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Topology Seminar

The Topology Seminar has talks on a variety of topics in topology, including algebraic geometry, chromatic homotopy theory, configuration spaces, continuum theory, functor calculus, graph cohomology, homotopical algebra, Nielsen fixed-point theory, operads, simplicial sets, spaces of knots and links, span theory, and topological groups.
For more information contact Robin Koytcheff.

Spring 2019

For the Spring 2019 semester we will meet 1:00 - 2:00 on Fridays in 208 Maxim Doucet Hall.

  • 25 January 2019
    Connectivity and growth in the homology of graph braid groups
    Ben Knudsen
    Harvard University
    Abstract: I will discuss recent work with An and Drummond-Cole showing that the homology of configuration spaces of graphs exhibits eventual polynomial growth, an analogue of classical homological and representation stability results for manifolds. We compute the degree of this polynomial in terms of an elementary connectivity invariant, in particular verifying an upper bound conjectured by Ramos. Along the way, we uncover a new “edge stabilization” mechanism and a family of spectral sequences arising from a small chain model first introduced by Swiatkowski.
  • 1 February 2019 (TIME CHANGE 1:30-2:30)
    The Taylor tower for the space of knots and additive finite-type knot invariants
    Robin Koytcheff
    UL Lafayette
    Abstract: Any space of embeddings has a sequence of Taylor approximations which comes from Goodwillie–Weiss functor calculus. In joint work with Budney, Conant, and Sinha, we studied this Taylor tower for the space of long knots in 3-space and showed that it is an additive finite-type (a.k.a. Vassiliev) invariant. Our result suggests that the tower could be a universal finite-type invariant over the integers. We will discuss a multiplicative structure on the tower which is compatible with stacking long knots and which was a key ingredient in our work.
  • 7 February 2019
    Manifold calculus of functors
    Paul Arnaud Songhafouo Tsopméné
    University of Regina
    Abstract: Manifold calculus, due to Goodwillie and Weiss, goes back to around 1999. It is concerned with the study of functors from the poset O(M) of open subsets of a manifold M to the category of topological spaces. Manifold calculus has been used by many authors to understand the topology of a variety of embedding spaces. This talk will go over three things. First I will explain the concept of manifold calculus by highlighting some analogies with the ordinary calculus. Next I will present a combinatorial model for manifold calculus. Lastly, if time permits, I will say few words about the classification of homogeneous functors from O(M) into any (simplicial) model category.
  • 15 February 2019
    Configuration spaces as weak operads, II
    Philip Hackney
    UL Lafayette
    Abstract: Last October, Robin Koytcheff showed us how one can compactify configuration spaces to yield an E_n operad. The goal of this talk (which is based on arXiv:1707.05027) is to explain how configuration spaces can be considered directly as (a part of) an "operad up-to-homotopy."
  • 22 February 2019
    Manifold calculus of functors for r-immersions
    Ismar Volić
    Wellesley College
    Abstract: Manifold calculus of functors can be used for studying spaces of embeddings, including spaces of r-immersions, which are immersions where no more than r – 1 points are allowed to equal (embeddings are thus 2-immersions). I will give some background on manifold calculus of functors and then present some recent work on understanding the manifold calculus Taylor tower that approximates the space of r-immersions. Manifold calculus in this context supplies interesting connections to combinatorial topology, such as the structure of certain subspace arrangements as well as Tverberg-like problems, so some time will be devoted to these topics. This is joint work with Franjo Šarčević.
  • 8 March 2019
    Configuration spaces as weak operads, III
    Philip Hackney
    UL Lafayette
    Abstract: The goal of this talk is to explain how configuration spaces can be considered directly as (a part of) an "operad up-to-homotopy."
  • 22 March 2019
    Spaces of diffeomorphisms and spaces of knots
    Robin Koytcheff
    UL Lafayette
    Abstract: I will discuss results on spaces of diffeomorphisms of low-dimensional manifolds and how these can be used to deduce results about spaces of knots.
  • 29 March 2019
    Spaces of diffeomorphisms and spaces of knots, part II
    Robin Koytcheff
    UL Lafayette
    Abstract: Last week, I surveyed results on diffeomorphisms of disks and spheres, and I explained how the (3-dimensional) Smale Conjecture shows that the space of long unknots is contractible. This week, I will discuss how it can be used to calculate homotopy types of spaces of some more interesting knots.
  • 26 April 2019
    A Model For S^{-g}
    Thomas Credeur
    UL Lafayette
    Abstract: For a fixed prime p, it is well known that the K(n)-local Spanier-Whitehead dual of the nth Morava E-theory, E_n, is equivalent to the -n^2 suspension of E_n. This equivalence is equivariant with respect to the action of the Morava stabilizer group in the homotopy category, but is not equivariant at the point-set level. More recent work of Beaudry, Goerss, Hopkins, and Stojanoska has introduced a "dualizing spectrum" S^{-g}, which has the property that the K(n)-local dual of E_n is equivariantly equivalent to S^{-g} \wedge E_n. In this talk we give a model for the spectrum S^{-g}, and show that the smash product of E_n with this model is equivalent to the -n^2 suspension of E_n.

