- 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. - 2018–4–25:
- Filtered D-modules and filtered Omega-modules. Texas Tech University Algebra and Number Theory Seminar. Abstract: I will discuss the Koszul duality between modules over the ring of differential operators and modules over the algebraic de Rham complex, for a smooth algebraic variety over a field of characteristic 0. We extend the previous work on this topic (including, most recently, the work of Positselski) by showing that the equivalences between appropriately derived categories of filtered D- and Omega-modules (without any quasi-coherence properties or boundedness conditions on filtrations) can be refined to a pair of monoidal Quillen equivalences with a new model structure. This model structure is well-adapted to geometric purposes, which is used in another paper to construct highly structured pushforward and pullback functors for D-modules.
- 2018–1–22, 2018–1–29, 2018–2–5, 2018–2–19, 2018–2–26, 2018–3–5, 2018–3–19:
- The Chern-Simons theory via bundle 2-gerbes. Texas Tech Geometry Seminar. Abstract: The Chern-Simons theory uses an action functional defined using a somewhat mysterious formula. In this series of talks we demonstrate how this formula arises in a completely canonical fashion from principal G-bundles with connection and bundle 2-gerbes with connection, the latter being a certain differential geometric object that will be explained in the talks. No prior knowledge of the Chern-Simons theory or bundle gerbes will be assumed or necessary.
- 2017–9–11, 2017–9–18, 2017–9–25, 2017–10–1, 2017–10–8, 2017–10–15, 2017–10–23, 2017–10–30:
- A very gentle introduction to functorial field theory. Texas Tech Geometry Seminar. Abstract: This series of talks will be an overview of functorial field theory, as introduced by Witten and Segal and later developed by Atiyah, Kontsevich, Schwarz, Freed, Resehtikhin, Turaev, Viro, Baez, Dolan, Hopkins, Lurie, and many others. We will emphasize the nontopological aspects, since these are more relevant for physics and have fewer expositions compared to the relatively abundant literature on TQFTs. No prior knowledge of field theory or physics will be assumed in the series.
- 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.
- 2016–12–5:
- Concordance theory for sheaves. Winter School on Bordism, L-theory, and Real Algebraic K-theory, Regensburg.
- 2016–4–28:
- De Rham modules and mixed Hodge modules. Arbeitsgruppenseminar Prof. Bunke, Regensburg.
- 2015–11–12:
- Hypercompleteness and convergence of Postnikov towers. Arbeitsgruppenseminar Prof. Bunke, Regensburg.
- 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.
- 2015–2–23:
- Smooth Rezk categories and 1-dimensional field theories. Seminar on Factorization Algebras, Göttingen, 2014–2015.
- 2015–2–3:
- Factorization algebras and Goodwillie Calculus. Seminar on Factorization Algebras, Göttingen, 2014–2015.
- 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).
- 2014–9–9:
- Concordance theory of homotopy sheaves. A short talk at the Modular Invariants in Topology and Analysis conference in Regensburg. Abstract: Starting from a homotopy sheaf of spaces (e.g., a simplicial presheaf that satisfies descent) on the site of smooth manifolds we define its associated concordance presheaf and prove that it is representable in the sense of the enriched Yoneda lemma. 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. We then explore various applications of this theorem, including functorial field theories, classifying spaces of categories, and higher geometric twists for K-theory. Joint work with Daniel Berwick-Evans (Stanford) and Pedro Boavida de Brito (Münster).
- 2014–7–25:
- Concordance theory of homotopy sheaves. Talk in CRCG Lunch Seminar by Chenchang Zhu. Abstract: Starting from a homotopy sheaf of spaces (e.g., a simplicial presheaf that satisfies descent) on the site of smooth manifolds we define its associated concordance presheaf and prove that it is representable in the sense of the enriched Yoneda lemma. 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. We then explore various applications of this theorem, including functorial field theories, classifying spaces of categories, and higher geometric twists for K-theory. Joint work with Daniel Berwick-Evans (Stanford) and Pedro Boavida de Brito (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.
- 2014–4–28:
- Introduction to model categories. An expository talk given at the Derived Algebraic Geometry reading group organized by Jakob Scholbach.
- 2013–4–18:
- String topology via Hochschild homology. An expository talk given at the Workshop on String Topology and Related Topics in Copenhagen organized by Richard Hepworth and Anssi Lahtinen.
- 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.
