http://www-math.umd.edu/research/seminars.html http://www-math.umd.edu/research/seminars.html Mon, 28 Aug 2023 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Fedya Manin (UCSB) -


Abstract: I will talk about the following question of Gromov: what closed manifolds can be efficiently wrapped with Euclidean wrapping paper? That is, for what M is there a 1-Lipschitz map $\mathbb R^n \to M$ with positive asymptotic degree? Gromov called such manifolds elliptic. We show that, for example, the connected sum of k copies of CP^2 is elliptic if and only if k ≤ 3. I will try to explain the intuition behind this example, how it extends to a more general dichotomy governed by the de Rham cohomology of M, and why ellipticity is central to the program of understanding the relationship between topology and metric properties of maps.

If I have time, I'll also explain why for a non-elliptic M, a maximally efficient map $\mathbb R^n \to M$ must have components at many different frequencies (in a Fourier-analytic sense), and even then it's at best logarithmically far from having positive asymptotic degree. This is joint work with Sasha Berdnikov and Larry Guth.
]]> http://www-math.umd.edu/research/seminars.html Mon, 11 Sep 2023 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Zhenkun Li (University of South Florida) - https://stanford.edu/~zhenkun/

Abstract: Instanton Floer homology is introduced by Floer in 1980s. It is a powerful invariant for 3-manifolds and knots and links inside them. In this talk, I will present a surgery formula for instanton theory, which describes the instanton Floer homology of a 3-manifold coming from Dehn surgeries along knots. This formula can be applied in computing the instanton Floer homology of surgery 3-manifolds and study the SU(2)-representations of fundamental groups of 3-manifolds. In particular, using this technique, we could prove that the fundamental group of 3-surgery along any non-trivial knots in S^3 admits an irreducible SU(2) representation, answering a question by Kronheimer and Mrowka proposed in 2004. This is a joint work with John Baldwin, Steven Sivek, and Fan Ye.
]]> http://www-math.umd.edu/research/seminars.html Mon, 18 Sep 2023 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Nathaniel Sagman (Luxemberg) -



Abstract: 

For S a closed surface of genus at least 2, Labourie proved that every Hitchin representation of pi_1(S) into PSL(n,R) gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for n=2,3), and explained that if true, then the space of Hitchin representations admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmuller space.
After giving the relevant background, we will discuss the geometry of large area minimal surfaces in symmetric spaces, and explain how such surfaces can give counterexamples to Labourie’s conjecture. We will then share some new questions about minimal surfaces.
]]> http://www-math.umd.edu/research/seminars.html Mon, 25 Sep 2023 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Wouter Van Limbeek (UIC) -

Abstract: The commensurator of a group consists of isomorphisms between its finite index subgroups. Geometrically, the commensurator encodes isometries between finite covers of a Riemannian manifold. I will discuss a question raised independently by Greenberg and Shalom: Can an infinite discrete subgroup of a simple Lie group have dense commensurator and not be a lattice? I will explain the surprising connections between this question and other long-standing open problems, and discuss recent progress on special cases of the question. This is joint work with (subsets of) Brody, Fisher, and Mj.
]]> http://www-math.umd.edu/research/seminars.html Mon, 02 Oct 2023 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Deeparaj Bhat (MIT) -


Abstract: We prove an exact triangle relating the knot instanton homology to the instanton homology of surgeries along the knot. As the knot instanton homology is computable in many instances, this sheds some light on the instanton homology of closed 3-manifolds. We illustrate this with computations in the case of some surgeries on the trefoil which are different from the analogous groups in other Floer theories such as Heegaard Floer and monopole Floer. Finally, we sketch the proof of the triangle. 
]]> http://www-math.umd.edu/research/seminars.html Mon, 09 Oct 2023 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Emily Riehl (JHU) -


