencils on ruled surfaces — Seraphina Eun Bi Lee <slee@math.harvard.edu>

Works of Donaldson and Gompf show that a closed, oriented 4-manifold admits a symplectic structure if and only if it admits the structure of a Lefschetz pencil. However, the question of how many Lefschetz pencils (or fibrations) a given symplectic 4-manifold admits remains open. Works of Park--Yun and Baykur construct 4-manifolds admitting arbitrarily large (but finite) numbers of Lefschetz pencils or fibrations of the same genus. In this talk, we will construct infinitely many inequivalent Lefschetz pencils of the same genus on ruled surfaces of negative Euler characteristic. In fact, our construction gives the first example of infinitely many inequivalent but diffeomorphic Lefschetz pencils and fibrations of the same genus. This is joint work with Carlos A. Serván.

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1. GeoGT

Injective metrics and affine hyperplane arrangements — Katherine Goldman <kat.goldman@mcgill.ca>

A complex affine hyperplane arrangement is a locally finite collection of affine hyperplanes (complex codimension-1 subspaces) in a finite dimensional complex affine space. Since these subspaces have complex codimension 1, the complement of their union is a connected manifold. It is a broad, longstanding problem with many connections to different areas of mathematics to determine the arrangements for which this manifold is aspherical (has contractible universal cover). A subset of this problem dating back to the 1970s, commonly attributed to Arnol’d, Brieskorn, Thom, and Pham, concerns arrangements arising from reflection groups in real affine space. One approach that has seen success is to construct a cell complex which is homotopy equivalent to this complement and endow it with some kind of ("singular") non-positive curvature. Along these lines, by showing that a specific cell complex (based on a construction of Falk) carries an injective metric, we show that a broad class of affine arrangements (including the infinite families of affine reflection arrangements, modulo a conjecture about $D_n$-type) have aspherical complement. In particular, this provides some of the first examples of infinite affine arrangements which have aspherical complement, but do not arise from reflection groups. This is joint work with Jingyin Huang.

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1. Dynamics

Interval maps mimicking circle rotations — Fryderyk Falniowski <falniowf@uek.krakow.pl>

We investigate the dynamics of maps of the real line whose behavior on an invariant interval is close to a rational rotation on the circle. We focus on a specific two-parameter family, describing the dynamics arising from models in game theory, mathematical biology and machine learning. If one parameter is a rational number, k/n, with k, n coprime, and the second one is large enough, we prove that there is a periodic orbit of period n. It behaves like an orbit of the circle rotation by an angle 2 π k/n and attracts trajectories of Lebesgue almost all starting points. We also discover numerically other interesting phenomena.

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1. GeoTop

Isotopy versus Equivariant Isotopy — Trent Lucas <trentl1@uci.edu>

Given a finite group action on a manifold, we discuss the following question: if two equivariant diffeomorphisms are isotopic, must they be equivariantly isotopic? In the case of closed hyperbolic surfaces, a remarkable theorem of Birman and Hilden says that the answer is “yes”: isotopy implies equivariant isotopy. By contrast, we show that in dimensions three and higher, there are many diffeomorphisms which are isotopic but not equivariantly isotopic. We will explain the new obstructions that arise in higher dimensions, as well as some applications and further questions that don’t arise in the world of surfaces.

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1. GeoGT
3. Plenaries

Large-scale geometry of right-angled Coxeter groups — Pallavi Dani <pdani@math.lsu.edu>

Right-angled Coxeter groups form an extremely accessible, yet remarkably rich class of objects in geometric group theory. They are defined by simple presentations: they are generated by involutions, with the only additional relations requiring certain pairs of generators to commute. Despite this elementary definition, they display an extraordinary range of geometric behaviors. Consequently, they have played a crucial role in the field, as a source of illuminating examples and counterexamples and as a testing ground for conjectures. In this talk, I will survey recent progress in understanding their large-scale geometry, focusing in particular on questions of quasi-isometry and commensurability. Along the way, I will illustrate some of the main tools and techniques used for establishing such results, many of which are applicable in more general settings.

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1. Continua

Local Variations of Shadowing — Ellie Stephens <ellie_stephens2@baylor.edu>

It is known that under the assumption of chain transitivity, shadowing is equivalent to other, weaker variations of shadowing. For example, a sequence of points in a continuum may act as a pseudo-orbit only on a thick set. We know that such a sequence can be shadowed on a different thick set under the assumption of chain transitivity and shadowing, but we lose information about where the pseudo-orbit begins. To address this, we study a form of shadowing in which the pseudo-orbit is shadowed on a thick set $T \subseteq \mathbb N$ such that $1 \in T$. We discuss the relationship of this form of shadowing with the standard shadowing property in the context of dynamical systems on continua.

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1. Applied & Data

Lower-bounding the Gromov-Hausdorff Distance Between Balls — Kushagri Sharma <kushagrisharma@ufl.edu>

We lower bound the Gromov-Hausdorff distance between Euclidean unit balls of different dimensions, $d_{GH}(B^m,B^n)$ for $m>n$. This is significant because the standard persistent homology lower bound is zero, since all balls possess trivial persistent homology. Our most powerful approach to lower bound the Gromov--Hausdorff distance between Euclidean unit balls of different dimensions leverages the Borsuk-Ulam theorem. We exploit the fact that any continuous map between a sphere and a ball of appropriate dimensions must identify antipodal points. This yields a positive metric distortion and a computable lower bound.

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1. GeoGT

Manifold models for hyperbolic graph braid groups — Saumya Jain <sjain15@lsu.edu>

Given a finite graph $\Gamma$, the associated *graph braid group* $B_n(\Gamma)$ is the fundamental group of the unordered $n$-point configuration space of $\Gamma$. Genevois classified which graph braid groups are Gromov hyperbolic and asked the question: When do these groups arise as $3$-manifold groups? In this talk, we give a partial answer for $B_3(\Theta_m)$ where $\Theta_m$ is the *generalized $\Theta$-graph*.

