Weekly Bulletin

n 2005, Britto, Cachazo, Feng and Witten (BCFW) gave a recurrence relation for computing scattering amplitudes in N = 4 super Yang–Mills theory. In 2013, Golden, Goncharov, Spradlin, Vergu and Volovich discovered in the scattering amplitudes of this theory a cluster algebraic structure. The amplituhedron A(n,k,m) is a geometric object, introduced by Arkani-Hamed and Trnka in 2013, conjectured to encode scattering amplitudes in planar N = 4 super Yang–Mills. In this talk, I will discuss the amplituhedron and how both the aforementioned BCFW recursion and cluster algebra structures are manifested in it. Based on joint works with Even-Zohar, Parisi, Sherman-Bennett, Tessler and Williams.

The relationship between computational models and dynamical systems has fascinated mathematicians and computer scientists since the earliest formulations of computation. In recent years, this connection has attracted renewed interest, driven by groundbreaking results showing that Turing machines can be simulated by the flow lines of solutions to the Euler and Navier–Stokes equations. In this talk, we propose a novel viewpoint on computability within dynamical systems, inspired by ideas from Topological Quantum Field Theory. We prove that any computable function can be realized as the flow of a volume-preserving vector field on a smooth bordism. Beyond providing a new computational model, this perspective reveals deep connections between the topological features of the flow, the existence of compatible contact-type geometric structures on the bordism, and the computational complexity of the function. Joint work with E. Miranda and D. Peralta-Salas

We will start talking about how to use neural networks to find unstable self-similar profiles in fluids PDE. These profiles are numerical, but are accurate up to machine precision We will then spend most of our time explaining how to use computer-assisted proofs (CAP) to rigorously prove singularity formation around those unstable solutions. The main ingredients are CAP operator norm bounds to reduce the linear problem to a finite-dimensional one, and CAP bounds on the linear propagator applied to the remaining finitely many modes. Lastly, we will briefly talk about recent work constructing the first counterexample of the Schiffer problem on the half-sphere, which is also the first contractible counterexample in any geometry Joint works with Google Deepmind, Yongji Wang, Ching-Yao Lai, Javier Gómez-Serrano, Tristan Buckmaster and Antonio J. Fernández

A Kakeya set is a compact subset of R^n that contains a unit line segment pointing in every direction. Kakeya set conjecture asserts that every Kakeya set has Minkowski and Hausdorff dimension n. We prove this conjecture in R^3 as a consequence of a more general statement about union of tubes. This is joint work with Josh Zahl.

<p>In 2019, I introduced the concept of self-similarity as a tool towards the proof of Furstenberg's slicing conjecture. I will discuss an extension of the theory to higher dimensions, obtained in joint work with E. Corso, and the higher-rank Furstenberg slicing estimate that follows.</p>

The group AUT(A^n) of polynomial automorphisms of the space is not a simple group, as it contains the closed subgroup SAUT(A^n) of Jacobian 1. The group SAUT(A^2) is known to be abstractly not simple by a result of Danilov of 1974. The group SAUT(A^n) was however claimed to be topologically simple (no non-trivial closed normal subgroups) by Shafarevich in 1981. We will here prove this claim in dimension n=2 and give some ideas about the situation in higher dimension. This is over an infinite field, the case of finite fields being dramatically different.

Frequency-domain models of wave propagation and vibration phenomena often give rise to large-scale linear systems and (potentially nonlinear) eigenvalue problems, whose solution is essential for tasks such as resonance analysis, design optimization, or uncertainty quantification. High-fidelity simulations, however, are typically too costly to be used directly in such applications. Model reduction and surrogate modeling thus play a central role. In this talk, we focus on rational approximation techniques that able to adaptively target the spectral features of interest. Building on ideas from reduced basis methods, we describe the "minimal rational interpolation" strategy, which adaptively selects frequency samples to construct accurate surrogates without oversampling. This allows us to approximate resonances efficiently and reliably. We further discuss how the same perspective can be leveraged to address parametric eigenvalue problems, where additional parameters are introduced to model, e.g., material or geometry variations or uncertainties. In such problems, eigenvalue manifolds may exhibit crossings, bifurcations, or even discontinuities as parameters vary. Our approach combines adaptive rational approximation with contour-integration-based eigensolvers to synthesize these manifolds in a robust way. This talk will cover joint work with Alessandro Borghi (TU Berlin), Ralf Hiptmair (ETH Zurich), and Ilaria Perugia (U Vienna).

