One can learn a lot about the stable homotopy groups of spheres by understanding the homotopy groups of interesting finite CW-spectra and their periodic self-maps. For example, Mark Mahowald showed that the spectrum $Y= \mathbb{RP}^2 \wedge \mathbb{CP}^2$ admits a periodic self-map, which can be used to produce an infinite family in the chromatic layer one of the $2$-primary stable homotopy groups of spheres. Mark Mahowald, used the spectrum $Y$ to prove the height $1$ prime $2$ telescope conjecture. In this talk, I will introduce a a $2$-local spectrum $Z$ which admits a $1$-periodic $v_2$-self-map and can be regarded as the height $2$ analogue of the spectrum $Y$ (joint with P.Egger). We will discuss some of its notable properties. I will discuss the calculation of the $K(2)$-local homotopy groups of $Z$. I will also discuss some of the key features of the $tmf$-resolution of $Z$ and what we need to analyze in the $tmf$-resolution to prove or to disprove the telescope conjecture at the chromatic height 2 prime 2 (joint with Beaudry, Behrens, Culver and Xu).
21 Oct 2019
Computing Minimal Homotopy Area
We study the problem of computing a homotopy from a planar curve to a point that
minimizes the total area swept, and provide structural and geometric properties
of these minimum homotopies. In particular, we prove that for any curve there
exists a minimum homotopy that consists entirely of contractions of
self-overlapping sub-curves. This observation leads to an (exponential time)
algorithm to compute the minimum homotopy area. Furthermore, we study various
properties of these self-overlapping curves.
The results presented are joint work with Parker Evans, Selcuk Karakoc, David
Millman, Brad McCoy, and Carola Wenk.
28 Oct 2019
COTORSION PAIRS AND A K-THEORY LOCALIZATION THEOREM
Cotorsion pairs were introduced in the ’70s as a generalization of projective and injective objects in an abelian category, and were mainly used in the context of representation theory. In 2002, Hovey showed a remarkable correspondence between compatible cotorsion pairs on an abelian category $\mathcal{A}$ and abelian model structures one can define on $\mathcal{A}$. These include, for example, the projective and injective model structures on chain complexes.
In this talk, we turn our attention to Waldhausen categories, and explain how cotorsion pairs can be used to construct Waldhausen structures on an exact category, with the usual class of admissible monomorphisms as cofibrations, and some freedom to choose the class of desired acyclic objects. This allows us to prove a new version of Quillen’s localization theorem, relating the K-theory of exact categories $\mathcal{A} \subset \mathcal{B}$ to that of a cofiber, constructed
through a cotorsion pair.
4 Nov 2019
Duality in homotopy theory
We explore some implications of a fact hiding in plain sight: Namely, the $n$-sphere has the remarkable property that the “swap” map $\sigma: S^n \wedge S^n \to S^n \wedge S^n$ can be “untwisted”: it is homotopic to $(-1)^n \wedge 1$. This simple fact remains true in equivariant and motivic contexts.
One consequence is a structural fact about symmetric monoidal $\infty$-categories with finite colimits and duals for objects: it turns out that any such category splits as the product of three canonical subcategories (for instance, one of these subcategories is characterized by being stable).
As another consequence, we show that for any finite abelian group $G$, the symmetric monoidal $\infty$-category of genuine finite $G$-spectra is obtained from finite $G$-spaces by stabilizing and freely adjoining duals for objects. This universal property vindicates one motivation sometimes given for studying genuine $G$-spectra: namely that genuine $G$-spectra (unlike naive $G$-spectra or Borel $G$-spectra) have a good theory of Spanier-Whitehead duality. We take steps toward a similar universal property for nonabelian groups and also in motivic homotopy theory.
11 Nov 2019
v_n torsion free H-spaces
For some years there have been (k-1)-connected irreducible H-spaces, Y_k, with no p-torsion in homology or homotopy. All p-torsion free H-spaces are products of these spaces and they show up regularly in the literature. Boardman and I have generalized theses spaces and theorems using (k-1) connected H-spaces, Y_k, that have no v_n torsion in homology or homotopy (to be defined). These spaces seem ripe for exploitation in the environment of chromatic homotopy theory.
18 Nov 2019
Loop space constructions of elliptic cohomology
Elliptic cohomology is a generalised cohomology theory related to elliptic curves, which was introduced in the late 1980s. An important motivation for its introduction was to help understand index theory for families of differential operators over free loop spaces. However, for a long time the only known constructions of elliptic cohomology were purely algebraic, and the precise connection to free loop spaces remained obscure. In this talk, I will summarise two constructions of complex analytic, equivariant elliptic cohomology: one from the K-theory of free loop spaces, and one from the ordinary cohomology of double free loop spaces. I will also describe a Chern character-type map from the former to the latter, as well as the relationship to Kitchloo's twisted equivariant elliptic cohomology theory.