Abstract: The fundamental theorem of category theory is the Yoneda lemma, which in its simplest form identifies natural transformations between represented functors with morphisms between the representing objects. The ∞-categorical Yoneda lemma is surprisingly hard to prove --- at least in the traditional set-based foundations of mathematics. In this talk we'll describe the experience of developing ∞-category theory in an alternate foundation system based on homotopy type theory, in which constructions determined up to a contractible space of choices are genuinely "well-defined" and elementwise mappings are automatically homotopically-coherently functorial. In this setting the proof the ∞-categorical Yoneda lemma is arguably easier than the 1-categorical Yoneda lemma. We'll end by posing the question as to whether similar foundations would be useful for other "higher structures." This is based on joint work with Mike Shulman and involves computer formalizations written in collaboration with Nikolai Kudasov and Jonathan Weinberger.
]]> http://www-math.umd.edu/research/seminars.html Mon, 16 Oct 2023 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Jing Tao (Oklahoma) -


Abstract: In the 1970s, Thurston generalized the classification of self-maps of the torus to surfaces of higher genus, thereby completing the work first initiated by Nielsen. This is known as the Nielsen-Thurston Classification Theorem. A well-known proof of this theorem is due to Bers, who reformulated the problem in terms of extremal quasiconformal maps between complex surfaces. In joint work with Camille Horbez, we revisit Bers's approach but from the point of view of hyperbolic geometry. This gives a new proof of the classification theorem, as well as new representatives for pseudo-Anosov homeomorphisms as extremal Lipschitz maps between hyperbolic surfaces. As another application, we also classify the isometries of the Thurston metric on Teichmuller space.
]]> http://www-math.umd.edu/research/seminars.html Mon, 30 Oct 2023 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Tian Yang (Texas AM) -

Abstract: In this talk, I will present an explicit formula we found for the Reidemeistertorsion of a closed oriented hyperbolic 3-manifold twisted by the adjoint action ofthe holonomy representation of the fundamental group of the manifold. This is tothe best of our knowledge the first known explicit formula for such quantity. Thisis a joint work with Ka Ho Wong.
]]> http://www-math.umd.edu/research/seminars.html Mon, 06 Nov 2023 15:00:00 EST Where: Kirwan Hall 3206
Speaker: Nir Gadish (Michigan) -

Abstract: The moduli space of genus g tropical curves with n marked points is a fascinating topological space, with a combinatorial flavor and deep algebro-geometric meaning. In the algebraic world, forgetting the n marked points gives a fibration whose fibers are configuration spaces of a surface, and Serre's spectral sequence lets one compute the cohomology "in principle". In joint work with Bibby, Chan and Yun, we construct a surprising tropical analog of this spectral sequence, manifesting as a graph complex and featuring the cohomology of compactified configuration spaces on graphs.
]]> http://www-math.umd.edu/research/seminars.html Mon, 13 Nov 2023 15:00:00 EST Where: Kirwan Hall 3206
Speaker: Paul Feehan (Rutgers University) - https://sites.math.rutgers.edu/~feehan/

Abstract: We describe an approach to Bialynicki-Birula theory for holomorphic C^* actions on complex analytic spaces and Morse-Bott theory for Hamiltonian functions for the induced circle actions. A key principle is that positivity of a suitably defined "virtual Morse-Bott index" at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum. Inspired by Hitchin’s 1987 study of the moduli space of Higgs monopoles over Riemann surfaces, we apply our method in the context of the moduli space of non-Abelian monopoles or, equivalently, stable holomorphic pairs over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and show that these indices are positive in a setting motivated by a conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type (including symplectic four-manifolds) obey the Bogomolov-Miyaoka-Yau inequality.
]]> http://www-math.umd.edu/research/seminars.html Mon, 20 Nov 2023 15:00:00 EST Where: Kirwan Hall 3206
Speaker: Rose Kaplan-Kelly (George Mason University) - https://sites.google.com/view/rosekaplan-kelly/

Abstract:  In this talk, we will consider a generalization of alternating links and their complements in thickened surfaces. We will define a family of these links with a specific type of right-angled structure on their complements (RGCR links) and show that this property is equivalent to the links having two totally geodesic checkerboard surfaces, and equivalent to a set of restrictions on the links' alternating projection diagrams. If time permits, we will then use these diagram restrictions to consider the commensurability classes of RGCR links.
]]> http://www-math.umd.edu/research/seminars.html Mon, 04 Dec 2023 15:00:00 EST Where: Kirwan Hall 3206
Speaker: Sanghoon Kwak (University of Utah) - https://www.samkwak.info/