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1. GeoTop

Mapping class groups and Freudenthal compactifications of infinite type surfaces — Jeremy Brazas <jbrazas@wcupa.edu>

Let $\textbf{MCG}(X)$ denote the group of isotopy classes of self-homeomorphisms of a space $X$. When $S$ is an orientable infinite type surface $S$ with no planar ends and without boundary, the extended mapping class group $\textbf{MCG}(S)$ is isomorphic to $\textbf{MCG}\left(\overline{S}\right)$ where $\overline{S}$ is the Freudenthal compactification of $S$. Using this identification, it follows that $\textbf{MCG}(S)$ canonically embeds into $\text{Out}\left(\pi_1\left(\overline{S}\right)\right)$. It remains open if $\textbf{MCG}(S)$ is isomorphic to $\text{Out}\left(\pi_1\left(\overline{S}\right)\right)$ in the spirit of the Dehn-Neilsen-Baer Theorem. A clear difficulty is the fact that $\overline{S}$ is not locally simply connected and $\pi_1\left(\overline{S}\right)$ is uncountable and not free. In this talk, we will show that $\overline{S}$ is the quotient of $\mathbb{D}^2$ by a countable edge-pairing on $\mathbb{S}^1$. This structural decomposition implies that $\overline{S}$ may constructed by attaching a single 2-cell to a one-dimensional Peano continuum. Using established technology for dealing with fundamental groups of one-dimensional Peano continua, we show that $\pi_1\left(\overline{S}\right)$ is the free product with amalgamation of two locally free groups along an infinite cyclic group. We also show that every automorphism $\phi:\pi_1\left(\overline{S}\right)\to\pi_1\left(\overline{S}\right)$ is induced by a continuous map $f:\overline{S}\to \overline{S}$ that restricts to a homeomorphism on the end set $\overline{S}\backslash S$.

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1. GeoGT

Metrizability of Furstenberg boundaries — Sumun Iyer <sumuni@andrew.cmu.edu>

Let G be a Polish group. A G-flow is a continuous action of G on a compact Hausdorff space. G is amenable if every G-flow has an invariant probability measure. G has metrizable universal minimal flow if every G-flow contains a metrizable G-flow. We consider the property of G having a metrizable Furstenberg boundary. This is a common weakening of amenability and having metrizable universal minimal flow. We prove a characterization of G having metrizable Furstenberg boundary in the spirit of Kechris-Pestov-Todorcevic, Bartosova, Moore, and Zucker. We show mapping class groups of infinite type surfaces never have a metrizable Furstenberg boundary. This strengthens a theorem of Long that such groups are not amenable. This is joint work with George Domat and Forte Shinko.

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1. GeoTop

Monodromy of curves in a simply connected surface (joint with Nick Salter) — Ishan Banerjee <banerjee.238@osu.edu>

We compute the image of the monodromy representation (as a subgroup of the mapping class group) associated to a complete linear system of curves in a simply connected smooth projective surface X under some ampleness hypotheses. It turns out to always be of finite index.

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1. GeoTop

Morse theory on moduli of curves — Changjie Chen <changjie.chen@umontreal.ca>

In 1997, Sarnak conjectured that the determinant of the Laplacian is a Morse function on the space of unit area Riemannian metrics on a given real surface, and hence induces a Morse function on its moduli space. Meanwhile, the systole function, defined as the length of a shortest essential closed geodesic with respect to the base Riemannian metric, is topologically Morse on the Teichmüller space of n-dimensional flat tori (due to Ash) and of Riemann surfaces of genus g with n marked points (due to Akrout), though it does not yield a classical Morse theory. In this talk, I will introduce a family of Morse functions, denoted sys_T, defined as weighted exponential averages of all geodesic-length functions, on the Deligne--Mumford compactification (M_{g,n} bar). These functions are compatible with the Deligne--Mumford stratification and the Weil--Petersson metric, and their critical points can be characterized by a combinatorial property named eutaxy. I will talk about the index gap theorem for sys_T and its homological consequences, in the form of a stability theorem for the homology of moduli spaces of stable curves. I will also briefly explain how sys_T connects to Sarnak’s conjecture.

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1. Applied & Data

Multi-parameter Čech complexes — Carl Ye <jye1@ufl.edu>

In "A Multicover Nerve for Geometric Inference" paper, Sheehy shows that filtering the barycentric decomposition of a Čech complex by the cardinality of the vertices recovers exactly the topology of k-covered regions among a collection of balls. We describe this construction and present ideas related to this.

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1. Applied & Data

MultiPersistence Topological Fusion with Vision Transformers for Skin Cancer Detection — Sayoni Chakraborty <sayoni.chakraborty@utdallas.edu>

Skin cancer is a common and potentially fatal disease where early detection is crucial, especially for melanoma. Current deep learning systems classify skin lesions well, but they primarily rely on appearance cues and may miss deeper structural patterns in lesions. We present TopoCon-MP, a method that extracts multiparameter topological signatures from dermoscopic images to capture multiscale lesion structure, and fuses these signatures with Vision Transformers using a supervised contrastive objective. Across three public datasets, TopoCon-MP improves in-distribution performance over strong pretrained CNN and ViT baselines, and in cross-dataset transfer, it maintains competitive performance. Ablations show that both multiparameter topology and contrastive fusion contribute to these gains. The resulting topological channels also provide an interpretable view of lesion organization that aligns with clinically meaningful structures. Overall, TopoCon-MP demonstrates that multipersistence-based topology can serve as a complementary modality for more robust skin cancer detection.

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1. GeoTop

Negatively Curved Einstein Metrics — Barry Minemyer <bminemyer@commonwealthu.edu>

Gromov and Thurston famously used hyperbolic branched cover manifolds to construct the first examples of manifolds which admit a pinched negatively curved metric, but do not admit any locally symmetric metric. Much more recently, Fine and Premoselli (n=4) and Hamenstadt and Jackel (n>4) proved that many of these hyperbolic branched covers admit negatively curved Einstein metrics. In this talk I will give an overview of these results and show how, in joint work with Lafont, we extended the construction of Fine and Premoselli to complex hyperbolic branched covers. This gives an explicit description of the first known negatively curved Kahler-Einstein metric on a manifold which does not admit a locally symmetric metric, whose existence was first proved by Guenancia and Hamenstadt.

