Talk Titles and Abstracts

Abstract: Thermodynamics understanding by Geometric model were initiated by all precursors Carnot, Gibbs, Duhem, Reeb, Carathéodory. It is only recently that Symplectic Foliation Model introduced in the domain of Geometric statistical Mechanics has opened the door for a solid  bedrock, giving a geometric definition of Entropy as invariant Casimir function on Symplectic leaves (coadjoint orbit of the Lie Group acting on the system and interpreted as level sets of Entropy).

As observed by Georges Reeb "Thermodynamics has long accustomed mathematical physics [see DUHEM P.] to the consideration of completely integrable Pfaff forms: the elementary heat dQ [notation of thermodynamicists] representing the elementary heat yielded in an infinitesimal reversible modification is such a completely integrable form. This point does not seem to have been explored since then." Notion of foliation in thermodynamics appears in  C. Carathéodory paper where horizontal curves roughly correspond to adiabatic processes, performed in the language of Carnot cycles.  The properties of the couple of Poisson manifolds was  previously explored by C. Carathéodory in 1935, under the name of “function groups, polar to each other”. This seminal work of C. Caratheodory leads to the concept of a Poisson structure which was first defined independently by Lichnerowicz and  Kirillov.

A symplectic foliation model of Thermodynamics has been defined by Jean-Marie Souriau based on "Lie Groups Thermodynamics" model. This model gives a cohomological characterization of Entropy, as an invariant Casimir function in coadjoint representation. The dual space of the Lie algebra foliates into coadjoint orbits identified with the Entropy level sets. In the framework of Thermodynamics, a symplectic bifoliation structure is associated to describe non-dissipative dynamics on symplectic leaves (on level sets of Entropy as constant Casimir function on each leaf), and transversal dissipative dynamics, given by Poisson transverse structure (Entropy production from leaf to leaf). The symplectic foliation orthogonal to the level sets of moment map is the foliation determined by hamiltonian vector fields generated by functions on dual Lie algebra. The orbits of a Hamiltonian action and the level sets of its moment map are polar to each other. The space of Casimir functions on a neighborhood of a point is isomorphic to the space of Casimirs for the transverse Poisson structure. Souriau’s model could be then interpreted by Libermann's foliations, clarified as dual to Poisson Gamma-structure of Haefliger, which is the maximum extension of the notion of moment in the sense of J.M. Souriau, as introduced by P. Molino, M. Condevaux and P. Dazord in papers of  “Séminaire Sud-Rhodanien de Géométrie ». The symplectic duality to a symplectically complete foliation, in the sense of Libermann, associates an orthogonal foliation. We conclude with link to Cartan foliation and Edmond Fedida works on Cartan's mobile frame-based foliation.

In the first part, we will present the theme "Statistical learning on Lie groups" [1,2] which extends statistics and machine learning to Lie groups based on the theory of representations and cohomology of Lie algebra. From the work of Jean-Marie Souriau on " Lie Groups Thermodynamics" [4] initiated within the framework of symplectic models of statistical mechanics, new geometric statistical tools have been developed to define probability densities (Gibbs density of Maximum Entropy) on Lie Groups or homogeneous manifolds for supervised methods, and the extension of the Fisher metric of Information Geometry for unsupervised methods (in metric spaces).

In a 2nd part, TINNs (Thermodynamics-Informed Neural Networks) [3,5] will be discussed for AI-augmented engineering applications. The geometric structures of TINNs are studied by their metriplectic flow (also called GENERIC flow) modeling non-dissipative dynamics (1st thermodynamic principle of energy conservation) and dissipative dynamics (2nd thermodynamic principle of entropy production). The Thermodynamics of Lie Groups of Souriau makes it possible to geometrically characterize the metriplectic flow by a structure of “webs” composed of symplectic foliations and transversely Riemannian foliations. From the symmetries of the problem, the coadjoint orbits of the Lie group generate via the moment map (geometrization of Noether's theorem) the symplectic foliation (defined as the level sets of entropy, where entropy is an invriant Casimir function on these symplectic leaves). The metric on symplectic leaves is given by the Fisher metric. The dynamics along these symplectic leaves, given by the Poisson bracket, characterizes the non-dissipative dynamics. The dissipative dynamics is then given by the transverse Poisson structure and a metric flow  bracket, with an evolution from leaf to leaf constrained by the production of entropy. The transverse foliation is a Riemannian foliation whose metric is given by the dual of Fisher metric (Hessian of Entropy).

