Monday, 10 December 2018

I will outline a very surprising connection between scattering amplitudes which are basic objects in quantum field theory to describe the processes of elementary particles, and the cells in the positive Grassmannian which were studied quite recently from the point of view of algebraic geometry and combinatorics. I will show that cluster algebras play an important role in this description.

I will introduce cluster algebras through the language of quivers. The goal of this lecture series is two-fold---first, to give plenty of explicit examples so that participants feel comfortable working with cluster algebras, and second to give classifications of both the finite type and finite mutation type cluster algebras. Along the way, we will see relationships with root systems and Dynkin diagrams, with associahedra, and with the combinatorics of triangulations of surfaces.

In the fist half, I explain basic notions and properties of cluster patterns for cluster algebras. In the second half, I explain the dilogarithm identities associated with the periodicity of a cluster pattern.

Tuesday, 11 December 2018

We apply noncommutative deformation theory to general moduli problems.

It is well known that ordinary deformation theory of modules applies to the theory of moduli, and that it solves problems in very special algebraic situations.

In most algebraic situations, e.g. geometric invariant theory, the ordinary deformation theory is not sufficient. Olav Arnfinn Laudal generalized the deformation functor

Def_{M} : ℓ → Sets,

which goes from the category of local artinian (pointed) k-algebras to the category of sets and where M is an A-module, to a noncommutative deformation functor

Def_{M} : a_{r} → Sets,

which goes from the category of r-pointed, not necessarily commutative, artinian k-algebras to the category of sets, and where M = {M1, . . . , Mr} is a family of r (right) A-modules. The study of this generalization is interesting in its own rights, and it turns out that it more or less solves the problems in geometric invariant theory (e.g. when an action of a group is not free).

Lecture 1 introduces noncommutative deformation theory, and the class of resulting algebras. Lecture 2 gives the details of computation of the resulting algebras, and ends with a result on local representability. In Lecture 3 we observe that the representations of the (noncommutative) algebras resulting from the noncommutative deformation theory can be glued together to a noncommutative scheme-theory, thereby unifying representation theory and algebraic geometry. In lecture 4, we introduce Noncommutative Geometric Invariant Theory (NGIT) and prove that this solves some particular problems when a group acts non-freely. Also, we give an example on the moduli of endomorphisms.

The lectures are included in a text which will be available on the conference cite or on ArXiv. The four lectures is the background for the dynamical applications of noncommutative algebraic geometry, which is the second part of the text.

I will describe the natural way in which quiver gauge theories arise in string theory.

I will introduce cluster algebras through the language of quivers. The goal of this lecture series is two-fold---first, to give plenty of explicit examples so that participants feel comfortable working with cluster algebras, and second to give classifications of both the finite type and finite mutation type cluster algebras. Along the way, we will see relationships with root systems and Dynkin diagrams, with associahedra, and with the combinatorics of triangulations of surfaces.

In the fist half, I explain basic notions and properties of cluster patterns for cluster algebras. In the second half, I explain the dilogarithm identities associated with the periodicity of a cluster pattern.

Wednesday, 12 December 2018

In the fist half, I explain basic notions and properties of cluster patterns for cluster algebras. In the second half, I explain the dilogarithm identities associated with the periodicity of a cluster pattern.

We apply noncommutative deformation theory to general moduli problems.

It is well known that ordinary deformation theory of modules applies to the theory of moduli, and that it solves problems in very special algebraic situations.

In most algebraic situations, e.g. geometric invariant theory, the ordinary deformation theory is not sufficient. Olav Arnfinn Laudal generalized the deformation functor

Def_{M} : ℓ → Sets,

which goes from the category of local artinian (pointed) k-algebras to the category of sets and where M is an A-module, to a noncommutative deformation functor

Def_{M} : a_{r} → Sets,

which goes from the category of r-pointed, not necessarily commutative, artinian k-algebras to the category of sets, and where M = {M1, . . . , Mr} is a family of r (right) A-modules. The study of this generalization is interesting in its own rights, and it turns out that it more or less solves the problems in geometric invariant theory (e.g. when an action of a group is not free).

