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Topology Festival Archive

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Festival cancelled due to COVID-19 crisis.

2021


2022

Tarik Aougab, Haverford College
Detecting covers, simple closed curves, and Sunada's construction ▾

Given a pair of finite degree (not necessarily regular) covers (p,X),(q,Y) of a finite type surface S, we show that the covers are equivalent if and only if the following holds: for any closed curve gamma on S, some power of gamma admits an embedded lift to X if and only if some power of gamma admits an embedded lift to Y. We apply this to study the well-known construction of Sunada which yields pairs of hyperbolic surfaces (X,Y) that are not isometric but that have the same unmarked length spectrum. In particular we show that the length-preserving bijection from closed geodesics on X to those on Y arising from the Sunada construction never sends simple closed geodesics to simple closed geodesics. We also show that length-isospectral surfaces arising from several of the most well-known manifestations of the construction are not simple length isospectral. Even more, we construct length-isospectral hyperbolic surfaces so that for each finite n, the set of lengths corresponding to closed geodesics with at most n self intersections disagree. This represents joint work with Maxie Lahn, Marissa Loving, and Nicholas Miller. 
Ian Biringer, Boston College
The space of subgroups of PSL(2,R) ▾

We’ll describe some features of the global topology of the space Sub(G) of all closed subgroups of G=PSL(2,R), equipped with the Chabauty topology. The quotient of the hyperbolic plane by the action of a discrete subgroup of G is a hyperbolic 2-orbifold, and we will mostly focus on sets of subgroups where the quotient orbifold has a fixed finite topological type, and their closures in Sub(G).
Aaron Calderon, Yale University
Long curves on hyperbolic surfaces ▾

In her thesis, Maryam Mirzakhani counted the number of simple closed curves on a (real) hyperbolic surface of bounded length. This breakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, symplectic geometry, surface topology, and homogeneous dynamics. In this talk, I’ll survey these connections as well as discuss answers to some qualitative strengthenings: what do long curves look like on a hyperbolic surface? And what do hyperbolic surfaces with long curves look like? Answers to these questions represent joint work with Francisco Arana-Herrera and James Farre, respectively.
Josh Greene, Boston College
Peg problems ▾

I will talk about joint work with Andrew Lobb related to Toeplitz's square peg problem, which asks whether every (continuous) Jordan curve in the Euclidean plane contains the vertices of a square. Specifically, we show that every smooth Jordan curve contains the vertices of a cyclic quadrilateral of any similarity class. I will describe the context for the result and its proof, which involves symplectic geometry in a surprising way.
Sara Maloni, University of Virginia
d-Pleated Surfaces ▾

Pleated surfaces are an important tool introduced by Thurston to study hyperbolic 3-manifolds and can be described as piecewise totally geodesic surfaces immersed in the 3-manifold and bent along a geodesic lamination. Bonahon generalized this notion to representations of surface groups in PSL_2(C) and described a holomorphic parametrization of the resulting open chart of the character variety in term of shear-bend cocycles. In this talk I will discuss joint work with Martone, Mazzoli and Zhang, where we generalize this theory to surface group representations in PSL_d(C). In particular, I will discuss the notion of d-pleated surfaces, and their holomorphic parametrization.
Dusa McDuff, Barnard College
Symplectic embeddings and recursive patterns ▾

One of the simplest measurements you can make of the "size" of a compact symplectic manifold X is its Gromov width that measures the capacity of the largest ball that embeds into it. More generally, one can study the size of the largest ellipsoid of a given eccentricity that embeds into X. This function of the eccentricity has been (partially) calculated for certain 4-dimensional targets, such as the 4-ball or its one-point blowup, and turns out to have intricate arithmetic properties. This talk, which will be aimed at a nonspecialized audience, will describe some recent work with Nicki Magill and Morgan Weiler about the properties of this function when the target is a ball that has been blown up once with weight b.
Arpon Raksit, Massachusetts Institute of Technology
Motivic filtrations on topological Hochschild homology ▾