Fall 2018

For the Fall 2018 semester we will meet 1:00 - 2:00 on Fridays in 208 Maxim Doucet Hall.

  • 31 August 2018
    A complete definition of E_\infty ring spaces, pt. I: E_\infty spaces
    Daniel Davis
  • 14 September 2018
    A complete definition of E_\infty ring space, pt. II
    Daniel Davis
  • 21 September 2018
    Spaces over operad pairs and the special case of E_\infty ring spaces
    Daniel Davis
  • 28 September 2018
    Spaces over operad pairs: the final leg in their definition and a few examples
    Daniel Davis
    Abstract: Building on the three previous talks, we give the definition of a space over an operad pair, which enables us to complete the definition of an E_\infty ring space. Then we give several examples of operad pairs and what algebraic structures on spaces these yield. Time permitting, the next goal is to understand how symmetric bimonoidal categories give E_\infty ring spaces in a canonical way.
  • 12 October 2018
    The state of the art on a few problems in chromatic homotopy theory
    Daniel Davis
  • 26 October 2018
    Manifold-theoretic E_n operads
    Robin Koytcheff
    Abstract: An E_n operad is an operad equivalent to the little n-disks operad. I will begin by recalling the definition of the little disks operad. The Fulton—Macpherson operad and Kontsevich operad are two E_n operads, both of which can be defined via a manifold-theoretic compactification of the configuration space of points in n-dimensional space. I will discuss this compactification, the operad structure maps, and some applications.
  • 2 November 2018
    Homotopy theory of algebras over operads
    Dmitri Pavlov
    Texas Tech University
    Abstract: Suppose O is an operad in simplicial sets, chain complexes, motivic spectra, etc. Consider the category Alg_O of algebras over this operad equipped with degreewise weak equivalences. We are interested in the following questions:
    1) Under what conditions on O does Alg_O present the "homotopically correct" category of algebras over an operad (e.g., in the sense of \infty-categories)?
    2) Under what conditions on O does Alg_O possess a model structure?
    3) Under what conditions on O and O' does a weak equivalence $O \to O'$ induce a Quillen equivalence $Alg_O \to Alg_O'$?
    We provide a complete answer to 1) and 3) in terms of an if-and-only-if criterion that is easy to verify in practice, and we also give a sufficient condition for 2) that is applicable to all known practical examples. Our criteria work in abstract monoidal model categories, such as simplicial sets, chain complexes, motivic spectra, topological spaces, and many others. Our work espouses the yoga of "synthetic" model category theory, which postulates that model structures and their properties can be most easily established inductively by tracing the construction of the category under consideration step-by-step, as opposed to constructing model structures directly in one step.
    The above is joint work with Jakob Scholbach (Muenster). If time permits, I will also discuss the case of coalgebras over operads, as well as Leinster-style homotopy algebras over operads.
  • 9 November 2018
    Configuration spaces as weak operads
    Philip Hackney
    Abstract: Two weeks ago, Robin Koytcheff told us one can compactify configuration spaces to yield an E_n operad. In this talk, I'll explain how configuration spaces can be considered directly as (a part of) an "operad up-to-homotopy." The beginning of the talk will be a brief introduction to the dendroidal objects of Moerdijk & Weiss. This talk is based on arXiv:1707.05027.
  • 16 November 2018
    Cellular E_2-algebras and the unstable homology of mapping class groups
    Alexander Kupers
    Harvard University
    Abstract: We discuss joint work with Soren Galatius and Oscar Randal-Williams on the application of higher-algebraic techniques to classical questions about the homology of mapping class groups. This uses a new "multiplicative" approach to homological stability -- in contrast to the "additive" one due to Quillen -- which has the advantage of providing information outside of the stable range.
  • 30 November 2018
    Transfer Maps and a Perfect Pairing of Spectra
    Thomas Credeur
    Abstract:Let G be a profinite group. Given any discrete G-spectrum X and a closed subgroup K of G, one can form the homotopy fixed point spectrum X^{hK}. Given an open subgroup K of G and an open subgroup H of K, there is a "wrong way" map in the stable homotopy category, the transfer map from X^{hH} to X^{hK}. Using these maps we can construct a transfer map from E_n^{hU} to E_n^{hG_n}, where G_n is the Morava stabilizer group and E_n is a Lubin-Tate spectrum. The transfer maps can then be used to construct a perfect pairing in the homotopy category. As a consequence of this perfect pairing, we will see that E_n^{hU} is self dual.
  • 14 December 2018
    Special meeting: 2:15 - 3:45 (talk proper: 2:45-3:45)
    Dependent paths in homotopy type theory
    Jonathan Steven Prieto Cubides
    Informatics PhD student
    University of Bergen (Norway)
    Abstract: We introduce some basic notions about HoTT to talk about dependent paths. We show a joint work with Marc Bezem that gives us two proofs about the geometrical intuition in HoTT behind the dependent paths, also called pathovers, which has been only mentioned but not proved before. We type-check in Agda the proofs and show some lemmas for the second one that makes a shorter proof.

