(Oxford University)

Oxford Mathematics Public Lectures

Can Mathematics Understand the Brain?' - Alain Goriely

The human brain is the object of the ultimate intellectual egocentrism. It is also a source of endless scientific problems and an organ of such complexity that it is not clear that a mathematical approach is even possible, despite many attempts. 

In this talk Alain will use the brain to showcase how applied mathematics thrives on such challenges. Through mathematical modelling, we will see how we can gain insight into how the brain acquires its convoluted shape and what happens during trauma. We will also consider the dramatic but fascinating progression of neuro-degenerative diseases, and, eventually, hope to learn a bit about who we are before it is too late. 

Alain Goriely is Professor of Mathematical Modelling, University of Oxford and author of 'Applied Mathematics: A Very Short Introduction.'

March 8th, 5.15 pm-6.15pm, Mathematical Institute, Oxford

Please email @email to register

In the game 'Set', players compete to pick out groups of three cards sharing common attributes. But how many cards must be dealt before such a group must appear? 
This is an example of a "cap set problem", a problem in Ramsey theory: how big can a set of objects get before some form of order appears? We will translate the cap set problem into a problem of geometry over finite fields, discussing the current best upper bounds and running through an elementary proof. We will also (very) briefly discuss one or two implications of the cap set problem over F_3 to other questions in Ramsey theory and computational complexity
 

Let A be an abelian variety over the function field K of a curve over a finite field of characteristic p>0. We shall show that the group A(K^{p^{-\infty}}) is finitely generated, unless severe restrictions are put on the geometry of A. In particular, we shall show that if A is ordinary and has a point of bad reduction then A(K^{p^{-\infty}}) is finitely generated. This result can be used to give partial answers to questions of Scanlon, Ziegler, Esnault, Voloch and Poonen.

Moduli spaces attempt to classify all mathematical objects of a particular type, for example algebraic curves or vector bundles, and record how they 'vary in families'. Often they are constructed using Geometric Invariant Theory (GIT) as a quotient of a parameter space by a group action. A common theme is that in order to have a nice (eg Hausdorff) space one must restrict one's attention to a suitable subclass of 'stable' objects, in effect leaving certain badly behaved objects out of the classification. Assuming no prior familiarity, I will elucidate the structure of instability in GIT, and explain how recent progress in non-reductive GIT allows one to construct moduli spaces for these so-called 'unstable' objects. The particular focus will be on the application of this principle to the GIT construction of the moduli space of stable curves, leading to moduli spaces of curves of fixed singularity type.
 

Manifolds with ordinary boundary/corners have found their presence in differential geometry and PDEs: they form Man^b or Man^c category; and for boundary value problems, they are nice objects to work on. Manifolds with analytical corners -- a-corners for short -- form a larger category Man^{ac} which contains Man^c, and they can in some sense be viewed as manifolds with boundary at infinity.
In this talk I'll walk you through the definition of manifolds with corners and a-corners, and give some examples to illustrate how the new definition will help.

Floer (co)homology, invariant which recovers periodic orbits of a Hamiltonian system, is the central topic of symplecic topology at present. Its analogue for open symplecic manifolds is called symplectic (co)homology. Our goal is to compute this invariant for big family of spaces called Nakajima's Quiver Varieties, spaces obtained as hyperkahler quotients of representation spaces of quivers.