Johns Hopkins Topology Seminar

Spring 2024: Thursdays 4:30–5:30pm Room: Maryland 110

Spring 2024
22 Jan 2024
Grigory Andreychev, Institute for Advanced Study
K-Theory of Adic Spaces
We will explain a new approach to the K-theory of analytic adic spaces via condensed mathematics. Its main advantage is that it allows us to define K-theory in that context as the (continuous) K-theory of a certain dualizable stable \infty-category associated to an analytic adic space (as opposed to previous constructions that produced the K-theory spectrum somewhat indirectly). In particular, it makes the study of its general properties much more feasible; for instance, it is possible to prove a very general descent statement without any restrictive hypotheses on the adic space. Moreover, one can prove an analog of the GRR theorem in this context.
1 Feb 2024
Rok Gregoric, Johns Hopkins University
Stone duality between condensed mathematics and algebraic geometry
One classical incarnation of Stone duality is an anti-equivalence between the categories of profinite sets and Boolean algebras respectively. In this talk, we will discuss how this can be extended to a fully faithful embedding of condensed sets into fpqc sheaves over any base field. The resulting algebro-geometric spaces are not as pathological as one might expect, and we will discuss how many properties of a condensed set reflect in terms of the algebro-geometric properties on the other side.
8 Feb 2024
Nathaniel Stapleton, University of Kentucky
On the rationalization of the K(n)-local sphere
Since the 70s it has been expected that the homotopy groups of the rationalization of the K(n)-local sphere are an exterior algebra over the p-adic rationals on classes in degrees 1-2i for i from 1 to n. Recently, Barthel, Schlank, Weinstein and I have discovered a proof of this result by making use of a result of Faltings. In this talk, I will give some background on the K(n)-local sphere, explain the reduction to arithmetic geometry, and then give an idea of the modern techniques in arithmetic geometry that go into the proof.
15 Feb 2024
Rin Ray, Universität Münster
Moduli Stack of Curves in Homotopy Theory
We construct ramified families of curves to explicitly model the Lubin-Tate action, the action of a formal group law on its deformation space, for a maximal finite subgroup G. We will see in many cases as a G-representation, this deformation space is a quotient of a regular representation of a finite cyclic group! This allows us to deduce information about the E2 page of the homotopy fixed point spectral sequence of the K(h, p)-local homotopy groups of spheres for height h=p^{k-1}(p-1_, for all such h and p simultaneously.
22 Feb 2024
Kendric Schefers, University of California, Berkeley
Decategorifying the singular support of coherent sheaves
The microlocal homology is a family of chain theories that interpolates between the Borel–Moore homology and singular cohomology of a complex variety in the case when that variety is singular and Poincaré duality fails. Such a device allows one to speak of the singular support of classes in Borel–Moore homology, which we show decategorifies the Arinkin–Gaitsgory singular support of coherent sheaves in a precise sense. The connection between these two singular support theories leverages a description of the microlocal homology in terms of the canonical perverse sheaf of vanishing cycles on shifted cotangent bundles, as well as the known relation between vanishing cycles and categories of matrix factorizations.
29 Feb 2024
Andrew Salch, Wayne State University
ell-adic topological Jacquet-Langlands duality
I will describe the construction of an "ell-adic topological Jacquet-Langlands (TJL) dual" of each finite CW-complex. One effect of the TJL dual is that the representations of the Morava stabilizer group, which arise in the height n Morava E-theory of a finite spectrum X, get turned into continuous actions of GL_n(Z_p) on the ell-adic TJL dual of X. On the level of homotopy groups, these actions of GL_n(Z_p) correspond, via the classical ell-adic Jacquet-Langlands correspondence, to the representations of the Morava stabilizer group on the Morava E-theory groups of X. When n=1, these representations of GL_1 have associated automorphic L-factors. I will explain how the product of those L-factors, i.e., the "automorphic L-function of a finite CW-complex X," analytically continues to a meromorphic function on the complex plane whose special values in the left half-plane recover the orders of the KU-local stable homotopy groups of X. Part of the construction of the TJL dual is a solution to an old problem on construction of E_\infty-ring realizations of the Lubin-Tate tower. I will describe that solution, which is joint work with Matthias Strauch.
7 Mar 2024
Lorenzo Riva, University of Notre Dame
Constructing (infinity,n)-categories with desired properties
In this talk I want to describe a specific example of a symmetric monoidal (infinity,3)-category of generalized spans with local systems (a la Haugseng) where the local systems compose with a push-pull formula similar to the Fourier-Mukai transform of algebraic geometry. The motivation comes from physics and functorial field theory but the general construction illustrates a way to answer the following recurring question: How do we put a (possibly higher) categorical structure on a complicated collection of objects, morphisms, etc. so that the result 1. satisfies some desired properties that might not look functorial at first glance, 2. allows us to carry computations, and 3. is generalizable? In the process we will also discuss models of higher categories, derived algebraic geometry, and a little bit of symplectic geometry.
14 Mar 2024
Marcy Robertson, University of Melbourne
Infinity cyclic operads, configuration spaces and Galois symmetries
The idea behind Grothendieck-Teichmüller theory is to study the absolute Galois group via its actions on (the collection of all) moduli spaces of genus g curves. In practice, this is often done by studying an intermediate object: The Grothendieck-Teichmüler group, GT. In this talk, I’ll describe an cyclic infinity operad of configurations on the cylinder which gives a nice homotopical approximation to the genus zero Teichmüller tower. Includes joint work with Luciana Basualdo Bonatto.
21 Mar 2024
28 Mar 2024
Amnon Neeman, University of Milan
Vanishing negative K-theory and bounded t-structures
We will begin with a quick reminder of algebraic K-theory, and a few classical, vanishing results for negative K-theory. The talk will then focus on a striking 2019 article by Antieau, Gepner and Heller - it turns out that there are K-theoretic obstructions to the existence of bounded t-structures. The result suggests many questions. A few have already been answered, but many remain open. We will concentrate on the many possible directions for future research.
4 Apr 2024
Maxime Ramzi, University of Copenhagen
Separability in homotopical algebra
Separable algebras are a generalization of étale algebras that can be defined in more general homotopical contexts, and have been studied in tensor-triangular geometry, partly due to their nice properties in this context. In this talk, I will attempt to explain some of these nice properties in tensor-triangulated categories, by showing that they come from surprising features of separable algebras in stable oo-categories, in particular showing that all separable up-to-homotopy algebras lift (almost) uniquely to homotopy coherent algebras. If time permits, I will also mention how the basics of the classical theory of separable algebras extend to homotopical algebra.
11 Apr 2024
Theo Johnson-Freyd, Dalhousie University
The universal target category
Hilbert's Nullstellensatz says that the complex numbers $\mathbf C$ satisfy a universal property among all $\mathbf R$-algebras: every not-too-large nonzero commutative $\mathbf R$-algebra maps to $\mathbf C$. Deligne proved a similar statement in categorical dimension 1: every not-too-large symmetric monoidal category of $\mathbf R$ maps to the category $sVec$ of super vector spaces. These statements help explain why quantum field theory involves imaginary numbers and fermions. I will describe the universal symmetric monoidal higher category that extends the sequence $\mathbf C, sVec, ...$. In particular, I will tell you what you have to add at each categorical dimension --- in other words, I will tell you about the ``$\infty$-categorical absolute Galois group'' of $\mathbf R$. This computation involves a version of surgery theory for topological quantum field theories instead of manifolds, and shows that this absolute Galois group is enticingly similar to the infinite piecewise-linear group PL. This talk is based on joint work in progress with David Reutter.
18 Apr 2024
Tasos Moulinos, Institute for Advanced Study
Towards an analogue of the filtered circle over the sphere
I will begin by describing past work with Robalo and Toën on the filtered circle. This is an abelian group stack whose category of representations corresponds to filtered complexes with a compatible S^1 action. By mapping this object into schemes, one recovers the HKR filtration on Hochschild homology. I will then describe some first steps toward lifting this construction to the setting of spectral algebraic geometry over the sphere spectrum, using the theory of synthetic spectra and the even filtration of Hahn-Raksit-Wilson.
2 May 2024
Adrian Clough, NYU Abu Dhabi

