2023 - 2024 Program

Title :        Stochastic modeling for describing crystallization droplets in water emulsion.

Speaker:  Mhamed Eddahbi . King Saoud University . Saudi Arabia.

Abstract: This paper introduces a new, stochastic mathematical model for the crystallization of emulsion in dispersed media. The mathematical model reads as a stochastic partial differential equation by combining the heat energy equation and the nucleation theory with specified drift and diffusion. We show the existence and uniqueness of the solution of the model by using techniques of stochastic partial differential equations. Numerical experiments are drawn to support the theoretical results. Moreover,  comparison of numerical results to experimental ones is provided.

Joint work with H. El-Otmany (France), A. Almualim (KSU, KSA), T. El Rhafiki (Morocco)

Title : Fixed point equation for the superdiffusive limit of the elephant random walk. ( slides)

Speaker:   Lucille Laulin. Bordeaux University. France.

Abstract : The Elephant Random Walk (ERW) was first introduced by two physicists in order to investigate the long-term memory effects in non-Markovian random walks. It is a one-dimensional discrete-time random walk on integers, which has a complete memory of its whole history and is influenced by a memory parameter. It was referred to as the ERW in allusion to the famous saying that elephants can remember where they have been. 

In this talk, we will start by giving an overview of the literature and the results. Then, we will mainly focus our attention on the superdiffusive regime (when the memory parameter is bigger than a critical constant). We will show that, thanks to a connection with Polya urns, its is possible to obtain a fixed point equation for the limiting random variable which is the key point for a better understanding of this random variable.

—> Joint work with Hélène Guérin and Kilian Raschel

Title : Nonexistence results for the semilinear wave equation in the scale-invariant case. 

Speaker:  Makram Hammouda. Imam Abdurrahman Bin Faisal University. KSA.

Abstract: We present some results on the blow-up of the semilinear wave equation with a scale-invariant damping in several situations (with a mass term or/and  a time-dependent speed of propagation). The novelty consists in a better characterization of the blow-up region. Our results are somehow in relationship with the well-known Glassey exponent. From the numerical viewpoint and aiming to understand the threshold between the blow-up and the global existence regions, we will give a taste on the numerical approximation applied to the linear problem. This will constitute a good start in proving some conjectures on the critical value of the nonlinearity’s exponent. 

Title : Gaussian Fluctuations of the Elephant Random Walk with Gradually Increasing Memory. 

Speaker:  Mohamed El Machkouri. Rouen University. France.

Abstract: The elephant random walk (ERW) model was introduced by Schütz and Trimper in 2004 with a view to study memory effects in a one-dimensional discrete-time nearest-neighbor walk on $\mathbb Z$ with a complete memory of its whole past. The name of the model is inspired by the traditional saying that elephants can always remember anywhere they have been. The memory of the walker is measured in terms of a parameter $p$ between zero and one and the model exhibits three regimes: diffusive regime ($0<p<3/4$), critical regime ($p=3/4$) and superdiffusive regime ($3/4<p<1$). The ERW has drawn a lot of attention in the last years and several theoretical results (law of large numbers, central limit theorem, law of the iterated logarithm,…) have been established for each of the three regimes. In 2022, Gut and Stadtmüller introduced an extension of the ERW model allowing the memory of the walker to gradually increase in time. For this new model, we establish central limit theorems in the three regimes (joint work with R. Aguech).

Title : On irreducibility and positivity improvement of norm continuous quantum Markov semigroups.

Speaker:  Skander Hachicha. King Faisal University. Saudi Arabia.

Abstract: We show that for a norm continuous quantum Markov semigroup irreducibility and positivity improvement for any time $t>0$ are equivalent under some general conditions that hold, in particular, in the finite-dimensional case. This result is applied to quantum Markov semigroups of weak coupling limit type. We also discuss the discrete-time case.

Title : Topological sensitivity analysis for the 3D nonlinear Navier-Stokes operator and applications.

