- 2018–11–2:
- Homotopy theory of algebras over operads.
Topology Seminar, University of Louisiana at Lafayette.
Abstract: Suppose O is an operad in simplicial sets,
chain complexes, motivic spectra, etc. Consider
the category Alg_O of algebras over this operad
equipped with degreewise weak equivalences. We
are interested in the following questions:

1) Under what conditions on O does Alg_O present the "homotopically correct" category of algebras over an operad (e.g., in the sense of \infty-categories)?

2) Under what conditions on O does Alg_O possess a model structure?

3) Under what conditions on O and O' does a weak equivalence $O \to O'$ induce a Quillen equivalence $Alg_O \to Alg_O'$?

We provide a complete answer to 1) and 3) in terms of an if-and-only-if criterion that is easy to verify in practice, and we also give a sufficient condition for 2) that is applicable to all known practical examples. Our criteria work in abstract monoidal model categories, such as simplicial sets, chain complexes, motivic spectra, topological spaces, and many others. Our work espouses the yoga of "synthetic" model category theory, which postulates that model structures and their properties can be most easily established inductively by tracing the construction of the category under consideration step-by-step, as opposed to constructing model structures directly in one step.

The above is joint work with Jakob Scholbach (Muenster). If time permits, I will also discuss the case of coalgebras over operads, as well as Leinster-style homotopy algebras over operads. - 2018–8–22:
- Homotopy theory of algebras over operads.
A 3-hour talk at the Pohang Operadic Workshop, IBS Center for Geometry and Physics, South Korea.
Abstract: Suppose O is an operad in simplicial sets, chain complexes, motivic spectra, etc.
Consider the category Alg_O of algebras over this operad equipped with degreewise weak equivalences.
We are interested in the following questions:

1) Under what conditions on O does Alg_O present the “homotopically correct” category of algebras over an operad (e.g., in the sense of ∞-categories)?

2) Under what conditions on O does Alg_O possess a model structure?

3) Under what conditions on O and O' does a weak equivalence O→O' induce a Quillen equivalence Alg_O → Alg_O'?

