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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 Philip Hackney.

Spring 2022

For the Spring 2022 semester we will usually meet 1:00 - 2:00 on Fridays. Some meetings will be on Zoom and others will be in-person. We may need to change the time and date on occasion to accomodate visitors.

For more information or connection details contact Philip Hackney.

  • 20 January 2022 (on Zoom at 4:40 pm Thursday)
    Operadic Unitarization
    Olivia Borghi (University of Melbourne)
    Abstract: In the 1990 paper "Quasi-Hopf Algebras" V.G. Drinfe'ld describes a notion of hopf algebra with a weakened coassociativity called a quasi-Hopf algebra. He builds upon this notion and defines several different iterations of quasi-Hopf algebras with weakened cocommutativity. These include triangular, quasitriangular and coboundary quasi-Hopf alegbras. Drinfe'ld further describes a process of "unitarization" whereby we can provide certain quasitiangular quasi-Hopf algebras (quantum groups) with coboundary qausi-Hopf algebra structures. Tanaka duality tells us there is a correspondence between these weakened Hopf algebras and monoidal categories. Using Donald Yau's theory of G-monoidal categories I build an operadic counterpart to this "unitarization" whereby we can provide certain braided monoidal categories with coboundary monoidal structures. I then build a theory of infinite dimensional G-monoidal categories to describe this operadic unitarization in that setting.
  • 11 February 2022 (on zoom)
    Smooth embeddings and their families
    Danica Kosanović (ETH Zürich)
    Abstract: Configuration spaces of manifolds are examples of spaces of embeddings, which can be employed for studying all other embedding spaces, via Goodwillie-Weiss-Klein calculus. We will discuss how certain classes in homotopy groups of configuration spaces give rise to nontrivial families of embeddings, that generalise lower central series of braid groups and Vassiliev-Gusarov-Habiro constructions in knot theory.
  • 25 February 2022 (on zoom)
    Open problems in graph models for operadic structures
    Philip Hackney (UL Lafayette)
    Abstract: Last Fall we gave a talk about a new way to express the category of undirected graphs that is used to model modular operads. We will briefly recall this, and then present several related open problems.
  • 4 March 2022 (on zoom)
    Quasi-2-Segal sets
    Matt Feller (University of Virginia)
    Abstract: There are various models of "categories up to homotopy," such as Segal spaces which are simplicial spaces satisfying a condition that encodes unique (up to homotopy) composition of morphisms. In the last decade, the study of simplicial spaces which satisfy a particular weaker algebraic condition was independently begun by both Dyckerhoff–Kapranov, who call them 2-Segal spaces, and Gálvez–Kock–Tonks, who call them decomposition spaces. Meanwhile, quasi-categories are an alternative model of up-to-homotopy categories whose theory has been massively developed in the past two decades, thanks largely due to Joyal and Lurie. With the ultimate hope of harnessing the technology for quasi-categories in the 2-Segal setting, we set out to define a 2-Segal version of quasi-categories as simplicial sets with a particular horn lifting condition. In particular, our goal is to describe an associated model structure on the category of simplicial sets.
  • 25 March 2022 (on zoom)
    Graphing, homotopy groups of spheres, and spaces of links and knots
    Robin Koytcheff (UL Lafayette)
    Abstract: We determine the homotopy groups of spaces of 2-component long links in a range that depends on the dimensions of the source manifolds and target manifold. Our result applies in roughly the triple-point-free range, if one incorporates the dimension of the homotopy group into the source manifold. Within this range, graphing maps identify each group with the direct sum of a homotopy group of a sphere and a group of isotopy classes of higher-dimensional knots. We also show that the homotopy groups of a sphere generally form a direct summand of the homotopy groups of a space of links. Finally, by joining components, we obtain a nontrivial map from a homotopy group of a sphere into the first nontrivial homotopy group of a space of long knots with odd codimension. These maps send the class of each Hopf fibration to a generator.
  • 1 April 2022 (on zoom)
    Grothendieck–Teichmüller theory and modular operads
    Luciana Basualdo Bonatto (Oxford)
    Abstract: The absolute Galois group of the rationals Gal(Q) is one of the most important concepts in number theory. Although we cannot explicitly describe more than two elements in this infinite group, we know it acts on well-known algebraic and topological objects in compatible ways. Grothendieck–Teichmüller theory uses these representations to study this Galois group. One of the most important representations comes from the compatible actions of Gal(Q) on all the profinite mapping class groups of surfaces. In this talk, we introduce an algebraic tool called an infinity modular operad and use it to construct an infinity modular operad of surfaces capturing the compatibility structure above. We show this admits an action of Gal(Q), translating the Grothendieck–Teichmüller program into the theory of infinity modular operads, which provides new ideas and tools to approach this problem. This is joint work with Marcy Robertson.
  • 8 April 2022 (on zoom)
    The homotopy-type of diffeomorphism groups of "small" manifolds, and their relation to spaces of string links
    Ryan Budney (University of Victoria)
    Abstract: I will describe a technique for probing the homotopy-type of diffeomorphism groups of "small" manifolds. These techniques are largely dimension-agnostic, although they only give new results in dimension 4 and up. We generate our diffeomorphisms via an understanding of the homotopy-type of spaces of string links, and we also detect our diffeomorphisms using embedding-calculus (alternatively, finite-type invariants) for different spaces of string links. This allows us to say that the mapping class group of S^1 x D^3 is not finitely generated. It also allows us to detect the Hatcher-Wagoner diffeomorphisms of S^1 x D^{n-1} for n>5, and it gives us some interesting mapping classes of S^1 x D^4.
  • 29 April 2022 (on zoom)
    Twisted arrow categories of operads and Segal conditions
    Sergei Burkin
    Abstract: Several homotopy coherent structures can be defined via Segal conditions on presheaves over categories such as the simplex category, Moerdijk–Weiss dendroidal category, Segal's category, and Hackney–Robertson–Yau categories. We introduce a uniform construction of such categories from operads. The construction at least partially explains how and why these categories appear in homotopy theory.

