Scientific Program

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Title:  Large Random Matrices, Coulomb and Riesz Gases: Equations and Control

Title: From Phase Separation in Heterogeneous Media to Learning Training Schemes for Image Denoising

Abstract: What do these two themes have in common? Both are treated variationally, both deal with energies of different dimensionalities, concepts of geometric measure theory prevail in both, and higher order penalizations are considered. Will learning training schemes for choosing these penalizations in imaging may be of use in phase transitions?


Phase Separation in Heterogeneous Media: Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations using a variational approach based on the gradient theory of phase transitions, the potential and the wells may have to depend on the spatial position, even in a discontinuous way, and different regimes should be considered.


In the critical case case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition and the wells are fixed, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. The supercritical case for fixed wells is also addressed, now leading to an isotropic interfacial energy. In the subcritical case with moving wells, where the heterogeneities of the material are of a larger scale than that of the diffuse interface between different phases, it is observed that there is no macroscopic phase separation and that thermal fluctuations play a role in the formation of nanodomains.


This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands) and Likhit Ganedi (Aachen University, Germany), USA), based on previous results also obtained with Adrian Hagerty (USA) and Cristina Popovici (USA).


Learning Training Schemes for Image Denoising: Due to their ability to handle discontinuous images while having a well-understood behavior, regularizations with total variation (TV) and total generalized variation (TGV) are some of the best known methods in image denoising. However, like other variational models including a fidelity term, they crucially depend on the choice of their tuning parameters. A remedy is to choose these automatically through multilevel approaches, for example by optimizing performance on noisy/clean image training pairs. Such methods with space-dependent parameters which are piecewise constant on dyadic grids are considered, with the grid itself being part of the minimization. Existence of minimizers for discontinuous parameters is established, and it is shown that box constraints for the values of the parameters lead to existence of finite optimal partitions. Improved performance on some representative test images when compared with constant optimized parameters is demonstrated.


This is joint work with Elisa Davoli (TU Wien, Austria), Jose Iglesias (U. Twente, The Netherlands) and Rita Ferreira (KAUST, Saudi Arabia).

Title: Real and sparse intersections.

Abstract: Models in science and engineering are frequently expressed as sets of solutions of parameterized systems of polynomial equations with fixed supports, giving rise to special families of affine algebraic varieties. However, these models are usually concerned with real solutions, or even positive real solutions, and the number of variables and parameters is often very large. This poses new challenges that existing techniques fail to overcome. I will describe some lower and upper bounds on the number of positive solutions obtained in recent years inspired by applications, and state some basic open problems.

Title: Saturation numbers for primes and almost primes

Abstract: The classical problems such as twin primes and Goldbach-Waring, of producing an abundance of primes and almost primes can be formulated in terms of saturation numbers. This allows for their investigation  more generally in terms of orbits of affine linear and nonlinear morphisms. We review some highlights and recent developments.

Title: 3d Quantum field theories and Langlands duality

Abstract: In joint works with Braverman and Finkelberg, I studied 3-dimensional supersymmetric quantum field theories (QFT) by mathematically rigorous approaches in recent years. As an application, we give a recipe to construct (partially conjectural) pairs of affine algebraic symplectic manifolds with group action, where the group for the second manifold is replaced by Langlands dual group. The definition of these pairs was motivated by the S-duality of boundary conditions of 4d gauge theory, studied by Gaiotto-Witten. This S-duality gave a new connection between 3d QFT and Langlands duality as pointed out by Gaiotto, and also by Ben-Zvi, Sakellaridis, and Venkatesh as relative Langlands duality.

Title: On the Cauchy problem for a class of parabolic quasilinear systems.

Abstract: We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies a Petrovskii condition of positivity of the real part of the eigenvalues. Local wellposedness is known since the work of Amann in the 90s, by a semi-group method. We shall revisit these results in the context of Sobolev spaces modelled on L2: in particular, we will see that if the Petrovskii condition is known not to be sufficient to ensure exponential decay in time for systems of ordinary differential equations, the quasilinear structure here nevertheless ensures the well-posedness of the system.

This is a collaboration with Ayman Moussa.

TBA.

Title: Forward and Inverse Problems in Nonlinear Acoustics.

Abstract: The importance of ultrasound is well established in the imaging of human tissue. In order to enhance image quality by exploiting nonlinear effects, recently techniques such as harmonic imaging and nonlinearity parameter tomography have been put forward. 

As soon as the pressure amplitude exceeds a certain bound, the classical linear wave equation loses its validity and more general nonlinear versions have to be used.

