L3

Solving completely $x+y-z=1$ in unknowns taken from the group generated by a variable $t$ with $1-t$ over a finite field is not so easy as might be expected. We present a generalization to arbitrary linear varieties and finitely generated groups (keywords effective Mordell-Lang). We also mention applications to (a) solving equations like $u_n+v_m+w_l+f_k=0$ in $n,m,l,k$ for given recurrences $u,v,w,f$; and to (b) finding the smallest order of non-mixing of a given algebraic ${\bf Z}^s$-action. This is joint work with Harm Derksen.

[This is a joint seminar with OASIS]

A formulation of quantum mechanics in terms of symmetric monoidal categories

provides a logical foundation as well as a purely diagrammatic calculus for

it. This approach was initiated in 2004 in a joint paper with Samson

Abramsky (Ox). An important role is played by certain Frobenius comonoids,

abstract bases in short, which provide an abstract account both on classical

data and on quantum superposition. Dusko Pavlovic (Ox), Jamie Vicary (Ox)

and I showed that these abstract bases are indeed in 1-1 correspondence with

bases in the category of Hilbert spaces, linear maps, and the tensor

product. There is a close relation between these abstract bases and linear

logic. Joint work with Ross Duncan (Ox) shows how incompatible abstract

basis interact; the resulting structures provide a both logical and

diagrammatic account which is sufficiently expressive to describe any state

and operation of "standard" quantum theory, and solve standard problems in a

non-standard manner, either by diagrammatic rewrite or by automation.

But are there interesting non-standard models too, and what do these teach

us? In this talk we will survey the above discussed approach, present some

non-standard models, and discuss in how they provide new insights in quantum

non-locality, which arguably caused the most striking paradigm shift of any

discovery in physics during the previous century. The latter is joint work

with Bill Edwards (Ox) and Rob Spekkens (Perimeter Institute).

In 2006, a bad field was constructed (together with Baudisch, Hils and Wagner) collapsing Poizat's green fields. In this talk, we will not concentrate on the general methodology for collapsing specific structures, but more on a specific result in algebraic geometry, a weaker version of the Conjecture on Intersection with Tori (CIT). We will present a model theoretical proof of this result as well as discuss the possible generalizations to positive characteristic. We will try to make the talk  self-contained and aimed for an audience with a basic acquaintance with Model Theory.

We consider the symmetric group S_n of degree n and an algebraically

closed field F of prime characteristic p.

As is well-known, many representation theoretical objects of S_n

possess concrete combinatorial descriptions such as the simple

FS_n-modules through their parametrization by the p-regular partitions of n,

or the blocks of FS_n through their characterization in terms of p-cores

and p-weights. In contrast, though closely related to blocks and their

defect groups, the vertices of the simple FS_n-modules are rather poorly

understood. Currently one is far from knowing what these vertices look

like in general and whether they could be characterized combinatorially

as well.

In this talk I will refer to some theoretical and computational

approaches towards the determination of vertices of simple FS_n-modules.

Moreover, I will present some results concerning the vertices of

certain classes of simple FS_n-modules such as the ones labelled by

hook partitions or two part partitions, and will state a series of

general open questions and conjectures.

In 1989, Happel raised the following question: if the Hochschild cohomology

groups of a finite dimensional algebra vanish in high degrees, then does the

algebra have finite global dimension? This was answered negatively in a

paper by Buchweitz, Green, Madsen and Solberg. However, the Hochschild

homology version of Happel's question, a conjecture given by Han, is open.

We give a positive answer to this conjecture for local graded algebras,

Koszul algebras and cellular algebras. The proof uses Igusa's formula for

relating the Euler characteristic of relative cyclic homology to the graded

Cartan determinant. This is joint work with Dag Madsen.