Fall 2021

For the Fall 2021 semester we will usually meet 11:00 - 12:00 on Fridays. Some meetings will be on Zoom and others will be in-person. We may need to change the time and date on occasion to accomodate visitors.

For more information or connection details contact Philip Hackney.

  • 10 September 2021 (on Zoom)
    Graph categories for operadic structures
    Philip Hackney
    In this talk, I'll give a brand new definition of several categories of graphs with loose ends. These categories have appeared previously in joint work with Robertson and Yau. Each is designed to control a particular operadic structure, like modular operads, cyclic operads, (wheeled) properads and so on, as well as their "higher" variants. This is analogous to the situation with the Moerdijk–Weiss dendroidal category Ω and the simplicial category Δ, which are useful for describing higher operads and higher categories.
    We will explain the Segal condition and the relationship with operadic structures, and also how the new descriptions facilitate comparison of structures.
    This talk will not require any background on generalized operads or operadic structures, nor will it require any knowledge of the graphical categories under discussion. Indeed, a key feature of the new definitions is that they can be understood from scratch, and nerve theorems allow one to define operadic structures as certain presheaves on these graph categories.
  • 17 September 2021 (on Zoom)
    Integrals, trees, and spaces of pure braids and string links
    Robin Koytcheff
    Abstract: The based loop space of configurations in a Euclidean space R^n can be viewed as the space of pure braids in R^{n+1}. In joint work with Komendarczyk and Volic, we studied its real cohomology using an integration map from a certain graph complex and recovered a result of Cohen and Gitler. Specifically, the map we studied is a composition of Kontsevich’s formality integrals and Chen’s iterated integrals. We showed that it is compatible with Bott-Taubes integrals for spaces of 1-dimensional string links in R^{n+1}. As a corollary, the inclusion of pure braids into string links in R^{n+1} induces a surjection in cohomology for any n>2. More recently, we showed that the dual of the integration map embeds the homotopy groups of the space of pure braids into a space of trivalent trees. We also showed that a certain subspace of these homotopy groups injects into the homotopy groups of spaces of k-dimensional string links in R^{n+k} for many values of n and k.
  • 24 September 2021 (on Zoom)
    Khovanov's categorification of the Jones polynomial
    Jake Sundberg
    Abstract: This talk focuses primarily on the Bar-Natan paper of the same name, which gives a “picture-rich glimpse” into Khovanov homology. In this construction, a 1+1-dimensional TQFT is applied to the cube of resolutions of a knot to transition from a topological setting to an algebraic one. This construction gives the homological object at the core of this talk, which is strictly more powerful than the Jones polynomial. A long exact sequence used in computing Khovanov homology will be discussed if time permits.
  • 1 October 2021 (on Zoom at 11:15)
    A Gillet-Waldhausen theorem for chain complexes of sets
    Maru Sarazola (Johns Hopkins)
    Abstract: In recent work, Campbell and Zakharevich introduced a new type of structure, called ACGW-category. These are double categories satisfying a list of axioms that seek to extract the properties of abelian categories which make them so particularly well-suited for algebraic K-theory. The main appeal of these double categories is that they generalize the structure of exact sequences in abelian categories to non-additive settings such as finite sets and reduced schemes, thus showing how finite sets and schemes behave like the objects of an exact category for the purpose of algebraic K-theory.
    In this talk, we will explore the key features of ACGW categories and the intuition behind them. Then, we will move on to recent work with Brandon Shapiro where we further develop this program by defining chain complexes and quasi-isomorphisms for finite sets. These satisfy an analogue of the classical Gillet–Waldhausen Theorem, providing an alternate model for the K-theory of finite sets.
  • 15 October 2021 (on Zoom at 3:30)
    A topological characterization of the Kashiwara–Vergne groups
    Marcy Robertson (University of Melbourne)
    Abstract: Solutions to the Kashiwara–Vergne equations in noncommutative geometry are a "higher dimensional" version of Drinfeld associators. In this talk we build on work of Bar-Natan and Dancso and identify solutions of the Kashiwara–Vergne equations with isomorphisms of (completed) wheeled props of "welded tangled foams" — a class of knotted surfaces in $\mathbb{R}^4$. As a consequence, we identify the symmetry groups of the Kashiwara–Vergne equations with automorphisms of our (completed) wheeled props.