Another characteristic property of ultrasound propagating in human tissue is frequency power law attenuation leading to fractional derivative damping models in time domain.

In this talk we will first of all dwell on modeling of nonlinearity on one hand and of fractional damping on the other hand. 

Then we will give an idea on the challenges in the analysis of the resulting PDEs and discuss some parameter asymptotics.

Finally, we address some relevant inverse problems in this context, in particular the above mentioned task of  nonlinearity parameter imaging, which leads to a coefficient identification problem for a quasilinear wave equation.

Title: The Value of Errors in Proofs - the fascinating journey from Turing's 1936 R \neq RE to the 2020 breakthrough of MIP* = RE.  

Abstract: A few years ago, a group of theoretical computer scientists posted a paper on the Arxiv with the strange-looking title "MIP* = RE", impacting and surprising not only complexity theory but also some areas of math and physics. Specifically, it resolved, in the negative, the "Connes' embedding conjecture" in the area of von-Neumann algebras, and the "Tsirelson problem" in quantum information theory. You can find the paper here https://arxiv.org/abs/2001.04383 

As it happens, both acronyms MIP* and RE represent proof systems, of a very different nature. To explain them, we'll take a meandering journey through the classical and modern definitions of proof. I hope to explain how the methodology of computational complexity theory, especially modeling and classification (both problems and proofs) by algorithmic efficiency, naturally leads to the generation of new such notions and results (and more acronyms, like NP). A special focus will be on notions of proof which allow interaction, randomness, and errors, and their surprising power and magical properties. 

The talk  requires no special background.

Title: Growth and Complexity functions in Algebra and Symbolic Dynamics.

Abstract: We will discuss history and recent results concerning (i) growth functions of groups, algebras, monoids and languages, (ii) complexity functions of infinite sequences.

Title: Geometric simplification in design and simulation.

Abstract: Removing geometric features from a complex domain is a standard procedure in computer-aided design (CAD) for simulation and manufacturing, known as defeaturing. Each CAD software implements its own algorithms to perform this task, which aims to simplify the meshing process, i.e, the creation of a geometric model suitable for simulating the governing equations of the physical phenomena at hand. Defeaturing is essential for a successful meshing process and enables faster simulations with reduced memory requirements.

However, the impact of geometric simplification on the accuracy of the computed solutions is often overlooked and has largely escaped rigorous mathematical analysis until now.

This study investigates the problem from a mathematical perspective and develops estimators capable of measuring the accuracy loss due to feature removal. Our goal is to create estimators that do not impose a significant computational burden but instead provide a straightforward measure of accuracy loss based solely on the numerical solution computed on the simplified geometry.

All results will be validated through numerical experiments, and we will conclude by outlining a roadmap for future developments.

Title: First and second-order Cucker-Smale models with non-universal interaction, time delay, and communication failures.

Abstract: We analyze first and second-order alignment models with non-universal interaction, time delay, and possible lack of connection between the agents. More precisely, we analyze the situation in which the system's agents do not transmit information to all the other agents, and also agents that are linked to each other can suspend their interaction at certain times. Moreover, we take into account possible time lags in the interactions. To deal with the considered "non-universal" and delayed connection, we assume that the digraph describing the interaction between the agents is strongly connected. Under a so-called persistence excitation condition, we establish the exponential convergence to consensus for both models.

Title: Bilinear control of parabolic evolution equations

Abstract: Despite the importance of control systems governed by bilinear controls for the description of phenomena that cannot be realistically modeled by additive controls, the action of multiplicative controls is generally not so widely studied as it happens for boundary and locally distributed controls. The main reasons of this fact might be found in the intrinsic nonlinear nature of such problems and furthermore, for controls that are scalar functions of time, in an ineluctable obstruction for proving results of exact controllability, which is contained in the celebrated work of Ball, Marsden and Slemrod, 1982.

In this talk I will present some results of stabilization and controllability of evolution equations of parabolic type via a scalar input bilinear control to some particular target trajectories called eigensolutions. Then, I will show applications of the abstract results to the heat equation, to some degenerate parabolic equations, to the Fokker-Planck equation and furthermore to diffusion equations on a network structure.

References:

[1] F. Alabau-Boussouira, P. Cannarsa, C. Urbani "Superexponential stabilizability of evolution equations of parabolic type via bilinear control'', Journal of Evolution Equations, vol. 21, pages 941-967 Springer (2021).

[2] P. Cannarsa, C. Urbani "Superexponential stabilizability of degenerate parabolic equations via bilinear control'', Inverse Problems and Related Topics, vol. 310, pages 31-45, Springer Singapore (2020). 

