STDC 2025 will feature plenary and semi-plenary talks from several notable researchers.
An n-dimensional rep-tile is a PL n-manifold X, embedded in $\mathbb{R}^n$, which can be decomposed as the union of mutually isometric manifolds similar to X which have non-overlapping interiors. For one astonishing example, all knot exteriors are homeomorphic to rep-tiles, by a 2021 result of Blair, Marley and Richards. I will give an isotopy classification of rep-tiles in all dimensions. I will also outline our proof, which is based on a technique called ball swapping. This is joint work with Ryan Blair, Patricia Cahn and Hannah Schwartz.
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Bounded cohomology is a functional-analytic analogue of group cohomology that is central to rigidity theory, dynamics, geometric topology, and geometric group theory. A major drawback is the failure of excision, which renders even basic computations currently out of reach. One of the few cases where non-trivial computations are possible is transformation groups with certain displacement properties that are classically used in homology and stable commutator length. I will introduce a new algebraic criterion that captures this, is satisfied in many interesting settings, and implies vanishing in all degrees and with a large class of coefficients. Based on joint work with Caterina Campagnolo, Yash Lodha, and Marco Moraschini
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We will examine three threads of inquiry in topology: convergence/compactness properties, spaces built out of other spaces (i.e. the space of real-valued continuous functions or the hyperspace of closed sets), and topological games. When a space is built out of another space, we can often translate the topological information from the first space to the second. For instance, open covers of the space can produce clustering sequences of real-valued functions. This topological information can be encoded through strategies in certain topological games. Working with Chris Caruvana and Steven Clontz, we have developed techniques for tying all of these threads together and have proven a wide array of connections between spaces and common constructions on those spaces. The general theory will be discussed and specific examples will be displayed.
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The joint action of $x\mapsto2x$ (mod 1) and $x\mapsto3x$ (mod 1) has a number of remarkable properties. Among them is that ever joint orbit is either finite or dense. Of course any minimal system has that property, but the x2,x3 system is special because it has a dense set of finite orbits that intermingle with dense orbits. In joint work with B. Kra and S. Schmieding, we abstract this property to what we call a chaotic almost minimal (CAM) system. I this talk I will discuss some properties of CAM systems, showing their similarities to and differences from the x2,x3 system.
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Persistent homology, the flagship method from the field of Topological Data Analysis, is a powerful tool for measuring shape and structure of data. In this talk, we explore methods for using this tool to detect homological changes in the underlying structure of dynamical systems. As a first step, we can simplify a vineyard of persistence diagrams into a CROCKER plot to provide visual representations of qualitative shifts in the structure of examples such as the Lorenz and Rossler systems. We can also construct a "homological bifurcation plot" to enable the identification of qualitative shifts, namely P-type (phenomenological) bifurcations, within stochastic dynamical systems, defined by structural changes in the probability density functions (PDF) of the state variables. The talk will explore the successful application of this method to stochastic oscillators, showcasing its effectiveness in algorithmically detecting P-bifurcations. This talk is based on joint work with many collaborators, including Firas Khasawneh, İsmail Güzel, Sunia Tanweer, Sarah Tymochko, Audun Myers, and David Muñoz.
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Let M be a compact, connected, orientable 3-manifold with non-empty boundary. In this talk, we study the classifying space for the diffeomorphism group of M fixing the boundary pointwise, and show that it has the homotopy type of a finite CW complex. This parallels analogous results of Gramain and Earle-Schatz for surfaces, and confirms a conjecture of Kontsevich for orientable 3-manifolds. The proof will take us on a crash course in 3-manifold topology, and will feature a combination of results on geometrization of 3-manifolds with a topological poset parametrizing embedded spheres in M. This is joint work with Rachael Boyd and Jan Steinebrunner.
