Week 01.05.2023 – 07.05.2023

These three lectures will aim to provide a pedagogical introduction to the dynamics of N=2 supersymmetric gauge theory and the work of Seiberg and Witten. We will assume only basic knowledge of supersymmetry.

Please visit https://lonti.weebly.com/spring-2023-series.html for more information.

The Thirring Model is a covariant quantum field theory of interacting fermions, sharing many features in common with effective theories of two-dimensional electronic systems with linear dispersion such as graphene. For a small number of flavors and sufficiently strong interactions the ground state may be disrupted by condensation of particle-hole pairs leading to a quantum critical point. With no small dimensionless parameters in play in this regime the Thirring model is plausibly the simplest theory of fermions requiring a numerical solution. I will review what is currently known focussing on recent simulations employing Domain Wall Fermions (a formulation drawn from state-of-the-art QCD simulation), to faithfully capture the underlying symmetries at the critical point, focussing on the symmetry-breaking transition, the critical flavor number, and the anomalous scaling of the propagating fermion.

All vector autoregressive processes have an associated order p\DSEMIC conditional on observations at the preceding p time points, the variable at time t is conditionally independent of the earlier history. Learning the order of the process is therefore important for its characterisation and subsequent use in forecasting. For example, the order can serve as a point of comparison between different data sets and informs the decomposition of the time series into latent processes, which provides information about the underlying dynamics. It is common to assume that a vector autoregressive process is stationary. A vector autoregression is stable if and only if the roots of its characteristic equation lie outside the unit circle, which constrains the autoregressive coefficient matrices to lie in the stationary region. Unfortunately, the geometry of the stationary region can be very complicated, and specification of a prior distribution over this region is difficult. In this work, the autoregressive coefficients are mapped to a set of transformed partial autocorrelation matrices which are unconstrained, allowing for easier prior specification, routine computational inference, and meaningful interpretation of the magnitude of the elements in the matrix. The multiplicative gamma process is used to build a prior distribution for the unconstrained matrices, which encourages increasing shrinkage of the partial autocorrelation parameters as the lag increases. Posterior inference is performed using Hamiltonian Monte Carlo via the probabilistic programming language Stan. Samples from the posterior distribution of the order of the process use a truncation criterion which is motivated by classical theory on the sampling distribution of the partial autocorrelation function. The work is applied in a simulation study to investigate the agreement between the posterior distribution for the order of the process and its known value, with promising results. The model and inferential procedur...

Conditional on the Riemann hypothesis, Selberg showed in 1943 that the average size of the squares of differences between consecutive primes less than x is O(log(x)^4). Unconditional results still fall far short of this conjectured bound: Peck gave a bound of O(x^{0.25+epsilon}) in 1996 and to date this is the best known bound obtained using only methods from classical analytic number theory.


In this talk we discuss how sieve theory (in the form of Harman's sieve) can be combined with classical methods to improve bounds on the number of short intervals which contain no primes, thus improving the unconditional bound on the mean square gap between primes to O(x^{0.23+epsilon}).

The twisted holography program (1812.09257) describes a duality that is conjectured to be a topologically twisted version of the familiar AdS_5/CFT_4 correspondence. In the bulk this involves the topological B-model on \mathbb{C}^3. Including the D-branes on which the dual CFT lives, there is a backreaction that deforms the bulk geometry.
Recently, twisted holography has made contact with the celestial holography program (2201.02595) which led to the first concrete 4d/2d dual pair in asymptotically flat spacetime (2208.14233). I will try to discuss the role of branes and backreactions in this context and potentially some extensions that my supervisor David Skinner and me have worked out (to appear).