- 2012–10–22:
- Two-dimensional Yang-Mills theory and string topology of classifying spaces. Oberseminar Topologie at the University of Münster.
- 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.
- 2012–9–19:
- Two-dimensional Yang-Mills theory and string topology of classifying spaces. Joint short talk with Daniel Berwick-Evans at 2012 Oberwolfach topology workshop.
- 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.
- 2012–4–18:
- Γ-spaces and group completion of simplicial monoids. Higher Algebraic K-theory seminar by Arthur Bartels and Johannes Ebert.
- 2012–1–18:
- Calabi-Yau objects and string topology. Reading Seminar on Lurie's paper “On the Classification of Topological Field Theories” at the University of Münster.
- 2011–10–26:
- 2-dimensional field theories. Reading Seminar on Lurie's paper “On the Classification of Topological Field Theories” at the University of Münster.
- 2011–10–24:
- Euclidean field theories and topological modular forms. Oberseminar Topologie at the University of Münster.
- 2011–10–18:
- 1-dimensional field theories. Reading Seminar on Lurie's paper “On the Classification of Topological Field Theories” at the University of Münster.
- 2011–5–16:
- Witten genus as the equivariant index of the Dirac operator on a loop space. A 25-minute talk at Atiyah-Singer Index Theorem Mini-Conference by 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.
- 2011–4–27:
- Jones index via a symmetric monoidal bicategory of von Neumann algebras. Student Topology Seminar at UC Berkeley by Peter Teichner.
- 2011–3–8:
- Examples of n-categories. A talk on Section 6.2 of The Blob Complex paper by Scott Morrison and Kevin Walker given at Hot Topics Course on the Blob Complex by Peter Teichner.
- 2011–3–3:
- (Preceded by 2011–2–14.) Loop group representations as twisted CFT's.
- 2011–2–24:
- (Preceded by 2011–2–17.) Cohen and Godin's construction of a string topology TQFT.
- 2011–2–17:
- (Followed by 2011–2–24.) Operations in String topology. Student String Topology Seminar at UC Berkeley by Constantin Teleman. Abstract: We will review the Chas-Sullivan construction of a BV structure on the homology of free loop space of a compact oriented manifold. Subsequent developments by Cohen and collaborators, Kevin Costello and others reﬁne this structure to a 2-dimensional field theory, which couples to homology (classes and even chains) on the moduli of smooth curves. We will survey some of these developments.
- 2011–2–14:
- (Preceded by 2011–2–10 and followed by 2011–3–3.) Weak versus twisted CFTs II: Basic constructions and the determinant line.
- 2011–2–10:
- (Followed by 2011–2–14.) Weak versus twisted CFTs I: Basic constructions. Conformal Field Theory Student Seminar at UC Berkeley by Peter Teichner.
- 2010–12–3:
- The Obstruction Complex. Witten Genus Seminar by Daniel Berwick-Evans at the Max Planck Institute for Mathematics.
- 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.
- 2010–10–18:
- 2|1-dimensional Euclidean field theories and noncommutative L_p-spaces.
A 10-minute talk at Higher Index Theory
and Differential K-Theory School at the University of Göttingen.

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 several new 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. - 2010–9–20:
- 2|1-dimensional Euclidean field theories. A 30-minute talk at Student Topology Seminar by Matthias Kreck at Max Planck Institute for Mathematics.
- 2010–8–17:
- Tomita-Takesaki theory for fermions. Workshop on operator algebras and conformal field theory at the University of Oregon by André Henriques.
- 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. - 2010–5–20:
- (Preceded by 2010–5–6.) Yamagami's formalism in modular theory. Student Subfactor Seminar at UC Berkeley by Michael Hartglass. Abstract: I will discuss some practical applications of Yamagami's formalism such as Sakai's Radon-Nikodym theorem and intuitive explanation of Rieffel-van Daele approach to modular theory. After that we will take a look at L^p-modules and Connes' fusion.
- 2010–5–6:
- (Followed by 2010–5–20.) A conceptual approach to Tomita-Takesaki’s modular theory, Connes’ fusion, and noncommutative L^p-spaces. Student Subfactor Seminar at UC Berkeley by Michael Hartglass. Abstract: Tomita-Takesaki’s modular theory and Connes’ fusion tensor product have a reputation of technical and intimidating subjects in some circles. In this talk I will demonstrate that such claims are untrue and all these subjects can be treated in a nice, mostly algebraic way, using the algebraic framework first introduced by Yamagami in 1992. Imagine that proofs of such properties as Connes’ cocycle condition and other similar statements for cocycle derivatives, modular automorphisms, spatial derivatives, operator valued weights, Connes’ fusion etc. suddenly become one-liners!
