L3

We will first review the construction of N =1

supersymmetric Yang-Mills theory in three dimensions. Then we will

construct a superloop space formulation for this super-Yang-Mills

theory in three dimensions.Thus, we will obtain expressions for loop

connection and loop curvature in this superloop space. We will also

show that curvature will vanish, unless there is a monopole in the

spacetime. We will also construct a quantity which will give the

monopole charge in this formalism. Finally, we will show how these

results hold even in case of deformed superspace.

Central extensions of a finite group G correspond to 2-cocycles on G, which give rise to an abelian cohomology group known as the Schur

multiplier of G. Recently, the Schur multiplier was defined in a much more

general setting of a monoidal category. I will explain how to twist algebras by categorical 2-cocycles and will mention the role of

such twists the theory of quantum groups. I will then describe an approach to twisting rational Cherednik algebras by cocycles,

and will discuss possible applications of this new construction to the representation theory of these algebras.

I shall give a non-technical survey of Pure Inductive Logic, a branch of Carnap's Inductive Logic which was

anticipated early on in that subject but has only recently begun to be developed as an area of Mathematical Logic. My intention

is to cover its origins and aims, and to pick out some of the key concepts which have emerged in the last decade or so.

Using the Borel-Schur algebra, we construct explicit characteristic-free resolutions for Weyl modules for the general linear group. These resolutions provide an answer to the problem, posed in the 80's by A. Akin and D. A. Buchsbaum, of constructing finite explicit and universal resolutions of Weyl modules by direct sums of divided powers. Next we apply the Schur functor to these resolutions and prove a conjecture of Boltje and Hartmann on resolutions of co-Specht modules. This is joint work with I. Yudin.

Masser recently proved a bound on the number of rational points of bounded height on the graph of the zeta function restricted to the interval [2,3]. Masser's bound substantially improves on bounds obtained by Bombieri-Pila-Wilkie. I'll discuss some results obtained in joint work with Gareth Boxall in which we prove bounds only slightly weaker than Masser's for several more natural analytic functions.

The study of difference algebraic geometry stems from the efforts of Macintyre and Hrushovski to

count the number of solutions to difference polynomial equations over fields with powers of Frobenius.

We propose a notion of multiplicity in the context of difference algebraic schemes and prove a first principle

of preservation of multiplicity. We shall also discuss how to formulate a suitable intersection theory of difference schemes.

The Outer Model Programme investigates L-like forcing  extensions of the universe, where we say that a model of Set Theory  is L-like if it satisfies properties of Goedel's constructible universe of sets L. I will introduce the Outer Model Programme, talk  about its history, motivations, recent results and applications. I  will be presenting joint work with Sy Friedman and Philipp Luecke.