These machine-learning tools on Lie groups and TINNs are addressed in two European projects: a European network COST CaLISTA [6] and the European Marie-Curie action MSCA CaLIGOLA [7].

References:

[1] Barbaresco, F. (2022) Symplectic theory of heat and information geometry, chapter 4, Handbook of Statistics, Volume 46, Pages 107-143, Elsevier, https://doi.org/10.1016/bs.host.2022.02.003 https://www.sciencedirect.com/science/article/abs/pii/S0169716122000062 

[2] Barbaresco, F. (2023). Symplectic Foliation Transverse Structure and Libermann Foliation of Heat Theory and Information Geometry. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, Cham. https://doi.org/10.1007/978-3-031-38299-4_17 ; https://link.springer.com/chapter/10.1007/978-3-031-38299-4_17  

[3] Barbaresco F. (2022). Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Non-Linear Lindblad Quantum Master Equation, submitted to MDPI special Issue "Geometric Structure of Thermodynamics: Theory and Applications", 2022

[4] Souriau, J.M. (1969). Structure des systèmes dynamiques. Dunod. http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm   

[5] Cueto E. and Chinesta F. (2022), Thermodynamics of Learning of Physical Phenomena, arXiv:2207.12749v1 [cs.LG] 26 Jul 2022

[6] European COST network CA21109 : CaLISTA  - Cartan geometry, Lie, Integrable Systems, quantum group Theories for Applications ; https://site.unibo.it/calista/en 

[7] European HORIZON-MSCA-2021-SE-01-01 project CaLIGOLA - Cartan geometry, Lie and representation theory, Integrable Systems, quantum Groups and quantum computing towards the understanding of the geometry of deep Learning and its Applications; https://site.unibo.it/caligola/en

Given a random m*n*l tensor T, r is said to be a typical rank if Pr(rk(T)=r) > 0. For many real tensor formats, there is a single typical rank, but it is known, for example, that the typical ranks of real 3*3*5 tensors are 5 and 6. Here, we link the rank probabilities of a Gaussian 3*3*5 tensor to the probability of having real lines on a random cubic surface. As a consequence, we get a bound on the expected number of real lines on such a surface.

For 1 < m,n, it is known that in many cases, the typical ranks of real m*n*l tensors, where l = (m-1)(n-1)+1, are l and l+1. We provide a geometric proof of this fact. The key ingredient of this proof also illustrates how to determine the tensor-rank decomposition of order-three tensors by solving a system of polynomial equations. Calculating such decompositions, a problem that can be NP-hard, is central to many fields including signal processing or machine learning. This is joint work with Paul Breiding and Andrea Rosana.

Neural networks have recently gathered a lot of traction in the algebraic statistics community because of their rich algebraic and geometric properties. Previous research has largely focused on the case of linear activation functions, often composed with linear functions belonging to a certain class. In this talk I aim to explain how nonlinear functions can enter the algebraic game, and in particular introduce polynomial and rational functions. This is based on ongoing works with S. Sorea and with J. Li, E. Robeva, J. Rodriguez, Sonja Petrovic, and M. Zubkob.

Mixed-integer nonlinear programming (MINLP) is a powerful framework for modeling and solving many real-world optimization problems including scheduling, network design, and bin packing. A state-of-the-art method for solving MINLP problems is branch-and-bound, which iteratively splits the problem into smaller subproblems until an optimal solution to the problem is found. But already for linear problems it is well known that symmetries can negatively impact the performance of branch-and-bound methods, since symmetric subproblems will be repeatedly explored without providing new information. Most of the existing approaches for detecting symmetries focus on permutation symmetries that exchange the order of entries in a solution vector. For many problems classes, however, permutation symmetries do not capture all symmetries. For example, when packing objects into rectangular containers, also reflection symmetries of the container can be taken into account.