With an example borrowed from the research of Professor Hugo Duminil-Copin, we will illustrate how counting can shed light on the behavior of complex physical systems, while simultaneously revealing the need to sometimes go beyond what numbers tell us in order to unveil all the mysteries of the world around us.

We provide a theoretical framework for a wide class of generalized posteriors that can be viewed as the natural Bayesian posterior counterpart of the class of M-estimators in the frequentist world. We call the members of this class M-posteriors and show that they are asymptotically normally distributed under mild conditions on the M-estimation loss and the prior. In particular, an M-posterior contracts in probability around a normal distribution centered at an M-estimator, showing frequentist consistency and suggesting some degree of robustness depending on the reference M-estimator. We formalize the robustness properties of the M-posteriors by a new characterization of the posterior influence function and a novel definition of breakdown point adapted for posterior distributions. We illustrate the wide applicability of our theory in various popular models and illustrate their empirical relevance in some numerical examples. This is joint work with Juraj Marusic and Cindy Rush

We discuss recent developments around Hilbert's Sixth Problem about the passage from kinetic models to fluid equations. We employ the technique of slow spectral closure to rigorously establish the existence of hydrodynamic manifolds in the linear regime and derive new non-local fluid equations for rarefied flows independent of Knudsen number. We show the divergence of the Chapman--Enskog series for an explicit example and apply machine learning to learn the optimal hydrodynamic closure from DSMC data. The new dynamically optimal constitutive laws are applied to a rarefied flow problem. We then show how this geometric rational from dynamical systems theory can be applied analogously to the CEV model in finance and demonstrate how arbitrage opportunities can be related to bifurcations in market elasticity on the Fokker—Planck level.

Phase transitions mark sudden, dramatic changes in the behavior of complex systems. To explore the mathematics behind these phenomena, we will focus on two of the most fundamental models in statistical physics: the Ising model and percolation. Originally introduced to capture ferromagnetism and the flow through porous media, these models have since evolved into a rich mathematical framework and a versatile tool for understanding abrupt shifts across diverse settings. In this talk, I will survey key advances in their study and highlight the broader insights they offer into the nature of phase transitions in statistical physics.

In this talk, we will explore the rich interplay between two-dimensional critical percolation models and the six-vertex model, a classical integrable random height model. By leveraging the remarkable symmetries and emergent structures that arise in the large-scale behavior of these systems, we will discuss how the so-called Coulomb Gas Formalism may be placed on rigorous mathematical foundations in this context. This perspective opens new pathways toward a deeper mathematical understanding of the phase transition of these models. The presentation is intended to be accessible to a broad mathematical audience.

We discuss a Plateau problem based on capillarity theory in which soap films are described as sets with small volume v that satisfy a spanning condition. Existence and interior regularity are understood for fixed v>0 and for the limiting Plateau problem, and so several questions arise regarding boundary regularity and the nature of the convergence to minimal surfaces as v approaches zero. We present ongoing joint work in these directions with Francesco Maggi (UT Austin) and Daniel Restrepo (Johns Hopkins).

Data attribution methods aim to quantify how training examples shape model predictions, supporting applications in interpretability, unlearning, and robustness. The dominant tools in practice are influence functions (IF) and Newton step (NS) approximations, yet their theoretical guarantees and practical accuracy have remained poorly understood. In this talk, I will present new analytic techniques that uncover the scaling laws of the approximation error of IF and NS. Our results improve on prior analyses both by establishing asymptotically sharper bounds and by avoiding dependence on the global strong convexity parameter, which is often prohibitively small in practice. These insights not only explain long-standing empirical observations—such as why and when NS is more accurate than IF—but also guide the design of new methods. As an application, I will present rescaled influence functions (RIF), a simple, drop-in replacement for IF that matches the efficiency of IF while achieving the accuracy of NS. I will discuss both theoretical advances and empirical results on real-world datasets. Together, these contributions provide a first principled understanding of data attribution methods and demonstrate how to turn this understanding into more reliable tools.

In this talk, I will explain the construction of Lagrangian classes for perverse sheaves in cohomological Donaldson-Thomas theory, whose existence was conjectured by Joyce. The two key ingredients are a relative version of the DT perverse sheaves and a hyperbolic version of the dimensional reduction theorem. As a special case, we recover Borisov-Joyce/Oh-Thomas virtual classes in DT4 theory. As an application, I will explain how to construct cohomological field theories for gauged linear sigma models. This is joint work in progress with Adeel Khan, Tasuki Kinjo, and Pavel Safronov.