Fall 2013

Mathematics 280x

Bridgeland Stability Conditions

Catalog Number: 90433
Instructor: Hiro Lee Tanaka
Half course (fall term). Tu., Th., 10–11:30.
EXAM GROUP: 12, 13
The basics of Bridgeland stability conditions for stable oo-catagories will be covered. The courses ultimate goal is to represent Hall (co)algebra-like structures as a co/sheaf on a Ran space.

• See the rough syllabus (updated as the class goes on) of the topics we'll cover. You will also see what topics your team will have to talk about, and on which days.

Teams

Below are the talk teams:

• Team Edward (basics of Coh(X)).
  1. Basics of Coh(P^1. (Jeremy Hahn, Krishanu Sankar, Aaron Mazel-Gee)
     1. Definition of line bundles O(k) and computation of global sections
     2. Definition and computations of Ext(O(k),O(k'))
     3. Grothendieck Splitting Theorem
     4. Serre Duality. Why the canonical sheaf is O(-2)
     5. Exceptional Collections
     6. Generalities on P^n
     7. Mukai pairing
  2. Stability conditions for an elliptic curve (Omar Antolín Camarena)
     1. Definition of numerical Grothendieck group
     2. Action of universal cover of GL+(2,R) (Reference: Bridgeland, last two pages of original paper)
     3. Stability conditions on a possibly singular Weierstrass curve (Reference: Burban-Kreussler, Derived categories of...)
  3. Walls between stability conditions for P^1 (Charmaine Sia)
     1. Understanding what happens across different walls (Reference: Okada, arXiv:0411220v3)
     2. Example of a heart where Db(heart) does not equal DbCohP^1 (Reference: Bayer, Utah lecture notes)
     3. Some generalities on DbCoh(X) for X any smooth curve (Reference: Macri, Some Examples)
  4. Unassigned members:
     1. Netanel Blaier
     2. Gabriel Bujokas
     3. Saul Glasman
• Team Jacob (basics of oo-categories).
  1. Basics of oo-categories. (References: Higher Topos Theory, Joyal Barcelona lectures.)
     1. Simplicial sets and nerves of categories, motivation of weak Kan condition.
     2. Morphism spaces in oo-categories
     3. Geometric realization and singular chains, Kan complexes as spaces and as oo-groupoids
     4. Simplicial nerve. Getting a oo-category from a Top-enriched or Kan-enriched category.
     5. Getting a oo-category from a simplicial model category
     6. Initial and Terminal objects
     7. Limits/colimits
  2. Stable oo-categories. (References: Higher Topos Theory, DAG I or Higher Algebra Chapter I, Cohn)
     1. Chain complexes and Dold-Kan correspondence
     2. dg nerve
     3. Definition of stability
     4. Translation/shift functor
     5. LES of Ext as a consequence of LES of Serre fibration (Ext^i as negative homotopy groups)
     6. Homotopy category of a stable oo-category is triangulated
     7. t-structures for stable oo-categories
  3. Team Members:
     1. Justin Campbell
     2. Chenglong Yu
     3. Yi Xie
     4. Danny Shi
• Team Apple (applications).
  
  1. McKay Correspondence (Tobias Barthel)
     1. Bridgeland survey article
     2. Craw lecture notes, especially Section 6.
  2. Polynomial Stability Conditions (Lukas Brantner)
     1. Bayer
  3. Kontsevich-Soibelman wall crossing formulas for simple quivers (Gijs Heuts)
     1. Kontsevich-Soibelman pre-print
     2. Kontsevich-Soibelman digest
     3. Kontsevich-Soibelman lectures
     4. Joyce-Song
     5. Gaiotto-Moore-Neitzke
     6. Aganagic-Ooguri-Vafa-Yamazaki
  4. Matthew's research and MMP (Matthew Woolf)
  5. Quadratic Differentials
     • Ivan Smith, Quiver algebras as Fukaya categories
     • Bridgeland-Smith, Quadratic Differentials as Stability Conditions
     • Howie Masur, Closed Trajectories for Quadratic Differentials with Applications to Billiards
     • Dimitrov-Haiden-Katzarkov-Kontsevich, Dynamical systems and categories
  6. Unassigned: Eric Wofsey

Notes

Every week we'll have two audience members take notes. Their notes will become available (at the latest) on the Tuesday of the week after. You can click on the names of the notetakers to download the notes. Also, my notes are often prepared ahead of time, so depending on how the class time goes, I may cover material different from the stuff I prepared. (For instance, in lecture 4, I never planned to talk about the definition of the Fukaya category, but realized I should, so I gave a brief sketch of it.) As such, you should try reading the students' notes, too, in case I covered something not in my original notes.