    The first part of this talk will be accessible to a general audience and I will not assume familiarity with the Kashiwara-Vergne equations, Drinfeld associators or wheeled props. Includes joint work with Z. Dancso and I. Halacheva.
  • 22 October 2021 (on Zoom at 3:30)
    Torsion in Khovanov Homology, Part I
    Jake Sundberg
    Abstract: Przytycki discovered infinite families of links and knots whose Khovanov homology contains torsion different than $\mathbb{Z}_2$. We examine the methodology used to find these families and discuss some counterexamples to the Przytycki–Sazdanović braid conjecture that were found using these techniques.
  • 29 October 2021 (on Zoom at 2:00)
    Torsion in Khovanov Homology, Part II
    Jake Sundberg
    Abstract: A homologically thin knot/link is one whose Khovanov homology over a ring R is supported on two adjacent diagonals. As an introduction to the topic, several related concepts are defined and examples are given. We work toward proving a theorem which gives conditions for which all torsion of knots/links which are homologically thin over an interval is $\mathbb{Z}_2$-torsion. If time permits, the Khovanov homology of closed 3-braids will be discussed.
  • 5 November 2021 (on Zoom at 11:00 am)
    Classifying model structures on finite total orders
    Scott Balchin (MPIM Bonn)
    Abstract: To do abstract homotopy theory in a category of interest, one may assign to it the data of a Quillen model structure. This naturally raises the question of the possibility of classifying all model structures on a fixed category. In this talk I will report on joint work with Ormsby, Osorno and Roitzheim where we classify all model structures on the category [n]. In the process we will see many well known combinatorial invariants, and tantalizing links to the theory of equivariant homotopy theory. The talk will involve plenty of examples, and no prior knowledge will be assumed!
  • 19 November 2021 (on Zoom at 11:00 am)
    Torus knots and links
    Jake Sundberg
    Abstract: The primary goal of this talk is to classify knots and links which can be drawn on the surface of a torus with no points of intersection. To enrich the discussion, some necessary background on Seifert surfaces and matrices will be provided. The Khovanov homology of the torus links T_{2k, 2kn} and T'_{2k, 2kn}, where the latter is a variant of the former with exactly half of the components' orientations reversed.

Spring 2020

For the Spring 2020 semester we will meet 1:00 - 2:00 on Fridays in 208 Maxim Doucet Hall.

  • 31 January 2020
    Pre-Talbot Seminar I: an example of ambidexterity in chromatic homotopy theory
    Daniel Davis
    Abstract: In unpublished and transformative work, Hopkins and Lurie developed and studied a phenomenon known as ambidexterity (expressed in the framework of infinity-categories): when ambidexterity occurs, a certain limit of a diagram of a particular type of spectra is equivalent to a colimit of the same diagram. We will give an example of ambidexterity that was known before the work of Hopkins-Lurie, and time permitting, we will discuss a relationship of this kind of example to work in a paper in preparation by Thomas Credeur, Drew Heard, and Davis.
  • 7 February 2020
    Pre-Talbot Seminar II: another (pre-)example of ambidexterity and the theorem itself
    Daniel Davis
    Abstract: Last time, we gave a contemporary presentation of the main result of Hovey-Sadofsky and an application of it in a paper of Behrens, which we generalized (just an exercise). In this talk, we give another application of it that is used in Credeur-Davis-Heard (in preparation) and we consider an interesting pattern that appears in a corollary of that work's main result. We present the ambidexterity theorem and possibly make some comments about BG and the simplicial replacement of a diagram of K(n)-local spectra.
  • 14 February 2020
    Pre-Talbot Seminar III: considering the finite group case of ambidexterity further
    Daniel Davis
    Abstract: Let K be a finite group and X a K(n)-local spectrum upon which K acts. We review the "naive definition" of the morphism rho : BK \rightarrow Sp_{K(n)} that is associated to X. The speaker hasn't yet nailed down the exact content of the rest of the talk, but possible topics include using cosimplicial replacement to understand the homotopy fixed point spectrum, noting its relationship to BK, using simplicial replacement as a way to present the homotopy orbit spectrum, and noting relationships between these two replacements as a way of making this case of ambidexterity a little less mysterious.