Speaker:  Hassine Maatoug. Monastir university, Tunisia.

Abstract :  In this talk, we are concerned with the detection of objects immersed in a fluid flow domain from boundary measurements of the velocity field. To solve this ill-posed geometric inverse problem, we develop an identification approach based on the Kohn-Vogelius formulation and the topological gradient method. The inverse problem is reformulated as a topology optimization one.  In the theoretical part, we consider the 3D Navier-Stokes equations as a model problem and we derive a topological sensitivity analysis for a design function with respect to the insertion of a small obstacle inside the fluid flow domain. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is examined. The performed mathematical framework can be applied for a large class of design functions and arbitrarily shaped geometric perturbations. In the numerical part, we develop a simple and fast detection algorithm. The unknown obstacle is located and reconstructed using the leading term of the shape function variation.  As application, we apply the performed numerical algorithm for identifying rigid bodies immersed in the Mediterranean sea.

Title: An Improved Lower Bound on the Largest Common Subtree of Random Leaf-Labeled Binary Trees .

Speaker: Ali Khezeli, Inria, Paris, France.

Abstract: It is known that the size of the largest common subtree (i.e., the maximum agreement subtree) of two independent random binary trees with n given labeled leaves is of order between n^(0.366) and n^(1/2). We improve the lower bound to order n^(0.4464) by constructing a common subtree recursively and by proving a lower bound for its asymptotic growth. The construction is a modification of an algorithm proposed by D. Aldous by splitting the tree at the centroid and by proceeding recursively.

Title: Universality for the Longest Increasing Subsequence.

Speaker:  Mohammed Slim Kammoun

Abstract: It is known, from the work of Baik, Deift, and Johansson, that Tracy-Widom fluctuations occur in the context of the longest increasing subsequence of uniform permutations. In this presentation, we extend this result to encompass the Ewens distribution and, more generally, a class of random permutations with conjugacy-invariant distribution. Furthermore, we establish the convergence of the first components of the associated Young tableaux to the Airy Ensemble. This outcome provides a partial solution to a conjecture by Bukh and Zhou concerning the longest common subsequence of i.i.d. permutations.

Title : The blow-up rate for some nonlinear evolution equations in the log non-scaling invariance case

Speaker:  Mohamed Ali Hamza. Imam Abdurrahman Bin Faisal University.

Abstract: In this talk, we will discuss about some evolution equations with logarithmic nonlinearity in the  ``subcritical'' regime.  We show that the blow-up rate of any singular solution to the problem is given by the ODE solution associated.  In other terms, all blow-up solutions in the ``subcritical'' range are Type I solutions.  This will constitute a good start in proving that  the scale invariance property is not crucial in deriving the blow-up rate.

Title :  Persistence for a class of order-one autoregressive processes and  Mallows-Riordan polynomials.        

Speaker:  Kilian Raschel , CNRS, Laboratoire Angevin de recherches en Mathématiques. Anger University, France.

Abstract: Abstract. We establish exact formulae for the (positivity) persistence probabilities of an autoregressive sequence with symmetric uniform innovations in terms of certain families of polynomials, most notably a family introduced by Mallows and Riordan as enumerators of finite 

labeled trees when ordered by inversions. The connection of these polynomials with the volumes of certain polytopes is also discussed. 

Two further results provide factorizations of general autoregressive models, one for negative drifts with continuous innovations, and one for positive drifts with continuous and symmetric innovations. The 

second factorization extends a classical universal formula of Sparre Andersen for symmetric random walks. Our results also lead to explicit asymptotic estimates for the persistence probabilities. This is a 

joint work with Gerold Alsmeyer, Alin Bostan and Thomas Simon (Adv. Appl. Math., 2023).

Title: Large-scale analysis  for closed stochastic networks with finite capacity nodes and reservation (slides).

Speaker:   Christine Fricker, DI-ENS et INRIA Paris. France.

