2024 - 2025 Program

Title: Asymptotic of consecutive patterns in permutations and matchings 

Speaker: Khaydar Nurligareev, LIB, University of Burgundy. (slides, video)

Abstract: The first part of this talk is devoted to consecutive patterns in permutations. Our attention is focused on very tight patterns (aka Hertzsprung patterns) where consecutive permutation entries appear in consecutive positions. We start by recalling the enumerative results of Myers and Claesson and then move on to the asymptotics that we found using Borinsky’s approach. Along the way, we establish the complete asymptotics of self-overlapping permutations that play an important role in the study of consecutive patterns in permutations.

In the second part, we study the behavior of so-called endhered (end-adhered) patterns in matchings, which was motivated by connections to RNA secondary structures with allowed pseudoknots. Here, by a matching of size n, we mean a configuration of 2n points on a line that are consecutively labeled with integers from 1 to 2n and connected into disjoint pairs by n edges. An endhered pattern of size p consists of p edges, such that the set of starting points is an interval, and so is the set of ending points. In the case of p = 2, we show that the corresponding bivariate exponential generating function has a closed exact form, which allows us to obtain the asymptotic behavior by simple means. In the general case, for obtaining enumerative results we apply the Goulden-Jackson cluster method, while the asymptotics come from Borinsky’s approach.

This talk is based on the ongoing work with Célia Biane and Sergey Kirgizov. 

Title: How Probability Helped Achieve Ultra-Efficient Algorithms for Pattern Matching 

Speaker: Tatiana Starikovskaya, École normale supérieure, Paris, France. (slides, video)

Abstract: In the fundamental problem of pattern matching, one is given two strings, usually referred to as a pattern and a text, and must decide whether the pattern appears as a substring of the text. In 1970, Knuth, Morris, and Pratt gave the first linear-time and linear-space algorithm for pattern matching. One can argue that this algorithm is optimal: one has to use linear time to read the input and one has to use linear space to store the input. In this talk, we will discuss how probability helped surpass these barriers. 

Title: Tamari intervals and blossoming trees 

Speaker: Wenjie Fang, LIGM, University Gustave Eiffel, France. (slides)

Abstract: The Tamari lattice, defined on Catalan objects, has attracted quite some attention in combinatorics in the recent years due to its connections with various topics. In this talk, we will present the enumerative study of its intervals, which are in bijection with other well-studied objects such as planar maps. Our focus will be on a recently discovered simple bijection between Tamari intervals and the blossoming trees (Poulalhon and Schaeffer, 2006) encoding planar triangulations, using a new meandering representation of such trees. Its specializations to the families of synchronized, Kreweras, new/modern, and infinitely modern intervals give a combinatorial proof of the counting formula for each family. Compared to (Bernardi and Bonichon, 2009), our bijection behaves well with the duality of Tamari intervals, enabling also the counting of self-dual intervals. 

Title: Numerical Analysis and simulations of a model of biofilm growth in porous media 

Speaker:  Azhar Saeed Alhammali, Imam Abdurrahman Bin Faysal University, KSA. (video, slides)

Abstract: We consider a mathematical and computational model for biofilm growth and nutrient utilization in porous media. The model is a system of two coupled nonlinear diffusion--reaction partial differential equations (PDEs). One of the PDEs can be characterized as a parabolic variational inequality (PVI) due to a constraint on the volume of the biofilm. Solutions to PVIs have low regularity which limits the numerical scheme to low order. We approximate the the model using mixed finite element method with the lowest order Raviart-Thomas elements. We show the well-posedness of the discrete problem, derive rigorous error estimates, and present numerical experiments in 1D, 2D that confirm the predicted estimates. We also illustrate the behavior of biofilm and nutrient dynamics in simple and in complex porescale geometries. This is a joint work with Prof. Malgorzata Peszynska from Oregon State University, US, and Dr. Choah Shin from Ab Initio Software, US. 

Title: The Affine (k,s,b)-Urn Scheme: Multicolor Affine Urn Models with Multiple Drawings. 

Speaker: Joshua Sparks, Department of Statistics, George Washington University, USA. (video, slides)

Abstract: The urn model has a rich history in representing probabilistic phenomena within the real world. Many results have come from the study of these structures that evolve from a single ball drawn at a given stage, but difficulties arise with the analysis when we take the leap from one ball to multiple balls sampled during a single event. A class which navigates around this hurdle is that of the affine (k,s,b)-urn model, a structure where the replacement criteria is based on a linear combination of the balls sampled within each draw. In this talk, we explore the affine urn model and its inherent “core matrix”, which dictates the urn’s progression through its linear replacement criteria and provides a path of study similar to its single-draw analogue. We address this relationship for when the core matrix index is “small” and “critical”, while distinguishing the paths created when the index is “large”, and connect our results to real-world structures.