Spring 2018

For the Spring 2018 semester we will meet at 1:00 on Fridays in 208 Maxim Doucet Hall.

  • 26 January 2018
    Graph complexes, formality, and configuration space integrals for spaces of braids, part I
    Robin Koytcheff
    Abstract: This series of talks will culminate in a report on recent work joint with Komendarczyk and Volic, where we connect two integration-based approaches to the cohomology of spaces of braids. This first part will include an introduction to the configuration space of distinct points in Euclidean space, as well as the related braid groups. I will discuss the basic algebraic topology of these configuration spaces and some of the underlying geometric intuition.
  • 2 February 2018
    Graph complexes, formality, and configuration space integrals for spaces of braids, part II
    Robin Koytcheff
  • 16 February 2018
    Graph complexes, formality, and configuration space integrals for spaces of braids, part III
    Robin Koytcheff
  • 23 February 2018 (1:30-2:30 TIME CHANGE)
    Graph complexes, formality, and configuration space integrals for spaces of braids, part IV
    Robin Koytcheff
    Abstract: I will continue to talk about cohomology of the space of loops in a space X, in terms of the bar construction on the cochains (or cohomology) of X. I will briefly discuss the notion of a Koszul algebra, as well as Chen’s iterated integrals for the case where X is a manifold. If time permits, I will begin discussing configuration space integrals and related cochain complexes of graphs.
  • 16 March 2018
    Graph complexes, formality, and configuration space integrals for spaces of braids, part V
    Robin Koytcheff
    Abstract: I will define Chen’s iterated integrals, which are a way of realizing the cohomology of the loop space of a manifold via differential forms. I will also discuss configuration space integrals. One variant of these was used to prove the formality (over the rationals) of configuration spaces. Another variant produces cohomology classes in spaces of knots and links. In joint work with Volic and Komendarczyk, we establish a relationship involving these three types of integrals.
  • 23 March 2018
    Homotopy fixed points for trivial group actions
    Daniel Davis
  • 19 April 2018
    Goodwillie-Weiss manifold calculus
    Victor Turchin
    Kansas State
    Abstract: The goal of this talk is to explain a bit more deeply an approach to studying embedding spaces which will be briefly mentioned in the Colloquium talk and which is called Goodwillie-Weiss calculus of functors.
  • 27 April 2018
    Another homotopy theory: the category of complete Segal spaces
    Daniel Davis

Fall 2017

For the Fall 2017 semester we will meet at 1:00 on Fridays in 208 Maxim Doucet Hall.

