# Reading Seminar on Gromov--Witten and Derived Category

### Spring 2015

**Oganizers:** Y.P. Lee, JWB 305 and
A. Bertram, M. Shoemaker, N. Tarasca

**Meeting**

**Time.** Fridays 14:00-16:00

**Room.** JWB 208

**Topics:**

**First Topic**

**Title:** Stability conditions on Threefolds

**Speaker:** Cristian Martinez

**Abstract:**
Stability conditions were introduced by Bridgeland as an effort to
understand results of Douglas on \Pi-stability for D-branes in string
theory. However, even when the motivating example is that of a Calabi-Yau
threefold, there is not known way to produce a stability condition on a
general CY3.
It has been stablished by Bayer, Bertram, Macri, and Toda, that once
certain Bogomolov-Gieseker type inequality on the Chern classes of a
"stable" object is known then it is possible to construct a family of
stability conditions on the threefold. This conjectural inequality is
often referred as the hardest problem in the area of stability conditions.
It has been proven for the projective space, the quadric threefold in P^4,
and recently on abelian threefolds and on Calabi-Yau threefolds obtained
as (the crepant resolution of) a finite quotient of an abelian threefold.
In this series of lectures I will discuss some generalities on stability
conditions and some known results on surfaces. Then introduce the
Bogomolov-Gieseker--type inequality conjecture, and show the proof for the
case of the projective space. Then move to study the paper by Bayer,
Macri, and Stellari that proves the conjecture for the case of abelian
threefolds.
These are some references:
- T. Bridgeland. Spaces of Stability Conditions.
- A. Bayer, E. Macri, Y. Toda. Stability conditions on threefolds I: Bogomolov-Gieseker type inequalities.
- A. Bayer, A. Bertram, E. Macri, Y. Toda. Stability conditions on threefolds II: An application to Fujita's conjecture.
- E. Macri. A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space.
- A. Bayer, E. Macri, P. Stellari. The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds.

**Second Topic**

???

**Presentation**
- (1/16) Bridgeland stability: Cristian Martinez
- (1/23) Bridgeland stability: Cristian Martinez
- (1/30) Bridgeland stability: Cristian Martinez
- (2/6) Bridgeland stability: Cristian Martinez
- (2/13) Bridgeland stability: Cristian Martinez
- (2/20) On Bayer--Macri--Stellari: Cristian Martinez
- (3/6) On Bayer--Macri--Stellari: Cristian Martinez
- (3/13) On Bayer--Macri--Stellari: Cristian Martinez
- (3/27) On Bayer--Macri--Stellari: Cristian Martinez
- (4/3) On Bayer--Macri--Stellari: Cristian Martinez
- (4/10) On Bayer--Macri--Stellari: Cristian Martinez

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