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Topology Festival Archive

Festival Speakers 1982-2018






































Festival cancelled due to COVID-19 crisis.



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!


David Gay, University of Georgia, Malott 251
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, Malott 251
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, Malott 251
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, Malott 251
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, Malott 251
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, Malott 251
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, Malott 251
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, Malott 251
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.