Topological Hochschild homology (THH) is an invariant of associative ring spectra, closely related to algebraic K-theory. Bhatt–Morrow–Scholze defined a "motivic filtration" on the THH of ordinary commutative rings, after completion at a prime number p, relating it to invariants (new and old) in p-adic Hodge theory. These new structures have led to many advancements in the study of algebraic K-theory and in arithmetic geometry. In this talk, I will try to give an accessible overview of this landscape of invariants, and then discuss a new construction of the motivic filtration on THH, with the advantages that it is quite simple to state and applies not only to ordinary commutative rings but a broad class of commutative ring spectra; the new work is joint with Jeremy Hahn and Dylan Wilson.
Emily Riehl, Johns Hopkins University
Absolute lifting diagrams and formal category theory ▾

Left and right lifting diagrams, like the dual left and right (Kan) extensions, can be defined in any 2-category. In this talk, we'll make the case that absolute lifting diagrams are particularly expedient for formal category theory, using them to provide a streamlined version of the classical proof that left adjoints preserve colimits and right adjoints preserve limits. This is joint work with Dominic Verity.
Genevieve Walsh, Tufts University
Planar boundaries and parabolic subgroups ▾

Relatively hyperbolic groups generalize geometrically finite Kleinian groups acting on real hyperbolic space H^3. The boundaries of relatively hyperbolic groups generalize the limit sets of Kleinian groups. Since the boundary of H^3 is S^2, the limit set of every Kleinian group is planar. Can every relatively hyperbolic group with planar boundary be realized a Kleinian group? The answer is no, and we will give illustrative examples to show the many ways this can fail. However, we prove that if G is one-ended, (G,P) is relatively hyperbolic, and the boundary of (G,P) is planar and without cut points, then all the peripheral subgroups are surface groups. This is consistent with G being a Kleinian group. We also formulate a conjecture about the general situation (which is an extension of the Cannon Conjecture). This is joint work with Chris Hruska.
Jenny Wilson, University of Michigan
The high-degree rational cohomology of the special linear group ▾

In this talk I will describe some current efforts to understand the high-degree rational cohomology of SL_n(Z), or more generally the cohomology of SL_n(R) when R is a number ring. Although the groups SL_n(R) do not satisfy Poincare duality, they do satisfy a twisted form of duality, called (virtual) Bieri--Eckmann duality. Consequently, their high-degree rational cohomology groups are governed by an SL_n(R)-representation called the Steinberg module. The key to understanding these representations is through studying the topology of certain associated simplicial complexes. I will survey some results, conjectures, and ongoing work on the Steinberg modules, and the implications for the cohomology of the special linear groups. This talk includes work joint with Brück, Kupers, Miller, Patzt, Sroka, and Yasaki. The talk is geared for topologists and will not assume prior expertise on the cohomology of arithmetic groups!



2023

David Gay, University of Georgia
On the smooth mapping class group of the 4-sphere ▾

I will talk about a group which might be the trivial group, in which case either the results I will discuss are pointless or are a useful step in the direction of proving triviality! But the group might not be trivial, which is one of the many examples of how little we really understand about dimension 4. The group in question is the smooth mapping class group of the 4-sphere, namely the group of smooth isotopy classes of orientation-preserving self-diffeomorphisms of S^4. Using ideas going back to Cerf I will explain why every element of this group can be described by loops of 2-spheres in connected sums of S^2XS^2's, that under favorable conditions (which might always hold), this description can be reduced to a loop of circles in S^1XS^3 and that the subgroup of the smooth mapping class group that arises under these favorable conditions is at worst a cyclic group of order 2. The last part involves finding explicit generators and relations for the subgroup in question.
Robert E. Gompf, University of Texas at Austin
Contact and Engel Topology ▾

Contact structures, like foliations, are intimately sensitive to the topology of their ambient manifolds, and have been used to prove deep results in 3-manifold topology. The contact condition is an example of a "topologically stable" condition for k-plane distributions on n-manifolds. Such conditions were classified by Cartan at the turn of the previous century. The one topologically stable condition that is still poorly understood is the Engel condition. Engel structures are 2-plane fields that only exist on 4-manifolds. It is still unknown what, if anything, they might tell us about 4-manifold topology. After discussing background, we will take a first step in this direction by exploring an Engel analogue of transverse knots in contact 3-manifolds.
Thomas Haettel, IUT Montpellier
Group actions on injective metric spaces ▾