Abstract: The Culler--Vogtmann's Outer space $CV_n$ is a space of marked metric graphs, and it compactifies to a set of $F_n$-trees. Each $F_n$-tree on the boundary of Outer space is equipped with a length measure, and varying length measures on a topological $F_n$-tree gives a simplex in the boundary. The extremal points of the simplex correspond to ergodic length measures. By the results of Gabai and Lenzhen--Masur, the maximal simplex of transverse measures on a fixed filling geodesic lamination on a complete hyperbolic surface of genus $g$ has dimension $3g-4$. In this talk, we give the maximal simplex of length measures on an arational $F_n$-tree has dimension in the interval $[2n-7, 2n-2]$. This is a joint work with Mladen Bestvina, Jon Chaika, and Elizabeth Field.
]]> http://www-math.umd.edu/research/seminars.html Fri, 08 Dec 2023 14:00:00 EST Where: Kirwan Hall 3206
Speaker: Piotr Pstrągowski (Harvard University) - https://people.math.harvard.edu/~piotr/
Abstract: A phenomenon in topology is said to be stable if it occurs in all sufficiently high dimensions. As discovered by Quillen over five decades ago, such phenomena are closely related to number theory, and can often be described in terms of arithmetic objects known as formal groups. Unfortunately, in general this dictionary is not quite one-to-one, and many periodicities one sees on the arithmetic side become broken and more complex in the world of topology. In this talk, I will describe a solution to an old conjecture of Franke that the arithmetic - topology correspondence can be refined to an equivalence of categories when the ambient prime is sufficiently large.

]]> http://www-math.umd.edu/research/seminars.html Mon, 22 Jan 2024 14:30:00 EST Where: Kirwan Hall 3206
Speaker: Sungkyung Kang (Oxford University)

We say that two smooth 4-manifolds are exotic if they are homemorphic but not diffeomorphic. Wall's theorem, proven in 1964, says that when the given 4-manifolds are simply-connected, they are always diffeomorphic after sufficiently many stabilizations, i.e. connected-summing with S^2 \times S^2, but whether we need more than one stabilization remained mysterious for almost sixty years. In this talk, I will present the first example of an exotic pair of simply-connected 4-manifolds with a common boundary which stays exotic after one stabilization. This talk is based on my work arXiv:2210.07510, which involves a careful argument involving bordered-involutive techniques in Heegaard Floer theory, pioneered by arXiv:2202.12500 and arXiv:2207.11870 by K. and K.-Park.
]]> http://www-math.umd.edu/research/seminars.html Mon, 29 Jan 2024 14:00:00 EST Where: Kirwan Hall 1308
Speaker: Brian Udall (Rice University) 
Abstract: We consider the collection of parabolically geometrically finite (PGF) subgroups of mapping class groups, which were defined by Dowdall-Durham-Leininger-Sisto. These are generalizations of convex cocompact groups, and the class of PGF groups contains all finitely generated Veech groups as well as certain free products of multitwist groups. We will see some basic motivations and properties of these groups, as well as discuss a combination theorem for PGF groups generalizing the combination theorem of Leininger-Reid for Veech groups. This allows one to build many more examples of PGF groups, including Leininger-Reid surface groups.
]]> http://www-math.umd.edu/research/seminars.html Mon, 12 Feb 2024 15:00:00 EST Where: Kirwan Hall 3206
Speaker: Johanna Mangahas (University at Buffalo) - https://www.nsm.buffalo.edu/~mangahas/


