Organizers: Carmen Rovi and Bena Tshishiku
Suppose a finite group acts on a closed manifold M. Given two equivariant homeomorphisms of M, if we know that they are isotopic, can we conclude that they are equivariantly isotopic? An important theorem of Birman-Hilden and MacLachlan-Harvey says the answer is "yes" if M is a hyperbolic surface; Margalit-Winarski asked whether the same is true when M is a 3-manifold. We answer Margalit-Winarski's question for a wide class of group actions on 3-manifolds; this includes a 3-manifold analog of the hyperelliptic involution, which we can understand particularly well via a connection with geometric group theory.
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We show that cubulated hyperbolic groups with spherical boundary of dimension 3 or at least 5 are virtually fundamental groups of closed, orientable, aspherical manifolds, provided that there are sufficiently many quasi-convex, codimension-1 subgroups whose limit sets are locally flat subspheres. The proof is based on ideas used by Markovic in his work on Cannon's conjecture for cubulated hyperbolic groups with 2-sphere boundary. This is joint work with Merlin Incerti-Medici.
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In this talk, we will consider a generalization of alternating links and their complements in thickened surfaces. In particular, a family of generalized alternating links which each correspond to a Euclidean or hyperbolic tiling and have a right-angled complete hyperbolic structure on their complement. We will determine the arithmeticity of these links and find their pairwise commensurability. This is joint work with David Futer.
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The smooth fine curve graph of a surface is an analogue of the fine curve graph that only contains smooth curves. It is natural to guess that the automorphism group of the smooth fine curve graph is isomorphic to the diffeomorphism group of the surface. But it has recently been shown that this is not the case. In this talk, I will give several more examples with increasingly wild behavior and give a characterization of this automorphism group for the particular case of continuously differentiable curves.
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In this talk, we provide new examples of 3-manifolds with weight one fundamental group and the same integral homology as the lens space $L(2k,1)$ which are not surgery on any knot in the three sphere. Our argument uses Furuta's 10/8-theorem, and is simple and combinatorial to apply. This is joint work with Piccirillo.
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Let $\mathrm{Poly}_n\mathbb{C}$ be the space of monic, squarefree, degree $n$ polynomials in one variable over $\mathbb{C}$. Ferrari's solution to the quartic equation gives rise to a holomorphic map $R\colon\mathrm{Poly}_4\mathbb{C}\to\mathrm{Poly}_3\mathbb{C}$. We show that every holomorphic map $\mathrm{Poly}_n\mathbb{C}\to\mathrm{Poly}_m\mathbb{C}$ for $m\leq n$ is equivalent in a certain sense to a constant map, the identity map, or Ferrari's map $R$. This is joint work with Jeroen Schillewaert.
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I will show that the space of pointed infinite volume hyperbolic 3-manifolds is connected but not path connected. This space is equipped with the geometric topology, in which two pointed manifolds are close if they are almost isometric on large neighborhoods of their basepoints. The proof of connectivity will be an application of the density theorem for Kleinian groups. I will then use a combination of results on representations of Kleinian groups and Chabauty spaces of subgroups to construct an infinite family of path components of this space.
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Thurston's Hyperbolic Dehn Filling Theorem is a seminal result in the theory of 3-manifolds. Given a single noncompact finite-volume hyperbolic 3-manifold M, the theorem provides a construction for a countably infinite family of closed hyperbolic 3-manifolds converging to M in a geometric sense. The theorem is a major source of examples of 3-manifolds admitting hyperbolic structures, and closely connects the topology of a 3-manifold to the analysis of the character variety of its fundamental group in PSL(2, C). In this talk, we discuss some analogs and generalizations of Thurston's theorem in the context of general (arbitrary-rank) semisimple Lie groups. We will explain how our results provide a way to construct new examples of Anosov and relatively Anosov representations into higher-rank Lie groups; time permitting, we will also discuss upcoming joint work with Jeff Danciger, which applies our results to construct exotic new examples of convex cocompact and geometrically finite groups acting on complex hyperbolic 3-space.
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In this talk, I will present recent advancements in the study of smooth mapping class groups of 4-manifolds. Our work focuses on diffeomorphisms arising from Dehn twists along embedded 3-manifolds and their interaction with Seiberg-Witten theory. These investigations have led to intriguing applications across several areas, including symplectic geometry (related to Torelli symplectomorphisms), algebraic geometry (concerning the monodromy of singularities), and low-dimensional topology (involving exotic diffeomorphisms). This is collaborative work with Hokuto Konno, Jianfeng Lin, and Juan Munoz-Echaniz.
