Lectures
Hans
Adler
"Strong theories and weight"
Alessandro Berarducci
"Cohomology of definable sets in o-minimal expansions of groups" (joint work with A. Fornasiero)
Ayse Berkman
"Curtis-Tits type of theorems in groups of finite Morley rank"
Jeff Burdges
"Our current picture of torsion in connected groups"
Paola D'Aquino
"Schanuel's conjecture and exponential fields"
Grigory Garkusha
"Reconstructing schemes from abelian categories"
Ivo Herzog
"Definable subspaces of finite-dimensional representations"
Gareth Jones
"Locally polynomially bounded structures"
Moshe Kamensky
"Model theory of fibre functors"
Angus MacIntyre
"Towards decidability of the theory of pseudofinite-dimensional representations of sl_2(C)"
Margarita Otero
"Torsion in definable groups"
Mike Prest
"From abelian groups to 2-categories: interpretations between additive categories"
Gena Puninski
"Towards the decidability of theories of modules over finite commutative rings"
Cédric Rivière
"Existentially closed difference-differential fields" (joint work in progress with Nicolas Guzy)
Philipp Rothmaler
"Inbetween pure projective and Mittag-Leffler"
"Strong theories and weight"
Abstract:
It is not clear what weight (with respect to forking) should be in a non-simple theory, but as result of recent work by Shelah, Usvyatsov and Onshuus it is becoming increasingly clear what it means for a non-simple theory to have finite weight. I will call a theory strong if a certain constant from Shelah's book is countable. A simple theory is strong if and only if every type has finite weight. A theory is strongly dependent if and only if it is strong and dependent (i.e. does not have the independence property). As a corollary, a theory is strongly stable (i.e. stable and strongly dependent) if and only if it is stable and every type has finite weight. Examples for strong theories are superstable theories and o-minimal theories.
It is not clear what weight (with respect to forking) should be in a non-simple theory, but as result of recent work by Shelah, Usvyatsov and Onshuus it is becoming increasingly clear what it means for a non-simple theory to have finite weight. I will call a theory strong if a certain constant from Shelah's book is countable. A simple theory is strong if and only if every type has finite weight. A theory is strongly dependent if and only if it is strong and dependent (i.e. does not have the independence property). As a corollary, a theory is strongly stable (i.e. stable and strongly dependent) if and only if it is stable and every type has finite weight. Examples for strong theories are superstable theories and o-minimal theories.
Alessandro Berarducci
"Cohomology of definable sets in o-minimal expansions of groups" (joint work with A. Fornasiero)
Abstract:
We prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated and invariant under elementary extensions and expansions of the language. We also study the cohomology of the intersection of a definable decreasing family of definably compact sets under the additional assumption that the o-minimal structure expands a field.
We prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated and invariant under elementary extensions and expansions of the language. We also study the cohomology of the intersection of a definable decreasing family of definably compact sets under the additional assumption that the o-minimal structure expands a field.
Ayse Berkman
"Curtis-Tits type of theorems in groups of finite Morley rank"
Abstract:
Theorem. Let G be a simple group of Lie type of rank ≥ 3 over any field with a fundamental root system Π. For each α ∈ Π, let Kα be the corresponding root subgroup of G and for α ≠ ± β ∈ Π, let Kα,β := [Kα,Kβ]. Then any amalgamated product of Kα,β's amalgamated along Kα's is a central extension of G.
In my talk, I shall present some applications to generic simple groups of finite Morley rank and black box groups of Lie type. Also I shall mention another possible application: An on-going project (with A. Borovik) which is aimed at classifying connected groups of finite Morley rank acting on an abelian group with a pseudoreflection subgroup. If G acts on an abelian group V , then a subgroup R of G is called a pseudoreflection subgroup if R is abelian, V = [V,R]⊗CV(R) and R acts transitively on the non-zero elements of [V,R].
Theorem. Let G be a simple group of Lie type of rank ≥ 3 over any field with a fundamental root system Π. For each α ∈ Π, let Kα be the corresponding root subgroup of G and for α ≠ ± β ∈ Π, let Kα,β := [Kα,Kβ]. Then any amalgamated product of Kα,β's amalgamated along Kα's is a central extension of G.