Lecture 1 introduces noncommutative deformation theory, and the class of resulting algebras. Lecture 2 gives the details of computation of the resulting algebras, and ends with a result on local representability. In Lecture 3 we observe that the representations of the (noncommutative) algebras resulting from the noncommutative deformation theory can be glued together to a noncommutative scheme-theory, thereby unifying representation theory and algebraic geometry. In lecture 4, we introduce Noncommutative Geometric Invariant Theory (NGIT) and prove that this solves some particular problems when a group acts non-freely. Also, we give an example on the moduli of endomorphisms.

The lectures are included in a text which will be available on the conference cite or on ArXiv. The four lectures is the background for the dynamical applications of noncommutative algebraic geometry, which is the second part of the text.

I will introduce cluster algebras through the language of quivers. The goal of this lecture series is two-fold---first, to give plenty of explicit examples so that participants feel comfortable working with cluster algebras, and second to give classifications of both the finite type and finite mutation type cluster algebras. Along the way, we will see relationships with root systems and Dynkin diagrams, with associahedra, and with the combinatorics of triangulations of surfaces.

I will describe the natural way in which quiver gauge theories arise in string theory.

Thursday, 13 December 2018

The aim of this series of lectures is to introduce the earliest version of categorification of cluster algebras, via the Caldero Chapoton map. The plan will be as follows:

1 - Representations of quivers and their category

2 - Auslander-Reiten theory

3 - Euler characteristic of module Grassmannians

4 - The Caldero-Chapoton map

This series of lectures will be useful as an introduction to the series of lectures given by B. Keller and the series of lectures given by P.G. Plamondon.

We apply noncommutative deformation theory to general moduli problems.

It is well known that ordinary deformation theory of modules applies to the theory of moduli, and that it solves problems in very special algebraic situations.

In most algebraic situations, e.g. geometric invariant theory, the ordinary deformation theory is not sufficient. Olav Arnfinn Laudal generalized the deformation functor

Def_{M} : ℓ → Sets,

which goes from the category of local artinian (pointed) k-algebras to the category of sets and where M is an A-module, to a noncommutative deformation functor

Def_{M} : a_{r} → Sets,

which goes from the category of r-pointed, not necessarily commutative, artinian k-algebras to the category of sets, and where M = {M1, . . . , Mr} is a family of r (right) A-modules. The study of this generalization is interesting in its own rights, and it turns out that it more or less solves the problems in geometric invariant theory (e.g. when an action of a group is not free).

Lecture 1 introduces noncommutative deformation theory, and the class of resulting algebras. Lecture 2 gives the details of computation of the resulting algebras, and ends with a result on local representability. In Lecture 3 we observe that the representations of the (noncommutative) algebras resulting from the noncommutative deformation theory can be glued together to a noncommutative scheme-theory, thereby unifying representation theory and algebraic geometry. In lecture 4, we introduce Noncommutative Geometric Invariant Theory (NGIT) and prove that this solves some particular problems when a group acts non-freely. Also, we give an example on the moduli of endomorphisms.

The lectures are included in a text which will be available on the conference cite or on ArXiv. The four lectures is the background for the dynamical applications of noncommutative algebraic geometry, which is the second part of the text.

Friday, 14 December 2018

It is well known that ordinary deformation theory of modules applies to the theory of moduli, and that it solves problems in very special algebraic situations.

In most algebraic situations, e.g. geometric invariant theory, the ordinary deformation theory is not sufficient. Olav Arnfinn Laudal generalized the deformation functor

Def_{M} : ℓ → Sets,

Def_{M} : a_{r} → Sets,

Lecture 1 introduces noncommutative deformation theory, and the class of resulting algebras. Lecture 2 gives the details of computation of the resulting algebras, and ends with a result on local representability. In Lecture 3 we observe that the representations of the (noncommutative) algebras resulting from the noncommutative deformation theory can be glued together to a noncommutative scheme-theory, thereby unifying representation theory and algebraic geometry. In lecture 4, we introduce Noncommutative Geometric Invariant Theory (NGIT) and prove that this solves some particular problems when a group acts non-freely. Also, we give an example on the moduli of endomorphisms.