We provide a complete answer to 1) and 3) in terms of an if-and-only-if criterion that is easy to verify in practice, and we also give a sufficient condition for 2) that is applicable to all known practical examples. Our criteria work in abstract monoidal model categories, such as simplicial sets, chain complexes, motivic spectra, topological spaces, and many others. The above is joint work with Jakob Scholbach (Münster). If time permits, I will also discuss the case of coalgebras over operads, as well as Leinster-style homotopy algebras over operads. - 2017–7–27:
- Extended QFTs are local. Higher Structures Lisbon 2017. Abstract: An extended QFT (not necessarily topological) is a functor from the n-category whose objects are 0-dimensional manifolds and k-morphisms are k-dimensional bordisms with corners to some target n-category, e.g., E_{n−1}-algebras. “Extended” here refers to the fact that one starts at 0-dimensional manifolds, in contrast to the more traditional definition of Atiyah, which can start at d-dimensional manifolds for some d>0. In typical applications bordisms are also equipped with a map to some manifold (more generally, higher stack) X. The category of extended QFTs over X is denoted by QFT(X). We prove that QFT(X) is a (higher) sheaf with respect to X. In the QFT world this property is known as locality and thus our result can be reformulated by saying that extended QFTs are local. We then combine this result with a result from another paper of ours and show that concordance classes of QFTs over X are in bijection with the homotopy classes of maps from X to a certain classifying space of QFTs, for which we give an explicit formula. This result is an important step toward the Stolz-Teichner conjecture, which claims that concordance classes of 2|1-dimensional Euclidean QFTs over X are in bijection with TMF_0(X), where TMF is the cohomology theory induced by the spectrum of topological modular forms of Hopkins and Miller.
- 2017–2–7:
- Concordances of geometric objects and representability of associated cohomology theories. Colloquium Talk, Department of Mathematics and Statistics, Texas Tech University. Abstract: We prove that concordance classes of fully extended quantum field theories are representable by a (unique) classifying space. The core of our proof is a refined version of the Brown representability theorem: given a sheaf of spaces on the site of smooth manifolds, we show that the concordance classes extracted from this sheaf are representable by homotopy classes of maps into a unique classifying space. As an added benefit, we get a concise rederivation (essentially in one line) of a large variety of classical representability results for de Rham cohomology, singular cohomology, vector bundles, K-theory, Chern character as a morphism of E-infinity ring spectra, Quinn's model for cobordism, equivariant de Rham theory and equivariant K-theory, Haefliger structures, etc. This project is a part of a larger research program that aims to extend these results to the setting of differential and equivariant cohomology, part of which is joint work with Daniel Berwick-Evans, Pedro Boavida de Brito, and Alexander Kahle.
- 2015–11–4:
- Abstract Simons—Sullivan construction for generalized differential cohomology. Oberseminar Globale Analysis, Regensburg.
- 2015–6–9:
- Concordance theory for homotopy sheaves. Trimester Seminar at the Homotopy theory, manifolds, and field theories trimester program at the Hausdorff Research Institute for Mathematics, Bonn. A video of the talk is available.
- 2014–11–18:
- Concordance theory for homotopy sheaves. Topology Seminar at Stanford University. Abstract: We establish a Brown representability-type result for concordance spaces of homotopy sheaves on the site of smooth manifolds. In particular, we obtain that concordance classes of sections of any homotopy sheaf are classified by the homotopy classes of maps into a classifying space, which is unique up to a contractible choice, extending a previous result by Ib Madsen and Michael Weiss. We then explore various applications of this theorem, including functorial field theories, classifying spaces of categories (generalizing results of Ieke Moerdijk and Michael Weiss), higher geometric twists for K-theory, and a new proof of the de Rham theorem. Joint work with Daniel Berwick-Evans (Stanford) and Pedro Boavida de Brito (Louvain).
- 2014–11–13:
- Rectification of homotopy coherent algebraic structures to strict ones. K-theory/Motivic Homotopy Theory Seminar at Ohio State University. Abstract: Why are homotopy coherent associative (i.e., A-infinity) monoids in simplicial sets always equivalent to strictly associative ones, whereas homotopy coherent commutative (i.e., E-infinity) monoids are not? In this talk I will discuss a necessary and sufficient criterion that allows one to answer this and many similar questions about rectification of homotopy coherent structures (specifically, those given by operads) to strict ones with relative ease. Apart from simplicial sets, the underlying category can also be the category of topological spaces, chain complexes, simplicial presheaves, and more generally, any model category satisfying some mild additional conditions. Of particular interest is the case of algebraic structures on structured spectra, for example, symmetric spectra, for which concrete applications force us to consider symmetric spectra in categories more general than simplicial sets, for example, motivic spaces of Morel and Voevodsky. It turns out that the criterion mentioned above is automatically verified under some mild conditions on the underlying category (satisfied, for example, by simplicial sets and motivic spaces), which means that homotopy coherent algebraic structures in symmetric spectra can always be rectified to strict ones. If time permits, we will discuss applications to Deligne cohomology, Toën-Vezzosi homotopical algebraic geometry, Goerss-Hopkins obstruction theory, enriched categories, operads, factorization algebras. Joint work with Jakob Scholbach (Münster).
- 2014–5–22:
- Concordance theory of homotopy sheaves. Talk in Regensburg. Abstract: Starting from a homotopy sheaf of spaces on the site of smooth manifolds we define its concordance sheaf and prove that it is representable. We then explore the applications of this theorem to field theories, classifying spaces of categories, and higher geometric twists for K-theory. Joint work with Daniel Berwick-Evans and Pedro Boavida de Brito.
- 2014–5–6:
- Tomita-Takesaki theory via modular algebras. NCGOA 2014, Vanderbilt University. Abstract: We show that the Haagerup-Yamagami modular algebra of a von Neumann algebra M is the free complex-graded extended von Neumann algebra generated by M in degree 0 and the predual of M in degree 1. Apart from providing very simple proofs of the fundamental identities involving modular automorphism groups, Connes' Radon-Nikodym cocycle derivatives, and other similar objects, this approach allows us to extend the Tomita-Takesaki theory to settings other than von Neumann algebras, for example, to smooth stacks.
- 2013–2–13:
- Two-dimensional Yang-Mills theory and equivariant TMF. Workshop on Field Theories with Defects in Hamburg organized by Daniel Roggenkamp, Ingo Runkel, and Christoph Schweigert. Abstract: We construct examples of functorial field theories that are both fully local (i.e., go all the way down to points) and nontopological (i.e., depend on the underlying geometry of bordisms, e.g., the volume form or the metric). The significance of such examples comes from the fact that all known examples of two-dimensional field theories that are written down in any considerable detail are either nonlocal (e.g., constructions of Segal and Pickrell) or topological (e.g., all examples related to the cobordism hypothesis). Secondly, the Stolz-Teichner program implies that such field theories give classes in the equivariant version of TMF (topological modular forms). Thus we obtain the first explicit examples of nontrivial (equivariant) elliptic objects. (Joint work in progress with Daniel Berwick-Evans.)
- 2012–11–6:
- Two-dimensional Yang-Mills theory and string topology as local Segal-style functorial field theories. Part II: Classical two-dimensional Yang-Mills theory and its quantization. Oberseminar Topologie at the University of Bochum.
- 2012–10–16:
- Two-dimensional Yang-Mills theory and string topology as local Segal-style functorial field theories. Part I: Overview of functorial field theories. Oberseminar Topologie at the University of Bochum.
- 2012–6–4:
- Differential cohomology and smooth topological field theories. FRG Conference on Topology and Field Theories at the University of Notre Dame. A video of the talk is available. Abstract: We will discuss a characterization of differential cohomology and related functors in terms of topological field theories fibered over the site of smooth manifolds. This can be seen as the first step toward a smooth version of the cobordism hypothesis. Joint work in progress with Daniel Berwick-Evans, Stephan Stolz, and Peter Teichner.
- 2011–5–3:
- Jones index via a symmetric monoidal bicategory of von Neumann algebras. Notre Dame Topology Seminar. Abstract: I will describe a new symmetric monoidal structure on the bicategory of von Neumann algebras, bimodules and intertwiners, which is motivated by conformal and Euclidean field theories. I will then demonstrate how the bicategorical formalism of shadows of 1-morphisms and traces of 2-morphisms developed by Ponto and Shulman yields the Jones index in a purely categorical way.
- 2010–12–1:
- Bivariant 0|1-dimensional field theories and de Rham homology and cohomology. University of Utrecht talk organized by André Henriques. Abstract: This talk is an introduction to bivariant field theories in the Stolz-Teichner program. I will discuss the easiest non-trivial case, namely 0-dimensional bivariant field theories with one supersymmetry. It turns out that the resulting bi-cycles are combinations of currents and forms, as in the de Rham homology and cohomology. Finally, I will give hints as to what these simple field theories might teach us about higher dimensional ones, in particular K-homology and KK-theory.
- 2010–8–6:
- 2|1-dimensional Euclidean field theories and noncommutative L^p-spaces.
FRG Workshop on mathematical 2D-field theory and the algebraic topology of closed manifolds
at Stony Brook University.