In my talk, I shall present some applications to generic simple groups of finite Morley rank and black box groups of Lie type. Also I shall mention another possible application: An on-going project (with A. Borovik) which is aimed at classifying connected groups of finite Morley rank acting on an abelian group with a pseudoreflection subgroup. If G acts on an abelian group V , then a subgroup R of G is called a pseudoreflection subgroup if R is abelian, V = [V,R]⊗CV(R) and R acts transitively on the non-zero elements of [V,R].
Jeff Burdges
"Our current picture of torsion in connected groups"
Abstract:
We'll discuss two themes concerning torsion in connected groups of finite Morley rank: In the absence of unipotent torsion, all torsion is toral, Sylow p- subgroups are conjugate, and a nontrivial Weyl group has an involution. On the other hand, unipotent torsion also tends to either produce a bad fields in positive characteristic or else directly influence the generic element of the group, at least in a minimal simple group.
We'll discuss two themes concerning torsion in connected groups of finite Morley rank: In the absence of unipotent torsion, all torsion is toral, Sylow p- subgroups are conjugate, and a nontrivial Weyl group has an involution. On the other hand, unipotent torsion also tends to either produce a bad fields in positive characteristic or else directly influence the generic element of the group, at least in a minimal simple group.
Paola D'Aquino
"Schanuel's conjecture and exponential fields"
Abstract:
We will present some consequences of Schanuel's cojecture in the real exponential field and in the complex exponential field, and more abstractly in the pseudoexponential fields introduced by Zilber (joint work with Giuseppina Terzo).
We will present some consequences of Schanuel's cojecture in the real exponential field and in the complex exponential field, and more abstractly in the pseudoexponential fields introduced by Zilber (joint work with Giuseppina Terzo).
Grigory Garkusha
"Reconstructing schemes from abelian categories"
Abstract:
I shall expose what we have done with Mike Prest on reconstructing affine and projective schemes from associated abelian categories. Using properties of the Ziegler and Zariski topologies one classifies finite torsion theories of modules resulting a bijection between the latter and triangulated categories of perfect complexes. Also, some further generalizations will be given.
I shall expose what we have done with Mike Prest on reconstructing affine and projective schemes from associated abelian categories. Using properties of the Ziegler and Zariski topologies one classifies finite torsion theories of modules resulting a bijection between the latter and triangulated categories of perfect complexes. Also, some further generalizations will be given.
Ivo Herzog
"Definable subspaces of finite-dimensional representations"
Abstract:
Let k be an algebraically closed field of characteristic 0, and denote by L a finite-dimensional semisimple Lie algebra over k. If φ(v) is a positive-primitive formula in the language of modules over the universal enveloping algebra U(L) then the subset φ(V) defined by φ in an L-representation V is a subspace of V, considered as a vector space over k. If φ(V) defines a sum of weight spaces of V, for every finite-dimensional representation V, then there is a positive-primitive formula φ- that defines an orthogonal complement of φ(V), for every finite-dimensional V. The talk will be devoted to a proof of this fact, as well as an explanation of the conjecture that if φ(V) defines the 1-simplex of weights whose boundary consists of the highest weight space, and one of its conjugates under a simple reflection, then φ(V) must be a minimal linearly bounded formula.
Let k be an algebraically closed field of characteristic 0, and denote by L a finite-dimensional semisimple Lie algebra over k. If φ(v) is a positive-primitive formula in the language of modules over the universal enveloping algebra U(L) then the subset φ(V) defined by φ in an L-representation V is a subspace of V, considered as a vector space over k. If φ(V) defines a sum of weight spaces of V, for every finite-dimensional representation V, then there is a positive-primitive formula φ- that defines an orthogonal complement of φ(V), for every finite-dimensional V. The talk will be devoted to a proof of this fact, as well as an explanation of the conjecture that if φ(V) defines the 1-simplex of weights whose boundary consists of the highest weight space, and one of its conjugates under a simple reflection, then φ(V) must be a minimal linearly bounded formula.
