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

We give an alternative, statistical physics based proof of the Ω(d2^{-d}) lower bound for the maximum sphere packing density in dimension d by showing that a random configuration from the hard sphere model has this density in expectation. While the leading constant we achieve is not the best known, we do obtain additional geometric information: we prove a lower bound on the entropy density of sphere packings at this density, a measure of how plentiful such packings are. This is joint work with Felix Joos and Will Perkins.

CAT(0) spaces are defined as having triangles that are no fatter than Euclidean triangles, so it is no surprise that under special conditions  you find pieces of the Euclidean plane appearing in CAT(0) spaces. What is surprising though is how weak these special conditions seem to be. I will present some well known results of this phenomenon, along with detailed sketch proofs.

We give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over Z. This gives an analogue for K3 surfaces of Deligne's description of the category of ordinary abelian varieties over a finite field, and refines earlier work by N.O. Nygaard and J.-D. Yu. Two important ingredients in the proof are integral p-adic Hodge theory, and a description of CM points on Shimura stacks in terms of associated Galois representations. References: arXiv:1711.09225, arXiv:1707.01236.

I will start with briefly describing the HISH ( Holography Inspired Hadronic String) model and reviewing the fits of the spectra of mesons, baryons, glue-balls and exotic hadrons. 

I will present the determination of the hadron strong decay widths. The main decay mechanism is that of a string splitting into two strings. The corresponding total decay width behaves as $\Gamma =\frac{\pi}{2}A T L $ where T and L are the tension and length of the string and A is a dimensionless universal constant. The partial width of a given decay mode is given by $\Gamma_i/\Gamma = \Phi_i \exp(-2\pi C m_\text{sep}^2/T$ where $\Phi_i$ is a phase space factor, $m_\text{sep}$ is the mass of the "quark" and "antiquark" created at the splitting point, and C is adimensionless coefficient close to unity. I will show the fits of the theoretical results to experimental data for mesons and baryons. I will examine both the linearity in L and the exponential suppression factor. The linearity was found to agree with the data well for mesons but less for baryons. The extracted coefficient for mesons $A = 0.095\pm  0.01$  is indeed quite universal. The exponential suppression was applied to both strong and radiative decays. I will discuss the relation with string fragmentation and jet formation. I will extract the quark-diquark structure of baryons from their decays. A stringy mechanism for Zweig suppressed decays of quarkonia will be proposed and will be shown to reproduce the decay width of  states. The dependence of the width on spin and symmetry will be discussed. I will further apply this model to the decays of glueballs and exotic hadrons.

We study the spectrum of local operators living on a defect in a generic conformal field theory, and their coupling to the local bulk operators. We establish the existence of universal accumulation points in the spectrum at large s, s being the charge of the operators under rotations in the space transverse to the defect. Our main result is a formula that inverts the bulk to defect OPE, analogous to the Caron-Huot formula for the four-point function of CFTs without defects.

This is the continuation of a discussion of  Ezra Miller's paper https://arxiv.org/abs/1709.08155.

This is the first part of a discussion of  Ezra Miller's paper https://arxiv.org/abs/1709.08155.

I am interested in the moduli spaces of heterotic vacua. These are closely related to the moduli spaces of stable holomorphic bundles but in which the base and bundle vary simultaneously, together with additional constraints deriving from string theory. I will first summarise some pre-Brexit results we have derived. These include an explicit Kaehler metric and Kaehler potential for both the moduli space and its first cousin, the matter field space. I will secondly describe new, post-Brexit work in which these results are encased within an elegant geometry, which we call a universal heterotic geometry. Beyond compelling aesthetics, the framework is surprisingly useful giving both a concise derivation of our pre-Brexit results as well as some new results.