Abstract: A conjecture by Stolz and Teichner states that concordance classes of 2|1-dimensional Euclidean field theories are in bijective correspondence with cohomology classes of the cohomology theory TMF (topological modular forms). Here a field theory is a functor from the bicategory of 2|1-dimensional Euclidean bordisms to the bicategory of von Neumann algebras, L^p-bimodules, and their morphisms.

A significant amount of labor is required to make the definitions of the two bicategories mentioned above precise. Most of the talk will be devoted to a rigorous definition of the algebraic bicategory of von Neumann algebras, L^p-bimodules, and their morphisms, which involves proving several theorems about noncommutative L^p-spaces.

If time permits, I will also explain how the study of 2|1-dimensional Euclidean field theories naturally leads to consider such interesting structures as one-parameter semigroups of bimodules and two-parameter semigroups of bimodule endomorphisms further parametrized by the moduli space of elliptic curves. - 2009–10–20:
- Tensor products of noncommutative L_p-spaces and equivalences of categories of L_p-modules. Oberseminar C*-Algebren at the University of Münster by Joachim Cuntz and Siegfried Echterhoff.
Abstract:
In the first part of this talk I will introduce Haagerup's
theory of noncommutative $L_p$-spaces using the nice algebraic formalism
of modular algebras by Yamagami.
(Here $L_p=L^{1/p}$, in particular, $L_0=L^\infty$ and $L_{1/2}=L^2$.)
Then I will discuss some interesting properties of the resulting $L_p$-spaces,
in particular I will prove the following theorem:
$L_p(M)\otimes_M L_q(M)=L_{p+q}(M)$ for an arbitrary von Neumann algebra~$M$
and arbitrary complex $p$ and $q$ with nonnegative real parts.
Equality here means isometric isomorphism of $M$-$M$-bimodules.

In the second part of the talk I will describe $L_p$-modules by Junge and Sherman, which are the noncommutative analogs of modules of $p$-sections of bundles of Hilbert spaces over a measurable space. The special cases $p=0$ and $p=1/2$ correspond to the well-known cases of Hilbert W*-modules and Connes' correspondences. I will prove that W*-categories of $L_p$-modules for all values of~$p$ are equivalent to each other. After that I will explain how Connes' fusion (and its generalized version), which originally had very technical definition, can be described easily in this algebraic formalism.