L2

: With motivation from string compactifications, I will present work on the use of machine learning methods for the computation of geometric and topological properties of Calabi-Yau manifolds.

In this talk, I will describe some of the ways in which mathematical modelling contributed to the Covid-19 pandemic response in New Zealand. New Zealand adopted an elimination strategy at the beginning of the pandemic and used a combination of public health measures and border restrictions to keep incidence of Covid-19 low until high vaccination rates were achieved. The low or zero prevalence for first 18 months of the pandemic called for a different set of modelling tools compared to high-prevalence settings. It also generated some unique data that can give valuable insights into epidemiological characteristics and dynamics. As well as describing some of the modelling approaches used, I will reflect on the value modelling can add to decision making and some of the challenges and opportunities in working with stakeholders in government and public health.        

One of the major insights gained from holographic duality is the relation between the physics of black holes and quantum chaotic systems. This relation is made precise in the duality between two dimensional JT gravity and random matrix theory.  In this work, we generalize this to a duality between AdS3 gravity and a random ensemble of approximate CFT's.  The latter is described by a combined tensor and matrix model, describing the OPE coefficients and spectrum of a theory that approximately satisfies the bootstrap constraints.   We show that the Feynman diagrams of the random ensemble produce a sum over 3 manifolds that agrees with the partition function of 3d gravity.  A crucial element of this dictionary is the Virasoro TQFT, which defines the bulk gravitational path integral via the cutting and sewing relations of topological field theory.  Time permitting, we will explain why this TQFT has gravitational edge modes degrees of freedom whose entanglement gives rise to gravitational entropy.

I shall review some aspects of the relationship between scale and conformal invariance in 2-dimensional sigma models.  Then, I shall explain how such an investigation is related to the Perelman's ideas of proving the Poincare' conjecture.  Using this, I shall demonstrate that scale invariant sigma models  with B-field coupling and  compact target space  are conformally invariant. Several examples will also be presented that elucidate the results.  The talk is based on the arXiv paper 2404.19526.

In this talk I will discuss asymmetric orbifolds and will focus on their application to toroidal compactifications of heterotic string theory. I will consider theories in 6 and 4 dimensions with 16 supercharges and reduced rank. I will present a novel formalism, based on the Leech lattice, to construct ‘islands’ without vector multiplets.

Congruent numbers are natural numbers which are the area of right angled triangles with all rational sides. This talk will investigate conjectures for the density of congruent numbers up to some value $X$. One can phrase the question of whether a natural number is congruent in terms of whether an elliptic curve has non−zero rank. A theorem of Coates and Wiles connects this to whether the $L$-function associated to this elliptic curve vanishes at $1$. We will mention the conjecture of Keating on the asymptotic density based on random matrix considerations, and prove Tunnell’s Theorem, which connects the question of whether a natural number is a congruent number to counting integral points on varieties. Finally, I will hint at some future work I hope to do on non-vanishing of the $L$-functions.

In 1735, Euler observed that $ζ(2) = 1 + \frac{1}{2²} + \frac{1}{3²} + ⋯ = \frac{π²}{6}$. This is related to the famous identity $ζ(−1) "=" 1 + 2 + 3 + ⋯ "=" \frac{−1}{12}$. In general, values of the Riemann zeta function at positive even integers are equal to rational numbers multiplied by a power of $π$. The values at positive odd integers are much more mysterious; for example, Apéry proved that $ζ(3) = 1 + \frac{1}{2³} + \frac{1}{3³} + ⋯$ is irrational, but we still don't know if $ζ(5) = 1 + \frac{1}{2⁵} + \frac{1}{3⁵} + ⋯$ is rational or not! In this talk, we will explain the arithmetic significance of these values, their generalizations to Dirichlet/Dedekind L−functions, and to L−functions of elliptic curves. We will also present a new formula for $ζ(3) = 1 + \frac{1}{2³} + \frac{1}{3³} + ...$ in terms of higher algebraic cycles which came out of an ongoing project with Lambert A'Campo.

Khintchine's Theorem is one of the cornerstones in metric Diophantine approximation. The question of removing the monotonicity condition on the approximation function in Khintchine's Theorem led to the recently proved Duffin-Schaeffer conjecture. Gallagher showed an analogue of Khintchine's Theorem for multiplicative Diophantine approximation, again assuming monotonicity. In this talk, I will discuss my joint work with L. Frühwirth about a Duffin-Schaeffer version for Gallagher's Theorem. Furthermore, I will give a broader overview on various questions in metric Diophantine approximation and demonstrate the deep connection to both analytic and combinatorial number theory that is hidden inside the proof of these statements.