Abstract: The work is motivated by car-sharing systems like Autolib’ in Paris (2011-2017) where, for charging,  electric cars are parked in stations with small capacity. We consider a simple model for this system as a set of M/M/1/K queues where customers (cars) move between stations.  To prevent users to look for a parking space at the end of the trip in a saturated area, users can reserve the destination space at the beginning of the trip, or even before the trip.  We present both cases. The performance of the system is measured by  the probability that a node has no cars or no parking space available. The main issue  is the impact of the reservation. The study  is based on analysis when the system gets large. The probabilistic techniques are presented in the talk. It is a joint work with C. Bourdais and H. Mohamed.

Speaker:  Hélène Guérin. UQAM. Quebec, Canada. 

Title: On different stochastic epidemic models with varying infectivity and susceptibility. ( slides, video)

Abstract: Stochastic epidemic models with varying infectivity and waning immunity have been recently introduced and studied. Those models are more realistic than classical SIRS-type models, but also more difficult to study from a mathematical point of view since the underlying process is no more a Markov process. I will present results obtained by Forien et al. (2022) on the equilibrium of such a models, and by Foutel-Rodier et al. (2023) when a vaccination policy is taken into account. Then I will focus on a new model, introduced in collaboration with  Arsène Brice Zotsa-Ngoufack, using the same framework, but with memory of past infections: at each new infection, the new infectivity and susceptibility depend on their previous values. The main questions related to such models are the long time behavior of the epidemic, and the existence of an endemic equilibrium.

References:

Forien, Pang, Pardoux, Zotsa-Ngoufack, Stochastic epidemic models with varying infectivity and susceptibility, to appear in Annals of Applied Probability, arxiv: 2210.04667.

Foutel-Rodier, Charpentier, Guérin,  Optimal vaccination policy to prevent endemicity: A stochastic model, arXiv:2306.13633

Guérin, Zotsa-Ngoufack,  Stochastic epidemics models with varying infectivity and susceptibility, and memory of the previous infections, work in progress.

Title :  Generalized Optimized Schwarz Methods in arbitrary non-overlapping subdomain partitions

Speaker:  Xavier Claeys. Sorbonnes University. Paris, France.

Abstract: Optimized Schwarz Methods (OSM) stand among the most popular substructuring domain decomposition strategies for the simulation of wave propagation in harmonic regime. Considering arbitrary non-overlapping subdomain partitions with such methods, the presence of so-called cross points, where three or more subdomains could be adjacent, have raised serious practical and theoretical issues.

We will describe a novel approach to OSM that provides a systematic and robust treatment of cross points as well as a complete analytical framework. A salient new feature is the use of a non-local exchange operator to enforce transmission conditions and maintain subdomain coupling. The associated theory covers several pre-existing variants of OSM,  including Després' original algorithm, and yields new convergence bounds. We shall also present numérical results botn in 2D and 3D.

Title :  Combinatorics of nondeterministic walks.(slides, video)

Speaker:  Michael Wallner. Institute of Discrete Mathematics And Geometry, Wien, Austria. 

Abstract: Motivated by the study of networks involving encapsulation and decapsulation of protocols, we introduce nondeterministic walks, a new variant of one-dimensional discrete walks. The main difference to classical walks is that its nondeterministic steps consist of sets of steps from a predefined set such that all possible extensions are explored in parallel. In the first part of the talk, we discuss in detail the nondeterministic Dyck step set {{-1}, {1}, {-1,1}} and Motzkin step set {{-1}, {0}, {1}, {-1,0}, {-1,1}, {0,1}, {-1,0,1}}, and show that several nondeterministic classes of lattice paths, such as nondeterministic bridges, excursions, and meanders are algebraic. The key concept is the generalization of the ending point of a walk to its reachable points, i.e., a set of ending points. In the scond part, we extend our results to general step sets: We show that nondeterministic bridges and several subclasses of nondeterministic meanders are always algebraic. 