Fall 2019

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

View the seminar scheduling calendar (ics calendar file).

  • 20 September 2019
    An introduction to factorization homology
    Robin Koytcheff
    Abstract: Factorization homology is a construction that takes a manifold and an E_n-algebra as an input and produces an algebraic object as output. The possible outputs include various structures in topology, algebra, and mathematical physics, such as the homology of a space, the Hochschild homology of an algebra, and certain observables in a TQFT. The construction essentially encodes global phenomena that can be built out of local ones. In this talk, I will explain what an E_n-algebra is and discuss some examples of factorization homology.
  • 27 September 2019
    Group Actions and Cogroup Coactions in ∞-topoi
    Jonathan Beardsley
    Georgia Institute of Technology
    Abstract: In this talk, I will describe joint work with Maximilien Péroux on understanding Koszul duality in ∞-topoi. An ∞-topos is a particularly well behaved higher category that behaves like the category of compactly generated spaces. Particularly interesting examples of ∞-topoi are categories of simplicial sheaves on Grothendieck topologies. The main theorem of this work is that given a group object G of an ∞-topos, there is an equivalence of categories between the category of G-modules in that topos and the category of BG-comodules, where BG is the classifying object for G-torsors. In particular, given any pointed space X, the space of loops on X, denoted ΩX, can be lifted to a group object of any ∞-topos, so if X is in addition a connected space, there is an equivalence between objects of any ∞-topos with an ΩX-action, and objects with an X-coaction (where X is a coalgebra via the usual diagonal map). This is a generalization of the classical equivalence between G-spaces and spaces over BG for G a topological group.
  • 11 October 2019
    Learning Seminar: Factorization homology
  • 18 October 2019
    Learning Seminar: Factorization homology
  • 25 October 2019 (ROOM CHANGE Maxim Doucet Hall room 311)
    Broken techniques for disappearing things
    Hiro Lee Tanaka
    Texas State University
    Abstract: I will discuss stacks classifying families of broken lines and broken cycles. These stacks give new ways to organize algebraic structures, and have enticing applications to symplectic geometry. For example, in joint work with Jacob Lurie, we've shown that families of broken lines classify non-unital A_∞ structures while giving a clear pathway to enrich Lagrangian Floer theory over spectra (which are more powerful invariants than chain complexes).
  • 1 November 2019
    An introduction to left Kan extensions
    Daniel Davis
    Abstract: Factorization homology can be defined as the left adjoint to a particular functor between "structured functor categories." This left adjoint is obtained by forming the "operadic left Kan extension." This is a "structured" version of the higher categorical left Kan extension, which is an extension of classical left Kan extensions, which are the 1-categorical gadgets that we study in this talk.
  • 8 November 2019
    Examples of left Kan extensions, a model for them, and right Kan extensions
    Daniel Davis
    Abstract: We unpack the title a little further: under hypotheses that are often satisfied, we give a colimit that yields the left Kan extension. Also, we briefly note the adjunction that can be viewed as the definition of right Kan extensions and, time permitting, we will present a few other facts about these gadgets.
  • 15 November 2019
    An isovariant Elmendorf's theorem
    Sarah Yeakel
    MSRI / UC Riverside
    Abstract: An isovariant map is an equivariant map which preserves isotropy groups. Isovariant maps show up in equivariant surgery theory and in other settings when homotopy theory is applied to geometry. For a finite group G, we will discuss a cofibrantly generated model structure on the category of G-spaces with isovariant maps, along with an isovariant version of Elmendorf's theorem and possible applications to questions in fixed point theory.
  • 22 November 2019
    Learning Seminar: Factorization homology
  • 6 December 2019
    Learning Seminar: Factorization homology

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