Week 03.11.2024 – 09.11.2024

Scattering amplitudes provide crucial theoretical input in collider and gravitational wave physics, and at the same time exhibit a remarkable mathematical structure. These lectures will introduce essential concepts and modern techniques exploiting this structure so as to efficiently compute amplitudes and their building blocks, Feynman integrals, in perturbation theory. We will start by decomposing gauge theory amplitudes into simpler pieces based on colour and helicity information. Focusing on tree level, we will then show how these may be determined from their analytic properties with the help of Britto-Cachazo-Feng-Witten recursion. Moving on to loop level, we will define the the class of polylogarithmic functions amplitudes and integrals often evaluate to, and explain their properties as well as relate them to the universal framework for predicting their singularities, known as the Landau equations. Time permitting, we will also summarise the state of the art in the calculation of the aforementioned singularities, and their intriguing relation to mathematical objects known as cluster algebras.

We are interested in the recurrence and transience of a branching random walk in Z^d indexed by a critical Galton-Watson tree conditioned to survive. When the environment is homogeneous, deterministic, and if the offspring distribution has a finite third moment, it is known to be recurrent for d at most 4, and transient for d larger than 4. In this talk we consider an environment made of random conductances, and we prove that, if the conductances satisfy suitable technical assumptions, the same result holds. The argument is based on the combination of a 0-1 law and a truncated second moment method, which only requires to have good estimates on the quenched Green's function of a (non-branching) random walk in random conductances.

Gromov used convex integration to prove that any closed
orientable three-manifold equipped with a volume form admits three
divergence-free vector fields which are linearly independent at every
point. We provide an alternative proof of this using geometric
properties of eigenspinors in three dimensions. In fact, our proof
shows that for any Riemannian metric, one can find three
divergence-free vector fields such that at every point they are
orthogonal and have the same non-zero length.

Zipf’s law describes the empirical size distribution of the components of many systems in natural and social sciences and humanities. We show, by solving a statistical model, that Zipf’s law co- occurs with the maximization of the diversity of the component sizes. The law ruling the increase of such diversity with the total dimension of the system is derived and its relation with Heaps’ law is discussed. As an example, we show that our analytical results compare very well with linguistics and population datasets

I will revisit the problem of defining an invariant notion of brane tension, analogous to the ADM mass, in a theory of gravity. I will propose two natural definitions, a gravitational and an inertial tension, in terms of asymptotic data akin to that of a Defect CFT.
I will illustrate these definitions with various examples, and present the evidence why for supersymmetric branes the two tensions must be equal.

An operator $T$ acting on a Hilbert space $H$ is said to be a two isometry if $${T^*}^2T^2 -2T^* T+I_H= 0,$$ where $T^*$ denote the adjoint of $T$ and $I_H$ is the identity operator. S. Richter proved that an analytic cyclic two-isometry can be viewed as a shift operator on certain Dirichlet spaces. In this talk, we will present some advances in the study of Dirichlet spaces. We will also discuss several natural open problems related to these spaces, focusing on the description of invariant subspaces. Additionally, we will examine estimates of the reproducing kernel and the concept of capacities associated with Dirichlet spaces.

The Frequentist, Assisted by Bayes (FAB) framework aims to construct confidence regions that leverage information about parameter values in the form of a prior distribution. FAB confidence regions (FAB-CRs) have smaller volume for values of the parameter that are likely under the prior, while maintaining exact frequentist coverage. This work introduces several methodological and theoretical contributions to the FAB framework. For Gaussian likelihoods, we show that the posterior mean of the parameter of interest is always contained in the FAB-CR. As such, the posterior mean constitutes a natural notion of FAB estimator to be reported alongside the FAB-CR. More generally, we show that for a likelihood in the natural exponential family, a transformation of the posterior mean of the natural parameter is always contained in the FAB-CR. For Gaussian likelihoods, we show that power law tails conditions on the marginal likelihood induce robust FAB-CRs, that are uniformly bounded and revert to the standard frequentist confidence intervals for extreme observations. We translate this result into practice by proposing a class of shrinkage priors for the FAB framework that satisfy this condition without sacrificing analytical tractability. The resulting FAB estimators are equal to prominent Bayesian shrinkage estimators, including the horseshoe estimator, thereby establishing insightful connections between robust FAB-CRs and Bayesian shrinkage methods.