Previous Semesters
Fall 2023
18 Sep 2023
German Stefanich, University of Texas at Austin
Tannaka duality for algebraic 2-groups
The Tannakian formalism is a web of statements which, in its most basic form, attempts to recover a group G from its symmetric monoidal category of representations. The goal of this talk is to explain a Tannaka duality result that applies to groups in Artin stacks. In this case, the usual category of representations of G will be replaced by a certain symmetric monoidal 2-category of categories acted on by G.
25 Sep 2023
Lars Hesselholt, Nagoya University
Dirac geometry
Whatever it is that animates anima and breathes life into higher algebra, this something gives the homotopy groups of a commutative algebra in spectra the structure of a commutative algebra in the symmetric monoidal category of graded abelian groups. Being commutative, these algebras form the affine building blocks of a geometry, which we call Dirac geometry. Informally, Dirac geometry constitutes a square root of equivariant geometry for the multiplicative group, and more concretely, the grading exhibits the hallmarks of spin in that it distinguishes symmetric and anti-symmetric behavior and provide the coherent cohomology of Dirac schemes and Dirac stacks with half-integer Serre twists. This is joint work with Piotr Pstragowski.
2 Oct 2023
Sam Gunningham, Montana State University
Skein theory and the geometric Langlands program
Skein modules are certain families of vector spaces spanned by embedded links or graphs in a 3-manifold M, modulo certain local relations. They can be thought of both as an obstruction to defining a polynomial invariant of knots in M and as an invariant of the 3-manifold M. In this talk, I will survey some history, recent results, and work in progress on skein modules, motivated by their role in the geometric Langlands program. I will discuss joint work with (some subsets of) David Ben-Zvi, David Jordan, Pavel Safronov, Monica Vazirani, and Haiping Yang.
9 Oct 2023
Alexei Oblomkov, University of Massachussets, Amherst
Kapustin-Witten (KW) and Kapustin-Rozanksy-Witten (KRS) theory path to knot homology.
My talk is based on the joint project with Lev Rozansky. As a motivation, I will interpret KRS theory as KW theory with defects. Then I will explain how we rigorously construct 3D TQFT, which is a KRS theory with targets Hilbert scheme of points on a plane. Defects in the last theory encode a knot in the three-space and we rediscover the HOMFLY-PT homology of a link as a part of the TQFT. The construction is very flexible and allows us to construct the annular $gl_n$-homology as well as a categofified quantum super-group $gl(m|n)$ link invariant.
16 Oct 2023
Masaki Natori, University of Tokyo
Bulk-edge correspondence and an alternative proof of Bott periodicity theorem via Quot schemes
The study of topological matter is one example of the application of algebraic topology in physics. The bulk-edge correspondence, which asserts that the bulk index of a topological matter coincides with its edge index, has been formulated mathematically in various forms and has been proven. In this talk, we focus on the formulation by Shin Hayashi and provide a new proof of bulk-edge correspondence for type A two-dimensional free fermionic gapped systems. Additionally, in this proof attempt, we got an alternative proof of the Bott periodicity theorem using configuration space, a Quot scheme in algebraic geometry.
23 Oct 2023
Andrew Manion, North Carolina State University
Decategorified Heegaard Floer theory and actions of both E and F
I will outline a relative of decategorified bordered sutured Heegaard Floer homology in which the vector spaces assigned to surfaces S with corners have actions of the full psl(1|1) (with both E and F) for smooth circles in the boundary of S, and discuss a pairing theorem for these vector spaces when gluing along a circle. The vector spaces, like their relatives from decategorified Heegaard Floer theory, also admit actions of psl(1|1)^+ for intervals in the boundary of S and satisfy a gluing theorem when gluing along intervals; over F_2 these new vector spaces combine into an open-closed TQFT valued in the category of F_2-algebras with bimodules as morphisms. If time permits I will speculate about potential relationships with psl(1|1) Chern-Simons theory as recently studied from other directions in mathematics and physics.
30 Oct 2023
Ravi Shankar, James Madison University
Highly Connected 7-manifolds, Non-negative Curvature and the Linking Form
Closed manifolds admitting non-negative sectional curvature are not very well understood and it is, at present, quite difficult to obtain examples with interesting topology. All known examples depend in some way on two basic facts: First, compact Lie groups admit a bi-invariant metric (hence, non-negative curvature) and, second, Riemannian submersions do not decrease sectional curvature. In a recent paper, we (S. Goette, M. Kerin and K. Shankar) constructed an infinite, 7-dimensional family of 2-connected, 7-manifolds which have the cohomology of an S^3-bundle over S^4. This family includes all S^3-bundles over S^4, as well as all homotopy 7-spheres. In a subsequent paper we showed, by computing a secondary cohomology invariant, that this family also includes manifolds with non-standard linking form i.e., 2-connected, 7-manifolds which are cohomology S^3-bundles over S^4 admitting non-negative curvature, but not even homotopy equivalent to any S^3-bundle over S^4 (drawing on past work with N. Kitchloo). In this talk we will present the construction of these new examples. This is joint work with Goette and Kerin.
6 Nov 2023
Diego Manco Berrío, University of Oregon
Coherence for pseudo symmetric multifunctors and applications to $K$-theory
Multicategories where introduced by Elmendorf and Mandell in homotopy theory as an alternative way to encode multiplicative structures in the absence of symmetric monoidal structures. In a sense, they allow us to talk about multilinear maps even when we can't talk about tensor products. This has enabled several definitions of multiplicative structure preserving K-theory as a multifunctor taking structured categories to spectra. This is a way to produce E_\infty ring spectra, modules and algebras. In this talk I'll introduce pseudo symmetric multifunctors, weaker versions of multifunctors introduced by Yau, including a new coherence theorem. This makes clear that they preserve symmetric multiplicative structures like E_n algebras (n=1,...,\infty). In particular, Mandell's inverse K-theory preserves E_n algebras. I'll mention some other examples and conjectures.
13 Nov 2023
Slava Krushkal, University of Virginia
Towards a (3+1)-dimensional TQFT from TMF
I will discuss work in progress, joint with Sergei Gukov, Lennart Meier, and Du Pei, concerning a construction of a 4-manifold invariant using the theory of topological modular forms, and TQFT properties of this invariant. This is a mathematical construction related to a particular instance of the Gukov-Pei-Putrov-Vafa program associating an invariant of 4-manifolds to certain 6-dimensional superconformal field theories.
27 Nov 2023
Arpon Raksit, Massachussets Institute of Technology
THH(Z) and the image of J
I will try to overview some ideas from chromatic homotopy theory, including the "redshift" and "blueshift" phenomena therein, and then describe a result, obtained in joint work with Sanath Devalapurkar, that illustrates these phenomena, relating the two objects named in the talk's title.
4 Dec 2023
Deven Manam, Northwestern University
The Drinfeld and Quillen formal groups
The Cartier–Witt stack, introduced recently by Drinfeld and Bhatt–Lurie, is a geometric object whose quasicoherent sheaves provide a natural target category for prismatic cohomology. Over this stack, Drinfeld has defined a certain formal group involved in the construction of prismatic Chern classes, and conjectured that this formal group arises as the completion of an algebraic group of a specific form. I will explain how to identify this formal group with the Quillen formal group on a sheaf of spectra constructed from topological Hochschild homology, and how this identification relates to Drinfeld's conjecture. I will also sketch how this conjecture yields explicit complexes computing prismatic cohomology in certain cases.
11 Dec 2023 (starting 5:15pm)
Peter Haine, University of California, Berkeley
Exodromy beyond conicality
Let (X, P) be a stratified space. The most general version of the exodromy equivalence between constructible sheaves on (X, P) and functors out of the exit-path ∞-category of (X, P) currently available requires that the stratification of X be conical. Unfortunately, many stratifications naturally arising in geometry are not conical. In this talk, we’ll discuss joint work with Mauro Porta and Jean-Baptiste Teyssier that allows us to extend the exodromy theorem to a much larger class of stratified spaces. Examples include: stratifications that can be locally refinement by conical stratifications, subanalytic stratifications of real analytic spaces, and algebraic stratifications of real varieties. The main tools for our work come from topos theory; our main result follows from a stability theorem for the class of stratified ∞-topoi whose subcategories of constructible objects “admit an exit-path description”.

Spring 2023
30 Jan 2023
Venkata Sai Narayana Bavisetty, University of Illinois Urbana-Champaign
Picard groups in homotopy theory
The Picard group of a ring R is an invariant that has been well studied by both algebraic geometers and number theorists. This classical invariant readily generalizes to the setting of symmetric monoidal categories. The Picard group of the chromatic localizations of spectra have been actively investigated starting with the observation of Hopkins that the Picard group of the $K(n)$-local category of spectra is non-trivial and contains interesting elements. In this talk, I will discuss how one can use descent techniques to construct non-trivial invertible elements.
6 Feb 2023
Catherine Ray, Northwestern University
Moduli Spaces of Curves Model the Spheres
We construct ramified families of curves to explicitly model the Lubin-Tate action, the action of a formal group law on its deformation space, for a maximal finite subgroup G. We will see that as a G-representation, this deformation space is a quotient of a regular representation of a finite cyclic group! This allows us to compute a significant chunk of the E2 page of the homotopy fixed point spectral sequence of the K(h, p) - local homotopy groups of spheres for height h=p^{k-1}(p-1), for all such h and p simultaneously. Thus, we resolve a 40 year old computational stalemate.
13 Feb 2023
Maximilien Péroux, University of Pennsylvania
Trace methods for coHochschild homology
Algebraic K-theory is a fundamental invariant for rings but challenging to compute. The Dennis trace displays (topological) Hochschild homology as an approximation of K-theory that is computationally effective. A dual invariant for coalgebras called (topological) coHochschild homology exists and in this talk, I show how it connects to K-theory. We provide a dual notion of the Hattori-Stallings trace in the context of coalgebras and we outline how a Dennis trace could be defined for topological coHochschild homology. This would lead to new computationally effective tools for K-theory.
20 Feb 2023
Sanath Devalapurkar, Harvard University
Generalized equivariant cohomology and Soergel bimodules, II
In this continuation of the first talk, we will discuss how our setup naturally leads us to the theory of Soergel bimodules, which I will describe a high-level overview (and chromatic analogue) of. Then, I will describe the Kostant slice, which explains how the theory of Soergel bimodules is related to Bezrukavnikov-Finkelberg's derived geometric Satake equivalence; I will also briefly explain how the geometric Satake equivalence fits into the picture of geometric Langlands.
21 Feb 2023 (5:30-6:30)
Sanath Devalapurkar, Harvard University
Generalized equivariant cohomology and Langlands duality
The (derived) geometric Satake equivalence plays a central role in the geometric Langlands program: roughly, it describes the category of constructible sheaves of C-vector spaces on Bun_G(S^2) in terms of the Langlands dual group G^. In this talk, I will describe some ideas connecting chromatic homotopy theory to the derived geometric Satake equivalence. Namely, we will describe the category of local systems of A-modules on Bun_G(S^2), where A is an elliptic cohomology theory, in Langlands dual terms.
27 Feb 2023
Nikolai Konovalov, University of Notre Dame
Algebraic Goodwillie spectral sequence
The spectral Lie operad is the Koszul dual operad to the cocommutative cooperad in the category of spectra. A spectral Lie algebra is an algebra over the spectral Lie operad. M. Behrens and J. Kjaer constructed so-called Dyer-Lashof-Lie power operations acting on the mod-p homology groups of a spectral Lie algebra. However, they computed relations between these operations only for $p=2$. In my talk, I will explain how to compute desired relations for each prime by using functor calculus in the category of simplicial restricted Lie algebras. The latter category might be thought as an algebraic approximation of the category of spaces, and so, algebraic calculations may be also helpful in understanding of the topological Goodwillie spectral sequence.
6 Mar 2023
David Chan, Vanderbilt University
Distributivity in equivariant algebra
Equivariant cohomology theories are robust invariants for studying spaces with an action by a group G. In contrast to classical cohomology theories, these invariants do not take values in abelian groups. Instead, there are several possible algebraic structures, known collectively as incomplete Mackey functors, which provide a home for these additive theories. Similarly, multiplicative equivariant cohomology theories can take values in one of several different generalizations of rings. In this talk I will discuss examples of these different algebraic structures, and some recent work on the interplay between the various multiplicative and additive structures that arise in equivariant topology.
13 Mar 2023
Yutao Liu, University of Chicago
Computing equivariant homotopy with a splitting method
Computations in equivariant homotopy theory are usually hard, especially when the acting group becomes larger and non-abelian. In this talk, I will introduce a new computational method by splitting the category of equivariant spectra into smaller pieces after applying certain localizations. It provides an inductive way to compute equivariant objects acted on by large groups and works well for both abelian and non-abelian groups. Examples including equivariant Eilenberg-Maclane spectra and equivariant complex cobordism will be provided in details.
27 Mar 2023
Rune Haugseng, Norwegian University of Science and Technology
Lax transformations and mates through fibrations
A lax natural transformation that is given objectwise by right adjoints has an oplax “mate” transformation that is given objectwise by the corresponding left adjoints. (A concrete special case of this is that the left adjoint of a lax monoidal functor has a canonical oplax monoidal structure.) I will explain how these notions can be interpreted in terms of (co)cartesian fibrations in the special case where we consider functors from an infinity-category to the (infinity,2)-category of infinity-categories, and discuss how we can then prove the mate correspondence by manipulating fibrations. This is joint work with Fabian Hebestreit, Sil Linskens, and Joost Nuiten. Time permitting, I will also mention an upgrade of this result where the source can be an arbitrary (infinity,2)-category (which also makes it possible to generalize the target via Yoneda); this is work in progress with Fernando Abellán García and Andrea Gagna.
28 Mar 2023 (5:00-6:00)
Sebastian Wolf, University of Regensburg
Proper morphisms of infinity topoi
The goal of this talk to study proper morphisms of infinity topoi and in particular characterize them as those geometric morphisms that, in a suitable sense, commute with filtered colimits. In order to make this precise, I will introduce the theory of "internal higher categories", developed by me and Louis Martini during the past two years.
3 Apr 2023
Lucy Yang, Harvard University
Categorical dynamics on stable module categories
Topological entropy is a measure of the complexity of a continuous self-map of a compact topological space. Categorical entropy generalizes this measure to exact endofunctors of triangulated categories. In this work we ask: How can categorical entropy serve to quantify growth in stable homotopy theory, and how can the methods of stable homotopy theory aid in understanding and computing categorical entropy? We show that the categorical polynomial entropy of a twist functor on the stable module category of a finite connected graded Hopf algebra A over a field is at least one less than the Krull dimension of H*(A;k), generalizing a computation of Fan, Fu, and Ouchi. Along the way, we will see how a stable homotopy theoretic-perspective permits us to make this refinement.
10 Apr 2023
Andrew Senger, Harvard University
Equivariant power operations and fixed points of Morava E-theories
Let $E_n$ denote a height n Morava E-theory, and let G denote a finite subgroup of its Morava stabilizer group. In this talk, I will describe a new approach to the computation of the homotopy fixed points spectral sequence of $E_n ^{hG}$, based on equivariant power operations. In particular, I will show how one may compute the homotopy of $E_n ^{hC_2}$ completely from scratch---without the use of Real bordism $MU_R$ or any other external input. Time permitting, I will state some hopes and conjectures about the odd-primary case.
17 Apr 2023
Ningchuan Zhang, University of Pennsylvania
Profinite descent for Picard spaces and groups in K(n)-local chromatic homotopy theory
The study of Picard groups in homotopy theory was initiated by Hopkins-Mahowald-Sadofsky. They gave a general framework to compute Picard groups of the category of $K(n)$-local spectra. In the past decade, significant progress has been made in our understandings of Picard groups with new tools from higher category theory and equivariant homotopy theory. In this talk, we first review the basics of Picard groups in homotopy theory. Then we import the Devinatz-Hopkins construction to establish a descent spectral sequence to compute Picard groups of Galois extensions of the $K(n)$-local sphere. One key point in the proof is the limit topology on Picard groups of $K(n)$-local $E_\infty$ ring spectra. As an application, we compute Picard groups of all Galois extensions of the $K(1)$-local spheres at all primes. This is joint work in progress with Guchuan Li.
24 Apr 2023
Niny Arcila-Maya, Duke University
Decomposition problem for topological Azumaya algebras with an involution of the first kind
Topological Azumaya algebras (TAAs) over a space are topological shadows of more complicated algebraic Azumaya algebras defined over schemes. Tensor product is a well-defined operation on TAAs. Hence given a TAA $\mathcal{A}$ of degree $mn$ over $X$, where $m$ and $n$ are positive integers, it is a natural question to ask whether $\mathcal{A}$ can be decomposed according to this factorization of $mn$. A similar question arises for TAAs with an involution of the first kind. In this talk, I present results about what conditions should $m$, $n$, and $X$ satisfy so that a TAA with or without an involution can be decomposed.