- 2010–3–4:
- (Preceded by 2010–2–25.) Spanier-Whitehead duality and Milnor-Spanier-Atiyah duality. Student String Topology Seminar at UC Berkeley by Kevin Lin and Dmitri Pavlov. Abstract: I will define Thom spaces and spectra, explain Spanier-Whitehead duality, and state Milnor-Spanier-Atiyah duality. I will also sketch a proof of the fact that the Spanier-Whitehead dual of a smooth manifold M is equivalent to the Thom spectrum of the negative tangent bundle of M as a ring spectrum. In particular, I will explain how to obtain the usual intersection product in (shifted) homology as the homology of the multiplication map of the ring spectrum mentioned above.
- 2010–2–25:
- (Followed by 2010–3–4.) Symmetric spectra and Atiyah duality. Student String Topology Seminar at UC Berkeley by Kevin Lin and Dmitri Pavlov. Abstract: In this talk I will introduce some preliminary material necessary for Cohen-Jones construction of string topology in terms of stable homotopy theory. I will define the symmetric monoidal category of symmetric spectra and give some examples, including Thom spectra and Atiyah duality. In particular, I will explain how to obtain the usual intersection product in (shifted) homology as the homology of the multiplication map of a certain ring spectrum.
- 2009–11–2:
- (Preceded by 2009–10–26 and 2009–10–19.) Generalized Connes fusion tensor product for L_p-modules. Student Topology Seminar at Max Planck Institute by Peter Teichner and Matthias Kreck.
- 2009–10–28:
- Equivariant cohomology (sections 3.4 and 3.5 in Jacob Lurie's Survey of Elliptic Cohomology). Elliptic Cohomology Seminar at Max Planck Institute by Ryan Grady.
- 2009–10–26:
- (Preceded by 2009–10–19 and followed by 2009–11–2.) Relative L_p-spaces, W*-categories, L_p-modules and their equivalences. Student Topology Seminar at Max Planck Institute for Mathematics by Peter Teichner.
- 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. - 2009–10–19:
- (Followed by 2009–10–26 and 2009–11–2.) Tensor products of noncommutative L_p-spaces and equivalences of categories of L_p-modules. Student Topology Seminar at Max Planck Institute by Peter Teichner.
Abstract: The category of commutative von Neumann algebras is contravariantly equivalent to the category
of measurable spaces.
Generalizing this we can define the category of noncommutative measurable spaces as the opposite
category to the category of noncommutative von Neumann algebras.
Surprisingly, many notions and theorems of classical measure theory have analogs in the noncommutative case.
Examples include Riesz' representation theorem, Radon-Nikodym theorem,
Hahn-Jordan decomposition, $L_p$-spaces, H\"older inequality, Fubini theorem, Haar measure, and many others.
(Here $L_p=L^{1/p}$, in particular, $L_0=L^\infty$ and $L_{1/2}=L^2$.)

In this talk I will give an introduction to a small piece of noncommutative measure theory. For those unfamiliar with the noncommutative geometry, the content of the talk can be best described by pointing out the analogous notions in the world of smooth manifolds: Bundles of densities and $p$-densities, integration, hermitian vector bundles, and their tensor products. In more technical terms I will introduce Haagerup's theory of noncommutative $L_p$-spaces using the nice algebraic formalism of modular algebras by Yamagami. In particular you'll find out how one can multiply two measures (and what one can do with the result). 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, which are the noncommutative analogs 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 the categories of $L_p$-modules for different values of $p$ are equivalent to each other. After that I will explain how Connes' fusion, which originally had very scary and technical definition, can be described very easily in this algebraic formalism. - 2009–10–14:
- (Preceded by 2009–10–12.) Orientations of the multiplicative and the additive groups. Sections 3.1 and 3.2 in Jacob Lurie's Survey of Elliptic Cohomology. Elliptic Cohomology Seminar at Max Planck Institute by Ryan Grady.
- 2009–10–12:
- (Followed by 2009–10–14.) Derived group schemes and orientations. Section 3.0 in Jacob Lurie's Survey of Elliptic Cohomology. Elliptic Cohomology Seminar at Max Planck Institute by Ryan Grady.