Title: Moments of the super diffusive elephant random walk with general step distribution. 

Speaker: Bálint Vető, Budapest University of Technology and Economics (BME), Hungary.(video, slides)

Abstract: The elephant random walk is a one-dimensional discrete-time random walk that repeats one of its previous steps chosen uniformly at random with probability alpha and the next step is sampled independently from the past with probability 1-alpha where alpha is the memory parameter. The step distribution of the original elephant random walk takes values +1 and -1. We consider the case of general step distribution with finite pth moment for some p>=2. In the superdiffusive regime where the memory parameter alpha>1/2, we prove that the displacement rescaled by n^alpha converges almost surely and in L^p. Further, we compute the first four moments of the limiting random variables. This extends the results obtained by Bercu. The talk is based on a joint work with József Kiss. 

Title: The joint node degree distribution in the Erdős-Rényi network

Speaker: Bushra Khaled Alarfaj, King Saud University, KSA. (video, slides)

Abstract:   The Erdős-Rényi random graph is the simplest model for node degree distribution and is one of the most widely studied. In this model, pairs of n vertices are selected and connected uniformly at random with probability p. Consequently, the degrees of a given vertex follow a binomial distribution. If the number of vertices is large, the binomial distribution can be approximated by a normal distribution using the Central Limit Theorem. This approximation holds for each node independently. However, since the degrees of nodes in a graph are not independent, we aim to test whether the degrees per node collectively in the Erdős-Rényi graph have a multivariate normal distribution (MVN). We test for MVN based on two considerations: independent and dependent degrees. Our results are obtained from the percentages of rejected statistics from the chi-square test, the p-values of the Anderson-Darling test, and a cumulative distribution function (CDF) comparison. We observe that the approximation to MVN appears valid when np>=10. Additionally, we compare the maximum likelihood estimate of p in the MVN distribution, considering both independence and dependence. The estimators are assessed using bias, variance, and mean square error.

Title: Local limits of descent-biased permutations and trees.

Speaker: Stephan  Wagner ,  Graz University of Technology, Austria . (video, slides)

Abstract: In this talk, we will discuss two related probabilistic models of permutations and trees biased by their number of descents. Here, a descent in a permutation is a pair of consecutive elements where the first is greater than the second. Likewise, a descent in a rooted tree with labelled vertices is a pair of a parent vertex and a child such that the label of the parent is greater than the label of the child. For some nonnegative real number q, we consider the probability measures on permutations and on rooted labelled trees of a given size where each permutation or tree is chosen with a probability proportional to q^{number of descents}. In particular, we determine the asymptotic distribution of the first elements of permutations under this model. Different phases can be observed based on how q depends on the number of elements n in our permutations. Using the results on permutations, one can also characterize the local limit of descent-biased rooted labelled trees. 

Title: An Airy limit law for the balance of labeled and unlabeled galled trees. 

Speaker: Bernhard  Gittenberger, Institute of Discrete Mathematics and Geometry, TU Wien, Austria. (video, slides)

Abstract:    Phylogenetic trees are an important model for classical evolution in biology and their balance is intuitively understood as the tendency of the descendant taxa of any internal node to split into clades of similar size. There measures for the balance of phylogenetic trees were studied for decades.

In some cases, for instance evolution of bacteria, reticulate events may occur.  Biologists therefore searched for models that cover reticulate evolution and came up with numerous models of phylogenetic networks. One of those models is called galled trees.

We generalize the Sackin index, an important measure for the balance of phylogenetic trees, in two ways to galled trees and investigate its behavior for galled trees and two of its subclasses, namely simplex galled trees and normal galled trees. All classes will be considered in both the labelled and the

unlabeled setting. We show that in all cases the mean of the Sackin index for a uniformly sampled galled tree is asymptotically equal to $\mu n^{3/2}$ for an explicit, class-dependent constant $\mu$. Moreover, distribution of the scaled Sackin index converges weakly as well as with all its moments to the Airy distribution. 

Title:  Distribution of binomial coefficients modulo a prime.

Speaker: Hsien-Kuei Hwang, Institute of Statistical Science, Academia Sinica, Taiwan. (video, slides)

Abstract: The summatory function of the number of binomial coefficients not divisible by a prime is known to exhibit regular periodic oscillations, yet identifying the less regularly behaved minimum of the underlying periodic functions has been open for almost all cases. We propose an approach to identify such minimum in some generality, solving particularly a previous conjecture of B. Wilson [Asymptotic behavior of Pascal’s triangle modulo a prime, Acta Arith. 83 (1998), pp. 105–116]. This talk is based on joint work with Svante Janson and Tsung-Hsi Tsai (https://arxiv.org/abs/2408.06817).  

