Semi-plenary Talks

As AI models are trained on ever-expanding datasets, the ability to remove the influence of specific data from trained models has become essential for privacy protection and regulatory compliance. Unlearning addresses this challenge by selectively removing parametric knowledge from the trained models without retraining from scratch, which is critical for resource-intensive models such as Large Language Models (LLMs). However, existing unlearning methods often severely degrade model performance by removing more information than necessary when attempting to “forget” specific data. We introduce a mathematical framework based on information-theoretic regularization that can accommodate different types of machine unlearning, such as feature unlearning and data point unlearning. Our theoretical analysis reveals intriguing connections between machine unlearning, information theory, optimal transport, and extremal sigma algebras. For LLMs, we propose Forgetting-MarI, an unlearning framework that provably removes only the additional (marginal) information contributed by the data to be unlearned, while preserving the information supported by the data to be retained. Extensive experiments confirm that our approach outperforms current state-of-the-art unlearning methods, delivering reliable forgetting and better preserved general model performance across diverse benchmarks. This advancement represents an important step toward making AI systems more controllable and compliant with privacy and copyright regulations without compromising their effectiveness. We will also discuss applications in machine learning driven scientific discovery. This is joint work with Shizhou Xu, Yuan Ni, and Stefan Broecker.

I will first present a positivity preserving splitting scheme for some nonlinear stochastic heat equations driven by multiplicative space-time white noise in dimension 1, which converge with strong rate 1/4.
Second, I will show how similar techniques can be used to design domain preserving schemes for finite dimensional systems of stochastic differential equations: I will present a general class of integrators of strong order 1/2 and weak order 1. I will also present an integrator of strong order 1 for systems driven by one-dimensional noise, in the spirit of the Milstein scheme.
Finally, I will show how splitting domain preserving schemes can be obtained for a generalized version of the stochastic Nagumo equation.
This talk is based on joint works with David Cohen, Gijs Custers and Johan Ulander.

Many methods to describe systems rely on the assumption that unobserved variables in the system are mutually independent. Examples include independent component analysis and causal discovery procedures such as LiNGAM. Independence is a strong assumption. We show that it can be replaced by weaker restrictions, which say that the mean of a variable (rather than the whole distribution) is unchanged by conditioning on another variable. We demonstrate the identifiability of component analysis and causal discovery under these weaker mean-independence restrictions.
Just as independence-based procedures correspond to low-rank tensor decomposition problems, mean-independence is a different structured tensor decomposition. Parameter estimation methods in the mean-independent models compute eigenvectors of tensors and test the vanishing of polynomials. Based on joint work with Geert Mesters, Alvaro Ribot, and Piotr Zwiernik.

Supersingular elliptic curve ell-isogeny graphs over finite fields offer a setting for a number of quantum-resistant cryptographic protocols. The security analysis of these schemes typically assumes that these graphs behave randomly. Motivated by this assertion, we explore structural properties of these graphs. We detail the behaviour, governed by congruence conditions on p, of the ell-isogeny graph over the field F_p when passing to the spine, i.e. the subgraph induced by the F_p-vertices in the full ell-isogeny graph. We describe the diameter of the spine and offer numerical data on the number of vertices, over both F_p and its algebraic closure, in the center of the ell-isogeny graph. Our plots of these counts exhibit a wave-shaped pattern which supports the assertion that centers of supersingular ell-isogeny graphs exhibit the same behavior as those of random (ell+1)-regular graphs. This is joint work with Sarah Arpin and Taha Hedayat.

I will survey recent progress in the understanding of the optimization dynamics for important tasks for Machine Learning, or high dimensional statistics. We will see how these very high-dimensional dynamics are in fact ruled by the so-called “effective dynamics” of much lower dimensional systems. This dynamical dimension reduction is related to the BBP spectral transition of Random Matrix Theory, appearing dynamically along the algorithm path. I will illustrate these phenomena in multi-spike Tensor PCA, XOR, and classification of Gaussian mixtures with multi-layer neural nets.
This talk is based on joint works with Reza Gheissari (Northwestern), Jiaoyang Huang (Wharton), Aukosh Jagannath (Waterloo), and on joint works with Cedric Gerbelot (ENS Lyon) and Vanessa Piccolo (EPFL).