[3] F. Alabau-Boussouira, P. Cannarsa, C. Urbani "Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control", Nonlinear Differ. Equ. Appl., vol. 29, pages 38 (2022).

[4] P. Cannarsa, A. Duca, C. Urbani "Exact controllability to eigensolutions of the bilinear heat equation on compact networks", Discrete and Continuous Dynamical Systems - Series S, vol. 15, No. 6, pages 1377-1401 (2022).

[5] F. Alabau-Boussouira, P. Cannarsa, C. Urbani "Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker-Planck equation'', Comptes Rendus Mathématiques, vol. 362, pp. 511-545 (2024).

Title: Long-range phase transitions

Abstract: Phase transitions are a classical topic of investigation. They represent a complex phenomenon which needs to be attacked with different methodologies and different perspectives. I will discuss some rigidity and symmetry results for a long-range phase coexistence equation, their close relation with surfaces of minimal perimeter and a famous conjecture by Ennio De Giorgi.

Title: Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures.

Abstract: The Lott-Sturm-Villani theory of CD(K, N) metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense via optimal transport, though extremely successful, has been shown not to directly apply to sub-Riemannian geometries. Nonetheless, still using optimal transport tools, some entropy inequalities have been proved to hold in the case of the Heisenberg group and more in general in sub-Riemannian manifolds.  In this talk we survey the known results and motivate a new approach we propose aiming to unify Riemannian and sub-Riemannian synthetic Ricci lower bounds, introducing suitable curvature dimension conditions.


A main novelty is that we consider metric measure spaces endowed with a gauge function, and we allow general distortion coefficients

(joint with Andrea Mondino (Oxford) and Luca Rizzi (SISSA)).

Title: Quaternionic linear operators, S- spectrum and notions of trace

Abstract:We discuss the notion of S-spectrum for quaternionic linear operators and we show the peculiarities of notion of spectrum and eigenvalues in this noncommutative framework.  We then introduce an appropriate notion of trace in the setting of quaternionic linear operators, arising from the well-known companion matrices. We use it to define the quaternionic Fredholm determinant of trace-class operators in Hilbert spaces, and show that an analog of the classical Grothendieck-Lidskii formula. We discuss the particularities of the problem, which include, among others, the definition of an appropriate trace that differs from the usual one.

This is a joint work with P. Cerejeiras, F. Colombo, A. Debernardi Pinos, U. Kähler.

Title: On higher order hyperbolic equations with multiplicities.

Abstract: This is a survey talk on the well-posedness of higher order hyperbolic equations with multiplicities.

 We will discuss how the analysis of non-diagonalisable systems helps to understand higher order equations and we will provide explanatory examples.

Title: Algebraic interpretation of Cartan structures on Lie algebroids

Abstract: Several authors have introduced numerous notions and objects in an attempt to capture what they consider to be a geometric structure. Klein’s approach focused on the role of groups. A Klein space, or Klein geometry, is defined as a homogeneous space of a Lie group G, meaning it is a smooth manifold M on which the Lie group G acts smoothly, effectively, and transitively. On the other hand, Cartan’s perspective on geometry considered manifolds M, modeled locally on a Klein geometry. Among experts, a Cartan geometry is often viewed as a Klein geometry deformed by curvature; conversely, a Klein geometry is seen as a flat Cartan geometry [4].


In [1], Blaom demonstrates how to study Cartan geometries within the framework of Lie algebroids. Specifically, if by symmetry we refer to a smooth action G × M → M of a Lie group G on a smooth manifold M, this action manifests an infinitesimal counterpart in which the Lie algebra g of G acts on M. This action can be better understood through the concept of a Lie algebroid. Since not all Lie algebroids can be integrated into a Lie groupoid [2], Blaom asserts that Lie algebroids, endowed with compatible connections, provide the appropriate setting to study Cartan geometries.


In this talk, we will introduce the notion of Cartan Lie Rinehart algebras, which serve as the algebraic counterparts to the Cartan algebroids of Blaom. We will begin with a brief review of Lie Rinehart algebras, including several key examples. Among the most important are differential operators defined on general modules over a commutative algebra. We will demonstrate how these operators can be used to study connections on Lie Rinehart algebras. Additionally, we will explore how the structure of Cartan Lie Rinehart algebras integrates into the framework of matched pair Lie Rinehart algebras. This represents a slight generalization of the “matched pair of structures”, a well-known concept in higher Lie theory and Hopf algebra theory. These results are part of joint work with A. Tiraboschi [3].