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1. GeoTop
3. Plenaries

Nielsen realization problems in the Zimmer program — Bena Tshishiku <bena_tshishiku@brown.edu>

The Zimmer program seeks to classify smooth actions of arithmetic groups, like SL(n,Z), on compact manifolds. Separately, the Nielsen realization problem asks when a subgroup of a mapping class group Mod(M) can be realized by a group of diffeomorphisms of M. In many natural situations, the mapping class group is closely related to an arithmetic group, and the realization problem is tied to the Zimmer program. I will discuss examples of this connection and describe some recent results and open questions.

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1. Continua

Non-existence of a Common Model for a Class of Indecomposable Continua — Eiichi Matsuhashi <matsuhashi@riko.shimane-u.ac.jp>

In 1971, Bellamy proved that every continuum can be embedded in an indecomposable continuum as a retract. In 2017, together with Fukaishi, we showed that any continuum 𝑍 can be embedded as an open retract with Cantor set fibers in an indecomposable continuum, which is obtained as the closure of a countable union of topological copies of 𝑍. Building on this construction, together with Ortega, we investigate, for a fixed continuum 𝑍, the class of all indecomposable continua arising in this manner. We present the result that this class admits no common model, in the sense that there exists no single continuum admitting continuous surjections onto all members of the class. If time permits, we will also discuss other classes of continua that admit no common models.

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1. Dynamics

Observable attractors and typical dynamics on the interval — Piotr Oprocha <oprocha@icloud.com>

In this talk we will compare three versions of attractors with large (in the sense of Lebesgue measure) basins of attraction: Milnor, statistical, and physical (in the sense of Ilyashenko). The emphasis will be put on typical continuous (i.e. in topology of uniform convergence) dynamical systems on the unit interval. We will go beyond what is known so far about characteristics of these attractors. We will also explain why in the typical family the attractors depends continuously on the map with respect to the Hausdorff metric. The talk is based on joint work with Magdalena Forys-Krawiec, Jana Hantakowa and Michal Kowalewski

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1. General & ST

On hyperconvexity in partial metric spaces — Dariusz Bugajewski <ddbb@amu.edu.pl>

The notion of hyperconvexity of metric spaces was introduced by Aronszajn and Panitchpakdi in 1956 in their study of extensions of uniformly continuous mappings between metric spaces. From the topological point of view, a hyperconvex space is an absolute retract via a nonexpansive retraction. By the theorem of Nachbin and Kelley, hyperconvex real Banach spaces can be treated as Stonian spaces C(K) of all real-valued continuous functions on extremally disconnected compact Hausdorff spaces K. On the other hand, the notion of a partial metric space was introduced by Matthews in 1994. He showed, roughly speaking, how metric-like tools can be extended to non-Hausdorff topologies, and he also indicated some applications of this class of spaces in the study of the denotational semantics of programming languages. Further applications of partial metrics can also be found in the geometry of Banach spaces. In this talk, we present several different approaches to defining hyperconvexity in partial metric spaces. In particular, we show that the analogue of the Aronszajn–Panitchpakdi notion of hyperconvexity fails to possess certain key properties present in the classical metric setting. Finally, we outline some perspectives for further research.

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1. General & ST
3. Plenaries

On some new star selection principles among covering properties and separability — Davide Giacopello <davide.giacopello@unime.it>

We present some properties recently introduced by Bal and Bhowmik: the $R$-, $H$-, and $M$-star Lindelöfness. These properties are defined via selection principles involving the star operator and lie between covering properties (in particular, star-covering properties) and certain selective strengthenings of separability. This dual perspective leads to several implications and connections among known properties, some of which we present. Additionally, we provide some examples. In particular, we prove that there exists a Tychonoff M-star Lindelöf space of cardinality $\frak{c}$ which is not $R$-star Lindelöf answering a question posed by Bal and Bhowmik.

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1. Dynamics

On the Holomorphic and Random Dynamics for some examples of higher rank Free Groups generated by Hénon type maps — Andres Quintero-Santander <aequinte@iu.edu>

We study the Holomorphic and Random Dynamics of some rank 2 free groups generated by two Hénon type maps. For these simply constructed examples we prove that the Fatou set is non-empty and that the stationary measures are supported on a compact set. With some further care this allows us to construct examples having no stationary measures. These examples illustrate the types of phenomena that may arise when studying holomorphic group actions on non-compact manifolds.

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1. Dynamics
3. Plenaries

On the Mandelbrot set and the MLC Conjecture — Dzmitry Dudko <dzmitry.dudko@stonybrook.edu>

The Mandelbrot set encodes the dynamical dependence of quadratic polynomials on a parameter. The MLC Conjecture (asserting that the Mandelbrot set is locally connected) is a rigidity property that yields a satisfactory topological description of the Mandelbrot set. In this talk, we describe the historical motivation for the conjecture, explain how it became a central topic in Renormalization Theory (analyzing first-return maps to small neighborhoods of special points), and outline some of the ideas behind the most recent advances toward MLC and related questions.

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1. GeoGT

Order-Preserving Braids via the Burau Representation — Jonathan Johnson <jcj055@shsu.edu>

I will discuss a new sufficient condition for when a braided link, a braid closure together with its braid axis, is bi-orderable meaning the fundamental group of its exterior admits an order invariant under both left and right multiplication. In 2006, Perron and Rolfsen provided a condition which ensures that an automorphism of a free group preserves a bi-order of the free group and shows that many fibered 3-manifolds are bi-orderable, including many exteriors of braided links. Recent work of Khanh Le and I provide another new criterion, via the Burau representation, for a free group automorphism to be order-preserving. Using the new criterion, we produce new examples of bi-orderable braided link groups including some examples produced from braids whose underlying permutation is a full cycle which answers in affirmative a question of Kin and Rolfsen. This work is partially supported by the NSF grant DMS-2213213.

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1. Dynamics

Parabolic Implosion via Blaschke Product — Ricky Simanjuntak <rsimanju@iu.edu>

Classic theory of Parabolic Implosion deals with perturbation around parabolic point using Lavaurs theory. Here I will present a new approach using perturbation and rescaling limit of Blaschke product. As a consequence I will show a necessary and sufficient condition for continuity of Julia set around $z^2 + z$, allowing only movement within main hyperbolic component.

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1. Continua

Period Bounds and Induced Systems in the Hyperspace of Continua in One Dimension — Domagoj Jelic <djelic@pmfst.hr>

Given a self-map $f$ of a compact metric space $X$, one can associate to it the induced mappings $\overline{f}$ and $\tilde{f}$ on the hyperspace $2^X$ of compact subsets of $X$ and on the hyperspace $C(X)$ of continua in $X$, respectively, both defined in a natural way. Within this framework, it is natural to investigate the relationship between individual and collective dynamics. In this talk, we address the following question. Let $f$ be a self-map of a topological tree $T$, and let $x$ be a periodic point of $f$ with period $p$. What are the possible periods of periodic points of $\left(C(T), \tilde{f}\right)$, that is, of periodic subtrees containing $x$? We then discuss the significance of this result for the study of further properties of the system $\left(C(T), \tilde{f}\right)$. In particular, using this result, we show that the induced system is always almost equicontinuous and we characterize its Birkhoff center. \emph{The talk is based on a joint work with Piotr Oprocha.}

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1. Continua

Persistent Recurrence and Inverse Limits of Unimodal Maps — Lori Alvin <lori.alvin@furman.edu>

Given a unimodal map, the recurrent critical point $c$ is reluctantly recurrent if there exists a $\delta > 0$ such that for every $\ell\in \mathbb{N}$ there is a backward orbit $\overline{x} = (x_{-\ell},\ldots, x_{-2},x_{-1},x)$ in $\omega(c)$ such that $B(x,\delta)$ has a monotonic pull-back along $\overline{x}$; otherwise we say that $c$ persistently recurrent. Given a unimodal map $f$ with an infinite kneading sequence, it is known that the collection of endpoints for the inverse limit space $\varprojlim \{[c_2,c_1],f \}$ is precisely the collection of folding points if and only if $c$ is persistently recurrent. We revisit this known result and also show that when $c$ is infinitely recurrent and longbranched, then it is not possible for $c$ to be persistently recurrent.

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1. Continua

Plane continua, canals and dead ends — Rene Gril Rogina <rene.gril@um.si>

Given a continuum $X$ in the Euclidean plane, a canal of $X$ is a way of "approaching" the continuum from outside of $X$ or the bounded components of its complement. Often we search for simple dense canals, which are canals and also rays with $X$ as their remainder. While some things are known about planar continua with embeddings that admit such canals, there are still open questions on this topic. In this talk, we first define canals and then "dead ends", which are used in a construction to obtain new planar continua and new embeddings of these continua, all of which have canals with the desired properties. This is joint work with my PhD advisor Jernej Činč. This work was co-financed by the Slovenian Research and Innovation Agency (ARIS) under Contract No. SN-ZRD/22-27/0552.

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1. Dynamics

Prime periods on the interval — Gabriel Fuhrmann <gabrielfuhrmann@gmail.com>

Given two continuous self-maps f and g on the interval which have all periodic orbits in common (that is, O(x)={x,f(x),...,f^(p-1)(x)} is a p-periodic orbit of f if and only if it is a p-periodic orbit of g but a priori, f may permute the elements of O(x) in a different fashion than g does), it is natural to ask whether f=g on the closure of the periodic points (which is known to coincide with the closure of the recurrent points!). We show this is the case wherever orbits with prime periods are dense. Specifically, we show that mixing interval maps are uniquely determined by (the location of) their periodic orbits.

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1. GeoGT

Pseudo-Anosov subgroups of surface bundles over tori — Junmo Ryang <jr95@rice.edu>

In 2002, Farb and Mosher introduced the notion of convex cocompactness in the mapping class group to capture coarse geometric information of the associated surface group extensions. Convex cocompact subgroups are necessarily finitely generated and purely pseudo-Anosov, but it is an open question whether the converse is true. Several partial results are known in certain settings, however. For example, work of Dowdall, Kent, Leininger, Russell, and Schleimer give a positive answer for subgroups of fibered 3-manifold groups (aka surface-by-cyclic extensions) naturally embedded in punctured mapping class groups via the Birman exact sequence. We present a generalization of this in the setting of surface-by-abelian extensions.

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1. GeoTop

Pseudo-isotopy versus isotopy for homeomorphisms of 4-manifolds — Daniel Galvin <daniel.galvin@austin.utexas.edu>

Pseudo-isotopy is an equivalence relation on homeomorphisms that lies between isotopy and homotopy. Classifying homeomorphisms of 4-manifolds up to pseudo-isotopy is a potentially tractable problem, whereas isotopy classifications currently elude us outside of the simply-connected case. I will explain a program to understand some of this difference using the smooth invariants of Hatcher-Wagoner and Igusa. A result is the construction of many examples of homeomorphisms that are pseudo-isotopic to the identity but not isotopic to the identity on a range of 4-manifolds, including the 4-torus. This is joint work with Isacco Nonino.

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1. GeoGT

Quasi-isometric embeddings of Ramanujan complexes — Hyein Choi <hc71@rice.edu>

Euclidean buildings (a.k.a. affine buildings and Bruhat-Tits buildings) are considered as a p-adic analogue of symmetric spaces. We show that there is no quasi-isometric embedding between the symmetric space of SL(n,R) and the Euclidean building of SL(n,Q_p). Generalizing this, we distinguish Ramanujan complexes constructed by Lubotzky-Samuels-Vishne as finite quotients of Euclidean buildings of PGL(n,F_p((y))) up to quasi-isometric embeddings. These complexes serve as high dimensional expanders with fruitful applications in mathematics and computer science.

