KIAS Winter School on derived categories
and wall-crossing
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| Abstracts | Home > Abstracts |
Mini-course : Categorical resolution of singularities.
In the mid 60's, Hironaka proved what is considered one of the most important result in complex algebraic geometry : the existence of reolutions of singularities. However, the existence of minimal resolutions of singularities is still a wide open problem.
In this course, I will explain how homological and categorical methods provide a new point of view on this problem. In particular, I will describe a way to produce minimal resolutions which is far out of reach of the classical geometric and birational methods.
This course will be divided in three lectures:
_basics on triangulated categories and semi-orthogonal decompositions,
_some examples of singularities and their resolutions,
_construction of minimal categorical resolutions.
References :
_Bondal-Orlov, Derived categories of coherent sheaves, ICM talk 2002.
_Alexander Kuznetsov, Lefschetz decompositions and categorical resolutions of singularities, Selecta Mathemarica, 2008.
_Michel Van den Bergh, Three dimensional flops and non-commutative rings, Duke Mathematical Journal, 2004.
There will be two strands to the talks that I give. The first will concern the cohomological hall algebra, which is a rich algebraic structure that one can associate to a three Calabi-Yau category of the sort that Sven will be talking about. I'll explain the background for this, starting with equivariant cohomology, and end up taking about the structure of this device and some ways to make it amenable to calculation. The other stand strand of my talks will be more directly concerning motivic Donaldson-Thomas theory. I will introduce lambda rings and wall crossing identities. At the end I will explain how the two strands relate to each other, explaining the link between lambda rings, motivic Donaldson-Thomas invariants and cohomological Hall algebras.
Stacks and moduli problems:
We start our journey towards Donaldson-Thomas invariants by recalling
some classical moduli problems from algebraic geometry and
representation theory. The need for groupoid valued funtors will bring
us directly to the notion of stacks, and we close the first lecture by
describing classical moduli problems using quotient stacks associated to
group actions.
Rings of motives and the motivic Hall algebra:
We introduce Grothendiecks ring of (naive) motives and generalize this
to a relative situation. If there is time, a small chapter will present
Grothendiecks dream of motives and related constructions, putting our
ring of motives into the right context. After that we need to modify
motives of varieties in order to generalize the theory to quotient
stacks and to varieties with certain discrete group actions. By applying
all the structure to moduli stacks, we finally introduce the motivic
Hall algebra together with its Hall product, a very important object in
Donaldson-Thomas theory.
The motive of vanishing cycles:
This talk is going to be quite technical. Starting from the classical
sheaf of vanishing cycles as presented in Ben Davison's first talk, we
aim at lifting this into the motivic world. As the classical definition
cannot be applied directly, we need to take a detour involving arc
spaces. In addition to the rather technical definition we also provide
very useful rules needed in practical computations. A couple of examples
will illustrate the theory.
Motivic Donaldson-Thomas invariants:
In our last talk, we finally present the definition of motivic
Donaldson-Thomas invariants involving all the structure developed in the
previous talks. A couple of examples will be given, and we try to answer
the question up to which extend Donaldson Thomas invariants give some
new insight into classical moduli problems.
Donaldson-Thomas theory with orientifolds:
The orientifold construction from string theory takes as input a theory of
oriented strings and outputs a theory of unoriented strings. I will
describe some basic aspects of this construction and explain why a
mathematician may be interested. In particular, it is expected that there
exists a modification of Donaldson-Thomas theory that includes
orientifolds. I will explain this modification in the setting of quivers.
A key role is played by a certain representation of the Hall algebra,
which serves as a model for the space of BPS states in the orientifolded
theory.
In the mid 60's, Hironaka proved what is considered one of the most important result in complex algebraic geometry : the existence of reolutions of singularities. However, the existence of minimal resolutions of singularities is still a wide open problem.
In this course, I will explain how homological and categorical methods provide a new point of view on this problem. In particular, I will describe a way to produce minimal resolutions which is far out of reach of the classical geometric and birational methods.
This course will be divided in three lectures:
_basics on triangulated categories and semi-orthogonal decompositions,
_some examples of singularities and their resolutions,
_construction of minimal categorical resolutions.
References :
_Bondal-Orlov, Derived categories of coherent sheaves, ICM talk 2002.
_Alexander Kuznetsov, Lefschetz decompositions and categorical resolutions of singularities, Selecta Mathemarica, 2008.
_Michel Van den Bergh, Three dimensional flops and non-commutative rings, Duke Mathematical Journal, 2004.
Mini-course : Donaldson-Thomas theory and wall-crossing I
There will be two strands to the talks that I give. The first will concern the cohomological hall algebra, which is a rich algebraic structure that one can associate to a three Calabi-Yau category of the sort that Sven will be talking about. I'll explain the background for this, starting with equivariant cohomology, and end up taking about the structure of this device and some ways to make it amenable to calculation. The other stand strand of my talks will be more directly concerning motivic Donaldson-Thomas theory. I will introduce lambda rings and wall crossing identities. At the end I will explain how the two strands relate to each other, explaining the link between lambda rings, motivic Donaldson-Thomas invariants and cohomological Hall algebras.
Stacks and moduli problems:
We start our journey towards Donaldson-Thomas invariants by recalling
some classical moduli problems from algebraic geometry and
representation theory. The need for groupoid valued funtors will bring
us directly to the notion of stacks, and we close the first lecture by
describing classical moduli problems using quotient stacks associated to
group actions.
Rings of motives and the motivic Hall algebra:
We introduce Grothendiecks ring of (naive) motives and generalize this
to a relative situation. If there is time, a small chapter will present
Grothendiecks dream of motives and related constructions, putting our
ring of motives into the right context. After that we need to modify
motives of varieties in order to generalize the theory to quotient
stacks and to varieties with certain discrete group actions. By applying
all the structure to moduli stacks, we finally introduce the motivic
Hall algebra together with its Hall product, a very important object in
Donaldson-Thomas theory.
The motive of vanishing cycles:
This talk is going to be quite technical. Starting from the classical
sheaf of vanishing cycles as presented in Ben Davison's first talk, we
aim at lifting this into the motivic world. As the classical definition
cannot be applied directly, we need to take a detour involving arc
spaces. In addition to the rather technical definition we also provide
very useful rules needed in practical computations. A couple of examples
will illustrate the theory.
Motivic Donaldson-Thomas invariants:
In our last talk, we finally present the definition of motivic
Donaldson-Thomas invariants involving all the structure developed in the
previous talks. A couple of examples will be given, and we try to answer
the question up to which extend Donaldson Thomas invariants give some
new insight into classical moduli problems.
Donaldson-Thomas theory with orientifolds:
The orientifold construction from string theory takes as input a theory of
oriented strings and outputs a theory of unoriented strings. I will
describe some basic aspects of this construction and explain why a
mathematician may be interested. In particular, it is expected that there
exists a modification of Donaldson-Thomas theory that includes
orientifolds. I will explain this modification in the setting of quivers.
A key role is played by a certain representation of the Hall algebra,
which serves as a model for the space of BPS states in the orientifolded
theory.