Colloquium, Dept. of Mathematics and Applied Math. -- Γενικό Σεμινάριο

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Colloquium

Statistical optics of complex systems: From beam propagation through atmospheric turbulence to thermalization and thermodynamics

Abstract: We present a statistical-optics view of light in complex media, ranging from structured beam transport through atmospheric turbulence to thermodynamics and thermalization of weakly nonlinear multimode optical systems. The complexity arises either from the fluctuations of the refractive index in atmospheric turbulence or from the combination of nonlinearity with the presence of a large number of modes in optical thermodynamics.

Abstract: Generative Adversarial Networks (GANs) represent one of the most influential families of generative models. This talk will introduce the core principles of GANs and their formulation as divergence minimization problems. By employing variational (duality-based) representations of divergences, the intractable optimization problem can be reframed as a tractable two-player zero-sum game. While GANs have achieved remarkable success, training instability and convergence issues remain major challenges. To address these, we propose a general framework for constructing tailored divergences that bridge $f$-divergences and integral probability metrics, combining the stability properties of both. I will highlight key results demonstrating improved convergence and performance in challenging scenarios, such as heavy-tailed distributions and image generation. Finally, I will discuss a novel variant of GANs that naturally gives rise to Generative Gradient Flows and a corresponding Generative Particle Algorithm.

Abstract: The Shannon-Stam inequality, also known as Entropy Power Inequality (EPI), is a celebrated result in information theory, which goes back to Shannon's seminal paper. Nevertheless, the equality case has only been characterized under finite second moment or additional regularity assumptions. In this talk, we show that there is equality in the Shannon-Stam inequality if and only if the random variables in question are Gaussian, assuming nothing else than finiteness of differential entropies.

To that end, we present necessary and sufficient conditions for the validity of de Bruijn's identity. The latter links the derivative of entropy under Gaussian perturbations to Fisher information. While it has been widely applied and rigorously derived under mild moment or regularity assumptions, the necessary and sufficient conditions ensuring the validity of the required exchanges of differentiation and integration have not been fully identified in the literature.

Finally, we discuss the question of stability in the Shannon-Stam inequality, motivated by connections with recent advances in convex geometry and additive combinatorics. After reviewing some existing negative results, we conclude with a qualitative stability theorem in the weak convergence sense, valid under very mild assumptions.

Abstract: The problem of classifying collections of objects (graphs, manifolds, operators, etc.) up to some notion of equivalence (isomorphism, diffeomorphism, conjugacy, etc.) is central in all areas of mathematics. Invariant descriptive set-theory provides a formal framework for measuring the intrinsic complexity of such classification problems and for deciding, in each case, which types of invariants are "too simple" to be used for a complete classification. In this talk I will discuss various forms of classification which naturally occur in mathematical practice and how dynamics can be used to establish negative anti-classification results.

Squeezing a fixed amount of gravitational mass to arbitrarily small scales in $U(1)$ symmetry

Abstract: We discuss joint work with N. Carruth, where we construct solutions to the Einstein vacuum equations on a domain of fixed size, whose past boundary is a bifurcate null surface emanating from a plane. The solutions form a 1-parameter family, whose incoming gravitational energy (mass) near the plane is of fixed size, yet the its support can be squeezed to an arbitrary degree around one line, without affecting the size of the domain in which we obtain existence. Interpreting the space-times as bursts of incoming gravitational waves, which are allowed to diffuse on a region of space-times of uniform size, these are the largest amplitude such waves (relative to the size of their support) that have been obtained. We will place this work in the context of dynamical formation of black holes, results on the Burnett conjecture, as well as the hoop conjecture.

Abstract: Wave-driven propulsion (WDP) is a little-known type of locomotion in which a floating body generates surface waves to push itself forwards. Some animals have evolved to use WDP when moving on the water surface, such as water striders and water snakes. Meanwhile, WDP technologies have the potential to revolutionise engineering applications, such as reduced fuel consumption in shipping and low-energy water robots that can clean up oil spills.

In this seminar I will explore a simple model for WDP based on the wave equation, showing how to derive the optimum forcing using variational calculus as well as via numerical optimization. I will also explore a more realistic model based on coupling the equations of motion of a floating raft to a quasi-potential flow model of the fluid, showing good comparison with experimental data. Finally, I will discuss some of the key open challenges in the area and the road ahead for the future.

O'Donovan, D., Bustamante, M., Devauchelle, O. and Benham, G.P., 2025. Achieving Optimal Locomotion using Self-Generated Waves. Accepted in Journal of Fluid Mechanics. https://arxiv.org/abs/2506.11961

Benham, G.P., Devauchelle, O. and Thomson, S.J., 2024. On wave-driven propulsion. Journal of Fluid Mechanics, 987, p.A44. https://arxiv.org/pdf/2310.12886

Benham, G.P., Devauchelle, O., Morris, S.W. and Neufeld, J.A., 2022. Gunwale bobbing. Physical Review Fluids, 7(7), p.074804. https://arxiv.org/abs/2201.01533

Abstract: The term geometric inverse problems typically refers to problems related to extracting information about unknown objects defined on a Riemannian manifold from external measurements. Examples of such objects include functions, tensor fields, connections on a vector bundle, or the Riemannian metric itself. The most fundamental among these problems is perhaps that of determining an unknown function on a manifold with boundary from its geodesic X-ray transform, that is, from its integrals along geodesics of the metric. As is natural to expect, the information one can extract about the unknown objects depends strongly on the geometry of the underlying space.

In this talk, we will focus on asymptotically hyperbolic manifolds, a class of non-compact manifolds that are generally of non-constant curvature but admit a certain structure at infinity that makes their curvature asymptotically approach a negative constant. They include as a special case the Poincaré model of hyperbolic space and have attracted interest in the last three decades due to their connections with conformal geometry as well as with theoretical physics. We will discuss some results regarding the X-ray transform acting on scalar functions and symmetric tensor fields, and comment on ongoing work concerning the X-ray transform for connections.