Representations from a metric point of view
In the talk I will discuss metric properties of quotient spaces of representations of compact groups
on Euclidean spaces.
Program discussion
Holomorphic Bundles
Multiple harmonic sums, multiple zeta values and other related values
In this talk I will survey the historical and recent development in the study of multiple harmonic sums, multiple zeta values and some other related values such as multiple zeta star values and their q-analogs, mainly from the number theory point of view. Starting with Euler's decomposition formula for double zeta values I will conclude with a sketch of the Two-one formula conjectured by Ohno and Zudilin when they were here at MPIM in 2007. Most part of the talk will be accessible to the general audience.
Topos Theory
A Special Isogeny of K3 Surfaces
I will discuss a correspondence relating two specific classes of complex algebraic K3 surfaces. The first class consists of K3 surfaces polarized by the rank-sixteen lattice H+E_7+E_7. The second class consists of K3 surfaces obtained as minimal resolutions of double covers of the projective plane branched over a configuration of six lines. The correspondence underlies a geometric two-isogeny of K3 surfaces.
Topology of algebraic varieties and perverse sheaves III
Tree homology and a conjecture of Levine
In a joint project with Rob Schneiderman and Peter Teichner, we have analyzed the structure of the set of links up to "Whitney tower concordance." A crucial aspect of this analysis is a conjecture of Jerry Levine, which relates an algebraically defined Lie ring to a combinatorially defined abelian group of labeled trees modulo the IHX relation. This conjecture is easy to verify after tensoring with a field of characteristic $0$, but the general statement is not so easy. In this talk, I will discuss the conjecture and how we proved it using discrete Morse theory for chain complexes.
http://people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html
Geometric filtrations of string links and homology cylinders (Semester on 4-manifolds)
Geometric filtrations of string links and homology cylinders
(Joint with Schneiderman and Teichner)
We analyze several filtrations on the groups of concordance classes of string links and homology
cylinders modulo homology bordism. In particular we compare the twisted Whitney tower
filtration of string links to the Johnson filtration of string links, showing that the associated
graded groups differ by finitely generated 2-torsion groups, corresponding to higher-order
Arf invariants. We also continue Jerry Levine's analysis of the Johnson filtration of homology
cylinders to the filtration defined by embedded surgery along claspers, completely characterizing
the difference up to some unknown 2-torsion.
http://people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html
Intersection theory of Whitney towers
http://people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html
Intersection theory of Whitney towers
http://people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html
Toward multiple zeta values cycles
We will review the setting of Bloch-Kriz cycle complex over the
projective line minus three points.
We will then show how to recover in this context the algebraic cycles
associated to the classical polylogarithms
using a pull-back by the multiplication on the affine line.
For a low weight example, we will explain how a twisted multiplication
map allows to build more general cycles.
We will conclude by showing how a multiple zeta value arises from
the Gangl-Goncharov-Levin seesaw process.
Adjunctions and Localizations for quasicategories
Definition of adjunction as Cartesian/Co-cartesian map to Delta^1. How to get functors from this.
Example coming from Presheaf categories (discussion starts in HTT at the bottom of page 357 and continues into Prop 5.2.6.3. (needs Prop. 5.2.4.2).
The other main examples will be localization functors, as in Section 5.2.7. Describe the definition.
Then skip ahead to Section 5.5.4. The goal will be to talk about localizations at a strongly saturated class of morphisms. Try to avoid the technical details about presentable quasicategories and size issues. One goal is to describe Prop. 5.5.4.15.
Relate these to the Bousfield localizations we saw earlier.
https://docs.google.com/document/d/1llPzCxlhWSucYymRenInm861KK-QVJxQS0MeEQZdjE8/edit
Kan extensions for quasicategories
We only need Kan extensions along inclusions of subquasicategories [HTT 4.3.2]. Lurie introduces the more general notion of p-Kan extension. For us D’ will be the terminal quasicategory and p the unique map.
We will need HTT Prop 4.3.2.8, and Cor. 4.3.2.16
We need the example of Kan extensions along the Yoneda embedding into presheaves of spaces. Specifically we need Thm 5.1.5.6 (which uses Lma 5.1.5.5)
https://docs.google.com/document/d/1llPzCxlhWSucYymRenInm861KK-QVJxQS0MeEQZdjE8/edit
$\Theta_n$-spaces
- Describe the category $\Theta_n$.
- $\Theta_n$-spaces and their properties.
- The comparison map to iterated complete Segal spaces.
https://docs.google.com/document/d/1llPzCxlhWSucYymRenInm861KK-QVJxQS0MeEQZdjE8/edit
Barwick’s Iterated Complete Segal Spaces and Simpson’s Segal n-Categories
Barwick’s model is described in Lurie’s survey of the cobordism hypothesis, and also in Barwick-Schommer-Pries. You should describe it as a Bousfield localization.
We described Segal n-categories last semester.
How do these compare?
Why iterated complete Segal spaces fail to be Cartesian.
https://docs.google.com/document/d/1llPzCxlhWSucYymRenInm861KK-QVJxQS0MeEQZdjE8/edit
Introduction to Quasicategories
Basic concepts and definitions
terminal/initial objects
limits and colimits
simple examples
https://docs.google.com/document/d/1llPzCxlhWSucYymRenInm861KK-QVJxQS0MeEQZdjE8/edit
Topological Categories and comparisons
https://docs.google.com/document/d/1llPzCxlhWSucYymRenInm861KK-QVJxQS0MeEQZdjE8/edit
homotopy coherent nerve
classification diagram (see Rezk’s “A model for the homotopy theory...”)
Quasicategories from model categories.
In particular it would be really great if you can state Thm 4.2.4.1 (or maybe Prop 4.2.4.4 instead) in Lurie’s HTT, and perhaps sketch the proof. This is the result that related limits/colimits in quasicategories with classical homotopy colimits in simplicial categories.
(note: Chris is absent for this talk)
Complete Segal Spaces
https://docs.google.com/document/d/1llPzCxlhWSucYymRenInm861KK-QVJxQS0MeEQZdjE8/edit
Introduce Bousfield localizations of presheaf model categories.
example: An ordinary category as a complete Segal space
Other nice properties (inner homs, simplicial enrichment, homotopy hypothesis, etc).
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