Gareth Jones
"Locally polynomially bounded structures"
Abstract:
I will define locally polynomially bounded structures and give examples. I then give a characterization of definable closure in these structures. This leads to a description of definable functions. I will finish with some applications. For example, every definable function is piecewise infinitely differentiable. This is joint work with Alex Wilkie.
I will define locally polynomially bounded structures and give examples. I then give a characterization of definable closure in these structures. This leads to a description of definable functions. I will finish with some applications. For example, every definable function is piecewise infinitely differentiable. This is joint work with Alex Wilkie.
Moshe Kamensky
"Model theory of fibre functors"
Abstract:
Let $C$ be the category of (finite dimensional) representations of an affine algebraic group $G$. The Tannakian formalism provides an answer to the following question: What additional structure is required on $C$ in order to reconstruct the group $G$? I will attempt to describe a model theoretic interpretation of this theorem, and describe a simple proof, using the notion of internality.
Let $C$ be the category of (finite dimensional) representations of an affine algebraic group $G$. The Tannakian formalism provides an answer to the following question: What additional structure is required on $C$ in order to reconstruct the group $G$? I will attempt to describe a model theoretic interpretation of this theorem, and describe a simple proof, using the notion of internality.
Angus MacIntyre
"Towards decidability of the theory of pseudofinite-dimensional representations of sl_2(C)"
Abstract:
It will be shown how the prototypical decision problems arising in trying to make Herzog's Selecta paper effective are directly connected to subtle problems about genus zero plane curves over the rationals, and to recent deep work of Bilu and Tichy. (Joint ongoing with Sonia L'Innocente).
It will be shown how the prototypical decision problems arising in trying to make Herzog's Selecta paper effective are directly connected to subtle problems about genus zero plane curves over the rationals, and to recent deep work of Bilu and Tichy. (Joint ongoing with Sonia L'Innocente).
Margarita Otero
"Torsion in definable groups"
Mike Prest
"From abelian groups to 2-categories: interpretations between additive categories"
Abstract:
The theme of interpretability of one category of additive structures in another has been a running theme through the development of the model theory of modules. I will discuss concrete examples, some very general theorems and point out that there's an awful lot we don't know.
The theme of interpretability of one category of additive structures in another has been a running theme through the development of the model theory of modules. I will discuss concrete examples, some very general theorems and point out that there's an awful lot we don't know.
Gena Puninski
"Towards the decidability of theories of modules over finite commutative rings"
Abstract:
We will give an account of a recent progress on when the theory of all modules over a finite commutative ring is decidable. It seems to be that in this case the decidability problem can be solved completely as soon as this question is answered for finite quotients of Gelfand-Ponomarev algebras over finite fields.
We will give an account of a recent progress on when the theory of all modules over a finite commutative ring is decidable. It seems to be that in this case the decidability problem can be solved completely as soon as this question is answered for finite quotients of Gelfand-Ponomarev algebras over finite fields.
Cédric Rivière
"Existentially closed difference-differential fields" (joint work in progress with Nicolas Guzy)
Abstract:
The main goal of this work is to determine whether the theory of fields equipped with finitely many derivations and an automorphism which pairwise commute has a model companion and then to study this latter. Despite that we are not able from now to give a positive or negative answer to this question, we can characterize the existentially closed models of this theory and obtain some algebraic results about them.
The main goal of this work is to determine whether the theory of fields equipped with finitely many derivations and an automorphism which pairwise commute has a model companion and then to study this latter. Despite that we are not able from now to give a positive or negative answer to this question, we can characterize the existentially closed models of this theory and obtain some algebraic results about them.
Philipp Rothmaler
"Inbetween pure projective and Mittag-Leffler"
Abstract:
I will report on joint work with Guil Asensio, Izurdiaga, and Torrecillas and discuss the situation when every finite tuple of a module freely realizes a positive primitive formula.
I will report on joint work with Guil Asensio, Izurdiaga, and Torrecillas and discuss the situation when every finite tuple of a module freely realizes a positive primitive formula.
