Global Homotopy Theory and Orbifolds (Fall 2019)

We will be running a seminar on orbifolds and global homotopy theory, starting from "what is an orbifold?" and "what is equivariant homotopy theory?", in hopes of eventually finding the connections between the two topics.

We will meet on Mondays 4-5PM and Tuesdays 1130AM-1230PM in room 622.

The sourcecode was shamlessly copied from the Gauge theory learning seminar .

The seminar is co-organized by Inbar Klang and Lea Kenigsberg. 

Resources:

(a) Equivariant Homotopy Theory by J.P.C Greenless and J.P. May

(b) Orbifolds and Stringy Topology by Alejandro Adem, Johann Leida, Yongbin ruan

(c)  Global Homotopy theory by Stefan Schwede

(d)  Orbifolds as Groupoids: an Inroduction by Ieke Moerdijk

(e)  Equivariant Stable Homotopy theory Lecture notes from a course taught by Andrew Bulmberg

(d)  Talbot Notes See also refrences within.

Schedule 

Date & Time

Speaker

Title

References

October 7th, 3 PM Andrew Blumberg What is the point of equivariant stable homotopy theory?  
October 7th, 4 PM John Morgan Introduction to Orbifolds (b) ch 1.
October 8th, 11:30 AM Semon Rezchikov Orbifolds in Floer Homology  
October 14th, 4 PM John Morgan Introduction to Orbifolds - continued  
October 15th, 11:30 AM Davis Lazowski Unstable equivariant homotopy theory  
October 21st, 4:10 PM Lea Kenigsberg Cohomology and morphisms of orbifold groupoids  
October 22nd, 11:30 AM Roy Magen Overview of Spectra  
October 28th, 4:10 PM Alex Pieloch Orbifold K-Theory  
October 29th, 11:30 AM Lea Kenigsberg Equivariant stable homotopy theory  
Nov 11th, 4:05 PM Semon Rezchikov Chen Ruan Cohomology  
Nov 12th, 11:30 AM Inbar Klang Model Categories  
Nov 18th, 4:05 PM Roy Magen Orthogonal G-Spectra  
Nov 19th, 11:30 AM Inbar Klang Diagram Spectra  
Nov 26th, 11:30 AM Andrew Blumberg Linear Isometries Operad  
Dec 10th, 11:30 AM Lea Kenigsberg From orbifolds to global spaces  

 

Abstracts

October 7th

Andrew Blumberg, what is the point of equivariant stable homotopy theory?

Equivariant stable homotopy theory is the study of (co)homology theories for spaces with an action of a compact Lie group. These theories can be organized into a category, the equivariant stable category. Unfortunately, the equivariant stable category is remarkably complicated. This talk will try to give a gentle introduction that explains why the level of complexity that arises is an unavoidable consequence of elementary desiderata and what problems the technology helps you solve.

John Morgan, an Introduction to Orbifolds

The definition in terms of coordinate charts; the groupoid definition. Isomorphism of orbifolds. The local group at a point. The fundamental group and covering orbifolds; Sheaves on orbifolds, the sheaf of smooth functions, the tangent sheaf, the sheaf of differential forms. Smooth maps of orbifolds.

October 8th

Semon Rezchikov, Orbifolds in Floer theory

I will give a basic introduction to the Kuranishi theory approach to moduli spaces, which underlies Gromov-Witten theory, Donaldson invariants, the Floer homology theories, and much more. I will focus on describing the basic geometric structures that moduli spaces acquire from this approach, and will try to explain the appeal of orbifold cohomology theories to Floer theorists, without going into the formidable technical difficulties involved in the existing theories of virtual fundamental cycles. Some comparison to the algebraic geometry literature will be provided.

October 14th

John Morgan, Intorduction to Orbifolds-continued

October 15th

David Lazowski, Unstable equivariant homotopy theory

We'll introduce the basics of unstable equivariant homotopy theory, such as G-CW complexes, representation spheres and Bredon cohomology.

October 21

Lea Kenigsberg, Cohomology and morphisms of orbifold groupoids

I will describe singular and deRham cohomolgy of orbifold groupoids, and discuss morphisms and their classifications. 

October 22

Roy Magen, Overview of Spectra

We will define spectra and go over basic properties and examples. Some ∞-categorical perspectives may also be discussed if time permits.

October 29th

Lea Kenigsberg, Equivariant stable homotopy theory

I will carefully state and prove the equivariant Freudenthal suspension theorem, and discuss transfer maps and the Wirthmuller isomorphism.

November 12th

Inbar Klang, Model Categories

We'll introduce the basic definitions of model categories, and discuss examples and useful results.

November 18th

Roy Magen, , Overview of Orthogonal G-Spectra

We will define the category of orthogonal G-spectra and some structures on it. In particular: the smash product, the homotopy "groups", the model structures (up to 3 as time permits), and indexed monoidal products. We will mention some basic properties. The word "proof" will appear very few times.