Geometric numerical integration is designed to preserve relevant geometric structures of a dynamical system. For Hamiltonian or Lagrangian systems, notions such as energy or symmetry preservation, symplecticity, and Poisson structures immediately come to mind. However, in most realistic situations, systems are subject to external forces which, in general, destroy the geometric invariants of the associated free system.
In the first part of the talk, we will explain how techniques from discrete mechanics for free systems [2,4] can be adapted to obtain qualitative properties of forced systems through variational integration [1,4]. In particular, we will discuss variational error analysis and the evolution of the energy for forced systems [5,6].
Finally, there are important classes of externally forced systems that intrinsically possess geometric properties worth preserving. A particularly relevant example is given by mechanical systems with double-bracket dissipation, especially systems governed by Euler–Poincaré equations with this special type of dissipation. In this setting, the underlying coadjoint orbit on which the system evolves remains invariant, while the energy decreases along trajectories [3].
In particular, we will present a new geometric integrator based on an adaptation of discrete variational integrators exactly preserving the coadjoint orbits. Finally, we will analyze geometric integrators for metriplectic systems, which model thermodynamic consistency through the conservation of energy together with entropy production.
References:
[1] Barbero-Liñán M, Caruso M and Martín de Diego D: Discretization maps and forced mechanical systems, Preprint
[2] Barbero-Liñán, María; Martín de Diego, David: Retraction maps: a seed of geometric integrators, Found. Comput. Math. 23 (2023), no. 4, 1335–1380.
[3] Bloch A., Ferraro S, Martin de Diego D and Bharadwaj S: Discrete variational calculus for double-bracket dissipation, arXiv:2604.26049 [math.NA]
[4] Marsden J E and West M: Discrete mechanics and variational integrators, Acta Numer. 10 (2001) 357-514.
[5] Martín de Diego D and Sato Martín de Almagro, R T: Variational order for forced Lagrangian systems, Nonlinearity 31 (2018) no. 8, 3814-3846.
[6] Martín de Diego, D and Sato Martín de Almagro R T: Variational order for forced Lagrangian systems II. Euler-Poincaré equations with forcing, Nonlinearity 33 (2020) no. 8, 3709-3738.

Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. In this talk we present a systematic analysis of both local and global errors arising from arbitrary splitting methods applied to unitary problems. The estimates we derive are formulated in terms of norms of commutators, thus exhibiting the so-called commutator-scaling property, of particular interest in the simulation of quantum systems. In addition, they can, under suitable assumptions, be extended to certain classes of unbounded operators. Special attention is devoted to the case where only two operators are involved and also to time-symmetric schemes. The theoretical results are illustrated by obtaining explicit error bounds for some representative schemes.

We address real-time Bayesian inverse problems governed by time-shift-invariant wave equations, with particular focus on tsunami inference and optimal experimental design. Efforts are underway to instrument subduction zones with ocean bottom acoustic pressure sensors to provide tsunami early warning. Our goal is to create a physics-based early-warning system that employs this pressure data, along with the 3D coupled acoustic–gravity wave equations, to infer the earthquake-induced spatiotemporal seafloor motion in real time. The Bayesian solution of this inverse problem then provides the seafloor forcing to forward propagate the tsunamis toward populated areas along coastlines and issue forecasts with quantified uncertainties.
In the context of the Cascadia Subduction Zone, a single forward wave propagation requires 1 hour on a supercomputer. The Bayesian inverse problem, with a billion uncertain parameters, formally requires hundreds of thousands of adjoint wave propagations; thus real time inference appears to be intractable. We propose a novel approach to enable exact solution of the inverse and prediction problems in real time. The key is to exploit the time-shift-invariance of the parameter-to-observable map, which permits FFT diagonalization and fast GPU implementation. We demonstrate that tsunami inverse problems with a billion parameters can be solved exactly in a fraction of a second.
This fast Bayesian inversion capability is then exploited to solve the optimal experimental design problem of placement of seafloor pressure sensors to maximize expected information gain in predictive quantities of interest. Time permitting, we discuss data-driven prior
construction, goal-oriented dimension reduction, and construction of fast surrogates for nonlinear shallow water equation-based tsunami predictions, which are more accurate in shallower waters.
This work is joint with Stefan Henneking, Sreeram Venkat, Bowen Shi, and Yuhang Li at UT Austin, and Alice Gabriel at UCSD.

In this talk, I will discuss a particular inverse problem arising from Optimal transport (OT). OT is now a central modeling tool to compare and couple probability distributions, with applications spanning economics, imaging, generative modeling, and computational biology. In many modern pipelines, however, the transport cost is not known a priori and must be inferred from data. This leads to inverse optimal transport (iOT): recover the ground cost (or metric parameters) from an observed optimal coupling. On the other hand, modern computational pipelines typically exploit an entropic regularization variant (eOT) of OT. I will discuss iOT in a regime that is both mathematically delicate and practically unavoidable: the entropic regularization level is small (approaching the unregularized OT model), while the coupling is observed through a finite number of samples.

Universality is one of the central principles of random matrix theory: many spectral statistics are insensitive to the fine details of the model and instead depend only on a small number of structural parameters. A striking recent development is that this phenomenon extends beyond classical matrix ensembles to highly constrained sparse models. In particular, although the adjacency matrices of fixed-degree sparse random graphs have dependent entries and are generated from far fewer independent random variables than Wigner matrices, their spectral statistics can nevertheless agree with those of the Gaussian Orthogonal Ensemble, the canonical model of chaotic spectral behavior. In this talk, we will discuss some of these recent results and explain the main ideas behind this progress. Central to the analysis are combinatorial analogues of self-consistent equations, edge-switching arguments that recover approximate independence, and new approaches that move beyond local laws to establish different forms of universality through more directly combinatorial methods.