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A closed manifold $M$ of dimension at least $5$ has only finitely many smooth structures. Moreover, the product structure theorem states that the smooth structures on such an $M$ are in bijection with smooth structures on the product $M\times\mathbb{R}$. In this talk, I will describe a construction that gives rise to infinitely many equivariant smooth structures of a closed $G$-manifold $M$ which become isotopic after taking a product with $\mathbb{R}$.
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The shake genus is the main tool used to detect exotic traces. This is a powerful tool to construct exotic traces, however it has some limitations. We will discuss several desirable properties of exotic traces that are inaccessible using the shake genus. By moving past needing to use the shake genus, we will be able to construct novel examples of exotic traces.
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A Lefschetz fibration $M^4 \to S^2$ is a generalization of a surface bundle which also allows finitely many nodal singular fibers. The Arakelov--Parshin rigidity theorem implies that holomorphic Lefschetz fibrations of genus $g \geq 2$ admit only finitely many holomorphic sections. In this talk, we will show that no such finiteness result holds for smooth or symplectic sections by giving examples of genus-$g$ ($g \geq 2$) Lefschetz fibrations with infinitely many homologically distinct sections. This is joint work with Carlos A. Serván.
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Surfaces in 4-space can be described using tuples of b-string tangles called multiplane diagrams. In this talk, we will discuss local modifications for multiplane diagrams that affect the embedded surface in a controlled way. This talk will explore such operations in the context of bridge multisections. We show a uniqueness result for multiplane diagrams representing isotopic surfaces. If time permits, we will show that any n-valent graph with an n-edge coloring is the spine of a bridge multisection of an unknotted surface.
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Given a parametrized Legendrian $\Lambda$ in a contact manifold $(M, \xi)$, there is a well-defined operation called Legendrian surgery, which produces a new contact manifold $(M(\Lambda), \xi(\Lambda))$. The contactomorphism type of the surgered manifold depends only on the Legendrian isotopy class of the initial Legendrian. Given a loop of Legendrians $\Lambda_t$, it is also possible to realize a 1-parameter family of Legendrian surgeries. From this, we naturally obtain a bundle over the circle with fiber $(M(\Lambda), \xi(\Lambda))$. The non-triviality of the bundle depends on the contact isotopy class of a gluing contactomorphism, which we call the “Legendrian surgery contactomorphism.” Its contact isotopy class depends only on the homotopy class of the given loop of Legendrians within the space of parametrized Legendrians. The obvious realization problem is: which contactomorphisms of a given contact manifold are Legendrian surgery contactomorphisms? In this talk, I will address this question by showing that every formally trivial contactomorphism arises as a Legendrian surgery contactomorphism associated with a certain loop of Legendrians in some overtwisted contact manifold with controlled topology. As a consequence, in 3-dimensional contact topology, we will deduce the existence of formally contractible but non-contractible loops of loose Legendrians in every overtwisted contact 3-manifold. This is a joint work in progress with Fabio Gironella.
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The Prym representations of the mapping class group are an important family of representations that come from abelian covers of a surface. They are defined on the level-$\ell$ mapping class group, which is a fundamental finite-index subgroup of the mapping class group. One consequence of our work is that the Prym representations are infinitesimally rigid, i.e. they can not be deformed. We prove this infinitesimal rigidity by calculating the twisted cohomology of the level-$\ell$ mapping class group with coefficients in the Prym representation, and more generally in the $r$-tensor powers of the Prym representation. Our results also show that when $r\ge 2$, this twisted cohomology does not satisfy cohomological stability, i.e. it depends on the genus $g$.
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A quartic del Pezzo surface $X$ is an intersection of two degree $2$ hypersurfaces in $$\mathbb P^4$$. So rational points on $X$ correspond to solutions of a pair of homogeneous quadratic equations in $5$ variables. I will discuss joint work with R. Das, B. Lehmann, and S. Tanimoto, using topological methods to determine statistics of rational points on $X$ (over the function field $\mathbb F_q(t)$)
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Geometric rigidity theory aims to determine a geometric object with the smallest amount of data possible. For instance, one could ask whether the volume or length set of a manifold determines its metric. In this talk, I will motivate and present some results related to length spectrum and volume rigidity for negatively curved surface amalgams, natural generalizations of negatively curved surfaces.
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Arithmetic manifolds admit many 'Hidden' symmetries. In this talk, we want to discuss the inverse problem: if a object admit many symmetries, is it arithmetic? We will invest the question from variety of aspects: algebra, differential geometry and coarse geometry and answer the question for non-compact negatively curved manifolds.
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