References:


[1] Anthony D. Blaom. Geometric structures as deformed infinitesimal symmetries. Trans. Amer. Math. Soc., 58(8):3651–3671, 2006.

[2] Marius Crainic and Rui Loja Fernandes. Integrability of Lie brackets. Ann. of Math. (2), 157(2):575–620, 2003.

[3] Jesus Alonso Ochoa Arango and Alejandro Tiraboschi. Algebraic interpretation of Cartan structures on Lie algebroids. Work in progress.

[4] R. W. Sharpe. Differential geometry, volume 166 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. Cartan’s generalization of Klein’s Erlangen program, With a foreword by S. S. Chern.

Title: Multiscale Interface Coupling of PDEs and ODEs for Tissue Perfusion

Abstract: In biomechanics, local phenomena, such as tissue perfusion, are strictly related to the global features of the whole blood circulation. We propose a heterogeneous model where a local, accurate, 3D description of tissue perfusion by means of poroelastic equations is coupled with a systemic 0D lumped model of the remainder of the circulation. This represents a multiscale strategy, which couples an initial boundary value problem to be used in a specific tissue region with an initial value problem in the rest of the circulatory system. We discuss well-posedness analysis for this multiscale model, as well as solution methods focused on a detailed comparison between functional iterations and an energy-based operator splitting method and how they handle the interface conditions.

Title: Data Representation by Neural Networks: A control perspective.

Abstract: In this lecture I shall present some recent work in cooperation with Dr. Kang Liu (FAU).  We investigate several non-convex optimization problems involving shallow neural networks (NN), with the aim of representing data sets. The problems are convexified using a "mean-field" technique and the lack of relaxation gap is proved employing classical "representer theorems". We also derive generalization bounds providing insight into the choice of optimal parameters. We also derive efficient numerical algorithms, and analyze the analogy with classical control problems for parabolic equations.

Title: Control Systems on Wasserstein Spaces

Abstract: Dynamical systems involving large number of agents can be approached by considering moving sets of agents instead of union of vector-valued time-dependent paths of individual agents.


In some social sciences models, like the evacuation one, it may be interesting to assign to sets their measures and to work with metric spaces of Borel probability measures, the so called Wasserstein spaces. Here translation of vectors is replaced by the push-forward operation on measures leading  to  natural  extensions of notions of nonsmooth analysis to this metric space.


The dynamics of measures can be described then via controlled continuity (transport) equations. For Lipschitz kind dynamics, some cornerstone results of classical control theory known in the Euclidean framework have their analogues in Wasserstein spaces.


In this talk I will present necessary and sufficient conditions from [1] for the existence of solutions to state-constrained control systems.  Then I will recall more elaborate  results from [2] for measures with compact support that  were applied  to investigate uniqueness of solutions to Hamilton-Jacobi-Bellman  equations.


References:


[1] Bonnet B.,  Frankowska H., On the Viability and Invariance of Proper Sets under Continuity Inclusions in Wasserstein Spaces, SIAM Journal on Mathematical Analysis,  to appear.

[2] Badreddine Z. , Frankowska H., Solutions to Hamilton-Jacobi equation on a Wasserstein space,  Calculus of Variations and PDEs,  81: 9, 2022.

Title: Pseudoexponentials and covers: model theory meets some arithmetic geometry.

Abstract: I will revisit some of the attempts from the past three decades to understand some analytic functions by using model theory, mostly around the Zilber school at Oxford. Along the way, I will describe how some of the methods (mainly, quasiminimality) has spilled over from pseudoexponentiation to other analytic functions such as the j invariant. I will mention some connections to number theory. In the last part I will describe recent joint work with Baldwin and Cruz, as well as some new openings.

Organizers:

Duván Cardona (Chair)
Ghent Analysis and PDE Center, Belgium.
Board member of ISAAC
Scientific Director ICMAM  Latin America

Brian Grajales (Co-chair)
University of Campinas, Brazil.
Executive Director ICMAM  Latin America

Genni Fragnelli
University of Tuscia, Italy

Karina Navarro Gonzalez
Remote Lab of the Ghent Analysis and PDE Center in Armenia.

Yerevan State University

Yuriko Pitones
Universidad Autónoma Metropolitana, Mexico.

Milton Manuel Aguirre
Universidad de Sao Paulo, Brazil.

Local organizers

Diana Bueno
Pontificia Universidad Javeriana, Cali-Colombia.


Pedro Fernández Espinosa
Universidad de Caldas, Colombia. 

ICMAM Latin America 2024 has been supported by the following institutions/societies: