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.

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.

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.

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.

In this joint work with Jakob Bj��rnberg, Peter M��rters 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.

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.

Fuzzy sphere regularization has proven to be a useful tool for deriving CFT data in a number of theories, most notably the 3d Ising model. Previous studies of these fuzzy sphere models relied explicitly upon matching with data known from other methods such as the conformal bootstrap, and were thus limited to already well-studied theories. Following two recent articles 2409.02998 and 2409.08257 I will detail how it is possible to explore the emergence of conformality within the constraints of the fuzzy sphere models themselves by explicitly realizing the generators of the conformal algebra as fuzzy sphere operators, and discuss their results for fuzzy sphere Ising model.

A compact subset $K$ of the complex plane is said to be $W^{1,p}$ removable if any continuous real-valued function $f$ on $\mathbb{C}$ which is in the Sobolev space $W^{1,p}(\mathbb{C} \setminus K)$ is automatically also in the Sobolev space $W^{1,p}(\mathbb{C})$. This property is true for all values of $p$ for points and line segments, but false for sets with non-empty interior, and in general there is no simple condition to determine whether a given set is removable or not. In certain cases, there is a link between removability and how ���rough��� the set $K$ is. In particular, if $K$ is the graph of a function, it is known that its H��lder continuity is related to its removability. We will present new results on the removability of the graph of a one-dimensional Brownian motion on an interval and show that it is almost surely not $W^{1,p}$ removable for finite $p$, but is removable for $p = \infty$. This talk is based on joint work with Jason Miller.

How are bulk strings related to boundary Feynman diagrams? I will give an overview of my work with Rajesh Gopakumar on deriving the closed string dual to the simplest possible gauge theory, a Hermitian matrix integral. Working in the conventional 't Hooft limit, we extract topological string theories which replace the minimal string away from the double-scaling limit. We show how to exactly reconstruct both the closed string worldsheet and its embedding into the emergent target space, purely from the matrix Feynman diagrams. I'll close by embedding our results in the broader context of AdS/CFT.