The lectures are included in a text which will be available on the conference cite or on ArXiv. The four lectures is the background for the dynamical applications of noncommutative algebraic geometry, which is the second part of the text.

The aim of this series of lectures is to introduce the earliest version of categorification of cluster algebras, via the Caldero Chapoton map. The plan will be as follows:

1 - Representations of quivers and their category

2 - Auslander-Reiten theory

3 - Euler characteristic of module Grassmannians

4 - The Caldero-Chapoton map

This series of lectures will be useful as an introduction to the series of lectures given by B. Keller and the series of lectures given by P.G. Plamondon.

Monday, 17 December 2018

We will start by introducing the cluster category of an acylic quiver. We will show how it allows to categorify cluster variables and clusters in the associated cluster algebra. This leads to a natural explanation of Zamolodchikov periodicity for Dynkin quivers and, with a suitable generalization of the cluster category, for pairs of Dynkin quivers. We will then show how the choice of an initial seed leads to an orientation of the exchange graph, which then identifies with the Hasse diagram of a poset of torsion classes. We will conclude with a brief introduction to scattering diagrams, which are closely related to the poset of torsion classes.

We will first see how cluster characters are defined in the abstract setting of 2-Calabi-Yau triangulated categories. We will then see various applications to categorification of cluster algebras, including multiplication formulas. We will end with the study of generic cluster characters, and their application to the definition of generic bases for cluster algebras.

Tuesday, 18 December 2018

The aim of this series of lectures is to introduce the earliest version of categorification of cluster algebras, via the Caldero Chapoton map. The plan will be as follows:

1 - Representations of quivers and their category

2 - Auslander-Reiten theory

3 - Euler characteristic of module Grassmannians

4 - The Caldero-Chapoton map

This series of lectures will be useful as an introduction to the series of lectures given by B. Keller and the series of lectures given by P.G. Plamondon.

We will start by introducing the cluster category of an acylic quiver. We will show how it allows to categorify cluster variables and clusters in the associated cluster algebra. This leads to a natural explanation of Zamolodchikov periodicity for Dynkin quivers and, with a suitable generalization of the cluster category, for pairs of Dynkin quivers. We will then show how the choice of an initial seed leads to an orientation of the exchange graph, which then identifies with the Hasse diagram of a poset of torsion classes. We will conclude with a brief introduction to scattering diagrams, which are closely related to the poset of torsion classes.

We will first see how cluster characters are defined in the abstract setting of 2-Calabi-Yau triangulated categories. We will then see various applications to categorification of cluster algebras, including multiplication formulas. We will end with the study of generic cluster characters, and their application to the definition of generic bases for cluster algebras.

I will describe the natural way in which quiver gauge theories arise in string theory.

Thursday, 20 December 2018

We will start by introducing the cluster category of an acylic quiver. We will show how it allows to categorify cluster variables and clusters in the associated cluster algebra. This leads to a natural explanation of Zamolodchikov periodicity for Dynkin quivers and, with a suitable generalization of the cluster category, for pairs of Dynkin quivers. We will then show how the choice of an initial seed leads to an orientation of the exchange graph, which then identifies with the Hasse diagram of a poset of torsion classes. We will conclude with a brief introduction to scattering diagrams, which are closely related to the poset of torsion classes.

I will describe the natural way in which quiver gauge theories arise in string theory.

We will first see how cluster characters are defined in the abstract setting of 2-Calabi-Yau triangulated categories. We will then see various applications to categorification of cluster algebras, including multiplication formulas. We will end with the study of generic cluster characters, and their application to the definition of generic bases for cluster algebras.

Friday, 21 December 2018

1 - Representations of quivers and their category

2 - Auslander-Reiten theory

3 - Euler characteristic of module Grassmannians

4 - The Caldero-Chapoton map

This series of lectures will be useful as an introduction to the series of lectures given by B. Keller and the series of lectures given by P.G. Plamondon.