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1. GeoGT

Quasiconvex Subgroups of Acylindrically Hyperbolic Groups — Ping Wan <pwan5@uic.edu>

As a generalization of hyperbolic groups, the class of acylindrically hyperbolic groups includes many interesting examples and has has received considerable attention. In the world of hyperbolic groups, quasiconvex subgroups are important subjects. What would be a proper analog of quasiconvex subgroups in the context of acylindrically hyperbolic groups? In this talk, I will share my answer to this question and some more questions.

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1. GeoGT

Relative Quasi-Convexity in the Sageev Construction — Jagerynn Verano <jveran2@uic.edu>

Given a group *G* and a collection of codimension--one subgroups *H* of *G*, one can construct a CAT(0) cube complex on which *G* acts isometrically with no global fixed point. This is known as Sageev's construction. In this construction, codimension--one subgroups of *H* are commensurable with hyperplane stabilizers. By imposing certain conditions on *G* and *H*, one can promote the group action to a proper or cocompact one. A proper action is harder to obtain than a cocompact one. In this talk, we introduce a relative version of Groves--Manning's result on quasi-convexity in the Sageev construction. Let *(G,P)* be a finitely generated relatively hyperbolic group acting cocompactly and *P*-elliptically on a CAT(0) cube complex. In this setting, we show that vertex stabilizers are full relatively quasi-convex if and only if hyperplane stabilizers are full relatively quasi-convex.

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1. Dynamics

Renormalization, equipotential annuli, and the Hausdorff measure — Vladlen Timorin <vtimorin@hotmail.com>

For a complex single variable polynomial $f$ of degree $d$, let $K(f)$ be its filled Julia set, i.e., the union of all bounded orbits. Assume that $K(f)$ has an invariant component $K^{\*}$ on which $f$ acts as a degree $d^{\*}<d$ map. This is a simplest instance of _holomorphic polynomial-like renormalization_ (Douady-Hubbard): the dynamics of a higher degree (degree $d$) polynomial $f$ near $K^*$ can be understood in terms of a suitable lower degree (degree $d^{\*}$) polynomial to which the restriction $f{\|}_{K^{\*}}$ is conjugate. One can associate a certain Cantor-like subset $G’$ of the circle with $K^{\*}$; the latter is defined in a combinatorial way. We will describe a role the Hausdorff dimension of $G’$ and the respective Hausdorff measure play in geometry of $K^{\*}$.

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1. GeoTop

Root lattices and series valued invariants of plumbed 3-manifolds — Allison Moore <moorea14@vcu.edu>

Given a reduced plumbing tree and a spin-c structure, I will discuss how to construct a plumbed 3-manifold invariant in the form of a Laurent series twisted by a root lattice. Such a series is invariant under the Neumann moves on plumbing trees and the action of the Weyl group. These series-valued invariants generalize the Z-hat series of Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park and Ri. They are motivated by the study of the WRT invariants, and the work of Akhmechet-Johnson-Krushkal which found connections with lattice cohomology. Time permitting, I will also discuss a multivariable generalization of the root lattice-twisted series for knot complements and gluing formulas. This is joint work with N. Tarasca.

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1. General & ST
3. Plenaries

Selection Games with Compact Sets — Christopher Caruvana <chcaru@iu.edu>

Selection principles and their corresponding games involving points and open covers have a long and well-developed history. In this talk, we will highlight how many results for points and open covers have analogues involving compact sets and k-covers. Such examples will include traditional Menger/Rothberger variants, as well as various connections with the space of real-valued continuous functions with the topology of uniform convergence on compacta. Most of the results to be discussed come from joint work with Steven Clontz and Jared Holshouser.

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1. Applied & Data

Sheaf Laplacian Sparsification on Graphs — Minghua Wang <minghuaw@buffalo.edu>

Sheaf Laplacians generalize graph Laplacians to vector-valued node signals, enabling richer relational models but increasing computational cost. We present a spectral sparsification method for the $0$-dimensional sheaf Laplacian using leverage-style edge sampling from trace effective resistance with reweighting. The resulting sparse operator preserves the original quadratic form on $(\ker L_{\mathcal F})^\perp$ with high probability: for $\varepsilon\in(0,1)$ and $p_{\mathrm{fail}}\in(0,1)$, we obtain a $(1\pm\varepsilon)$ approximation with probability at least $1-p_{\mathrm{fail}}$. This gives a principled path to faster sheaf diffusion and scalable sheaf-based learning, and supports empirical study of the sparsity--accuracy tradeoff through tunable sampling.

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1. Dynamics

Simple Smale Flows on S^3 — Anthony Sloan <tonesval@outlook.com>

We discuss the linking structure of the attractor-repeller pairs in simple Smale flows on the 3-sphere in which the chaotic saddle set is modeled by four- band templates with twisted bands. We obtain new theorems which illustrate that the dynamics of simple Smale flows are sensitive to half-twists in the bands of the embedded template. Haynes and Sullivan showed that the attractor- repeller pair a∪r in a simple Smale flow with chaotic saddle set modeled by embedded template U^+ is either a Hopf link or a trefoil and meridian. By placing a single half-twist in a selected band of U^+, we obtain new templates that model chaotic saddle sets of Smale flows. For simple Smale flows on S^3 with chaotic saddle sets modeled by those templates, we find that such simple Smale flows are realizable and that a∪r must be a Hopf link, a figure-8 knot and meridian, or a trefoil and meridian. This is joint work with Michael Sullivan.

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1. GeoTop

Some fibrations of some pseudomanifold groups — Genevieve Walsh <genevieve.walsh@gmail.com>

3-dimensional pseudomanifolds are CW-complexes with the property that the link of each point is a closed, orientable surface. We give some interesting examples of these, show that there are many examples of these groups virtually algebraically fibering, and give some applications to higher-dimensional Coxeter groups. This is joint work with Lorenzo Ruffoni.