Fall 2022
29 Aug 2022
Ishan Levy , Massachussets Institute of Technology
The algebraic K theory of type 2 spectra
The algebraic K theory of the category of finite type n spectra is a fundamental object containing structural information about the stable homotopy category. However, until recently almost nothing was known about it for n>1, primarily because it is not the K theory of a connective ring. In this talk I will explain how, for n=2, it can be computed in terms of K theory of discrete rings and topological cyclic homology. In particular, we can read off the K groups in low degrees and find that there is an infinite family of 2-torsion classes in K_0 at the prime 2. I will also explain how to construct type 2 spectra representing these K_0 classes.
12 Sep 2022
Sil Linskens , University of Bonn
Global Spectra via Lax Limits
Global stable homotopy theory is a formalism which admits applications to orbifold cohomology, elliptic cohomology and importantly equivariant stable homotopy theory. Many important equivariant spectra admit global refinements, and these refinements have led to theoretical advances and have aided in calculations. In fact global homotopy theory is often described as "G-equivariant homotopy theory for all (compact Lie) groups G at the same time". In this talk I will present a result which makes this informal statement precise by explaining exactly how the infinity category of global spectra is built from the infinity categories of G-spectra for all G. Namely it is a partially lax limit of a diagram indexing the infinity categories of G-spectra, which is lax on the surjections and strict on the injections. (The work presented is joint with Denis Nardin and Luca Pol)
19 Sep 2022
Mikhail Khovanov , Columbia University
Foams, Tait colorings, and link homology
Several approaches to link homology utilize foams, which are cobordisms in 3D between planar graphs. To a planar graph one assigns a state space and to a foam -- a map between the state spaces of its boundaries. We review a construction of these state spaces which uses a rather subtle evaluation of closed foams. The resulting theory is connected to the SL(3) link homology, the four-color theorem, and the Kronheimer-Mrowka homology theory.
26 Sep 2022
Owen Gwilliam , University of Massachusetts, Amherst
E_n algebras from physics
What mathematicians and physicists mean by ``topological field theory'' do not match perfectly, as there are physical examples (like BF theory) that do not fit directly into the Atiyah-Segal framework. Nonetheless, if one constructs rigorously standard examples from physics (including BF and Chern-Simons theories), one obtains E_n algebras, a notion introduced to capture the structure of n-fold based loop spaces, and hence one opens up a new conduit of dialogue between physics and higher algebra. This talk will overview this relationship, some of its consequences, and some possibilities that it opens up. No background in physics will be needed to follow the talks; all examples can and will be explained in mathematical terms.
3 Oct 2022
William Balderrama , University of Virginia
The Real-oriented cohomology of infinite stunted projective spaces
Real-oriented cohomology theories $E\mathbb{R}$ are $C_2$-equivariant theories which are, informally, associated to the inversion on the formal group law of a complex-oriented cohomology theory $E$. Their spectra $ER$ of fixed points give higher chromatic analogues of real $K$-theory, and detect interesting $2$-torsion phenomena in stable homotopy theory. In this talk, I will discuss the structure of the $ER$-cohomology of the infinite stunted real projective spectra $\mathbb{R}P^\infty_j$ in the Landweber exact case, focusing on some tools that go into the computation and what it tells us about Real spectra and $C_2$-equivariant homotopy theory.
10 Oct 2022
Thomas Brazelton, University of Pennsylvania
Equivariant enumerative geometry
Classical enumerative geometry asks geometric questions of the form "how many?" and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these questions admit well-defined integral answers independent upon the initial parameters of the problem is Schubert’s principle of conservation of number. In this talk we will outline a program of "equivariant enumerative geometry", which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is an equivariant conservation result, which states that, under nice hypotheses, the sum of regular representations of the orbits of solutions to an equivariant enumerative problem is conserved. As an application, we explore the orbits of the 27-lines on an S_4-symmetric cubic.
7 Nov 2022
Sanath Devalapurkar, Harvard University
Generalized equivariant cohomology and Soergel bimodules, I
In this talk, I will explain some background for a talk on chromatic analogues of the geometric Satake equivalence. In particular, I will review the Atiyah-Segal completion theorem, and following Grojnowski, Greenlees, Lurie, Gepner-Meier, and others, I'll explain how this can be used as motivation to define generalized equivariant cohomology using derived algebraic geometry.
14 Nov 2022
Piotr Pstrągowski, Harvard University
Dirac geometry
The rings which appear naturally in derived contexts, for example as homology of differential graded algebras, are graded-commutative in the Koszul sense; that is, odd degree elements anticommute. In this talk, I will describe joint work with Lars Hesselholt where we develop algebraic geometry built out of such rings, which we call Dirac geometry. The latter can be considered as a natural extension of G_m-equivariant algebraic geometry where the Serre twist has a square-root. I will describe some applications to the theory of motives and stable homotopy theory.

Spring 2022

31 Jan 2022
Emma Phillips , University of New Hampshire
Title: An algebraic description of principal 2-group bundles
Abstract: 2-groups are a categorification of groups. We focus on 2-groups that are categorical central extensions of a finite group classified by a 3-cocycle, and study principal 2-group bundles over a manifold X. In this talk, we will introduce 2-groups with several examples. We'll then use a completely algebraic description of the bicategory of flat 2-group bundles on X to compare them to the classical moduli space of G-bundles on X. We'll also discuss how this relates to the Freed-Quinn line bundle from Chern-Simons theory. This is joint work with Dan Berwick-Evans, Emily Cliff, Laura Murray, and Apurva Nakade.
28 Feb 2022
Title: Multiplicative Motivic Infinite Loop Space Theory
Abstract: From a spectrum E one can extract its infinite loop space Omega^\infty E = X. The space X comes rich with structure. What exactly is all of this structure? The recognition principle for infinite loop spaces gives the following answer: the structure of an infinite loop space is exactly the structure of a grouplike E_\infty monoid. What's more, this identification respects multiplicative structures. In this talk, I'd like to discuss the analogue of this story in the setting of motivic homotopy theory. We'll review the symmetric monoidal recognition principle for motivic infinite loop spaces of Elmanto--Hoyois--Khan--Sosnilo--Yakerson, and discuss an extension of the theory that goes beyond symmetric monoidality to incorporate Bachmann--Hoyois' theory of norms.
7 Mar 2022
Title: Towards knot homology for 3-manifolds
Abstract: The Jones polynomial is an invariant of knots in R^3. Following a proposal of Witten, it was extended to knots in 3-manifolds by Reshetikhin--Turaev using quantum groups.

Khovanov homology is a categorification of the Jones polynomial of a knot in R^3, analogously to how ordinary homology is a categorification of the Euler characteristic of a space. It is a major open problem to extend Khovanov homology to knots in 3-manifolds.