Our results are obtained using generating functions, analytic combinatorics, and additive combinatorics. This is joint work with Élie de Panafieu and Mohamed Lamine Lamali.

Title : Machine Learning tool for solving Beam model "Timoshenko System with Thermoelasticity" . (slides, video)

Speaker:  Sabrine Chebbi. Tubingen University (Germany).

Abstract: In this presentation, I will introduce a novel machine learning tool that has garnered significant interest from researchers in recent years. This tool has demonstrated robustness and high performance in solving partial differential equations (PDEs). Specifically, we will explore the capabilities of Physics-Informed Neural Networks (PINNs) in providing approximate solutions for a complex beam model incorporating temperature and heat flux considerations. Furthermore, we will delve into utilizing this approximate solution to characterize the asymptotic behavior of the energy associated with the solution of the  model. 

Title : Rigid and non rigid time periodic solutions in  fluid dynamics. 

Speaker:  Pr.  Taoufik Hmidi. Rennes University, France.

Abstract:   In this talk I will discuss some aspects of the vortex motion for Euler equations in the planar case. In the first part, I will review some results on rigid time periodic solutions localized around stationary solutions. In the second part, I will explore the leapfrogging phenomenon  and discuss its rigorous derivation. Actually,  we show  that under suitable constraints, four concentrated vortex patches leapfrog  for any time. When observed from a translating frame of reference, the evolution of these vortex patches can be described as a  non-rigid time periodic motion. Our proof hinges upon two key components. First, we desingularize the symmetric four  point vortex configuration, which leapfrogs in accordance with Love's result, by concentrated vortex patches. Second, we use some tools from   KAM  theory  to effectively  tackle the small divisors problem and deal with the degeneracy in the time direction.

Title :   Rates of convergence in the central limit theorem for the elephant random walk with random step sizes.  

Speaker:  Xiequan Fan. School of mathematics and statistics Northeastern University at Qinghuangdao. China. ( slides)

Abstract: We introduce a generalization of the elephant random walk model. Compared to the usual elephant random walk, an interesting feature of this model is that the step sizes form a sequence of positive independent and identically distributed random variables instead of a fixed constant. For this model, we establish the law of the iterated logarithm, the central limit theorem, and we obtain rates of convergence in the central limit theorem with respect to the Kologmorov, Zolotarev and Wasserstein distances.  We also establish the Cramer moderate deviations for this model.  The talk is based on join work of Qi-Man Shao, Jerome Dedecker, Haijuan Hu and Florence Merlevede.  

 Title : Malthusian exponent and survival of a reinforced Galton-Watson process (slides).

Speaker:  Bastien Mallein, LAGA - Institut Galilée, University Paris13, France.  (slides)

Abstract: A reinforced Galton-Watson process with reproduction law ν and memory parameter q ∈ (0, 1) is a population model constructed at follows. At each step each individual either, with probability q , repeats the number of children of one of its forebears picked uniformly at random, or, with complementary property 1 − q , creates a number of children given by an independent variable sampled according to the law ν . We denote by Z_n the number of individuals alive at generation n. Although a simple modification of the Galton-Watson process, this process does not satisfy the branching property, which makes its analysis challenging. Notably, establishing a necessary and sufficient condition on (q, ν) for the survival of this process with positive probability remains an open question.

Thanks to explicit computations, we are able to estimate the growth rate of E(Z_n). Our approach relies on the analysis of urn processes via an associated transport equation, which owes much to the work of Flajolet and co-authors. This first moment estimate gives a necessary condition for the survival. To obtain a sufficient condition, we can study associated additive martingales. Using an encoding of the reinforced Galton-Watson process as an infinite-types Galton-Watson process, we introduce an additive martingale capturing some of the asymptotic  behavior of the reinforced Galton-Watson process. It gives a sufficient condition for the survival of this process. 

