Dmitri Pavlov's talks
- 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 Dan 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 hypthesis, 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 structrues 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.