- 2009–8–17:
- Stolz-Teichner program. Topology Seminar at St Petersburg Steklov Math Institute.
- 2009–4–7:
- (Preceded by 2009–3–31.) Continuation.
- 2009–3–31:
- (Followed by 2009–4–7.) Inductive reformulation of the cobordism hypothesis, a talk on the Section 3.1 of a paper by Jacob Lurie “On the Classification of Topological Field Theories (Draft)”. Hot Topics Course Spring 2009 (Spaces of TFTs) at UC Berkeley by Peter Teichner.
- 2008–12–11:
- The monoidal category of symmetric spectra. A 10-minute talk at Differential Topology Mini-conference at UC Berkeley by Peter Teichner.
- 2008–12–9:
- Smooth K-theory, a talk on a paper by Simons and Sullivan. Hot Topics Course Fall 2008 (Smooth Cohomology) at UC Berkeley by Peter Teichner.
- 2008–12–4:
- (Preceded by 2008–12–2.) Derived functors, model structures in rational homotopy theory. Rational Homotopy Theory Course at UC Berkeley by Constantin Teleman.
- 2008–12–2:
- (Followed by 2008–12–4.) Model categories, model structures on topological spaces (Serre and Hurewicz), simplicial sets (Kan), chain complexes (projective and injective), homotopy theory in model category, homotopy category of a model category, derived functors. Rational Homotopy Theory Course at UC Berkeley by Constantin Teleman.

Six talks given at Student Topology Seminar at UC Berkeley by Peter Teichner:

- 2008–11–5:
- Von Neumann algebras, their morphisms and preduals; commutative von Neumann algebras, their morphisms and preduals as measurable spaces, measurable maps, and finite densities (complex-valued measures); weights; traces and L_p-spaces for semifinite von Neumann algebras.
- 2008–11–12:
- Von Neumann algebras as noncommutative L_0-spaces and their preduals as noncommutative L_1-spaces; polar decomposition of L_p-spaces in commutative case and its generalization to noncommutative case; L_p-spaces as sets and arithmetic operations on them; unital *-algebra of affiliated operators of finite von Neumann algebra as Ore localization.
- 2008–11–19:
- Left Hilbert algebras; modular operator and modular conjugation; modular automorphisms; weights; bijective correspondence between faithful semifinite normal weights and full left Hilbert algebras.
- 2008–11–26:
- The unital *-algebra of measurable affiliated operators associated to a faithful semifinite normal trace on a semifinite von Neumann algebra; Connes cocycle derivative; crossed products; bounded modular algebra; the core of a von Neumann algebra, its canonical one-parameter automorphism group, and its canonical faithful semifinite normal trace; embedding of the modular algebra into the core; the unital *-algebra of measurable operators associated with the core and its embedding in the set of affiliated operators.
- 2008–12–3:
- The language of modular algebras; the modular algebra, the core, the unital *-algebra of measurable operators associated with the core, the set of affiliated operators of the core; L_p-spaces and the unbounded modular algebra; arithmetic operations on L_p-spaces and their polar decomposition; trace on L_1 and (quasi)norm on L_p-spaces; duality of L_p-spaces; quasi-Banach spaces and Aoki-Rolewicz theorem; tensor products and morphism spaces of L_p-spaces.
- 2008–12–10:
- Tensor products of L_p-spaces for von Neumann algebras.

- 2008–2–26:
- Computation of twisted equivariant K-theory of a compact Lie group. Hot Topics Course Spring 2008 (Twisted equivariant K-theory and the Verlinde algebra) at UC Berkeley by Peter Teichner.
- 2007–12–11:
- Stone-von Neumann theorem for Heisenberg algebras and groups. A 15-minute talk at Super symmetric field theories and generalized cohomology course mini-conference at UC Berkeley by Peter Teichner and Nicolai Reshetikhin.
- 2007–12–4:
- Closed string TCFT for Hermitian Calabi-Yau Elliptic Spaces, a talk on a paper by Costello, Tradler, and Zeinalian. Hot Topics Course Fall 2007 (Topological conformal field theory) at UC Berkeley by Peter Teichner.
- 2007–5–7:
- Some properties of Wu and Stiefel-Whitney classes. Topology Seminar at St Petersburg Steklov Math Institute.