Abstract: Stability for subgroups of finitely generated groups generalizes the property of quasiconvexity for subgroups of hyperbolic groups: they are quasi-isometrically embedded, and ambient-group quasi-geodesics between points in the subgroup fellow travel each other (from which it follows that the subgroup is hyperbolic, whereas the ambient group generally is not).  We are interested in when this property is recognized by an action of the larger group on some hyperbolic space, by which we mean the stable subgroup quasi-isometrically embeds into that space.  In well-studied settings such as mapping class groups, right-angled Artin groups, or more generally hierarchically hyperbolic groups, the hyperbolic space admitting the group's largest acylindrical action provides such a recognizing space for all stable subgroups.  Sometimes the corresponding result is true for relatively hyperbolic groups admitting largest acylindrical actions, but we provide a counterexample to show it is not true in general.  This is joint work with Sahana Balasubramanya, Marissa Chesser, Alice Kerr, and Marie Trin.
]]> http://www-math.umd.edu/research/seminars.html Mon, 19 Feb 2024 14:00:00 EST Where: Kirwan Hall 1308
Speaker: Fan Ye (Harvard University) - https://scholar.harvard.edu/fanye/home
Abstract: Since Floer's work in 1988, various Floer homologies have been constructed for closed 3-manifolds, knots, and sutured manifolds. In 2008, Kronheimer-Mrowka proposed a conjecture about isomorphisms among Floer homologies. In this talk, I will first introduce the history of the constructions in Floer theory and then introduce an approach to proving the isomorphisms. The idea is based on combinatorial version of Floer homology, which leads to an axiomatic construction of Floer homology. This work is joint with Baldwin, Li, and Sivek.
]]> http://www-math.umd.edu/research/seminars.html Mon, 26 Feb 2024 15:00:00 EST Where: Kirwan Hall 3206
Speaker: Marissa Kawehi Loving (University of Wisconsin-Madison) -https://sites.google.com/view/lovingmath/home


Abstract: In this talk, I will briefly introduce the notion of end-periodic homeomorphisms of infinite-type surfaces. My goal will be to illustrate the ways these homeomorphisms mimic the behavior of pseudo-Anosov homeomorphisms of finite-type surfaces by displaying interesting geometric, dynamical, and topological behavior. As part of this discussion, I will describe some joint work with Elizabeth Field, Autumn Kent, Heejoung Kim, and Chris Leininger (in various configurations) on the volume of end-periodic mapping tori.
]]> http://www-math.umd.edu/research/seminars.html Mon, 04 Mar 2024 15:00:00 EST Where: Kirwan Hall 3206
Speaker: Jiajun Yan (University of Virginia) - https://sites.google.com/view/jiajunyan/
Abstract: Non-compact hyperkähler spaces arise frequently in gauge theory. The 4-dimensional hyperkähler ALE spaces are a special class of non-compact hyperkähler spaces. They are in one-to-one correspondence with the finite subgroups of SU(2) and have interesting connections with representation theory and singularity theory, captured by the McKay Correspondence. 
In this talk, we first review the finite-dimensional construction of 4-dimensional hyperkähler ALE spaces given by Peter Kronheimer in his PhD thesis. Then we give a new gauge-theoretic construction of these spaces inspired by Kronheimer’s construction. Time permitting, we will discuss some future directions and ongoing work. 

]]> http://www-math.umd.edu/research/seminars.html Mon, 11 Mar 2024 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Ian Zemke (Princeton University/University of Oregon) - https://web.math.princeton.edu/~izemke/

Abstract: We will discuss the "bordered" perspective on the link surgery formula of Manolescu, Ozsvath and Szabo. This is a reformulation of the link surgery complex using the algebraic formulation of Lipshitz, Ozsvath and Thurston. To a 3-manifold with torus boundary components, we will describe a way of repackaging the link surgery formula to get a invariant of the complementary bordered 3-manifold which behaves nicely with respect to gluing. We will discuss the basic algebraic philosophy, as well as some applications to lattice homology and the link Floer homology of algebraic links. Parts of joint with B. Liu and M. Borodzik.
]]> http://www-math.umd.edu/research/seminars.html Mon, 25 Mar 2024 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Edgar A. Bering IV (San Jose State University) - https://sites.google.com/view/edgar-a-bering-iv

