Oxford

I will present a new formula for diffusion processes which involving

Ito integral for the transition probability functions. The nature of

the formula I discovered is very close to the Kac formula, but its

form is similar to the Cameron-Martin formula. In some sense it is the

Cameron-Martin formula for pinned diffusions.

Marstrand's Theorem is a one of the classic results of Geometric Measure Theory, amongst other things it says that fractal measures do not have density. All methods of proof have used symmetry properties of Euclidean space in an essential way. We will present an elementary history of the subject and state a version of Marstrand's theorem which holds for spaces whose unit ball is a polytope.

While the classification of crystals made up by just one atom per cell is well-known and understood (Bravais lattices), that for more complex structures is not. We present a geometric way classifying these crystals and an arithmetic one, the latter introduced in solid mechanics only recently. The two approaches are then compared. Our main result states that they are actually equivalent; this way a geometric interpretation of the arithmetic criterion in given. These results are useful for the kinematic description of solid-solid phase transitions. Finally we will reformulate the arithmetic point of view in terms of group cohomology, giving an intrinsic view and showing interesting features.

The first seminar will be given with the new students in

mind. It will begin with a brief overview of quantifier elimination and its

relation to the back-and-forth property.I shall then discuss complexity issues

with particular reference to algebraically closed fields.In particular,how much

does the height and degree of polynomials in a formula increase when a

quantifier is eliminated? The precise answer here gave rise to the definition

of a `generic' transcendental entire function,which will also be

discussed.