25 Nov 2019
2 Dec 2019
On the homotopy theory of stratified spaces
A natural question arises when working with intersection cohomology and other stratified invariants of singular manifolds: what is the correct stable homotopy theory for these invariants to live in? But before answering that question, one first has to identify the correct unstable homotopy theory of stratified spaces. The exit-path category construction of MacPherson, Treumann, and Lurie provides functor from suitably nice stratified topological spaces to "abstract stratified homotopy types” — ∞-categories with a conservative functor to a poset. Work of Ayala–Francis–Rozenblyum even shows that their conically smooth stratified topological spaces embed into the ∞-category of abstract stratified homotopy types. We explain how to go further and produce an equivalence between the homotopy theory of all stratified topological spaces and these abstract stratified homotopy types. We discuss how this new viewpoint provides a space for stratified homotopy invariants in algebraic geometry as well, which was the topic of recent work with Barwick and Glasman. This is the first step of work in progress with Barwick on understanding stable stratified homotopy invariants.
— Spring 2019 —
4 Feb 2019
The equivariant stable parametrized h-cobordism theorem
The stable parametrized h-cobordism theorem provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. For a compact smooth manifold M it gives a decomposition of Waldhausen's A(M) into QM_+ and a delooping of the stable h-cobordism space of M. I will talk about joint work with Malkiewich on this story when M is a smooth compact G-manifold.
11 Feb 2019
The algebra of stable 2-types
This talk will describe recent work using symmetric monoidal 2-categories to build algebraic models of stable homotopy 2-types (equivalently, 3-connected 6-types). We will describe a dictionary between homotopy-theoretic constructions among stable 2-types and algebraic constructions among symmetric monoidal 2-categories. As applications, we obtain a model for the 2-type of the sphere and the 2-type of algebraic K-theory spectra. This work is joint with Nick Gurski and Angélica Osorno.
18 Feb 2019
25 Feb 2019
Iterated Traces in Bicategories and Applications
Kate Ponto and Mike Shulman have developed a powerful categorical framework for defining traces in bicategories. This framework of "shadows" has wide application, in particular to algebraic K-theory and fixed point theory. In this talk I'll discuss another application: proving a very general Lefschetz fixed point theorem that recovers ones of Lunts, Shklyarov, and Cisinski-Tabuada. Time permitting, I'll discuss further applications to Topological Hochschild Homology (THH) and Hopkins-Kuhn-Ravenel theory. This is work joint with Kate Ponto.
4 Mar 2019
11 Mar 2019
Tangent ∞-categories and Goodwillie calculus
Goodwillie calculus is a set of tools in homotopy theory developed, to some extent, by analogies with ordinary differential calculus. The goal of this talk is to make that analogy precise by describing a common higher-category-theoretic framework that includes both the calculus of smooth maps between manifolds, and the calculus of functors, as examples. This framework is based on the notion of "tangent category" introduced first by Rosicky and recently developed by Cockett and Cruttwell in connection with models of differential calculus in logic. In joint work with Kristine Bauer and Matthew Burke (both at Calgary) we generalize to tangent structures on an (∞,2)-category and show that the (∞,2)-category of presentable ∞-categories possesses such a structure. This allows us to make precise, for example, the intuition that the ∞-category of spectra plays the role of the real line in Goodwillie calculus. As an application we show that Goodwillie's definition of n-excisive functor can be recovered purely from the tangent structure in the same way that n-jets of smooth maps are in ordinary calculus. If time permits, I will suggest how other concepts from differential geometry, such as connections, may play out into the context of functor calculus.
18 Mar 2019
25 Mar 2019
Classifying spaces for commutativity
In this talk I’ll give a brief summary of the most important results on the theory of classifying spaces for commutativity for a topological group G, denoted BcomG. The second part of the talk will treat TC structures: There is a natural inclusion of BcomG into the classifying space BG, and a natural question is when does the classifying map of a G-bundle lifts to BcomG. A homotopy class of such a lift is called “Transitionally Commutative” (TC) structure. In recent work with O. Antolín-Camarena, S. Gritschacher and D. Ramras we have constructed characteristic classes for the example cases of the orthogonal groups O(n) (but mostly O(2)), that give obstructions to such structures.
1 Apr 2019
The homotopy theory of homotopy presheaves
I will present a model category structure that encodes the homotopy theory of (small) homotopy functors, from a combinatorial model category to simplicial sets. The fibrant objects are the homotopy functors, i.e. functors that preserve weak equivalences. Next, I will explain how the homotopy theory of homotopy functors is homotopy invariant, i.e. a Quillen equivalence on domain categories induces a Quillen equivalence on homotopy functor categories. I will demonstrate the importance of this result with examples drawn from numerous fields, including spaces, spectra, chain complexes, simplicial presheaves, motivic spectra, infinity categories, and infinity operads. This is joint work with Boris Chorny.
8 Apr 2019
Morse theory on a point: Broken lines and associativity
I'll introduce a stack of Morse trajectories on a point. It turns out this stack classifies associative algebras in a large class of categories, and this is a first step toward constructing stable homotopy enrichments of invariants that people in mirror symmetry care about (Lagrangian Floer theory and, more generally, Fukaya categories). I'll begin with a basic review of Morse theory and give some feel for what this stack is doing. This is joint work with Jacob Lurie.