In this talk, I will explain forthcoming work towards solving this problem, joint with Leon Liu, David Reutter, Catharina Stroppel, and Paul Wedrich. Roughly speaking, our contribution amounts to the first instance of a braiding on 2-representations of a categorified quantum group. More precisely, we construct a braided (∞,2)-category that simultaneously incorporates all of Rouquier's braid group actions on Hecke categories in type A, articulating a novel compatibility among them.
28 Mar 2022
Luciana Basualdo Bonatto , University of Oxford
Title: Grothendieck-Teichmüller theory and modular operads
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.
4 Apr 2022
Najib Idrissi , Université de Paris
Title: Configuration spaces of surfaces
Abstract: Framed configuration spaces of a surface form a right module over the framed little disks operad. This rich algebraic structure has important consequences, for example for the computations of manifold calculus or factorization homology. Determining the homotopy type of this operadic right module remains however a difficult task. In this talk, I will explain how to compute the rational homotopy type for oriented compact surfaces. The result is a finite-dimensional purely combinatorial model. The proof involves several ingredients: Kontsevich’s formality, Tamarkin’s formality, and the cyclic formality of the framed little disks operad. (Joint work with Ricardo Campos and Thomas Willwacher.)
11 Apr 2022
Martina Rovelli , University of Massachusetts Amherst
Title: A Quillen adjunction between Theta_n-spaces and n-complicial sets
Abstract: I will report on work in progress with Viktoriya Ozornova towards proving the equivalence of models between two models of (\infty,n)-categories: Verity’s n-complicial sets and Rezk’s Theta_n-spaces. Precisely, we will construct a Quillen adjunction between the two model structures (which is conjecturally a Quillen equivalence). We will also discuss a crucial ingredient for the Quillen adjunction: the compatibility between the complicial nerve and the 2-point suspension construction.
18 Apr 2022
David Ayala , Montana State University
Title: Factorization homology in low-dimensions: 1-tangles in 3-spaces.
Abstract: Initial developments of factorization homology abstract the global observables of perturbative TQFTs, in which global observables are determined by point-observables/operators. In this talk, I'll discuss an upgrade of factorization homology tailored for a larger class of TQFTs, in which global observables are determined by point- and line-observables/operators — the discussion will be focused on low-dimensions, where it's related to Skein-modules. A key technical construction is an action of \Omega O(3) on each rigid braided-monoidal (\infty,1)-category. I find this construction fun, so will give space for it in the talk. This is joint work with John Francis.
25 Apr 2022
Hiro Lee Tanaka , Texas State University
Title: Localizations in symplectic geometry (or, toward a world where Fukaya categories help study stable homotopy theory)
Abstract: One of the great applications of infinity-categories is the ability to compute localizations without model categories. We apply this to prove a surprising result in symplectic geometry: A certain 1-category of symplectic manifolds localizes to an infinity-category that computes the homotopy type of the correct geometric mapping spaces! Even better, one can construct a further localization that (conjecturally) allows for purely symplectic constructions of localizations of the stable homotopy category. This is based on joint work with Oleg Lazarev and Zack Sylvan.
Fall 2021

4 Oct 2021
David Mehrle, Cornell University
When free algebras are not flat and other oddities of equivariant algebra
Abstract: In equivariant stable homotopy theory, we replace homotopy groups with homotopy Mackey functors. In many ways, the category of Mackey functors behaves like the category of abelian groups, but the analogy is not perfect. These differences can be a real headache for computations, but they also reflect how much richer the equivariant world can be. In this talk, we will explore several surprising differences between algebra with Mackey functors and algebra as you know it.
11 Oct 2021
Dev Sinha , University of Oregon
Title: Geometric cochains and the phenotypics of homotopy theory
Abstract: (joint with Greg Friedman and Anibal Medina) Interplay between discrete and continuous, combinatorial and geometric, digital and analog has always been at the heart of topology. This was expressed by Sullivan, who likened homotopy types with genetic codes, both being discrete data with continuous expressions, as he made remarkable progress in both homotopy theory and smooth topology. We are developing geometric cochains as a way, conjecturally, to provide an E-infinity algebra model - and thus homotopy model, by Mandell’s Theorem - for smooth manifolds through their differential topology. This would provide a phenotypical determination of the genetics of a manifold.

Geometric cochains are a smooth version of Chow theory, developed just in the last decade by gauge theorists such as Lipyanskiy and Joyce. The theory could have been defined alongside all of the classical cohomology theories, but there are technical obstacles. With manifolds with boundaries being needed for chain complex structure, and the natural product being intersection or more generally fiber product, one is quickly led to working with manifolds with corners. Moreover, the product requires transversality and thus is partially defined, as called for if it is to be commutative while modeling some inherently E-infinity algebra. In work being written, we set the foundations of this theory. Our original motivation for studying such a theory was pedagogical - teaching basic and intermediate algebraic topology - and we indicate some of those applications.

Recently posted work provides a proof of concept, where we bind the combinatorially defined cup product to the geometrically defined fiber product when both are in play - namely on a manifold with a smooth cubulation or triangulation. We do this through an almost-canonical vector field on a cubulated manifold, whose flow interpolates between the usual geometric diagonal and the Serre diagonal.

In the combinatorial setting, choices for resolving lack of commutativity at the cochain level give rise to an E-infinity structure. We think that choices for resolving lack of transversality give rise to a partially defined E-infinity structure on geometric cochains. In particular, we have an explicit conjecture for a partially defined action of the Fulton-MacPherson operad on geometric cochains. We hope to connect with experts on partially defined algebras and related matters to help resolve technicalities in this program.
18 Oct 2021
David Gepner , Johns Hopkins University
Elliptic cohomology and derived algebraic geometry
Abstract: Elliptic cohomology (a.k.a. topological modular forms) originated in mathematical physics, specifically in the study of elliptic genera and elliptic operators on the free loop space. In its modern incarnation, elliptic cohomology is a cohomology theory which is both powerful and computable. It is, in a certain precise sense, one level higher than topological K-theory, with which it shares many features, though it is simultaneously stronger and more mysterious.

In this talk we will explain Lurie's construction of elliptic cohomology via derived algebraic geometry and exploit this construction to make some computations in elliptic cohomology. In particular, we will show that the elliptic cohomology of the unitary groups can be calculated as the space of functions on the Hilbert scheme associated to a derived elliptic curve, and discuss how this is approximated by functions on symmetric powers of the underlying ordinary elliptic curve.
25 Oct 2021
Reuben Stern , Johns Hopkins University
Title: A Universal Property of Secondary Algebraic K-Theory
Abstract: Secondary algebraic K-theory is a relatively new and exciting invariant of rings/schemes that captures complex information undetected by primary K-theory. In this talk, I’ll discuss recent joint work with Aaron Mazel-Gee about formal properties of secondary K-theory and how our work paves the way to a secondary cyclotomic trace. A healthy tolerance for imprecision is required.
8 Nov 2021
Kiran Luecke , University of California, Berkeley
Title: From formal geometry to the dual Steenrod algebra (and not the other way around)
Abstract: Chromatic homotopy theory embraces the philosophy that formal geometry (the algebraic study of certain power series) exerts a huge amount of control over stable homotopy theory. One of the starting points of this philosophy is a theorem (due to Milnor) that the dual Steenrod algebra (a kind of automorphism group for mod 2 homology) coincides with the automorphism group of the formal additive group (the wonderful power series F(x,y)= x + y). To date, all proofs of this theorem require some amount of calculate-both-sides-and-see-that-they-agree type of arguments, making the connection between formal geometry and stable homotopy theory look kind of coincidental. In this talk I will present a remedy to the situation, where the dual Steenrod algebra is identified within formal geometry in an abstract/conceptual way, with the benefit of avoiding complicated spectral sequence calculations, Steenrod operations, and complicated formulas. In fact I will almost never even write an algebra by generators and relations. If there is time (no chance) I will discuss odd primes and other algebras of cohomology operations.
15 Nov 2021
Cary Malkiewich , Binghamton University
Title: The higher characteristic polynomial
Abstract: Let f be a self-map of a finite CW complex. The algebraic count of the periodic points of f can be organized to a power series, the Lefschetz zeta function. The same data is encoded by the characteristic polynomial of the action of f on homology. In this talk I'll discuss how the correct "higher" version of this zeta function is the Dennis trace map, but from K-theory of endomorphisms to topological restriction homology. Above pi_0, this captures algebraic counts of families of periodic points, and we can represent the information they capture pictorially. Much of this is joint work with Campbell, Lind, Ponto, and Zakharevich.
6 Dec 2021
Rhiannon Griffiths , Cornell University
Title: Cofibrantly Generated Model Structures for Functor Calculus
Abstract: Many objects of interest in algebraic topology may be viewed as functors, which often makes them difficult to analyze directly. Functor calculus provides a means by which to approximate a certain type of functor F with a tower of `degree n' functors under F that is analogous to the Taylor series of a function. However, each type of functor calculus differs in its notion of what it means for a functor to be degree n, and in the type of functor to which it applies.

I will present results necessary for developing a more universal framework for functor calculus. As a first example we show how the discrete functor calculus of Bauer, Johnson and McCarthy may be placed into the context of simplicial model categories, allowing for a direct comparison to Goodwillie's original functor calculus.

This is joint work with Julia Bergner, Lauren Bandklayder, Brenda Johnson and Rekka Santhanam.
Spring 2021

1 Mar 2021
Daniel Fuentes-Keuthan, Johns Hopkins
Goodwillie Towers of ∞-Categories and Desuspension
We reconceptualize the process of forming n-excisive approximations to ∞-categories, in the sense of Heuts, as inverting the suspension functor lifted to An-cogroup objects. We characterize n-excisive ∞-categories as those ∞-categories in which An-cogroup objects admit desuspensions. Applying this result to pointed spaces we reprove a theorem of Klein-Schwänzl-Vogt: every 2-connected cogroup-like A∞-space admits a desuspension.
8 Mar 2021
Categorified sheaf theory in topological field theory
In this talk I will give an overview of ongoing work on the theory of sheaves of higher categories in derived algebraic geometry. I will explain how this can be used to produce new examples of fully extended topological field theories - we will in particular encounter examples of field theories of relevance to the geometric Langlands program and three dimensional mirror symmetry.
Fall 2020
7 Sep 2020
21 Sep 2020
William Balderrama, University of Illinois at Urbana-Champaign
From power operations to E-infinity maps
A general heuristic in homotopy theory tells us that by understanding the operations which act naturally on the homotopy groups of a class of objects, one can build obstruction theories and so forth for working with these objects. For instance, in the setting of highly structured ring spectra, this heuristic leads one to obstruction theories built on top of power operations. In this talk I'll describe a general framework that makes it easy to set up these kinds of obstruction theories, focusing on the particular example of K(n)-local E-infinity algebras over a Morava E-theory. I'll explain how the picture one obtains is very pleasant at heights 1 and 2, and in particular can be applied to produce new E-infinity complex orientations.
28 Sep 2020
Ningchuan Zhang, University of Pennsylvania
Analogs of Dirichlet $L$-functions in chromatic homotopy theory

In the 1960’s, Adams computed the image of the $J$-homomorphism in the stable homotopy groups of spheres. The image of $J$ in $\pi_{4k-1}^s(S^0)$ is a cyclic group whose order is equal to the denominator of $\zeta(1-2k)/2$ (up to a factor of $2$). The goal of this talk is to introduce a family of Dirichlet J-spectra that generalizes this connection.

We will start by reviewing Adams’s computation of the image of $J$. Using motivations from modular forms, we construct a family of Dirichlet $J$-spectra for each Dirichlet character. When conductor of the character is an odd prime $p$, the $p$-completion of the Dirichlet $J$-spectra splits as a wedge sum of $K(1)$-local invertible spectra. These summands are elements of finite orders in the $K(1)$-local Picard group.