Title: Numerical Tools in Mathematical Research: Experiences and Insights. 

Speaker: Samir ben Hariz,  Le Mans University, France. (video, slides)

Abstract: The integration of numerical tools into mathematical and statistical research is revolutionizing the analysis and modeling of complex systems. This presentation reflects on personal experiences, with a focus on simulation as a helpful tool for understanding intricate phenomena. The discussion is organized around two key themes:  

1. Structural Break Detection in Models: Numerical tools are crucial for detecting and localizing structural breakpoints in time series and other complex mathematical frameworks. The presentation will showcase results obtained and highlight the robustness of recent algorithms.  

2. Simulation Methods for PDEs and SDEs:   Simulation techniques are indispensable for solving partial differential equations (PDEs) and stochastic differential equations (SDEs). This section will emphasize cutting-edge methods and introduce a novel approach leveraging Hermite's expansion of the transition density.  

This presentation highlights the transformative potential of numerical tools, particularly simulation, in addressing the complexities of mathematical models. It aims to motivate colleagues to adopt numerical approaches in their own research, driving innovation and enhancing their capacity to solve diverse scientific challenges.

Title: Mathematical modeling, analysis, and simulation of heat conduction in frozen soils in the Arctic.  

Speaker: Malgorzata Peszynska, Oregon State University, Corvallis, USA .  (Link to video)

Abstract: Most mathematicians are familiar with the (linear) heat equation, which features infinite smoothing of initial data which propagates ``infinitely fast''. This property is absent in free boundary problems such as Stefan problem modeling melting of ice, where the fluxes are discontinuous across the ice/water interface which moves with finite speed. We will start with an illustration of these properties. Next we discuss models in permafrost, a complex environment abundant in the Arctic with great importance to climate studies, and we present our recent work on homogenization from pore- to Darcy scale. We focus on the thermal and flow models, with an outlook towards coupled mechanical phenomena.  Our modeling, analysis, and simulation efforts aim at practical scenarios such as the freeze/thaw cycle in the Arctic subject to the changing environmental conditions. This is joint work with many students and collaborators to be named in the talk.  

Title: Coefficientwise total positivity of Hankel matrices in counting problems.

Speaker:  Bishal Deb, Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Sorbonne Université and Université Paris, France. (video, slides)

Abstract: A matrix M with real entries is said to be totally positive (TP) if all its square submatrices have non-negative determinants. We can extend this definition to matrices with polynomial entries where a polynomial is said to be coefficientwise non-negative if all coefficients are non-negative.

This talk will begin with a quick introduction to total positivity. We will then introduce Hankel matrices and provide a characterisation of totally positive Hankel matrices due to Gantmakher and Krein and show its relation to the Stieltjes moment problem and to continued fractions. Next, we will give examples of sequences occurring in combinatorics whose Hankel matrices are totally positive. We then generalise these notions to coefficientwise total positivity.

We will conclude this talk by mentioning some of the proof techniques in this area. If time permits, we will mention some of our recent results. No prerequisites will be necessary to follow the talk. 

Title:  On a population model with memory.

Speaker:  Jean Bertoin, Institut für Mathematik - Universität Zürich. (video, slides)

Abstract: Consider first a memoryless population model described by the usual branching process  with a given mean reproduction matrix on a finite space of types. Motivated by the consequences of atavism in Evolutionary Biology, we are interested in a modification of the dynamics where  individuals keep full memory of their forebears and procreation  involves the reactivation of a gene picked at random on the ancestral lineage. By comparing the spectral radii of the two mean reproduction matrices (with and without memory), we observe that, on average, the model with memory always grows at least as fast as the model without memory. The proof relies on analyzing a biased Markov chain on the space of memories, and the existence of a unique ergodic law is demonstrated through asymptotic coupling.

Title:  Cardinality estimation 

Speaker:  Robert Sedgewick, Department of Computer Science Princeton University, USA.(video, slides)

Abstract: This talk surveys decades of research on algorithms for estimating the number of different items in a datastream, from early algorithms of Flajolet and Martin to the recent algorithms of Lumbroso, Janson and Sedgewick. The story is a poster child for the use of probabilistic methods in algorithm science.

Title: BSDEs with central value reflection.

Speaker:  Youssef Ouknine, Cadi Ayyad University & Mohammed VI Polytechnic University, Morocco (slides, video).