15 Apr 2019
22 Apr 2019
Coalgebras and comodules in stable homotopy theory
We investigate how to use homotopy-theorical methods in order to study coalgebras and comodules, using model categories and ∞-categories. We explain how model categories fail to represent the correct homotopy theory of coalgebras in spectra, from joint work with Brooke Shipley. We also present current progress on rectification results for coalgebras and comodules in spectra over the Eilenberg-Mac Lane spectrum of a field.
29 Apr 2019
— Fall 2018 —
6 Sept 2018
K-theory of endomorphisms, Witt vectors, and cyclotomic spectra
There is an endofunctor of the category of categories which associates to a category C the category End(C) of endomorphisms of objects of C. If C is a stable infinity category then End(C) is as well, and the associated K-theory spectrum KEnd(C):=K(End(C)) is called the K-theory of endomorphisms of C. Using calculations of Almkvist together with the theory of noncommutative motives, we classify equivalence classes of endofunctors of KEnd in terms of a noncompeleted version of the Witt vectors of the polynomial ring Z[t], answering a question posed by Almkvist in the 70s. As applications, we obtain various lifts of Witt rings to the sphere spectrum as well as a more structured version of the cyclotomic trace via cyclic K-theory, as studied in recent work of Kaledin and Nikolaus-Scholze.
10 Sept 2018
The shadow of ∞-category theory in category theory
Any ∞-category has an underlying 2-category, in which the 2-morphisms are all invertible, which associates to any object of an ∞-category a presheaf of groupoids on a 2-category. Under appropriate conditions we can get a Whitehead-type theorem for the original ∞-category, in which homotopy groups are replaced by homotopy groupoids, and even a Brown representability theorem, constructing objects of the original ∞-category from this purely categorical data. These conditions hold notably for the ∞-categories of spaces and of small ∞-categories. If time allows, I'll describe joint work with Christensen proving that the 2-dimensional aspect is unavoidable for these theorems, even in the case of spaces.
17 Sept 2018
24 Sept 2018
1 Oct 2018
8 Oct 2018
Enriched fibrations and the relative nerve
The Grothendieck construction relates (pseudo)functors $B^{op} \to Cat$ with fibrations over $B$. In this talk, I will present an enriched version of this correspondence, which holds when the enriching category $V$ satisfies certain conditions. Applied to $V = sSet$, (the dual of) this result provides an alternative construction of Lurie's nerve of $B$ relative to a functor $B \to sCat \to sSet$, as well as a factorization of the operadic nerve. If time permits, I will discuss applications to coalgebras of an operad. This is joint work with Jonathan Beardsley.
15 Oct 2018
Cubical sets and higher category theory
I will report on the recent work joint with Voevodsky on using cubical
sets to gain a better understanding of a number of constructions in
higher category theory. This work is inspired by the use of cubical
sets in Homotopy Type Theory by Coquand and his group.
22 Oct 2018
No seminar (Kempf lectures)
29 Oct 2018
5 Nov 2018
The geometry of the cyclotomic trace
K-theory is a means of probing geometric objects by studying their vector bundles, i.e. parametrized families of vector spaces. Algebraic K-theory, the version applying to varieties and schemes, is a particularly deep and far-reaching invariant, but it is notoriously difficult to compute. The primary means of computing it is through its "cyclotomic trace" map K→TC to another theory called topological cyclic homology. However, despite the enormous computational success of these so-called "trace methods" in algebraic K-theory computations, the algebro-geometric nature of the cyclotomic trace has remained mysterious.
In this talk, I will describe a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry. This rests of a reinterpretation of genuine-equivariant homotopy theory in terms of stratified geometry (following Glasman and others). This represents joint work with David Ayala and Nick Rozenblyum. By the end of the talk, you will be able to take home with you a very nice and down-to-earth fact about traces of matrices. If there is time, I will also indicate joint work with Reuben Stern on a 2-dimensional version of this story that applies to "2-traces" for the 2-vector bundles of Baas--Dundas--Rognes.
12 Nov 2018
Localization in homotopy type theory
I will discuss a formulation of localization at a prime in homotopy type theory. The main goal of my talk is prove type-theoretic analogues of classical results on the effect of localization of spaces on algebraic invariants. The main theorem is that for a pointed, simply connected type, the natural $p$-localization map induces algebraic localization on all homotopy groups. I'll preface these results by summarizing key ideas of homotopy type theory and the theory of localization of spaces, and throughout my talk I will emphasize ways in which the type-theorietic story differs from the classical one. This is joint work with J. D. Christensen, E. Rijke, and L. Scoccola.
19 Nov 2018
26 Nov 2018
3 Dec 2018
The Linearization Conjecture
For G a nice profinite group (such as the Morava stabilizer groups), I will discuss the construction of a p-adic sphere which comes equipped with a natural action of G. The linearization conjecture predicts that this sphere is a kind of one point compactification of the p-adic Lie algebra of G. I will explain how to show that this holds when the action is restricted to certain finite subgroups of G and discuss an application to chromatic homotopy theory.