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1. Continua

Specification on the Lelek Fan — Van Nall <vnall@richmond.edu>

Recent work of Piotr Oprocha and his collaborators has provided a number of delicate examples of dynamical systems separating specification, shadowing, and periodic-point density, primarily in symbolic or totally disconnected spaces. The goal of the present paper is to demonstrated that similar - and in some cases sharper - separations occur on the Lelek fan, a continuum that can be embedded in a Cantor fan. Our constructions rely on Mahavier products of closed relations. By carefully choosing relations on the unit interval, we obtain Mahavier products that are homeomorphic to the Lelek fan whose associated shift maps display diverse dynamical behavior. This approach yields a unified framework for producing and analyzing examples on a familiar continuum.

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1. Continua

Spirals — Alejandro Illanes <illanes@matem.unam.mx>

A spiral is a compactification of the ray [0,1) with remainder a simple closed curve. In this talk we will discuss how spirals have appeared in important results of Continuum Theory, including some new results.

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1. GeoTop

Surgeries on knots and tight contact structures — Shunyu Wan <swan48@gatech.edu>

The existence and nonexistence of tight contact structures on the 3-manifold are interesting and important topics studied over the past thirty years. Etnyre-Honda found the first example of a 3-manifold that does not admit tight contact structure, and later Lisca-Stipsicz extended their result and showed that a Seifert fiber space admits a tight contact structure if and only if it is not the smooth (2n − 1)-surgery along the T(2,2n+1) torus knot for any positive integer n. Surprisingly, since then no other example of a 3-manifold without tight contact structure has been found. Hence, it is interesting to study if all such manifolds, except those mentioned above, admit a tight contact structure. Towards this goal, I will discuss the joint work with Zhenkun Li and Hugo Zhou about showing any negative surgeries on any knot in S^3 admit a tight contact structure.

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1. General & ST

The Cofinality Of Generating Familes — Thomas Gilton <tdgilton@gmail.com>

The Cofinality of Generating Familes (joint with Paul Gartside) Cofinal sets, which study what happens "eventually" in a given order, are ubiquitous in mathematics. In Topology, they play a particularly noticeable role. Consider, for example, a subspace $M$ of the reals. The topology can be captured by convergent sequences, and hence we can ask the question: how many sequences are needed to characterize the topology on $M$? As another example, consider questions (essentially about cofinalities) such as: how many compact sets cover $M$ (the "compact covering number" of $M$)? or how many compact subsets generate the topology on $M$ (the $k$-ness number of $M$)? In this talk, we will discuss recent work of Paul Gartside and the speaker on these questions. We will show how these questions about how many sequences (or compact sets) generate the topology of $M$ can be viewed through the lens of the Tukey order on relations and how, as such, we can apply set-theoretic techniques. We show how certain cardinal invariants are used to answer these topological questions. Finally, we discuss how (in light of recent work of James Cummings and the speaker on extender-based forcing and mod-finite scales) we can, among spaces with a fixed value of the compact covering number, the $k$-ness number can be quite varied.

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1. Applied & Data

The Geometric Latschev's Theorem: Euclidean Shape Reconstruction via Vietoris–Rips Shadow — Sushovan Majhi <s.majhi@gwu.edu>

The shadow of an abstract simplicial complex $\mathcal{K}$, whose vertices are in $\mathbb{R}^{N}$, is defined as the union of the convex hulls of its simplices. For a metric space $(S,d)$ at scale $\beta$, the Vietoris–Rips complex $\mathcal{R}_{\beta}(S)$ is the abstract simplicial complex where each $k$-simplex corresponds to $(k+1)$ points in $S$ with a diameter at most $\beta$. Latschev's theorem provides a qualitative guarantee for manifold reconstruction: for any closed Riemannian manifold $X$, there exists a scale $\epsilon_{0}>0$ such that for any $0<\beta \leq \epsilon_{0}$, there is a $\delta >0$ where any metric space $S$ within Gromov–Hausdorff distance $\delta$ of $X$ yields a Vietoris–Rips complex homotopy equivalent to $X$. Recently, Latschev's theorem has been quantified, allowing $X$ to be a more general geodesic space. When $X\subset \mathbb{R}^{N}$ is a Euclidean geodesic space (e.g., submanifold, graph), we address the theorem's geometric analog: under what conditions is the shadow of the Vietoris–Rips complex of a Hausdorff-close Euclidean sample $S\subset\mathbb{R}^N$ both homotopy equivalent and Hausdorff-close to $X$? Unlike the abstract complex, the shadow provides a geometric embedding within the host space, which is essential for the practical reconstruction of Euclidean shapes. In this talk, we discuss recent developments in answering this question and explore their implications for faithful reconstruction of low-dimensional submanifolds and Euclidean-embedded graphs.

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1. Applied & Data

The Shadow of Vietoris-Rips Complexes in Limits — Atish Mitra <amitra@mtech.edu>

For any abstract simplicial complex $K$ with the vertex set $K^{(0)}$ a Euclidean subset, its shadow, denoted $sh(K)$, is the union of the convex hulls of simplices of $K$. We consider the homotopy properties of the shadow of Vietoris--Rips complexes $K=Rips_\beta(X)$ with vertices from $\mathbb{R}^N$, along with the canonical projection map $p\colon Rips_\beta (X) \to sh(Rips_\beta(X))$. The study of the geometric/topological behavior of $p$ is a natural yet non-trivial problem. The map $p$ may have many "singularities", which have been partially resolved only in low dimensions $N\leq 3$. The obstacle naturally leads us to study systems of these complexes {$sh(Rips_{\beta}(S)) \mid \beta > 0, S\subset X$}. We address the challenge posed by singularities in the shadow projection map by studying systems of the shadow complex using inverse system techniques from shape theory, showing that the limit map exhibits favorable homotopy-theoretic properties. More specifically, leveraging ideas and frameworks from Shape Theory, we show that in the limit "$\beta \to 0$ and $S \to X$", the limit map "$\lim p$" behaves well with respect to homotopy/homology groups when $X$ is an ANR (Absolute Neighborhood Retract) and admits a metric that satisfies some regularity conditions. This results in limit theorems concerning the homotopy properties of systems of these complexes as the proximity scale parameter approaches zero and the sample set approaches the underlying space (e.g., a submanifold or Euclidean graph). This is joint work with Kazuhiro Kawamura and Sushovan Majhi.