Non-reversible Markov Chain Monte Carlo methods promise to accelerate convergence to stationarity by overcoming diffusive behaviour. However, standard mathematical approaches that have been developed for deriving quantitative bounds for convergence to equilibrium of reversible Markov processes usually do not apply or do not yield sharp bounds in the non-reversible case. In this talk, we discuss different recently developed techniques to derive both lower and upper bounds on relaxation times of non-reversible Markov processes.
Many non-reversible Markov processes related to MCMC methods (and not only those) can be viewed as lifts of an underlying reversible process. One can then ask for the existence of optimal lifts corresponding to maximal convergence acceleration. In this context, properties of the generator of the reversible process can be leveraged to derive bounds for a non-reversible lift. We will discuss several examples including lifted random walks, kinetic Langevin dynamics, Hamiltonian Monte Carlo, Event Chain Monte Carlo, and a related local time process of a self-repellent random walk. In some cases, lifts of optimal convergence order can be identified, whereas in other cases, the derivation of matching upper and lower bounds is still an open problem.

The resolvents of large random matrices, even very close to the imaginary axis, concentrate with a small fluctuation as the dimension increases. We extend this phenomenon to alternating products of resolvents and deterministic matrices using a new strategy consisting of an alternating tandem of two matrix flows. We focus on the applications, including (i) the precise identification of the blow up rate of the solution to a large system of ODE, (ii) eigenstate thermalisation hypothesis in physics and statistics and (iii) decorrelation phenomena of eigenvectors.
Based upon joint works with Zhigang Bao, Giorgio Cipolloni, Joscha Henheik, Oleksii Kolupaiev, Yuanyuan Xu.

Cryptography nestles where hard computational problems are found. But computational problems only tell half of the story: the hardness of factoring does not immediately suggest RSA.
It is now universally accepted that the fundamental problem upon which (supersingular) isogeny-based cryptography rests is the Endomorphism Ring Problem, or, equivalently, the Isogeny Problem. But how does one construct cryptographic schemes from it?
A generalization of Discrete Logarithm Cryptography, Cryptographic Group Actions have been a very successful paradigm for constructing isogeny-based protocols. However they only account for a small part of all proposed schemes. Outside of the framework lie several encryption and signature schemes that escape a simple formalization; among them SQIsign, possibly the most revered of all isogeny schemes.
In this talk I will introduce Forensic Categories, an abstract framework for modelling SQIsign-like primitives. Starting from the well known correspondence between quaternionic ideals and isogenies of supersingular curves, I will distill the key algorithmic properties that lead to SQIsign. The results is a category with a set of computational axioms sufficient to instantiate SQIsign-like primitives. We hope that this abstraction will both make SQIsign easier to use and lead to SQIsign-like signatures based on different computational problems.
Joint work with A. Basso, S. Patranabis, I. Radulescu and B. Wesolowski.

A quasicrystal is a structure that is not periodic yet still exhibits long-range order. One of the defining features of quasicrystals is their X-ray diffraction pattern, which shows isolated peaks — a hallmark of order previously associated only with periodic structures. Mathematically, quasicrystals are often modeled as discrete point sets, with their diffraction patterns represented by the absolute square of their Fourier transform, known as the diffraction measure. Interestingly, the support of this diffraction measure is typically dense, and only by ignoring contributions below a certain threshold (corresponding to experimental resolution) does it appear discrete. Fourier quasicrystals, however, are discrete point sets whose diffraction patterns are genuinely discrete, independent of experimental precision. To what extent aperiodic Fourier quasicrystals can exist, at least mathematically, has been a very active research topic over the last years. In this talk, we explain how Fourier quasicrystals can be constructed using certain real algebraic varieties with remarkable geometric, algebraic, and analytic properties. This is based on joint works with Alon, Kurasov, and Vinzant.

Image restoration in the presence of additive noise is well understood and can be addressed by a wide range of methods. In contrast, non-additive noise models are significantly more challenging due to their data-dependent nature. In this work, we focus on multiplicative noise and adopt a variational approach. While image denoising under multiplicative noise has been extensively studied from both theoretical and numerical perspectives, the combined setting of blur and multiplicative noise remains insufficiently understood, particularly at the theoretical level.
We propose a unified variational framework for image deblurring under multiplicative noise, that encompasses several well-established models, including Rudin-Osher, Aubert-Aujol, and so on. Within this framework, we establish well-posedness, convergence results, and error estimates under natural assumptions on the blurring operator.