In this talk, we present a novel mechanism for computing, next to permutation symmetries, also reflection and translation symmetries of MINLPs. As existing approaches for permutation symmetries, we introduce a suitable graph whose automorphisms correspond to symmetries of a MINLP. Since reflection symmetries may nontrivially interact with nonlinear functions of a MINLP, however, our graph is much more involved than the previously known symmetry detection graphs. We also briefly discuss generalizations of state-of-the-art symmetry handling methods to reflection symmetries. The talk is concluded by numerical experiments showing the effect of handling reflection symmetries in the MINLP solver SCIP.

Abstract: TBA 

Every day, cryptographic algorithms protect the transactions and sensitive data of countless individuals online. Technological advancements have resulted in the need for more efficient cryptographic primitives to both ensure the security of users and maintain the usability of systems. Advanced cryptographic protocols, such as zero-knowledge proofs, fully homomorphic encryption, and multi-party computation, rely on symmetric key primitives as their building blocks. While traditional designs like the AES block cipher and SHA3 hash functions are efficient on CPU architectures, they perform poorly within advanced protocols. This has led to the emergence of the arithmetization-oriented design paradigm, also known as algebraic designs, in the symmetric cryptography community. Many of these designs operate over finite fields of large order and when modeled as polynomial systems, contain non-trivial syzygy relations in addition to those resulting from the ring structure. Therefore, analyzing their security is challenging, as current approaches in the literature mostly rely on assumptions, such as the (semi-)regularity of the polynomial system, which is not necessarily true in the context of symmetric cryptography. In this presentation, we will discuss the state-of-the-art techniques  used to study the security of algebraic ciphers, along with the open problems and challenges faced by cryptographers.

Abstract: Bell's theorem, which states that the predictions of quantum theory cannot be accounted for by any classical theory, is a foundational result in quantum physics. In modern language, it can be formulated as a strict inclusion between two geometric objects: the Bell polytope and the convex body of quantum behaviours. Describing these objects leads to a deeper understanding of the nonlocality of quantum theory, and has been a central research theme is quantum information theory for several decades.

After giving an introduction to the topic, I will focus on the so-called translation-invariant Bell polytope. Physically, this object describes Bell inequalities of a translation-invariant system; mathematically it is obtained as a certain projection of the ordinary Bell polytope. Studying the facet inequalities of this polytopes naturally leads into the realm of tensor networks, combinatorics, and tropical algebra.

This talk is based on joint work in progress with Jordi Tura, Mengyao Hu, Eloic Vallée, and Patrick Emonts.

Abstract: In the Computational Illumination Optics group we develop mathematical models and numerical methods for optical design of illumination systems, based on the principles of geometrical optics. Thus, we describe light propagation in terms of rays.  In illumination optics, we can distinguish two branches, i.e., non-imaging and imaging optics. Non-imaging optics is concerned with the transfer of light from a source to a target domain, typically for ordinary lighting. On the other hand, in imaging, the purpose is to form a very precise image of an object minimizing aberrations. We collaborate with Signify on non-imaging optics and with ASML on imaging.

Fundamental in geometrical optics is Fermat’s principle, stating that the optical path length of a ray connecting two points assumes a stationary value. From this we can derive the well-knows laws of reflection and refraction, as well as the Hamiltonian system describing free propagation of light. These in turn are the basis for our mathematical models. In particular, for non-imaging optics applications, we have developed optimal transport models and corresponding numerical methods to compute an optical surface that converts a given source distribution into a desired target distribution. On the other hand, for imaging applications, we have developed a Lie-algebraic description of light propagation and quantification of aberrations. We will outline several of our mathematical models and numerical methods and indicate the connection with geometry.