Abstract: This work focuses on the well-posedness of a backward stochastic differential equation (BSDE) with jumps and central value reflection. The reflection constraint is applied to the real-valued function defined as the unique solution of the equation E(arctan(Yt − x)) = 0 at each time t ∈ [0, T]. The driver of the BSDE depends on the distribution of the component Y of the solution and follows a general quadratic-exponential structure, while the terminal value is assumed to be bounded. Using a fixed-point argument and BMO martingale theory, we establish the existence and uniqueness of local solutions, which are then combined to construct a global solution over the entire time interval [0, T]. 

Keyword: Central value reflection, BMO martingales, Jumps. A joint work with my Ph-D student Kaoutar Nasroallah. 

Title:  Notions of Non-Commutative Independence.

Speaker: Marwa Banna, New York University Abu Dhabi (slides,  video).

Abstract:  In the noncommutative realm, there are five distinct notions of independence: tensor, free, Boolean, monotone, and anti-monotone. In this talk, I will explore the interplay between noncommutative probability theory and random matrix theory, illustrating these notions of independence through matrix models. These notions of independence play a key role in studying joint distributions of noncommutative random variables, which in turn are crucial for analyzing the limiting distributions of the corresponding random matrix models. I will particularly highlight recent advancements in the context of monotone independence.

Much like in the classical setting, each notion of independence is associated with a central limit theorem (CLT). In the second part of the talk, i will showcase the CLT in each of these settings and discuss the corresponding quantitative Berry-Esseen estimates. Finally, i will shift the focus to the operator-valued framework, where i will present quantitative results related to operator-valued central limit theorems. 

 This talk is based on collaborations with Arizmendi, Mai, and Tseng.

Title:  Elephant random walk and 1/2. 

Speaker:  Shuo Qin , Beijing Institute of Mathematical Sciences and Applications, China (slides, video).

Abstract: We prove a conjecture by Bertoin that the elephant random walk on Z^d is transient in dimensions d ≥ 3, and show that it undergoes a phase transition in lower dimensions between recurrence and transience. By generalizing the results to the step-reinforced random walks, we find that there is a universal critical point 1/2 hidden in this model. 

Title:  Reinforced dynamics for interacting agents in cooperative or competitive environments. 

Speaker: Ida Germana Minelli, University degli Studi dell’Aquila, Italy.( slides, video)

Abstract: We study the evolution of opinions for a system composed by groups of interacting agents with a given ”attitude”, that can be cooperative or competitive. The dynamics has a reinforcement mechanism and each group can be influenced by other groups. We describe the system in terms of a population of interacting two-colors Polya urns where an agents’ opinion is represented by the fraction of balls of a given color in the corresponding urn. The case of cooperative interactions has been widely studied in the last decade. In particular, it has been proven that agents’ opinions synchronize, i.e., they converge a.s. to the same random limit. We will consider a model where agents may be also competitive and we will discuss its asymptotic behavior. For such a model, we allow eventually the presence in the population of urns or groups of urns that evolve independently and influence the global dynamics. We show that in this case the long time behavior of the system depends on the structure of the graph associated to the interaction weights. 

Title:  On the superzeta functions on function fields and Li's coefficients 

Speaker: Kamel Mazhouda, University of Sousse, Tunisia ( slides, viwww.youtube.com/watch?v=NGtv0VtrPL4 deo)

Abstract: In this talk, we study the superzeta functions on function fields as constructed by Voros in the case of the classical Riemann zeta function. Furthermore, we study special values of those functions, relate them to the Li coefficients, deduce some interesting summation formulas, and prove some results about the regularized product of the zeros of zeta functions on function fields.               This is joint work with Bllaca, Khmiri and Sodaïgui.

Ref: Bllaca, Kajtaz H.; Khmiri, Jawher; Mazhouda, Kamel; Sodaïgui, Bouchaïb, Superzeta functions on function fields, Finite Fields Appl. 95, Article ID 102367, 22 p. (2024).

Title:  Asymptotic behavior of propagation in the frog model with random initial configuration.

Speaker:  Naoki Kubota, College of Science and Technology, Nihon University, Chiba, Japan.(slides, video)

Abstract: We consider the frog model with random initial configurations on a multi-dimensional lattice. This model consists of two types of particles: active and sleeping. Active particles perform independent simple random walks on the lattice. On the other hand, although sleeping particles do not move at first, they become active and start moving when touched by active particles. Some sleeping particles are randomly assigned to any site of the lattice and we assume that only the origin has at least one active particle initially.

After the original active particles start moving, additional active particles are gradually generated and spread across the lattice, with time.

In this talk, we discuss the asymptotic behavior of the propagation of active frogs. It is easy to imagine that the speed of the spread of active particles is one of the key factors in observing their propagation. In particular, the average speed is determined by the so-called time constant, and this talk mainly focuses on recent results regarding the properties of the time constant.