Week 02.03.2025 – 08.03.2025

Quantum Chromodynamics (QCD) has been a profound source of inspiration for theoretical physics, driving the development of key concepts such as string theory, effective field theories, instantons, anomalies, and lattice gauge theories. In these lectures, I will explore two distinct regimes of QCD - its infrared (IR) and ultraviolet (UV) limits - and the theoretical tools used to study them.

In the IR regime, where perturbative techniques break down, Effective Field Theories (EFTs) provide a powerful framework. I will introduce the pion EFT as a tool to study non-linearly realized symmetries and soft theorems. In the UV regime, where QCD becomes amenable to perturbative analysis, I will discuss the Operator Product Expansion and renormalization group equations, focusing on their application to deep inelastic scattering, a cornerstone in the discovery of quarks and gluons.

These two regimes illustrate the richness of QCD and its pivotal role in shaping our understanding of fundamental physics.

The study of Ramsey properties of the binomial random graph G_{n,p} was initiated in the 80s by Frankl & Rödl and Łuczak, Ruciński & Voigt. In this area we are often interested in establishing the threshold function f(n) that governs G_{n,p} having a particular Ramsey-like property P or not, i.e. if p is sufficiently larger than f(n) then G_{n,p} asymptotically almost surely (a.a.s) has P, and if p is sufficiently smaller than f(n) then G_{n,p} a.a.s. does not have P.

In this talk, we will explore how combinatorial methods have been used to establish these thresholds for diagonal, off-diagonal and anti-Ramsey properties, including in recent joint work together with Natalie Behague, Candy Bowtell, Robert Hancock, Shoham Letzter and Natasha Morrison.

We introduce a non-probabilistic, path-by-path framework for continuous-time, path-dependent portfolio allocation. Extending the self-financing concept recently introduced in Chiu & Cont (2023), we characterize self-financing portfolio allocation strategies through a path-dependent PDE and provide explicit solutions for the portfolio value in generic markets, including price paths that are not necessarily continuous or exhibit variation of any order.

As an application, we extend an aggregating algorithm of Vovk and the universal algorithm of Cover to continuous-time meta-algorithms that combine multiple strategies into a single strategy, respectively tracking the best individual and the best convex combination of strategies. This work extends Cover’s theorem to continuous-time without probability.

In many situations in physics, the path of light is determined not only by spatial geometry but also by an underlying local density (e.g., mass concentration in general relativity, refractive index in optics). Consider a scenario where a Riemannian manifold in Euclidean space is shaped by a density function, with only a finite sample of points available. How can we infer the original metric and determine the manifold’s topology? This talk introduces a density-based method for estimating topological features from data in high-dimensional Euclidean spaces, assuming a manifold structure. The key to our approach lies in the Fermat distance, a sample metric that robustly infers the deformed Riemannian metric. Theoretical convergence results and implications in the homology inference of the manifold will be presented. Additionally, I will show practical applications in time series analysis with examples from real-world data. This talk is based on the article: X. Fernandez, E. Borghini, G. Mindlin, and P. Groisman. “Intrinsic Persistent Homology via Density-Based Metric Learning.” Journal of Machine Learning Research 24 (2023) 1-42.

Propagation and generation of "chaos" is an important ingredient for
rigorous control of applicability of kinetic theory, in general. Chaos
is here understood as sufficient statistical independence of random
variables related to the "kinetic" observables of the system. Cumulant
hierarchy of these random variables thus often gives a way of
controlling the evolution and degree of such independence, i.e., the
degree of "chaos" in the system. In this talk, we will consider two,
qualitatively different, example cases for which kinetic theory is
believed to be applicable: the stochastic Kac model with random velocity
exchange and the discrete nonlinear Schrodinger evolution (DNLS) with
suitable random, spatially homogeneous initial data. In both cases, we
set up suitable random variables and propose methods to control the
evolution of their cumulant hierarchies. The talk is based on joint
works with Sakari Pirnes and Aleksis Vuoksenmaa, and earlier works with
Matteo Marcozzi, Alessia Nota, and Herbert Spohn.

About fifty years ago, Gibbons and Hawking argued that the Euclidean gravitational path integral with suitable boundary conditions can be interpreted as a grand-canonical partition function. Classical gravitational solutions, including black holes, arise as saddles of this path integral, and from the saddle-point action one can extract the black hole entropy. In the talk, I will discuss some recent developments of these ideas. Working in five dimensions, we will see how imposing supersymmetric boundary conditions converts the partition function into an index. Then we will construct a class of saddles of this index which interpolates between supersymmetric black holes and horizonless microstate geometries. I will discuss how the saddle point action can be computed via equivariant localization. Finally, I will comment on the relevance of these findings for black hole microstate counting and holography.

Eigenfunctions of the Laplacian cannot vanish on a set of positive measure. Quantitative versions of this unique continuation are well-known on fixed Riemannian manifolds: the L² norm of an eigenfunction is bounded by its L² norm on a set of positive measure times a constant which grows exponentially with the frequency. This growing rate is sharp and reflects in observability properties for the heat equation.

In this talk, I will present recent results, in collaboration with M. Rouveyrol (Orsay) about non-compact hyperbolic surfaces. Quantitative unique continuation, and observability of the heat equation, hold under a necessary and sufficient condition of thickness of the observed set: it must intersect every large enough metric ball with a mass bounded from below, proportionally to the mass of the ball itself. The proof crucially uses the Logunov-Mallinikova estimates.