A metric space is called injective if any family of pairwise intersecting balls has a nonempty global intersection. Such injective metric spaces enjoy many properties typical of nonpositive curvature, notably the existence of a convex geodesic bicombing. We will show that many groups of geometric nature have an interesting action by isometries on an injective metric space, including Gromov-hyperbolic groups, braid groups, lattices in Lie groups and mapping class groups of surfaces.
Jean Lafont, Ohio State University
Cubulating strict hyperbolizations ▾

The strict hyperbolization process developed by Charney and Davis inputs a simplicial complex, and outputs a piecewise hyperbolic, locally CAT(-1) space. This process has been used to produce aspherical manifolds with various interesting properties. I'll briefly review the strict hyperbolization process, and then explain how to construct an action of the fundamental group of the resulting space on a CAT(0) cube complex with controlled cell stabilizers. This is joint work with Lorenzo Ruffoni (Tufts).
Maggie Miller, Stanford University
Slice obstructions from genus bounds in definite 4-manifolds ▾

Minimum genus bounds for surfaces representing specific homology classes in some 4-manifolds can be used to show that certain knots in S^3 are not slice. For example, one genus bound due to Bryan in the 1990s can be used to show that the (2,1)-cable of the figure eight knot is not slice, recovering a result of Dai—Kang—Mallick—Park—Stoffregen from last summer based on Heegaard Floer (and related) homology. I’ll talk about this construction, underlying motivation, and some interesting open questions. This is joint work with Paolo Aceto, Nickolas A. Castro, JungHwan Park and András Stipsicz.
Paige North, University of Pennsylvania
The univalence principle ▾

The Equivalence Principle is an informal principle asserting that equivalent mathematical objects have the same properties. For example, group theory has been developed so that isomorphic groups have the same group-theoretic properties, and category theory has been developed so that equivalent categories have the same category-theoretic properties (though sometimes other, ‘evil’ properties are considered). Vladimir Voevodsky established Univalent Foundations as a foundation of mathematics (based on dependent type theory) in which the Equivalence Principle for types (the basic objects of type theory) is a theorem. Later, versions of the Equivalence Principle for set-based structures such as groups and categories were shown to be theorems in Univalent Foundations.

In joint work with Ahrens, Shulman, and Tsementzis, we formulate and prove versions of the Equivalence Principle for a large class of categorical and higher categorical structures in Univalent Foundations. Our work encompasses bicategories, dagger categories, opetopic categories, and more.

Early versions and our generalization of the Equivalence Principle in Univalent Foundations rely on the fact that the basic objects -- the types -- can be regarded as spaces. That is, Univalent Foundations can be viewed as an axiomatization of homotopy theory and as such is closely related to Quillen model category theory. Univalent Foundations can also be viewed as a foundation of mathematics based not on sets, but on spaces. It is only the homotopical content of this foundation of mathematics that allows us prove something like the Equivalence Principle, something which is not possible in set-based foundations of mathematics, such as ZFC.
Margaret Symington , Mercer University
From K3 surfaces to spheres and back ▾

I will describe the beginnings of a project to understand and visualize singular integral affine structures on the two-sphere via rational polyhedra, integral affine unfoldings of them, and integral affine coordinate charts. Integral affine structures on the two-sphere are induced by special Lagrangian fibrations of K3 surfaces, which are studied in the context of mirror symmetry in dimension four. (The polyhedral presentations are motivated the Gross-Siebert approach to mirror symmetry on K3 surfaces.) More generally, the integral affine geometries are induced by almost toric structures on a symplectic K3 surface (symplectic four-manifold diffeomorphic to a quartic surface in CP3). I will explain how the integral affine structures arise, what some of them look like, and what they reveal about the fibered four-manifolds that produced them.
Liam Watson, University of British Columbia
Khovanov homology and Conway sphere ▾

I will describe some of my joint work with Claudius Zibrowius and Artem Kotelskiy, which recasts Bar-Natan’s invariants for Conway tangles in terms of immersed curves. This gives rise to some unexpected structural results, and I will attempt to place these in context by explaining what they tell us about Khovanov homology for knots admitting an essential Conway sphere.