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1. GeoTop

The existence of taut foliations with zero Euler class — Ying Hu <yinghu@unomaha.edu>

It is known that every oriented plane field on a closed 3-manifold is homotopic to an integrable one. However, this no longer holds if one requires the foliation to be taut. This leads naturally to the question of which second cohomology classes can arise as the Euler classes of co-oriented taut foliations on a given 3-manifold M. When M is a rational homology sphere, the second cohomology group is finite, and the zero class plays a distinguished role. In this talk, we present infinitely many rational homology 3-spheres, including small Seifert fibred, hyperbolic, and toroidal examples, that admit co-oriented taut foliations but do not admit any with vanishing Euler class. We will also discuss the implications of these examples in the context of the L-space conjecture. This is joint work with Steve Boyer, Cameron Gordon and Duncan McCoy.

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1. General & ST
3. Plenaries

Topological Methods in Cardinal Arithmetic — Todd Eisworth <eisworth@ohio.edu>

Saharon Shelah proved a remarkable theorem of cardinal arithmetic in 1989: if $\aleph_\omega$ is a strong limit cardinal, then $2^{\aleph_\omega}<\aleph_{\omega_4}$. His proof used the tools of pcf theory (a body of set-theoretic tools that are helpful in analyzing certain types of infinite products) but it was soon realized that once the basic ingredients of pcf theory are given, the rest of the argument is essentially topological. Our aim is to survey the topological aspects Shelah’s proof, and present some recent applications.

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1. Applied & Data

Topological data analysis on manifolds via de Rham–Hodge Theory — Zhe Su <zhs0011@auburn.edu>

Topological data analysis (TDA) provides powerful tools for understanding the structure of complex, high-dimensional data, yet most existing methods focus on points, graphs, or simplicial complexes. In this talk, I will present our recently developed de Rham–Hodge–based frameworks for analyzing data on manifolds. These methods provide effective and efficient ways to capture both the topological and geometric information of data and are well-suited for integration with machine learning tasks. I will demonstrate their usefulness through applications in mathematical biology, including protein–ligand binding affinity prediction, single-cell RNA velocity analysis, medical image classification, and B-factor analysis.

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1. Continua

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

A fan X is said to be Lelek-like if it has a dense set of endpoints. We show that there are uncountably many pairwise non-homeomorphic Lelek-like fans, each of which admits a topologically mixing non-invertible map as well as a topologically mixing homeomorphism.

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1. Dynamics

Topology of minimal spaces — Ľubomír Snoha <lubomir.snoha@umb.sk>

A metric space is called minimal if it admits a minimal (not necessarily invertible) map. The question of which metric spaces are minimal remains largely open and may be intractable in full generality. Numerous examples of specific minimal spaces are known -- those admitting a minimal homeomorphism, a minimal noninvertible map, or both. However, only a few general results identify minimal spaces within broad and significant classes, establish obstructions to minimality, or provide methods for constructing new minimal spaces from known ones. In this lecture, we will discuss a selection of classical and recent results that we find particularly important or interesting, highlighting those we especially like or find appealing.

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1. General & ST

Topology, Set Theory, and the $\pi$-Base — Steven Clontz <sclontz@southalabama.edu>

The [$\pi$-Base community database of topological counterexamples](https://topology.pi-base.org/) was recognized in Fall 2025 as the highest-voted [crowdsourced math project on Terence Tao's MathOverflow list](https://mathoverflow.net/questions/500720/list-of-crowdsourced-math-projects-actively-seeking-participants). While much can be done by simply modeling Objects/Spaces, Properties, and Theorems, without a notion of set theory and cardinality, we quickly find limitations, for example: - Several "open questions" on $\pi$-Base are equivalent to the Continuum Hypothesis - Thirteen properties on $\pi$-Base are just different cardinalities, with explicit theorems written to connect them. We will discuss the current plan to incorporate results from set theory into the $\pi$-Base, and seek input from potential users.

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1. Continua

Tree-like continua and the fixed-point property — Andrea Ammerlaan <ajammerlaan879@my.nipissingu.ca>

In 1980, David Bellamy constructed the first example of a tree-like continuum which does not have the fixed-point property. Several others have been constructed since, most recently in 2018 by Rodrigo Hern\'{a}ndez-Guti\'{e}rrez and Logan Hoehn. Their example is expressed as an inverse limit on trees, each of which is an arc with simple triods attached at select points. In this talk, I discuss the construction from Hern\'{a}ndez-Guti\'{e}rrez and Hoehn and give an overview of work towards a similar example where each factor space has branch points of lower degree. Joint work with Logan Hoehn.

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1. Dynamics

Triod twist cycles and circle rotations — Sourav Bhattacharya <sourav9221@gmail.com>

We study the problem of relating cycles on a *triod* *Y* to *circle rotations*. We prove that a *triod-twist* cycle *P*, the simplest cycle on a *triod* with a given *rotation number* ρ, is *conjugate* to *circle rotation*, by angle ρ, restricted to one of its cycles *Q*. Further, the conjugacy Ψ : *P* → *Q* is *piece-wise monotone* and its modality exceeds the modality *m* of *P* by *at-most* 3. This explicit bound *m*+3 serves as a *combinatorial distortion* principle, where the additive constant "+3" represents the "topological cost" imposed by the *valence* of the *branching point* *a*.

