Week 07.10.2024 – 13.10.2024

These lectures will review recent developments surrounding the infrared sector of gravity in (3+1)-dimensional asymptotically flat spacetimes (AFS). In the first part of the course we will introduce soft theorems which govern the low-energy scattering of massless particles such as photons and gravitons. We will explain how these are related to classical observables known as memory effects and discuss their application to computing infrared-finite collider observables and gravitational waveforms. In the second part, we will introduce the notion of asymptotic or large-gauge symmetries and use it to derive the infinite-dimensional asymptotic symmetry algebra of (3+1)-dimensional AFS, also known as the BMS algebra. We will show that the conservation laws associated with these symmetries are equivalent to the Weinberg soft graviton theorem. Time-permitting, we will discuss some implications of these ideas for non-AdS holography.

In this joint work with Jakob Björnberg, Peter Mörters and Daniel Ueltschi, we introduce a disordered version of the CRP in which tables have different weights (or fitnesses). When a new customer enters the restaurant, they choose to open a new table with probability proportional to a parameter $\theta$, or they sit at an occupied table with probability proportional to the weight of this table times the number of customers already sitting at this table. We show that, in this model, in probability, a proportion converging to one of all customers sit at the largest table. We also show that this is not true almost surely, but prove instead that, almost surely, a proportion converging to one of all customers sit at one of the two largest tables.

Given a quasi projective family S of complex algebraic varieties, its Hodge locus is the locus of points of S where the corresponding fiber admits exceptional Hodge classes (conjecturally: exceptional algebraic cycles). In this talk I will survey the many recent advances in our understanding of such loci, both geometrically and arithmetically, as well as the remaining open questions.

Most modern deep networks are overparameterized: the number of training
examples, P, is much smaller than the number of parameters, N. According
to classical learning theory, these kinds of overparameterized networks
should overfit, but they tend not to: increasing both depth and width
almost always decreases generalization error. While we don't have a
complete theory of why this happens, we do have a theory of why it should
not be surprising. The theory draws heavily on linear regression, y=wx,
where it's well known that generalization error can be small for
overparameterized models if the true weight, w, lies in the subspace
spanned by eigenvectors with large eigenvalue, and the eigenvalue spectrum
is sufficiently nonuniform. Our main contribution is to calculate the
eigenvalues spectrum of the linearized dynamics of deep networks and show
that for large N and P the spectrum is approximately power law -- at any
point in learning.

We demonstrate that crossing symmetry of S-matrices can be violated in theories with non-invertible symmetries. Focusing on integrable flows to gapped phases in two dimensions, we show that S-matrices derived previously from the bootstrap approach are incompatible with non-invertible symmetries along the flow. We present consistent alternatives, which however violate crossing symmetry and obey modified rules dictated by fusion categories. We also show how these modified crossing rules can be used to constrain the space of amplitudes with a given categorical symmetry.

The Brunn-Minkowski inequality is a fundamental geometric inequality, closely related to the isoperimetric inequality. It states that for (open) sets $A$ and $B$ in $\mathbb{R}^d$, we have $|A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}$. Here $A+B=\{a+b: a \in A, b \in B\}$. Equality holds if and only if $A$ and $B$ are convex and homothetic sets (one is a dilation of the other) in $\mathbb{R}^d$. The stability of the Brunn-Minkowski inequality is the principle that if we are close to equality, then A and B must be close to being convex and homothetic. We prove a sharp stability result for the Brunn-Minkowski inequality, establishing the exact dependency between the two notions of closeness, thus concluding a long line of research on this problem. This is joint work with Alessio Figalli and Peter van Hintum.

The mod p Langlands correspondence for GL_2(Q_p) relates irreducible mod p representations of this group, to mod p representations of the absolute Galois group of Q_p. In contrast with the situation over the complex numbers, all attempts at extending this correspondence to more general p-adic Lie groups have been so far unsuccessful, due to the lack of a classification of irreducible representations. In this talk we will discuss what can be gained by shifting attention from the irreducible objects to the whole category of representations. The results are joint work in progress with Matthew Emerton and Toby Gee, and with Bao V. Le Hung.