We will then introduce a spectral sequence to compute homotopy groups of the Dirichlet $J$-spectra. The $1$-line in this spectral sequence is closely related to congruences of certain Eisenstein series. This explains appearance of special values of Dirichlet $L$-functions in the homotopy groups of these Dirichlet $J$-spectra. Finally, we establish a Brown-Comenetz duality for the Dirichlet $J$-spectra that resembles the functional equations of the corresponding Dirichlet $L$-functions. In this sense, the Dirichlet $J$-spectra we constructed are analogs of Dirichlet $L$-functions in chromatic homotopy theory.

19 Oct 2020
Christopher Lloyd, University of Virginia
Calculating the nth Morava K-theory of the real Grassmannians using C4-equivariance
In this talk we will demonstrate how letting the cyclic group of order four act on the real Grassmannians can show the Atiyah-Hirzebruch spectral sequence calculating their nth Morava K-theory collapses. This uses chromatic fixed point theory coming from the classification of the equivariant Balmer spectrum of the cyclic groups. This work is joint with Nicholas Kuhn.
26 Oct 2020
Ang Li, University of Kentucky
A real motivic $v_1$ selfmap
We consider a nontrivial action of $C_2$ on the type 1 spectrum $Y:=S/2$ smash $S/\eta$. This can also be viewed as the complex points of a finite real-motivic spectrum. One of the $v_1$−self-maps of Y can be lifted to a $C_2$ equivariant self-map as well as a real-motivic self-map. Further, the cofiber of the self-map of the R-motivic lift of Y is a realization of the real-motivic Steenrod subalgebra A(1). This is joint work with Prasit Bhattacharya and Bertrand Guillou.
2 Nov 2020
Brandon Doherty, University of Western Ontario
Cubical models of (infinity,1)-categories
We describe a new model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We discuss the proof that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor, and a new theory of cones in cubical sets which is used in this proof. We also introduce the homotopy category and mapping spaces of a fibrant cubical set, and characterize weak equivalences between fibrant cubical sets in terms of these concepts. This talk is based on joint work with Chris Kapulkin, Zachery Lindsey, and Christian Sattler, arXiv:2005.04853.
9 Nov 2020
Hana Jia Kong, University of Chicago
The motivic Chow t-structure
In this talk, I will introduce the Chow t-structure on the motivic stable homotopy category over a general base field. This t-structure is a generalization of the Chow-Novikov t-structure defined on a p-completed cellular motivic module category in work of Gheorghe--Wang--Xu.
Moreover, we identify the heart of this t-structure with a purely algebraic category, and expand the results of Gheorghe-Wang-Xu to integral results on the entire motivic category over general base fields. This leads to computational applications on determining the Adams spectral sequences in the classical stable homotopy category, as well as that in the motivic stable homotopy category over C, R, and F_p. This is joint work with Tom Bachmann, Guozhen Wang and Zhouli Xu.
16 Nov 2020
Unstable modules with only the top k Steenrod operations
In this talk, I will introduce unstable modules with only the top k Steenrod operations at the prime 2. I will prove they have projective dimension at most k and establish some functors to relate such modules with the familiar unstable modules over the Steenrod algebra. In addition, I will talk about a generalization of the Lambda algebra which computes the Ext group from such modules to suspensions of the base field.
23 Nov 2020
7 Dec 2020
Yajit Jain, Northwestern University
Fiberwise Poincare–Hopf Theory and Exotic Smoothings of Fiber Bundles
We will discuss the Rigidity Conjecture of Goette and Igusa, which states that, after stabilizing and then rationalizing, there are no exotic smoothings of manifold bundles with closed even dimensional fibers. After motivating this work through classical results such as Smale's h-cobordism theorem, and later work such as that of Dwyer, Weiss, and Williams, we will discuss fiberwise Poincare–Hopf theory and the duality theorems that lead to a proof of the aforementioned conjecture.
Spring 2020
10 Feb 2020
A generalized Segal conjecture
The Segal conjecture is a surprising and highly nontrivial fact which enables many computations in equivariant homotopy theory. It can be seen as giving a simple formula for the cohomology of a finite group "with coefficients in the sphere spectrum." We will give an introduction to this conjecture and sample some computations that result from it. Following this, we will describe a generalized form of the Segal conjecture, building on work of Miller, Lunøe-Nielsen-Rognes, and Nikolaus-Scholze.
17 Feb 2020
J.D. Quigley, Cornell University
Tate blueshift for real oriented cohomology
This is joint work with Guchuan Li and Vitaly Lorman. The Johnson--Wilson spectra $E(n)$ play a fundamental role in chromatic homotopy theory. In the late 90's, Ando--Morava--Sadofsky showed that the Tate construction with respect to a trivial $\mathbb{Z}/p$-action on $E(n)$ splits into a wedge of $E(n-1)$'s. I will describe a $C_2$-equivariant lift of this result involving the Real Johnson--Wilson theories $E\mathbb{R}(n)$ studied by Hu--Kriz and Kitchloo--Lorman--Wilson. Our result simultaneously generalizes the work of Ando--Morava--Sadofsky (by taking underlying spectra) and a classical Tate splitting for real topological K-theory proven by Greenlees--May (by taking $C_2$-fixed points). I also plan to discuss a key technical ingredient, the parametrized Tate construction (developed by Quigley--Shah), which can be thought of as a "twisted" Tate construction for $C_2$-spectra.
24 Feb 2020
Foling Zou , University of Chicago
Nonabelian Poincare duality theorem in equivariant factorization homology
The factorization homology are invariants of $n$-dimensional manifolds with some fixed tangential structures that take coefficients in suitable $E_n$-algebras. In this talk, I will give a definition for the equivariant factorization homology of a framed manifold for a finite group $G$ by monadic bar construction following Miller-Kupers. Then I will prove the equivariant nonabelian Poincare duality theorem in this case. As an application, in joint work with Asaf Horev and Inbar Klang, we compute the equivariant factorization homology on equivariant spheres for certain Thom spectra.
2 Mar 2020
Arnav Tripathy, Harvard
A geometric model for complex analytic equivariant elliptic cohomology
A long-standing question in the study of elliptic cohomology or topological modular forms has been the search for geometric cocycles. Such cocycles are crucial for applications in both geometry and, provocatively, for the elliptic frontier in representation theory. I will explain joint work with D. Berwick-Evans which turns Segal's physically-inspired suggestions into rigorous cocycles for the case of equivariant elliptic cohomology over the complex numbers, with some focus on the role of supersymmetry on allowing for the possibility of rigorous mathematical definition. As time permits, I hope to indicate towards the end how one might naturally extend these ideas to higher genus.
9 Mar 2020
Arun Debray, UT Austin
Topological phases of matter and topological field theories
The theory of topological phases of matter is at the interface between condensed-matter physics and mathematics, amenable to study using algebraic topology and topological field theory. In this talk, I’ll describe how one studies these systems mathematically, delving into the easier invertible case as well as my work on a particular example in the noninvertible case. With the remaining time, I’ll discuss some open questions in this area.
16 Mar 2020
Fall 2019
23 Sep 2019
Mike Hill, UCLA
$\mathbb Z$-homotopy fixed points of Real and hyperreal spectra
Work of Kitchloo--Wilson and Kitchloo--Lorman--Wilson has shown how one can readily compute with the Real Johnson--Wilson theories. These higher chromatic height lifts of Atiyah's Real K-theory serve as approximations to the Fujii--Landweber Real bordism spectrum $MU_{\mathbb R}$ in the same way that the ordinary Johnson--Wilson theories approximate $MU$. Motivated by work of Bousfield on his unified K-theory, Mingcong Zeng and I studied the \(\mathbb Z\)-homotopy fixed points for these spectra, plugging them into an analogous framework. Viewing the problem slightly more generally, one can also very easily compute the \(\mathbb Z\)-homotopy fixed points for any of the norms of $MU_{\mathbb R}$ and the various chromatic localizations. Along the way, I'll present another way to use the slice filtration to study these kinds of questions.
30 Sep 2019
Carmen Rovi, Indiana University
Davis-Lueck equivariant homology in terms of $L$-theory
The $K$-theory $K_n(\mathbb{Z}G)$ and quadratic $L$-theory $L_n(\mathbb{Z}G)$ functors provide information about the algebraic and geometric topology of a smooth manifold $X$ with fundamental group $G = \pi_1(X, x_0)$. Both $K$- and $L$-theory are difficult to compute in general and assembly maps give important information about these functors. Ranicki and Weiss developed a combinatorial version of assembly by describing $L$-theory over additive bordism categories indexed over simplicial complexes. In this talk, I will present current work with Jim Davis where we define an equivariant version of Ranicki’s local / global assembly map and identify our local / global assembly map with the map on equivariant homology defined by Davis and Lueck. I will also mention some applications of our results.
7 Oct 2019
Christy Hazel, University of Oregon
Equivariant fundamental classes in $RO(C_2)$-graded cohomology
Let $C_2$ denote the cyclic group of order two. Given a manifold with a $C_2$-action, we can consider its equivariant Bredon $RO(C_2)$-graded cohomology. In this talk, we give an overview of $RO(C_2)$-graded cohomology in constant $\mathbb{Z}/2$ coefficients, and then explain how a version of the Thom isomorphism theorem in this setting can be used to develop a theory of fundamental classes for equivariant submanifolds. We illustrate how these classes can be used to understand the cohomology of any $C_2$-surface in constant $\mathbb{Z/2}$ coefficients, including the ring structure.
14 Oct 2019
Prasit Bhattacharya, University of Virginia
A $2$-local finite spectrum that admits $1$-periodic $v_2$-self-map
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
Brittany Fasy, Montana State University
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
Maru Sarazola, Cornell
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
Tim Campion, University of Notre Dame
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
Stephen Wilson, Johns Hopkins University
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
Matthew Spong, University of Melbourne
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.
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
Mona Merling, University of Pennsylvania
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
Niles Johnson, Ohio State University
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.
25 Feb 2019
Jonathan Campbell, Vanderbilt University
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.
11 Mar 2019
Michael Ching, Amherst College
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
David White, Denison University
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
Hiro Lee Tanaka, Texas State University
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
David Gepner, Purdue (THURSDAY!!! in Maryland 217)
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.
8 Oct 2018
Liang Ze Wong, University of Washington
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
Chris Kapulkin, University of Western Ontario
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)
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
Morgan Opie, Harvard
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
Aaron Royer (CANCELLED)
3 Dec 2018
Agnès Beaudry, University of Colorado
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.
Spring 2014
16 Sept 2013
A Classifying Space for Commutative in Lie Groups
The goal of this talk is to study the classifying space for commutative in a topological group G, BcomG. Such spaces are assembled from the different spaces of commuting elements in G. In particular we show the geometric role that these spaces play in terms of transitionally commutative bundles. Some cohomological computations will be provided for the classical Lie groups SU(n), U(n) and Sp(n). A decomposition as homotopy colimits will be provided for these spaces for the case of compact Lie groups and commutative K-theory is also introduced. This is a joint work with Alejandro Adem.
30 Sept 2013
Generalizations of Milnor’s Invariants
Brian Munson, US Naval Academy
In the 50s Milnor defined invariants of multi-component linked circles in three-space as a generalization of the linking number in order to capture higher-order linking phenomena not detected by the linking number, such as the linking of the Borromean rings. In the early 90s Koschorke generalized Milnor’s invariants to linking of higher-dimensional spheres in higher-dimensional Euclidean spaces. The invariants are no longer numbers, but rather elements of the stable homotopy groups of spheres. I will present a generalization of Koschorke’s work to linking of arbitrary manifolds in Euclidean space. Here the invariants are points in the zero space of a Thom spectrum, are induced by a map of spaces rather than defined on the group level, and they are a part of an EHP-like sequence which naturally arises in the study of the Taylor tower of the identity functor in Goodwillie’s homotopy calculus. A key feature of our generalization is that the invariants are “multirelative”. In particular, the classical linking number is really a relative invariant. Time permitting, geometric interpretations of the invariants will be discussed.
21 Oct 2013
Persistent Homology of Time-Delay Embeddings
Persistent homology is a topological method for measuring the shapes of spaces and features of functions. One of its most important applications is to point clouds, where shape is usually interpreted as the geometry of some implicit underlying object near which the point cloud is sampled. Time-delay embeddings, on the other hand, have been used mostly in the study of time series and dynamical systems to understand the nature of their attractors. In this talk we analyze the geometry and topology of time-delay embeddings through the lens of persistent homology. In particular, we study maximum persistence as a measure of periodicity at the signal level, present structural theorems for the resulting diagrams, and derive estimates for their dependency on the window size and embedding dimension. Time permitting, some biological applications will be presented.
28 Oct 2013
The Real K-Theory of Compact Lie Groups
Chi-Kwong Fok, Cornell
In this talk I will first briefly review the previous work on the complex K-theory and Atiyah’s Real K-theory of compact Lie groups. Then I will present a complete description of the ring structure of the equivariant Real K-theory of any compact, connected and simply-connected Lie group equipped with a Lie group involution. Along the way I will introduce the notion of Real equivariant formality, which is an important ingredient in obtaining the ring structure. Some applications and examples will also be discussed.