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1. GeoGT

Truncated braid groups — Ethan Dlugie <ethan_dlugie@brown.edu>

In the 1950s, Coxeter considered the quotients of braid groups given by adding the relation that all half Dehn twist generators have some fixed, finite order. He found a remarkable formula for the order of these groups in terms of some related Platonic solids. Despite the inspiring apparent connection between these objects, Coxeter's proof boils down to a finite case check that reveals nothing about the structure present. I'll explain recent work that gives an interpretation of the truncated 3-strand braid group that makes the connection with Platonic solids clear, using down-to-earth geometric and algebraic topological tools.

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1. MER

Tukey order and ultrafilters — Jonathan Cancino-Manríquez <mhacajoh@gmail.com>

We will discuss the Tukey order, focusing on ultrafilters on the natural numbers. After providing some context on Isbell's classical problem — asking for the number of equivalence classes in the Tukey order of ultrafilters — I will outline results that establish the consistent non-existence of basically generated ultrafilters, as well as the consistency of all ultrafilters having maximal Tukey type. This is joint work with Jindrich Zapletal.

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1. Continua

Unions of arcs which are fans — Goran Erceg <gorerc@pmfst.hr>

A fan is an arcwise-connected continuum that is hereditarily unicoherent and has exactly one ramification point. Many known examples of fans have been constructed as one-dimensional continua that are unions of arcs intersecting in exactly one point. In 1954, Borsuk proved that every fan is a one-dimensional continuum that can be expressed as a union of arcs intersecting in exactly one point. However, it is still unknown whether this property characterizes fans. In this talk, I will show under which additional assumptions every such union of arcs is indeed a fan. This is joint work with Iztok Banič, Alejandro Illanes, Ivan Jelić, Judy Kennedy, and Van Nall.

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1. Continua

Unique based path lifting fails under R-tree 'covers' of the unit disk — Paul Fabel <pfpoke@gmail.com>

We discuss and illustrate a key ingredient to a recent positive solution \[Brazas, Conner, F, Kent\] to the following problem posed by Jerzy Dydak in 2011. If the continuous surjection $\Pi :X\rightarrow D^{2}$ has unique based path lifting, must $\Pi$ be a homeomorphism, provided $X$ is a connected, locally path connected metric space and $D^{2}$ is the unit disk? The answer is "yes", but ruling out the possibility of a counterexample is nontrivial, and ultimately reduces to the question of whether $X$ could be a certain topological R-tree comprised of all $p$ based irreducible paths in $% D^{2}$. The irreducible paths $\alpha$ in $D^{2}$ are those such that every nonconstant subloop of $\alpha$ fails to lift to some loop in some dendrite. For example piecewise linear, and more geneally, piecewise irreducible paths in $D^{2}$ lift uniquely to $X,$ up to basepoint. The challenge is to exhibit a path in $D^{2}$ which does not lift uniquely to $X.$ Illustrating a method to do this is the main goal of the talk, and the tactic is as follows. Every dendrite is a quotient of a topological disk so that each point preimage intersects the boundary of the disk. However, mating two respective dendrite partitions of two unit half disks, reveals that the join of the quotients need not be a dendrite.

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1. GeoTop

Universal coefficients and Novikov homology — Kevin Schreve <kschreve@lsu.edu>

Kielak and Fisher have connected the $L^2$-Betti numbers (and their finite field variants) to the Novikov homology for a RFRS group $G$. This in turn relates vanishing of $F$-$L^2$-Betti numbers of $G$ to algebraic virtual fibering of $G$. We will give an example of a RFRS group $G$ which has vanishing top-dimensional Novikov cohomology with all field coefficients but not with $\mathbb{Z}$-coefficients.

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1. GeoTop

Unknotting number and L-space satellite operators — Hugo Zhou <hugozhou@umich.edu>

In a joint work with Daren Chen and Ian Zemke, we study the torsion order of Heegaard Floer homology under L-space satellite operators, which by a result by Alishahi-Eftekhary, leads to an unknotting number bound. The argument resembles the work of Hom-Lidman-Park; instead of using immersed curves, we use the L-space satellite formula by Chen-Zemke-Zhou.

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1. Applied & Data

Upper bounds for the connectivity of Vietoris--Rips complexes of spheres via the tight span — Arya Narnapatti <anarnapa@andrew.cmu.edu>

The Vietoris--Rips complex is a construction central to applied topology, including applications to geometric group theory, topological data analysis, and more. However, even for simple spaces such as spheres, their homotopy types are yet to be characterized. In this talk, I will present connectivity bounds for the Vietoris--Rips complexes of spheres $S^n$ in terms of covering properties of $\mathbb{R}P^n$. We leverage the connection to neighborhoods of $S^n$ in the tight span $E(S^n)$ (a.k.a hyperconvex hull) and tools from equivariant topology. These techniques generalize to the study of Vietoris--Rips complexes of antipodal metric spaces. This is joint work with Florian Frick.

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1. General & ST

Using the Pi-Base for Undergraduate Research — Daniel Leary <del2522@jagmail.southalabama.edu>

The $\pi$-Base is a database devoted to general topology, listing topological spaces, their properties, and theorems relating properties together. In this talk, I will show how I used the $\pi$-Base as an undergraduate student to discover interesting research problems. As an example, I will show how the $\pi$-Base led to me proving that a non-semiregular almost discrete space must be the disjoint union of the Sierpiński space and a discrete space.

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1. GeoGT

Švarc-Milnor actions and asymptotic dimension for big mapping class groups — George Shaji <georges@math.utah.edu>

In their paper, Branman, Domat, Hoganson and Lymann proved that if a topological group acts in a "nice" way on a simplicial graph, then the group has a well defined geometry that makes it quasi-isometric to the graph. These actions generalize a Svârc-Milnor action to the context of coarsely-boundedly (CB) generated Polish groups. We adapt these ideas to the context of locally bounded Polish groups and then construct an arc and curve model coarsely equivalent to Map(S) when Map(S) is locally bounded. We then use this model to show that the asymptotic dimension of Map(S) is infinite when S has a non-displaceable subsurface.

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