  • 8 September 2017
    An introduction to the Goerss-Hopkins Linearization Hypothesis and a connection to continuous G-spectra.
    Daniel Davis
  • 15 September 2017
    More about the Goerss-Hopkins Linearization Hypothesis and a connection to continuous G-spectra.
    Daniel Davis
  • 22 September 2017
    The Goerss-Hopkins Linearization Hypothesis and continuous G-spectra, part 3.
    Daniel Davis
  • 29 September 2017
    The Goerss-Hopkins Linearization Hypothesis and continuous G-spectra, part 4.
    Daniel Davis
  • 13 October 2017
    The Goerss-Hopkins Linearization Hypothesis and continuous G-spectra, part 5.
    Daniel Davis
  • 20 October 2017
    Finite-type invariants of knots, links, and string links
    Robin Koytcheff
    Abstract: Finite-type knot invariants (a.k.a. Vassiliev invariants) are an important class of invariants in that they conjecturally approximate all knot invariants and hence separate knots. They may also be defined for (closed) links and string links, and they are known to separate string links up to link homotopy. In other words, they are a complete invariant of string links where each component may pass through itself. This parallels (and is related to) a story about the kappa invariant, which conjecturally separates closed links up to link homotopy. In joint work with F. Cohen, Komendarczyk, and Shonkwiler, we showed that the kappa invariant separates string links up to link homotopy. In this talk, we will focus on the elementary, purely combinatorial description of finite-type invariants.
  • 27 October 2017
    Homotopy string links, configuration spaces, and the kappa invariant
    Robin Koytcheff
    Abstract: A link is an embedding of disjoint circles in space. A link homotopy is a path between two links where distinct components may not pass through each other, but where a component may pass through itself. In the 1990s, Koschorke conjectured that link homotopy classes of n-component links are distinguished by the kappa invariant. This invariant is essentially the map that a link induces on configuration spaces of n points. In joint work with F. Cohen, Komendarczyk, and Shonkwiler, we proved an analogue of this conjecture for string links. A key ingredient is a multiplication on maps of configuration spaces, akin to concatenation of loops in a space. This approach is related to recent joint work with Budney, Conant, and Sinha on finite-type knot invariants and the Taylor tower for the space of knots.
  • 17 November 2017
    The Simplicial Model Category Structure on Symmetric Spectra
    Thomas Credeur
  • 1 December 2017
    Symmetric Spectra and the Quillen Equivalence with Bousfield-Friedlander Spectra
    Thomas Credeur

Spring 2017

For the Spring 2017 semester we will meet at 1:00 on Fridays in 208 Maxim Doucet Hall.

  • 27 January 2017
    Genuine equivariant operads
    Luis Pereira
    University of Virginia
     
    Abstract: A fundamental result in equivariant homotopy theory due to Elmendorf states that the homotopy theory of $G$-spaces, with w.e.s measured on all fixed points, is Quillen equivalent to the homotopy theory of $G$-coefficient systems in spaces, with w.e.s measured at each level of the system. Furthermore, Elmendorf's result is rather robust: suitable analogue results can be shown to hold for, among others, the categories of (topological) categories and operads. However, it has been known for some time that in the $G$-operad case such a result does not capture the "correct" notion of weak equivalence, a fact made particularly clear in recent work of Blumberg and Hill discussing a whole lattice of "commutative operads with only some norms" that are not distinguished at all by the notion of w.e. suggested above. In this talk I will talk about one piece of a current joint project with Peter Bonventre which aims at providing a more diagrammatic understanding of Blumberg and Hill's work using a notion of $G$-trees, which are a somewhat subtle generalization of the trees of Cisinski-Moerdijk-Weiss. More specifically, I will describe a new algebraic structure, which we dub a "genuine equivariant operad", which naturally arises from the study of $G$-trees and which we conjecture to be the analogue of coefficient systems in the "correct" analogue of Elmendorf's theorem for $G$-operads.
  • 3 February 2017
    No meeting
  • 10 February 2017
    No meeting

Fall 2016

For the Fall 2016 semester we will meet at 1:00 on Fridays in 208 Maxim Doucet Hall.

  • 16 September 2016
    An introduction to infinity-categories and the example of small categories.
    Daniel Davis
  • 23 September 2016
    Infinity-categories: more on the example of small categories and infinity-categorical versions of basic category-theoretic notions.
    Daniel Davis
  • 30 September 2016
    Some examples of infinity-categorical concepts that build on category-theoretic notions.
    Daniel Davis
  • 14 October 2016
    The notions of join and overcategory in the setting of infinity-categories.
    Daniel Davis
  • 21 October 2016
    Simplicial nerves and the homotopy category of an infinity-category.
    Daniel Davis
  • 28 October 2016
    The homotopy category of an infinity-category and a nicer formulation of it.
    Daniel Davis
  • 4 November 2016
    Given an \infty-category C, there is an isomorphism h(C) \to \pi(C) of categories.
    Daniel Davis
  • 11 November 2016
    Given an \infty-category C, more on the category \pi(C), and equivalences in C.
    Daniel Davis

Topology Seminar Archive