4 Nov 2013
Exotic Elements in the K(n)-Local Picard Group
Drew Heard, Melbourne
Given a symmetric monoidal category we can study the group of invertible objects, known as the Picard group. For example the Picard group of the stable homotopy category is just the integers, generated by S1. The situation is more interesting when we consider the K(n)-local Picard group, where K(n) is the n-th Morava K-theory. I will review the basic theory, as well as outline work in progress in constructing so called ‘exotic’ elements of the Picard group at height n=p-1.
11 Nov 2013
Determinantal K-Theory and a Few Applications
Eric Peterson, Berkeley
Chromatic homotopy theory is an attempt to divide and conquer algebraic topology by studying a sequence of what we’d first assumed to be “easier” categories. These categories turn out to be very strangely behaved—and furthermore appear to be equipped with intriguing and exciting connections to number theory. I’ll describe the most basic of these strange behaviors, then I’ll describe an ongoing project which addresses a small part of the “chromatic splitting conjecture”.
18 Nov 2013
Unitary Embeddings of Finite Loop Spaces
In this talk I will discuss the existence of unitary embeddings for homotopical analogues of compact Lie groups, such as finite loop spaces and p-compact groups. The fusion systems of these objects are used to build suitable faithful representations of their Sylow subgroups, and obstruction theory to study whether they lift to unitary embeddings. Some examples and consequences will be provided. This is joint work with Natàlia Castellana.
25 Nov 2013
A New Proof of Strickland’s Theorem
Strickland proved that the scheme that classifies subgroups of order pk of the formal group associated to Morava E-theory is corepresented by the Morava E-theory of the symmetric group Spk (modulo a transfer ideal). In this talk we will apply the transchromatic generalized character maps to provide a new proof of this theorem. This is joint work with Tomer Schlank.
2 Dec 2013
Internal Languages for Higher Categories
Every category C looks locally like a category of sets and further structure on C determines what logic one can use to reason about these “sets”. For example, if C is a topos, one can use full (higher order) intuitionistic logic. Similarly, one expects that every ∞-category looks locally like an ∞-category of spaces. A natural question then is: what sort of logic can we use to reason about these “spaces”? It has been conjectured that such logics are provided by variants of Homotopy Type Theory, a formal logical system, recently proposed as a foundation of mathematics by Vladimir Voevodsky. After explaining the necessary background, I will report on the progress towards this conjecture.
4pm Thurs 5 Dec 2013
Three Hopf Algebras and Their Common Algebraic and Categorical Background
Ralph Kaufmann, Purdue / IAS
We discsuss the renormalization Hopf algebra of Connes and Kreimer, Gontcharov’s Hopf algebra for multi-zeta values and the Hopf algebra appearing in Baues’ double cobar construction. We show that these are a examples of a common algebraic framework. Moreover this framework is a manifestation of one of the properties of Feynman categories, which we briefly define and discuss at the end. These are a categorical universal frameworks for operad-like structures.
27 Jan 2014
Sheaves, Cosheaves and Topological Data Analysis
Justin Curry, UPenn / IAS
Sheaves and cosheaves, broadly interpreted, are data management tools with a local-to-global principle. A combinatorial description of (co)sheaves in terms of representations of a particular category (the entrance/exit path category) have enabled a streamlined development of applied (co)sheaf theory. These applications include, among many things, sensor networks, network coding/optimization problems, and topological data analysis (TDA). In this talk, I will focus on how sheaves and cosheaves offer insight into TDA, especially in level-set persistence. I will introduce an extended metric on the category of (co)sheaves, and explain how a derived perspective offers stability results.
10 Feb 2014
Bousfield Localization and Algebras over Operads
David White, Wesleyan
We give conditions on a monoidal model category M and on a set of maps S so that the Bousfield localization of M with respect to S preserves the structure of algebras over various operads. This problem was motivated by an example due to Mike Hill which demonstrates that for the model category of equivariant spectra, even very nice localizations can fail to preserve commutativity. As a special case of our general machinery we characterize which localizations preserve genuine equivariant commutativity. Our results are general enough to hold for non-cofibrant operads as well, and we will demonstrate this via a treatment of when localization preserve strict commutative monoids. En route we will introduce the commutative monoid axiom, which guarantees us that commutative monoids inherit a model structure. If there is time we will say a word about the generalizations of this axiom to other non-cofibrant operads, and about how these generalized axioms interact with Bousfield localization.
17 Feb 2014
GRT-Equivariance of Tamarkin’s Construction
Brian Paljug, Temple
Given two homotopy algebras and an infinity-morphism between them, it is natural to ask that, if we can modify the two homotopy algebras in some structured way, can we modify the infinity-morphism in some similar way, so as to preserve the new structures? In this talk we describe a situation in which the answer is yes, and indicate how it is possible. We will also give an application of these results, to show that Tamarkin’s construction of formality morphisms is equivariant with respect to the action of the Grothendieck-Teichmüller group.
24 Feb 2014
Recent Developments in Representation Stability
Tom Church, Stanford
I will give a gentle survey of the theory of representation stability, viewed through the lens of its applications. I will focus on the recent “second wave” of applications outside the domain of classical representation theory. These applications include: homological stability for configuration spaces of manifolds; understanding the stable (and unstable) homology of arithmetic lattices; uniform generators for congruence subgroups and “congruence” subgroups; and distributional stability for random squarefree polynomials over finite fields.
3 Mar 2014
Positive Scalar Curvature and Twisted Spin Cobordism
The basic question for my talk is whether a closed manifold admits a metric of positive scalar curvature and I will explain how this question reduces to calculations in certain cobordism rings, due to a result of Gromov and Lawson. Following an overview of these calculations in the case of Spin-cobordism, where they were carried out by Stolz and Führing, I want to address their generalisation to the case of twisted Spin-cobordism, which is ongoing joint work of Joachim and myself. In particular I will exhibit a generalisation of the Anderson-Brown-Peterson splitting and compute the mod 2 cohomology of the twisted, connective, real K-theory spectrum.
10 Mar 2014
Counting Curves and the Euler Class
Somnath Basu, Binghamton
We shall discuss the enumerative problem of counting the number of complex curves (in complex projective space of dimension 2) which pass through the requisite number of generic points and has a prescribed singularity at one point. Our exposition will be from a topological point of view via the Euler class. This is joint work with Ritwik Mukherjee.
Thurs 27 Mar 2014
in Remsen 101
A Manifestation of the Grothendieck-Teichmüller Group in Geometry
Inspired by Grothendieck’s lego-game, Vladimir Drinfeld introduced, in 1990, the Grothendieck-Teichmüller group GRT. This group has interesting links to the absolute Galois group of rationals, moduli of algebraic curves, solutions of the Kashiwara-Vergne problem, and theory of motives. My talk will be devoted to the manifestation of GRT in the extended moduli of algebraic varieties, which was conjectured by Maxim Kontsevich in 1999. My talk is based on the joint paper with Chris Rogers and Thomas Willwacher.
7 Apr 2014
An Approach to the Homotopy Groups of Behrens’ Spectrum Q(2) at the Prime 3
Don Larson, Penn State Altoona
In this talk I will describe an approach to computing the homotopy groups of a spectrum Q(2), originally constructed by Mark Behrens in an effort to clarify and extend previous work on the 3-primary K(2)-local sphere by Shimomura, Yabe, Goerss, Henn, Mahowald, and Rezk. The spectrum Q(2) is built using degree 2 isogenies of elliptic curves (hence the “2” in its name) and spectra related to TMF, and as such, is conjectured to have interesting number-theoretic properties that we will discuss. Finally, we will point out potential connections between Q(2) and the beta family in the 3-primary stable stems.
14 Apr 2014
Counting Electronic Excitations in Organic Systems Using Algebraic Topology
Mike Catanzaro, Wayne State
Excited electrons in organic semiconductors and insulators form metastable states, known as excitons. Excitons arise in many different physical and optical phenomena, such as photosynthesis. We formulate solutions to the Exciton Scattering equations (excitons) in terms of an intersection problem, and apply an index theorem to obtain a lower bound on the number of such excitons in a given system. This lower bound becomes exact when the molecule has long enough ‘arms’, or large translational symmetry.
21 Apr 2014
Towards a Cyclotomic Hodge Filtration
Loday’s Hodge filtration on the algebraic Hochschild homology of a commutative ring, a relative of the classical Hodge filtration on the de Rham complex of a smooth algebra, is analogous to the weight filtration on algebraic K-theory: the Adams operations act by prescribed scalars on its graded pieces. We’ll describe how to generalize this to a filtration by cyclotomic spectra of the topological Hochschild homology spectrum of an E-ring spectrum R. We’ll explain how this might ease calculations with topological cyclic homology and help us understand the coherent algebraic structures on THH and K-theory.
28 Apr 2014
Gauged Sigma Models and Equivariant Elliptic Cohomology
The universal elliptic cohomology theory, TMF, is expected to have a rich equivariant refinement related to categorified groups called 2-groups. I will start by defining some flavors of 2-group representation theory in the language of field theories, drawing on the toolbox developed for Chern-Simons theory of a finite group. This perspective leads to an intriguing relationship between characters of 2-group representations and a geometric model of equivariant TMF over the complex numbers built from the geometry of gauged sigma models.
Fall 2014
8 Sept 2014
Localizing the Adams and Adams-Novikov Spectral Sequences
Haynes Miller proved the n=1 case of the telescope conjecture at odd primes by computing π(v1-1S/p) explicitly. As a result of his work we understand the Adams spectral sequence for the Moore spectrum above a line of slope 1/(p2p–1). We will describe the analogue of Miller’s result for the sphere spectrum. When we try to set p=2 in our results we encounter problems. However, we can instead compute the η-localized Adams-Novikov spectral sequence. Of course, η-1π(S0)=0 but motivically there is an interesting question. We will compute η-1π∗,∗(S0,0) resolving a conjecture of Guillou and Isaksen, and understand the Adams-Novikov spectral sequence above a line of slope 1/5.
15 Sept 2014
Calculating the Adams Spectral Sequence for a Simplicial Algebra Sphere
While in the homotopy theory of simplicial algebras, the homotopy of ‘spheres’ is known, the unstable Adams spectral sequence is very far from degenerate. We’ll give some background on the setting, and discuss a method of calculating the E2-page of this spectral sequence.
22 Sept 2014
Lennart Meier, Virginia
Homotopy Fixed Points of Landweber Exact Spectra
Let E be a Landweber exact spectrum (like K-theory or elliptic homology) with an action by a finite group G. The talk is concerned with the following two questions:
  1. When is the norm map from the homotopy orbits to the homotopy fixed points an equivalence?
  2. When is the ∞-category of G-equivariant E-modules equivalent to that of EhG-modules?
At the end, I plan to generalize these questions (and answers) to the context of certain derived stacks.
29 Sept 2014
Aaron Mazel-Gee, Berkeley
Goerss-Hopkins Obstruction Theory for ∞-Categories
Goerss-Hopkins obstruction theory is a tool for obtaining structured ring spectra from algebraic data. It was originally conceived as the main ingredient in the construction of tmf, although it’s since become useful in a number of other settings, for instance in setting up a “naive” theory of spectral algebraic geometry and in Rognes’s Galois correspondence for commutative ring spectra. In this talk, I’ll give some background, explain in broad strokes how the obstruction theory is built, and then indicate how one might go about generalizing it to an arbitrary presentable ∞-category. This last part relies on the notion of a model ∞-category – that is, of an ∞-category equipped with a “model structure” – which provides a theory of resolutions internal to ∞-categories and which will hopefully prove to be of independent interest.
6 Oct 2014
Higher Associativity of Moore Spectra
Not much is known about homotopy coherent ring structures of the Moore spectrum Mp(i) (the cofiber of the pi self-map on the sphere spectrum S0), especially when i>1. Stasheff developed a hierarchy of coherence for homotopy associative multiplications called An structures. The only known results are that Mp(1) is Ap-1 and not Ap and that M2(i) are at least A3 for i>1. In this talk, techniques will be developed to get estimates of ‘higher associativity’ structures on Mp(i).
13 Oct 2014
Emily Riehl, Harvard
Toward the Formal Theory of Higher Homotopical Categories
One framework for stating and proving theorems in abstract homotopy theory uses quasi-categories (aka ∞-categories): for instance, the result of Francis that homology theories for topological n-manifolds are equivalent to n-disk algebras is formalized in this language. The foundational category theory of quasi-categories is developed in thousands of pages of dense mathematics by Joyal, Lurie, and others. Our project is to redevelop these foundations using techniques from formal category theory. We show that the accepted definitions (e.g., of equivalence, limits, adjunctions, cartesian fibrations) can be formulated inside the “homotopy 2-category” of quasi-categories. From this new perspective the proofs that they satisfy the expected relationships (e.g., that right adjoints preserve limits) mirror the classical categorical ones. Importantly, this 2-categorical work can also be applied to other homotopy 2-categories, e.g., for n-fold complete Segal spaces, which were used by Lurie to prove the Baez-Dolan cobordism hypothesis. This is joint work with Dominic Verity.
20 Oct 2014
Topological Analogs of the Radon Transform
We define topological analogs to the Radon transform using persistent homology and Euler characteristic curves. From these we construct metrics on the space of all embedded finite simplicial complexes in R3 or R2. This can be applied to shape recognition and morphology.
27 Oct 2014
Andrew Blumberg, Texas at Austin
The Algebraic K-Theory of the Sphere Spectrum, the Geometry of High-Dimensional Manifolds, and Arithmetic