Title:  Hidden processes and hidden Markov processes: classical and quantum (slides)

Speaker:  Abdessatar Souissi, Qassim University,   Saudi Arabia.

Abstract: This talk consists of 3 parts. The first part only considers classical processes and introduces two different extensions of the notion of hidden Markov process. In the second part, the notion of quantum hidden process is introduced. In the third part it is proven that, by restricting various types of quantum Markov chains to appropriate commutative sub–algebras (diagonal sub–algebras) one recovers all the classical hidden process and, in addition, one obtains families of processes which are not usual hidden Markov process, but are included in the above mentioned extensions of these processes. In this paper we only deal with processes with an at most countable state space.

A joint work with Luigi Accardi, El Gheteb Soueidy, Yun Gang Lu

Title :  Markov chains coming from card shuffling  (slides)

Speaker:  Matthieu Josuat-Verges, Research fellow at CNRS, affected to IRIF (Institute for Research in Fundamental Computer Science), Paris-Cité University, France.

Abstract: Riffle shuffle is a natural way to shuffle a deck of cards: cut in two blocks, and shuffle the two blocks by preserving their relative order.  Successive riffle shuffles give a Markov chain on the symmetric group that converges to the uniform distribution.  Bayer and Diaconis have precise results concerning the convergence rate.  I will outline the general results of Bidigare, Hanlon and Rockmore concerning a generalization in the context of hyperplane arrangement.  I will also explain the connection to my work on subalgebras of the descent algebras (joint work with Amy Pang).  

Title :  Lower bounds of $L^1$-norms of non-harmonic trigonometric polynomials ( slides)

Speaker:  Philippe Jaming, Université de Bordeaux 

Abstract: In this talk, we will present quantitative lower bounds of the $L^1$ norm of a non-harmonic trigonometric polynomial of the following form:

-- let $T > 1$; 

-- let $(\lambda_j){j\geq 0}$ be a sequence of non-negative real numbers with $\lambda{j+1}-\lambda_j\geq1$;

-- let $(a_j)_{j=0,\ldots,N}$ be a finite sequence of complex numbers. Then

$$C(T)\sum_{j=0}^N\frac{|a_j|}{j+1}\leq\frac{1}{T}\int_{-T/2}^{T/2}\left|\sum_{j=0}^Na_je^{2i\pi\lambda_jt}\right|\,\mathrm{d}t$$

where $C(T)$ is an explicit constant that depends on $T$ only. This provides a quantitative statement of a result by F. Nazarov. The $L^2$ analogue is Ingham's Inequality and the harmonic case ($\lambda_j$ integers) is McGehee, Pigno, Smith's solution of the Littlewood conjecture. We will also explain how these results apply in control theory of PDEs.

 Title : Height coupled trees. (slides)

Speaker:  Meltem Unel, Paris Saclay University, France. 

Abstract:  We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is of exponential form $e^{-\mu h}$. Defining the total weight for such trees of fixed size to be $Z^{(\mu)}_N$, we determine its asymptotic behaviour for large $N$, for arbitrary real values of $\mu$. Based on this we evaluate the local limit of the corresponding probability measures and find a transition at $\mu=0$ from a single spine phase to a multi-spine phase. Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for $\mu<0$ to the familiar quadratic growth at $\mu=0$ and to cubic growth for $\mu> 0$.

Title : Mathematics in artificial Intelligence: Contributions and challenges.

Speaker:  Lotfi Ben Romdhane (ISITCOM, Sousse).

Abstract: Artificial intelligence (AI) is revolutionizing industries and our lives at an unprecedented rate, and mathematics plays a fundamental role in this progress. In this talk, we explore the vital role of mathematics in AI and explore its main challenges. Deep learning, a type of machine learning algorithms that uses mathematical models inspired by the structure and function of the human brain, has achieved remarkable success in domains such as computer vision, natural language processing, and generative AI.  We will discuss also the main challenges faced in AI systems as the uncertainty and computational complexity of the used mathematical frameworks/models.