Bondal, Alexei: Lecture 1. Derived categries in algebra and geometry.
We define derived category of an abelian category and describe
homotopy theoretical origin of triangulated categories. We shall explain
basic invariants of triangulated categories. We give general overview of
derived categories of modules over algebras and of coherent sheaves on
algebraic varieties, and equivalences between the two, constructed via
strong generators. Categorical smoothness will also be defined.
 
Lecture 2. Exceptional collections and semiorthogonal decompositions.
We will define exceptional collections and semiorthogonal
decompositions in triangulated categories and describe the action of the
braid group on them. We will overview known examples of semiorthogonal
decompositions for the derived categories of coherent sheaves. We will
explain how the problem of constructing semiorthogonal decompositions is
related to the mirror symmetry.
 
Chen, Jiun-Cheng: Moduli space, Deligne-Mumford stack and derived categories in birational geometry
I will talk about my work, my joint work with D. Abramovich and my joint work with H-H Tseng on applications of moduli space,
Deligne-Mumford stack in studying birational geometry. These methods are not among the standard tools of a birational geometer. 
I will discuss the relation between flops, flips, moduli spaces and equivalence of derived categoris. I will
also discuss our work on globalizing the result by Bridgeland, King and Reid.
 
Cho, Cheol-Hyun: Counting holomorphic discs with Lagrangian boundary condition
We discuss various examples of counting holomorphic discs. The first example is the case 
when Lagrangian submanifold is a fixed point set of anti-symplectic involution. 
We provide another way to define a signed count of discs in this case 
(This was first proved by Welschinger, and called Welschinger invariants). 
The other examples are in the case of toric Fano manifolds. We set up certain counting problems 
and relate the totality of bubbling off discs in this counting with L-infinity algebra of 
Lagrangian submanfold (which is a symmetric version of A-infinity algebra of Fukaya, Oh, Ohta and Ono).

Kaledin, Dmitry: Non-commutative Hodge-to-de Rham degeneration and the Frobenius map
I will talk about my recent work math.AG/0511665, where an analog of

the Hodge-to-de Rham spectral sequence degeneration

is proved for non-commutative algebras, using an adaptation of the method of Deligne-Illusie

Karzarkov, Ludmil: Homological Mirror Symmetry (HMS) and applications

In this talk we will explain HMS conjecture and we will discuss a symplectic geometry

approach to classical questions in Algebraic Geometry.

Kuznetsov, Alexander: Derived categories of Fano 3-folds

Homological Projective Duality allows to describe derived categories of several Fano 3-folds.

A comparison reveals an unexpected relation between derived categories of 3-folds of different index.

On the geometric level this gives an identification of

some moduli spaces of vector bundles on these 3-folds.

Kuznetsov, Alexander: Introduction to Homological Projective Duality.

Homological Projective Duality is an extension of the classical projective duality relation 
in the context of derived categories. It allows to describe derived categories of linear sections 
of an algebraic variety if the Homologically Projectively Dual variety is known. 
In this talk I will discuss the definition and examples of Homological Projective Duality.

Logvinenko, Timothy: Derived McKay correspondence via pure sheaf transforms

Let G be a finite subgroup of SL_n(C) and let F be a Fourier-Mukai functor from the derived category 
of a scheme Y birational to the quotient singularity C^n/G to the equivariant derived category 
of C^n. Under an assumption that F sends point sheaves on Y to pure sheaves on C^n, 
I will talk about orthogonality conditions sufficient to make F into an equivalence of categories. 

Macrí, Emanuele: Introduction to stabilty conditions through some examples

We review the basic definitions of Bridgeland-Douglas stability conditions on derived categories

and give some examples of stability manifolds

Szendroi, Balazs: D-brains on ADE fibred surfaces
I show how to classify certain holomorphic D-branes, closely related to framed torsion-free sheaves, 
on threefolds fibered in ADE surfaces. For certain geometries, 
this recovers a quiver problem studied earlier 
by Cachazo-Katz-Vafa in the context of supersymmetric quiver gauge theory.

Toda, Yukinobu: On a certain generalization of spherical twists

I give a generalization of spherical twists, and describe the autoequivalences associated

to certain non-spherical objects. Typically these are obtained by deforming

the structure sheaves of (0, -2)-curves on threefolds,

or deforming P-objects introduced by D.Huybrecht and R.Thomas.

Uehara, Hokuto: Stability conditions and A_n-singularities on surfaces
We show the connectedness of the space of stability conditions on certain triangulated

categories associated with A_n-singularities on surfaces. My talk is based

on a joint work with A. Ishii and K. Ueda.

Yoshioka, Kota: Fourier-Mukai transform and the moduli of stable sheaves on abelian surfaces

Under suitable conditions, Fourier-Mukai transforms F preserves the stability condition

for a general stable sheaf E. In particular Fourier-Mukai transforms induce birational isomorphisms

of the moduli of stable sheaves on abelian surfaces. On the other hand, it is known that

we really need such conditions for the preservation of the generic stability.

In my talk, by replacing F(E) by another sheaf E' with v(E')=v(F(E)),

I construct a birational isomorphism of moduli spaces.