2024

Martin Bridson, Oxford University
Profinite rigidity, Grothendieck Pairs, and finiteness properties ▾

A finitely generated, residually finite group G is said to be profinitely rigid if the only finitely generated, residually finite groups with the same set of finite quotients as G are those that are isomorphic to G. More generally, one wants to know which properties P of groups are profinite invariants, i.e. if G has P and H has the same finite quotients as G, does H have P? For example, if G is a 3-manifold group, is H? If G is torsion-free, is H?

I will begin this talk with an overview of how the study of profinite rigidity has thrived in recent years due to a rich interplay between group theory, low-dimensional geometry and arithmetic. I'll then present recent work that underscores the importance of finiteness properties and the homology of groups in this context, describing results that exemplify extremes of rigid and non-rigid phenomena.

Bruno Martelli, Università di Pisa
Negatively curved spaces obtained via branched coverings over a torus ▾

In his seminal paper on hyperbolic groups, Gromov tries to extend a beautiful construction of Thurston from dimension 3 to arbitrary n: he builds of a non-positively curved manifold as a branched covering over the n-torus, ramified along codimension-2 subtori. This is particularly interesting in odd dimension n, because every generic fibration of the n-torus lifts to a fibration of the branched cover.

It was later noted by Bestvina that this construction does not yield a negatively-curved space when n>3, due to the presence of 2-flats in the universal cover. In this talk we show that Gromov's construction can be adjusted to produce negatively curved spaces (in fact, hyperbolic manifolds) in at least one specific case in dimension 5. We recover in this way some fibering hyperbolic 5-manifolds constructed with Italiano and Migliorini.

Viktor Ginzburg, University of California, Santa Cruz
Topological Entropy of Hamiltonian Systems and Persistence Modules ▾

Topological entropy is a fundamental invariant of a dynamical system, measuring its complexity. In this talk, we will discuss connections between the topological entropy of a Hamiltonian system, e.g., a geodesic flow, and the underlying filtered Morse or Floer homology viewed as a persistence module in the spirit of Topological Data Analysis. We will introduce barcode entropy — a Morse/Floer theoretic counterpart of topological entropy — and show that barcode entropy is closely related to topological entropy and that, in low dimensions, these invariants agree. For instance, for a geodesic flow on any closed surface the barcode entropy is equal to the topological entropy. The talk is based on joint work with Erman Cineli, Basak Gurel and Marco Mazzucchelli.
Eugenia Cheng, School of the Art Institute of Chicago and City, University of London
The Eckmann–Hilton argument in higher dimensional category theory ▾

The Eckmann–Hilton argument gives us circumstances in which commutativity arises from having two binary operations that interact in certain coherent ways. The argument is used to show that all higher homotopy groups are Abelian, but can be stated in generality as a piece of algebra: if a set is equipped with two monoid structures satisfying a type of interchange, then the two binary operations coincide and are commutative.

The Eckmann–Hilton argument arises in higher-dimensional category theory via degeneracy, that is, higher-dimensional structures where some of the lowest dimensions are trivial. More generally, commutativities arise when we consider cells with identity boundaries, as the cells can then commute past each other via interchange. Degenerate structures become key test cases for theories of higher category, enabling us to examine whether the appropriately weak forms of commutativity arise. For sets there is only strict commutavitiy, but for categories there is space for a weak form of commutativity called a braiding, and a stricter one called a symmetry. For 2-categories there is another possibility in between, and as we add in dimensions there are more and more nuanced possibilities for flavors of commutativity. These are arranged into the Periodic Table of Higher Categories, conjectured by Baez and Dolan. As we increase dimensions we also need to generalise the Eckmann–Hilton argument into higher dimensions, with more binary operations, and more interchange laws.

In this talk we will give an overview of the basic case of doubly-degenerate 2-categories, before moving on to the higher-dimensional argument showing how doubly-degenerate 3-categories give braided monoidal categories, as in my recent work with Alexander S. Corner. We will also introduce work in progress on $n$-degenerate $(n+1)$-categories.

Rhiannon Griffiths, Cornell University
The Homotopy Types of Higher Categories ▾

Grothendieck’s Homotopy Hypothesis states that homotopy n-types are modeled by n-groupoids, and by extension, that spaces are modeled by $\infty$-groupoids. Moreover, this equivalence is induced by the homotopy groups of both constructions.

While there are some geometric solutions to the Hypothesis, the most interesting solution would be to show that some completely algebraic structures are equivalent to topological ones. Unfortunately, the Homotopy Hypothesis is known to be false for strict higher groupoids, and fully weak ones are too complex to work with.