Waldhausen showed that the algebraic K-theory of the “spherical group ring” on the based loops of a manifold captures the stable concordance space of the manifold. In the simplest case, this result says that for high-dimensional disks, information about BDiff is encoded in K(S), the algebraic K-theory of the sphere spectrum. This talk explains recent work with Mike Mandell that provides a complete calculation of the homotopy groups of K(S) in terms of the homotopy groups of K(Z), the sphere spectrum, and a certain Thom spectrum.
3 Nov 2014
Chromatic Unstable Homotopy Theory
I will show how the Bousfield-Kuhn functor enables us to study monochromatic behavior of unstable homotopy theory using stable techniques. As an example, I will compute the K(2)-local unstable homotopy groups of the three sphere.
10 Nov 2014
The Homotopy Calculus of Categories and Graphs
There are several known examples of categories in which the Goodwillie derivatives of the identity functor have the structure of an operad, including based spaces, bounded-below differential graded Lie algebras over Q, and algebras over a symmetric spectral operad. I will show that this is also the case in the category of small categories. Additionally, I will discuss my efforts to compute the derivatives of the identity functor in two categories of graphs.
1 Dec 2014
Homological Algebra of Complete and Torsion Modules
Let R be a finite-dimensional regular local ring with maximal ideal m. The category of m-complete R-modules is not abelian, but it can be enlarged to an abelian category of so-called L-complete modules. This category is an abelian subcategory of the full category of R-modules, but it is not usually a Grothendieck category. It is well known that a Grothendieck category always has a derived category, however, this is much more delicate for arbitrary abelian categories. In this talk, we will show that the derived category of the L-complete modules exists, and that it is in fact equivalent to a certain Bousfield localization of the full derived category of R. L-complete modules should be dual to m-torsion modules, which do form a Grothendieck category. We will make this precise by showing that although these two abelian categories are clearly not equivalent, they are derived equivalent.
Spring 2015
26 Jan 2015
David Gepner, Purdue
Localization Sequences in the Algebraic K-Theory of Ring Spectra
The algebraic K-theory of the sphere spectrum encodes significant information in both homotopy theory and differential topology. Waldhausen’s chromatic convergence conjecture attempts to approximate K(S) by localizations K(LnS). The LnS are in turn approximated by the Johnson-Wilson spectra E(n)=BP<n>[vn–1], and K(BP<n>) is in principle computable. This would lead inductively to information about K(E(n)), and hence K(S), via the conjectural fiber sequence K(BP<n–1>) → K(BP<n>) → K(E(n)). In this talk, I will define the ring spectra of interest and construct some actual localization sequences in their K-theory. I will then use trace methods to show that the actual fiber of K(BP<n>) → K(E(n)) differs from K(BP<n–1>), meaning that the situation is more complicated than was originally hoped. This is joint work with Ben Antieau and Tobias Barthel.
2 Feb 2015
Coassembly in Algebraic K-Theory
The coassembly map allows us to approximate any contravariant homotopy-invariant functor by an excisive functor, i.e. one that behaves like a cohomology theory. We’ll apply this construction to Waldhausen’s algebraic K-theory of spaces, and its corresponding THH functor. The results are somewhat surprising: a certain dual form of the A-theory Novikov conjecture is false, but when the space in question is the classifying space BG of a finite p-group, coassembly on THH is split surjective after p-completion. Even better, we can show that the coassembly map links up with the more familiar assembly map to produce the equivariant norm. As a result, we get some splitting theorems after K(n)-localization, and a surprising connection between the Whitehead group and Tate cohomology. If there is time, we will also discuss related work on the equivariant structure of THH.
9 Feb 2015
Mike Hill, Virginia
Norms, Transfers, and Operads
I’ll discuss joint work with Blumberg in which we introduce a class of equivariant operads that parameterize infinitely homotopy commutative multiplications. This gives a language for describing what structure is preserved when we Bousfield localize commutative ring spectra while also allowing us to see how the transfer is encoded operadically.
16 Feb 2015
Schematic Homotopy Types of Operads
The rational homotopy type XQ of an arbitrary space X has pro-nilpotent homotopy type. As a consequence, pro-algebraic homotopy invariants of the space X are not accessible through the space XQ. In order to develop a substitute of rational homotopy theory for non-nilpotent spaces Toën introduced the notion of a pointed schematic homotopy type over a field k, (X×k)sch. In his recent study of the pro-nilpotent Grothendieck-Teichmüller group via operads, Fresse makes use of the rational homotopy type of the little 2-disks operad E2. As a first step in the extension of Fresse’s program to the pro-algebraic case we discuss the existence of a schematization of the little 2-disks operad.
23 Feb 2015
Combinatorial Models of Moduli Spaces CANCELLED
Ribbon graphs provide a powerful combinatorial tool in the study of the moduli space of Riemann surfaces. The theory of quadratic differentials in complex analysis gives a cellular decomposition of the moduli space indexed by ribbon graphs, and this allowed the computation of the Euler characteristic and Kontsevich’s proof of Witten’s intersection number conjecture. Costello found a different ribbon graph model in his work constructing the B-model counterpart to Gromov-Witten theory in terms of topological field theories. In this talk I will review these ideas and describe how to produce Costello-type combinatorial models of moduli spaces of many related classes of objects, such as unoriented, spin and r-spin surfaces, surfaces with G-bundles, and 3-dimensional handlebodies.
2 Mar 2015
T-Duality and the Chiral de Rham Complex
T-dual pairs are distinct manifolds equipped with closed 3-forms that admit isomorphism of a number of classical structures including twisted de Rham cohomology, twisted K-theory, and twisted Courant algebroids. An ongoing program is to study T-duality from a loop space perspective; that is, to identify structures attached to the loop spaces that are isomorphic under T-duality. In this talk, I’ll explain how the chiral de Rham complex of Malikov, Schechtman, and Vaintrob, gives rise to such structures. This is a joint work with Varghese Mathai.
9 Mar 2015
Mona Merling, Hopkins
Equivariant Algebraic K-Theory
The first definitions of equivariant algebraic K-theory were given in the early 1980’s by Fiedorowicz, Hauschild and May, and by Dress and Kuku; however these early space-level definitions only allowed trivial action on the input ring or category. Equivariant infinite loop space theory allows us to define spectrum level generalizations of the early definitions: we can encode a G-action (not necessarily trivial) on the input as a genuine G-spectrum. I will discuss some of the subtleties involved in turning a ring or space with G-action into the right input for equivariant algebraic K-theory or A-theory, and some of the properties of the resulting equivariant algebraic K-theory G-spectrum. For example, our construction recovers as particular cases equivariant topological real and complex K-theory, Atiyah’s Real K-theory and statements previously formulated in terms of naive G-spectra for Galois extensions. I will also briefly discuss recent developments in equivariant infinite loop space theory from joint work with Guillou, May and Osorno (e.g., multiplicative structures) that should have long-range applications to equivariant algebraic K-theory.
16 Mar 2015
—Spring vacation—
23 Mar 2015
Agnes Beaudry, Chicago
The Chromatic Splitting Conjecture at n=p=2
I will discuss why the strongest form of Hopkins’s chromatic splitting conjecture, as stated by Hovey, cannot hold at chromatic level n=2 at the prime p=2. More precisely, let V(0) be the mod 2 Moore spectrum. I will give a summary of how one uses the duality resolution of Bobkova, Goerss, Henn, Mahowald and Rezk to show that πkL1LK(2)V(0) is not zero when k is congruent to 5 modulo 8. I explain how this contradicts the decomposition of L1LK(2)S predicted by the chromatic splitting conjecture.
30 Mar 2015
Analysing Grothendieck Ring of Varieties Using K-Theory
The Grothendieck ring of varieties K0[Vk] is defined to be the free abelian group generated by varieties, modulo the relation that for a closed subvariety Y of X, [X]=[Y]+[X\Y]. This ring is used extensively in motivic integration. There are two important structural questions about the ring:
  1. If X and Y are two varieties such that [X]=[Y], are X and Y birationally isomorphic?
  2. Is the class of the affine line [A1] a zero divisor?
In a recent paper, Borisov constructed an example showing that the answer to question 2 is “yes”; in a beautiful coincidence this construction also produced varieties X and Y which show that the answer to question 1 is “no.” In this talk we will construct a spectrum K(Vk) such that π0K(Vk)=K0[Vk], and such that the higher homotopy groups contain further geometric information. We will then analyze the structure of this spectrum to show that Borisov’s coincidence is not a coincidence at all, and that in fact the kernel of multiplication by [A1] is generated by counterexamples to question 1.
6 Apr 2015
Equivariant Calculus of Functors
Let G be a finite group. I will define “J-excision” of functors on pointed G-spaces, for every finite G-set J. When J is the trivial G-set with n-elements we recover Goodwillie’s definition of n-excision. When J=G we recover Blumberg’s notion of equivariant excision. There are J-excisive approximations of homotopy functors which fit together into a “Taylor tree”. I will explain how “J-homogeneous” functors are classified by suitably equivariant spectra, and address some convergence issues of the Taylor tree.
3pm Mon 13 Apr 2015
Jack Morava, Hopkins
Kitchloo’s Category of Symplectic Motives
The geometric quantization program attempts to define a functor from classical mechanics (AKA symplectic geometry) to quantum systems; but it is known to have problems, associated to failure of transversality. Kontsevich has made progress in the related deformation quantization program, which is less intrinsically global.
Kitchloo [in arXiv:1204.5720] defines a stabilized category of symplectic manifolds, with morphism objects enriched over spectra, analogous to the algebraic geometers’ motives. He identifies its associated ‘motivic group’ as the (Hopf-Galois, ring) spectrum defined by the topological Hochschild homology of a Thom spectrum associated to a certain Lagrangian Grassmannian. After tensoring with Q, this is remarkably like the motivic group Kontsevich finds acting on his deformation quantization constructions.
20 Apr 2015
Jon Beardsley, Hopkins
Noncommutative Bialgebras in Spectra and Hopf-Galois Extensions
We briefly review the notions of infinity operads and infinity categories before describing definitions of non­commutative bi­algebras in spectra, allowing us to define Hopf-Galois extensions of non­commutative ring spectra. We give a number of interesting geometrically motivated examples of the latter objects.
27 Apr 2015
Vin de Silva, Pomona
Topological Persistence via Category Theory
Topological persistence originated as a strategy for measuring the topology of a statistical data set. The naive approach is to build a simplicial complex from the data and measure its homological invariants; but this approach is extremely sensitive to noise and is therefore unusable. The correct approach (made effective by Edelsbrunner, Letscher and Zomorodian in 2000) is to represent the data by a multiscale family of simplicial complexes, and to measure the homology as it varies across all scales. The resulting multiscale invariants, known as persistence diagrams, are provably robust to perturbations of the data (Cohen-Steiner, Edelsbrunner, Harer 2007).
In this talk, I will explain how these ideas may be expressed in the language of category theory. The basic concepts extend quite widely. In particular, I hope to explain how Reeb graphs and join-trees—well known constructions in data analysis—can be thought of as persistent invariants, enjoying some of the same properties as persistence diagrams. My collaborators in this work include Peter Bubenik, Jonathan Scott, Elizabeth Munch, Amit Patel.
Fall 2015
31 Aug 2015
Vitaly Lorman, Johns Hopkins
Real Johnson-Wilson Theories and Computations
Complex cobordism and its relatives, the Johnson-Wilson theories, E(n), carry an action of C2 by complex conjugation. Taking homotopy fixed points of the latter yields Real Johnson-Wilson theories, ER(n). These can be seen as generalizations of real K-theory and are similarly amenable to computations. We will outline their properties, describe a generalization of the η-fibration, and discuss recent computations of the ER(n)-cohomology of some well-known spaces, including CP.
7 Sept 2015
14 Sept 2015
21 Sept 2015
Ilya Grigoriev, University of Chicago
Characteristic classes of manifold bundles
For every smooth fiber bundle $f: E\to B$ with fiber a closed, oriented manifold $M^d$ of dimension $d$ and any characteristic class of vector bundles $p \in H^* \left( BSO(d) \right)$, one can define a ``generalized Miller-Morita-Mumford class" or ``kappa-class" $\kappa_p \in H^*(B)$. We are interested in the ideal $I_M$ of all the polynomials in the kappa classes which vanish for \textit{every} bundle with fiber diffeomorphic to $M$, as well as the algebraic structure of the quotient $R_M = \mathbb{Q}\left[\kappa_p\right]/I_M$ of the free polynomial algebra by this ideal. I will talk mainly about the case where the manifold is a connected sums of $g$ copies of $S^n \times S^n$, with $n$ odd. In this case, we can compute the ring $R_M$ modulo nilpotents, and show that the Krull dimension of $R_M$ is $n-1$ for all $g>1$. This is joint work with Søren Galatius and Oscar Randal-Williams.
28 Sept 2015
Ben Knudsen, Northwestern
Rational homology of configuration spaces via factorization homology
The study of configuration spaces is particularly tractable over a field of characteristic zero, and there has been great success over the years in producing complexes simple enough for explicit computations, formulas for Betti numbers, and descriptive results. I will discuss recent work identifying the rational homology of the configuration spaces of an arbitrary manifold with the homology of a Lie algebra constructed from its cohomology. The aforementioned results follow immediately from this identification, albeit with hypotheses removed; in particular, one obtains a new, elementary proof of homological stability for configuration spaces.
5 Oct 2015
Sander Kupers, Stanford
$E_n$ cells and homological stability
When studying objects with additional algebraic structure, e.g. algebras over an operad, it can be helpful to consider cell decompositions adapted to these algebraic structures. I will talk about joint work with Jeremy Miller on the relationship between $E_n$-cells and homological stability. Using this theory, we prove a local-to-global principle for homological stability, as well as give a new perspective on homological stability for various spaces including symmetric products and spaces of holomorphic maps.
12 Oct 2015
Don Davis, Lehigh University
Topological Complexity of Spaces of Polygons
The topological complexity of a topological space X is the number of rules required to specify how to move between any two points of X. If X is the space of all configurations of a robot, this can be interpreted as the number of rules required to program the robot to move from any configuration to any other. A polygon in the plane or in 3-space can be thought of as linked arms of a robot. We compute the topological complexity of the space of polygons of fixed side lengths. Our result is complete for polygons in 3-space, and partial for polygons in the plane.
19 Oct 2015
Angélica Osorno, Reed College
On equivariant infinite loop space machines
An equivariant infinite loop space machine is a functor that constructs genuine equivariant spectra out of simpler categorical or space level data. In the late 80's Lewis-May-Steinberger and Shimakawa developed generalizations of the operadic approach and the Gamma-space approach respectively. In this talk I will describe work in progress that aims to understand these machines conceptually, relate them to each other, and develop new machines that are more suitable for certain kinds of input. This work is joint with Anna Marie Bohmann, Bert Guillou, Peter May and Mona Merling.
26 Oct 2015
Dominic Verity, Macquarie University
The Unbearable Lightness of 2-Being
(Title with apologies to Milan Kundera)

