L3

The class of finite dimensional algebras over an algebraically closed field K

may be divided into two disjoint subclasses (tame and wild dichotomy).

One class

consists of the tame algebras for which the indecomposable modules

occur, in each dimension d, in a finite number of discrete and a

finite number of one-parameter families. The second class is formed by

the wild algebras whose representation theory comprises the

representation theories of all finite dimensional algebras over K.

Hence, the classification of the finite dimensional modules is

feasible only for the tame algebras. Frequently, applying deformations

and covering techniques, we may reduce the study of modules over tame

algebras to that for the corresponding simply connected tame algebras.

We shall discuss the problem concerning connection between the

tameness of simply connected algebras and the weak nonnegativity of

the associated Tits quadratic forms, raised in 1975 by Sheila Brenner.

This is based on joint work with Dave Jorgensen. Given a Gorenstein algebra,

one can define Tate-Hochschild cohomology groups. These are defined for all

degrees, non-negative as well as negative, and they agree with the usual

Hochschild cohomology groups for all degrees larger than the injective

dimension of the algebra. We prove certain duality theorems relating the

cohomology groups in positive degree to those in negative degree, in the

case where the algebra is Frobenius (for example symmetric). We explicitly

compute all Tate-Hochschild cohomology groups for certain classes of

Frobenius algebras, namely, certain quantum complete intersections.

(Joint work with P. Corvaja and D.

Masser.)

The topic of the talk arises from the

Manin-Mumford conjecture and its extensions, where we shall

focus on the case of (complex connected) commutative

algebraic groups $G$ of dimension $2$. The `Manin-Mumford'

context in these cases predicts finiteness for the set of

torsion points in an algebraic curve inside $G$, unless the

curve is of `special' type, i.e. a translate of an algebraic

subgroup of $G$.

In the talk we shall consider not merely the set of torsion

points, but its topological closure in $G$ (which turns out

to be also the maximal compact subgroup). In the case of

abelian varieties this closure is the whole space, but this is

not so for other $G$; actually, we shall prove that in certain

cases (where a natural dimensional condition is fulfilled) the

intersection of this larger set with a non-special curve

must still be a finite set.

We shall conclude by stating in brief some extensions of

this problem to higher dimensions.

We give an exposition of some results from matroid theory which characterise the finite pregeometries arising from Hrushovski's predimension construction as the strict gammoids: a class of matroids studied in the early 1970's which arise from directed graphs. As a corollary, we observe that a finite pregeometry which satisfies Hrushovski's flatness condition arises from a predimension. We also discuss the isomorphism types of the pregeometries of countable, saturated strongly minimal structures in Hrushovski's 1993 paper and answer some open questions from there. This last part is joint work with Marco Ferreira, and extends results in his UEA PhD thesis.

This is a joint work with Emmanuel Ullmo.

This work is motivated by J.Pila's strategy to prove the Andre-Oort conjecture. One ingredient in the strategy is the following

conjecture:

Let S be a Shimura variety uniformised by a symmetric space X.

Let V be an algebraic subvariety of S. Maximal algebraic subvarieties of the preimage of V in X are precisely the

components of the preimages of weakly special subvarieties contained in V.

We will explain the proof of this conjecture in the case where S is compact.

Let $x$ be a CM point in the moduli space $\mathcal{A}_g(\mathbb{C})$ of principally

polarized complex abelian varieties of genus $g$, corresponding to an

Abelian variety $A$ with complex multiplication by a ring $R$. Edixhoven

conjectured that the size of the Galois orbit of x should grow at least

like a power of the discriminant ${\rm Disc}(R)$ of $R$. For $g=1$, this reduces to the

classical Brauer-Siegel theorem. A positive answer to this conjecture

would be very useful in proving the Andr\'e-Oort conjecture unconditionally.

We will present a proof of the conjectured lower bounds in some special

cases, including $g\le 6$. Along the way we derive transfer principles for

torsion in class groups of different fields which may be interesting in

their own right.

(Joint with Ronnie Nagloo.) I investigate algebraic relations between sets of solutions (and their derivatives) of the "generic" Painlev\'e equations I-VI, proving a somewhat weaker version of ``there are NO algebraic relations".

After a short introduction to homogeneous relational structures (structures such that all local symmetries are global), I will discuss some different topics relating homogeneity to homomorphisms: a family of notions of 'homomorphism-homogeneity' that generalise homogeneity; generic endomorphisms of homogeneous structures; and constraint satisfaction problems.