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Dehn surgery is a classical operation in which one converts one three-manifold to another by first removing a solid torus, and then gluing it back in in a different way. The first operation is called "drilling" and the second "filling". Both of these operations have group-theoretic interpretations in the world of hyperbolic and relatively hyperbolic groups. I will explain those interpretations and applications related to the Cannon conjecture (a special case of Wall's conjecture about $PD(n)$ groups). The most recent work is joint with Groves, Haïssinsky, Osajda, Sisto, and Walsh.
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In geometric group theory, Rips complexes provide a natural construction to give higher structure to a Cayley graph. In topological data analysis, Rips complexes are used to reconstruct a sufficiently nice space from a sample. I will show different but related applications of Rips complexes and similar constructions to Borsuk-Ulam results, understanding Gromov-Hausdorff distances, and roots of zero-mean real-valued maps.
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In this talk, I will present recent results about the poset of hyperbolic structures on Thompson's group $F$. While the global structure of this poset is as simple as one would expect, the local structure turns out to be incredibly rich, in stark contrast with the situation for the $T$ and $V$ counterparts. I will focus on the subposet of quasi-parabolic hyperbolic structures, which contains uncountably many \emph{lamplike} structures, called so as they can be described combinatorially in terms of certain hyperbolic structures on related lamplighter groups. On the other hand, there are also many non-lamplike structures, showing the vastness and complexity of this poset. Lastly, I will talk about how these actions can be extended to more general Thompson's groups $F_n$ for $n \geq 2$. This is joint work with Francesco Fournier-Facio and Matthew C.B.Zaremsky.
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During the pandemic, Judy Kennedy and I, later joined by Goran Erceg, began investigating the dynamics of closed relations in dynamical systems, which we termed CR-dynamical systems. With travel restrictions in place, we established regular online meetings to collaborate on this research. Our initial focus was on fixed-point problems from the perspective of closed relations, which led us to explore broader dynamical properties and ultimately to introduce Mahavier dynamical systems. Despite being spread across different time zones--Judy in the US, Goran in Croatia, and I in Slovenia--we managed to coordinate meetings in the early evening for Goran and me, and at 1:30 PM for Judy. Since then, we have published numerous papers on Mahavier dynamical systems, with several more in progress. Once travel resumed, Goran and I visited Judy in the US twice, while Judy visited Slovenia and Croatia on many more ocations, allowing us to collaborate in person. While online meetings and screen-sharing have been invaluable, we recognize that nothing fully replaces in-person discussions. Our research has continued to gain momentum, and we are committed to furthering this long-term project. We believe our work is both fundamental and significant, and we remain excited about its potential. Along the way, we named our group the Topology Nerds, and later, Van Nall, Sina Greenwood, Rene Gril Rogina, Chris Mouron and Ivan Jeli\' c joined our efforts. In this talk, I will present an overview of the most important results achieved by the Topology Nerds group.
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After Ingram's conjecture was proven, new questions arose concerning tent functions. One of them is to identify all skew tent maps with their top vertices in the unit square whose (generalized) inverse limits are homeomorphic. In particular, it is interesting to identify the regions of top vertices in the unit square for which inverse limits are homeomorphic. In this talk, we revisit the skew tent maps problem and give some partial results.
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I will discuss two related questions concerning the two spaces in the title: the Čech-Stone remainder of the natural numbers $\mathbb N$, and the Čech-Stone remainder of the half-line $\mathbb H = [0,\infty)$. Both $\mathbb N$ and $\mathbb H$ are naturally ordered from left to right. These orders on $\mathbb N$ and $\mathbb H$ are reflected in their Čech-Stone remainders, in certain dynamical systems on $\mathbb N^\ast$ and in certain subcontinua of $\mathbb H^\ast$. Are these left-to-right aspects of $\mathbb N^\ast$ and $\mathbb H^\ast$ truly topological, or can either of the spaces be "reversed" via some self-homeomorphism?