Joint work with Dr. Emily Riehl, Johns Hopkins University.

For a few years now, Emily and I have been engaged in a project to study the extent to which the category theory of $(\infty,n)$-categories can be developed using some elementary tools from 2-category theory. This idea, which dates back to Andre Joyal's earliest work on quasi-categories, turns out to be remarkably and somewhat unbearably fruitful. Much to our own surprise, a very great deal of the category theory of such stuctures can be rendered directly from a study of {\em strict\/} 2-categories and bicategories of $(\infty,n)$-categories, using techniques largely developed by the Australian Category Theory School in the 1970s.

To date we have developed a largely 2-categorical theory of adjunctions, monadicity, cartesian fibrations, modules, limits and colimits, (pointwise) Kan extensions, final and initial functors, Yoneda's lemma, presheaf categories and many other important categorical structures besides. All of this work follows a mild re-working of a well trodden path of 2-being, which not only provides for a foundational redevelopment of the category theory of quasi-categories but is also couched in terms that is both model independent and amenable to application to $(\infty,n)$-categories and $(\infty,\infty)$-categories of various kinds.

One of the enduring themes in this work is the inspiration it draws from the {\em derivator\/} approach to abstract homotopy theory, as pioneered independently by Grothendieck and Heller. Such gadgets light up the theory of homotopy limits and colimits by observing that homotopy categories themselves can provide us with enough information so long as they come parameterised by {\em internal diagram types}. In this context a homotopy theory simply consists of a category fibred (or indexed) over the category of all small categories. The fibre of this structure over some small category $\mathbb{C}$ is thought of as the homotopy category of diagrams on $\mathbb{C}$ and adjunctions between fibres track the existance of homotopy Kan extensions. It is then the interplay between weak notions in the {\em external\/} world of each fibre and strong coherent notions in the internal world of abstract diagrams that allows one to develop an abstract homotopy theory to rival that available in th