Abstract: The curve graph $\mathcal{C}$ of a finite genus surface $\Sigma_g$ is a central object in the study of the mapping class group of $\Sigma_g$. It exhibits many remarkable combinatorial properties. One, despite being an infinite, locally infinite graph Bestvina, Bromberg and Fujiwara proved that it has finite chromatic number. Two, its structure can be probed by finite rigid subgraphs: subgraphs $X \subset \mathcal{C}$ such that any locally injective map $X \to \mathcal{C}$ is the restriction of a global automorphism of $\mathcal{C}$. The outer automorphism groups of finite-rank free groups are studied by analogy with mapping class groups, though the analogy is imperfect. One analog of the curve graph is the sphere graph of a connect sum of $S^1\times S^2$s. In this talk I will introduce the sphere graph and discuss recent work investigating analogous combinatorial structure: an upper bound on the chromatic number (joint with SJSU students B. Haffner, E. Ortiz, and O. Sanchez) and the construction of finite rigid subgraphs (joint with C. Leininger).
]]> http://www-math.umd.edu/research/seminars.html Thu, 28 Mar 2024 15:30:00 EDT Where: Kirwan Hall 2300
Speaker: Alexandra Kjuchukova (Notre Dame) -
Abstract: A knot K in $S^3$ is slice if it bounds a smooth disk in the
four-ball; and is ribbon if this disk can be chosen to have no local maxima
with respect to the radial height function on B^4. An old question of Fox
asks if every slice knot is ribbon.

Let K be a knot equipped with a dihedral quotient of \pi_1(S^3\K). I'll
explain how to extract an invariant of K from this data, using the
signature of a certain 4-manifold which is a branched cover of B^4 over a
properly embedded surface F with boundary K. I'll describe how this
invariant can obstruct K from bounding, in alternate cases, either ribbon
or slice disks in the four-ball. And, I'll give a necessary and sufficient
condition for the existence of such a surface.
]]> http://www-math.umd.edu/research/seminars.html Mon, 08 Apr 2024 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Ryan Dickmann (Georgia Tech) - https://sites.google.com/view/ryandickmann
Abstract: We will discuss the classification of the automatically continuous pure mapping class groups focusing on cases with noncompact boundary. We will also discuss the classification of general noncompact surfaces.
]]> http://www-math.umd.edu/research/seminars.html Mon, 22 Apr 2024 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Daniel Alvarez-Gavela (MIT/Brandeis) - https://danielalvarezgavela.com/


Abstract: In our paper "geomorphology of Lagrangian ridges" we showed how one may make a Lagrangian submanifold transverse to any Lagrangian distribution at the expense of introducing a certain combinatorial Lagrangian singularity which we called a "Lagrangian ridge". This result was essential to our existence theorem for arboreal skeleta of polarized Weinstein manifolds. I will explain a 1-parametric version of this story, which will be essential to our uniqueness theorem for arboreal skeleta of polarized Weinstein manifolds (up to positive Reidemeister moves). This is joint with Y. Eliashberg and D. Nadler.
]]> http://www-math.umd.edu/research/seminars.html Mon, 29 Apr 2024 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Jonathan Hanselman (Princeton University) - https://web.math.princeton.edu/~jh66/
Abstract: Satellite operations are a valuable method of constructing complicated knots from simpler ones, and much work has gone into understanding how various knot invariants change under these operations. We describe a new way of computing the (UV=0 quotient of the) knot Floer complex using an immersed Heegaard diagram obtained from a Heegaard diagram for the pattern and the immersed curve representing the UV=0 knot Floer complex of the companion. This is particularly useful for (1,1)-patterns, since in this case the resulting immersed diagram is genus one and the computation is combinatorial. In the case of one-bridge braid satellites the immersed curve invariant for the satellite can be obtained directly from that of the companion by deforming the diagram, generalizing earlier work with Watson on cables. This is joint work with Wenzhao Chen.
]]> http://www-math.umd.edu/research/seminars.html Mon, 06 May 2024 15:00:00 EDT Where: Kirwan Hall 3206
Speaker: Tom Mark (University of Virginia) - https://uva.theopenscholar.com/tom-mark

Abstract: Any contact structure on a closed 3-manifold can be obtained by a procedure called contact surgery, starting from a Legendrian link in the standard 3-sphere. If the contact surgery coefficients are all negative, then the resulting contact structure is fillable, meaning that it can be described as a suitably convex boundary of a compact symplectic 4-manifold. In the presence of positive surgery coefficients, geometric properties of the resulting contact structure are much less obvious. We focus on the case of contact surgery along a single knot K, and present a necessary and sufficient condition that some positive contact surgery along K results in a fillable contact structure. We will explore some additional problems and applications arising from this characterization, including a result on the smooth topology of knots admitting lens space surgeries for which we do not know a proof not involving contact topology. This work is joint with Bulent Tosun.
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