In this talk, I will present an operadic method for calculating the homotopy groups of algebraic higher groupoids. I will then use this to identity models algebraic of n-groupoid that are tractable enough to work with directly, but which escape the homotopy degeneracies that appear in the strict setting.
Morgan Weiler, Cornell University
Anchored symplectic embeddings of four-dimensional toric domains ▾

Symplectic geometry is a generalization of classical mechanics, in which position and momentum coordinates are paired. Mathematically, a symplectic manifold is an even-dimensional manifold carrying a "symplectic form" – a closed, nondegenerate 2-form. In two dimensions, symplectic geometry is equivalent to volume-preserving geometry, but in higher dimensions, Gromov proved in 1985 that an embedding from a finite-volume ball into an infinite-volume cylinder can only preserve the symplectic form if the ball embeds via the identity. Symplectic geometers have studied generalizations of Gromov's result ever since. In this talk, we will show that in many four-dimensional examples, requiring the complement of the embedding to contain a symplectic surface with fixed boundary conditions (the so-called "anchor") provides an even stronger restriction than the symplectic form alone. Our examples have a toric structure, and when symplectic embeddings between them are anchored we show they must also be toric. The main tool is the interplay between the action filtration and intersection number in embedded contact homology, which we will review. Joint work with Michael Hutchings, Agniva Roy, and Yuan Yao.
Ina Petkova, Dartmouth
Spectral GRID invariants and Lagrangian cobordisms ▾

Knot Floer homology is a powerful invariant of knots and links, developed by Ozsvath and Szabo in the early 2000s. Among other properties, it detects the genus, detects fiberedness, and gives a lower bound to the 4-ball genus. The original definition involves counting homomorphic curves in a high-dimensional manifold, and as a result the invariant can be hard to compute. In 2007, Manolecu, Ozsvath, and Sarkar came up with a purely combinatorial description of knot Floer homology for knots in the 3-sphere, called grid homology. Soon after, Ozsvath, Szabo, and Thurston defined invariants of Legendrian knots using grid homology. We show that the filtered version of these GRID invariants, and consequently their associated invariants in a certain spectral sequence for grid homology, obstruct decomposable Lagrangian cobordisms in the symplectization of the standard contact structure, strengthening a result of Baldwin, Lidman, and Wong. This is joint work with Jubeir, Schwartz, Winkeler, and Wong.
Kasia Jankiewicz, University of California Santa Cruz
Profinite properties of Artin groups ▾

Artin groups are a family of groups that generalize braid groups, and can be defined and studied from various perspectives: topologically, algebraically, and combinatorially. They are mysterious - a lot of basic questions about them remain unanswered. Artin groups are given by simple looking group presentations, are closely related to Coxeter groups, and arise as the fundamental groups of certain complex hyperplane arrangements. In this talk, I will focus on profinite properties of Artin groups. Informally, those properties tell us if the geometry and algebra of the group can be approximated by its finite quotients. I will mention some joint work with Kevin Schreve.
Noel Brady, University of Oklahoma
Isoperimetric inequalities and pushing fillings ▾

Suppose that a finitely presented group H is a normal subgroup of a hyperbolic group with free quotient. By a result of Gersten and Short the group H has a polynomial isoperimetric inequality. This is established by starting with a geometrically controlled disk filling in the ambient hyperbolic group of a loop in H and then pushing this filling into H. We describe other situations where one can push fillings with respect to height functions on spaces and give applications to subgroups of right-angled Artin groups and to Houghton groups.
Nathan Dunfield, University of Illinois Urbana-Champaign
Counting essential surfaces in 3-manifolds ▾

Counting embedded curves on a surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting surfaces in a 3-manifold, with the key difference that now the surfaces themselves have more intrinsic topology. As there are only finitely many essential surfaces of bounded Euler characteristic up to isotopy in an atoroidal 3-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Haken’s normal surface theory, we can characterize not just the rate of growth but show the exact count is a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples.

This is joint work with Stavros Garoufalidis and Hyam Rubinstein and the reference is our paper with the same title. The only background I will assume is the notion of a manifold, the genus of a surface, and a little about the fundamental group.