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Semi-Kelley continua were introduced by J. J. Charatonik and W.J. Charatonik in 1998, who proved that every Kelley continuum is semi-Kelley. Since then, this class of continua has been studied by several authors. The most important problem in this area is determining whether the hyperspaces of a semi-Kelley continua are contractible. In this talk, we present a positive partial answer to this question in the case of fans.
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The formation of topological data analysis (TDA) as a research area with dedicated meetings and funding happened during the years around the beginning of this millenium. A crucial step in this development was the introduction of persistent homology. The root system of this idea goes back to the dependent and independent work of a number of mathematicians, including Marston Morse. This talk recalls a few of the steps on my personal journey leading to this concept, and steps that expand the basic notion toward other branches of mathematics and applications outside of mathematics.
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The Rokhlin lemma is a finite approximation property that underpins a great many constructions in classical ergodic theory, including most spectacularly those at the basis of the Ornstein isomorphism theory for Bernoulli shifts. In the 1970s Ornstein and Weiss showed amenability to be the natural setting for finite approximation in dynamics by establishing a general form of the Rokhlin lemma in this setting, and this led, among other things, to a much broader recasting of the Ornstein isomorphism theory. Over the last couple of decades a growing interest in the interplay between dynamics and the geometric and analytic structure of groups has set the stage for a resurgence of applications of the Ornstein-Weiss Rokhlin lemma, not only in its original measure-theoretic incarnation but also as a versatile tiling principle that has turned out be intimately connected, on the topological side, to the remarkable recent successes in the Elliott classification program for separable nuclear C*-algebras. I will sketch a picture of these various developments at the interface of measure, topology, dynamics, geometric group theory, and operator algebras.
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The Borel Conjecture for closed manifolds implies that two closed aspherical manifolds with isomorphic fundamental group are homeomorphic. The Borel conjecture for compact aspherical manifolds with boundary states that a homotopy equivalence which is homeomorphism on the boundary is homotopic to a homeomorphism. Jonathan Hillman and I classify and prove the Borel Conjecture for all compact aspherical four manifolds with boundary with good (= elementary amenable) fundamental group. We classify all possible fundamental groups and all possible 3-manifold boundaries.
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For a polynomial on the Riemann sphere, infinity is a (super) attracting fixed point, and the filled Julia set is the set of points with bounded orbit. Consider the quadratic family $P_c(z)=z^2+c$. The Mandelbrot set $M$ is the set of parameters $c$ such that the filled Julia set of $P_c$ is connected. Computer experiments quickly reveal the existence of small homeomorphic copies of $M$ inside itself; the existence of such copies was proved by Douady and Hubbard. Each little copy is either primitive (with a cusp on the boundary of its main cardioid region) or a satellite (without a cusp). Lyubich proved that the primitive copies of $M$ satisfy a stronger regularity condition: they are quasiconformally homeomorphic to M. The satellite copies are not quasiconformally homeomorphic to $M$ (as we cannot straighten a cusp quasiconformally), but are they mutually quasiconformally homeomorphic? In joint work with C. Petersen we prove that the answer is negative in general, but positive in the case the satellite copies have rotation number with same denominator.
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The Milnor-Wood inequality, introduced in two landmark papers by John Milnor (1958) and John W. Wood (1971), is a striking result at the intersection of geometry, topology, and dynamics. It establishes sharp bounds on the Euler number of flat $\mathbb{S}^1$-bundles over surfaces, revealing deep connections between geometric curvature and topological invariants. Milnor’s original inequality highlights the boundedness of Euler invariants for flat bundles with "linear" structures, which Gromov later generalized using bounded cohomology. Wood extended Milnor's result to "non-linear" flat circle bundles, offering a perspective rooted in 1-dimensional dynamics. In the 1980s, Étienne Ghys posed the intriguing question of whether Wood’s inequality could be extended to flat-oriented $\mathbb{S}^3$-bundles. In this talk, we will also discuss the surprising ways in which inequality fails in higher-dimensional non-linear cases, showcasing the new calculations in the bounded cohomology of diffeomorphism groups.
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