Title: Mean field games with heterogeneity

Abstract: Mean field games (MFGs) are an extension of interacting particle systems, where the particles are interpreted as rational agents, offering applications in economics, social sciences, or computer science. They can be seen as the limits of large-population stochastic differential games with symmetric agents. In this work, we propose a method to incorporate heterogeneity into MFGs, thus relaxing the symmetry assumptions. We will present the concept of heterogeneous Markovian equilibria and provide a proof of their existence under standard conditions. Our definition of Nash Mean Field equilibrium in this context and our proof techniques are based on multi-valued mapping theory, which will be briefly introduced during the talk. Furthermore, we will demonstrate that this heterogeneous Markovian equilibrium closely approximates a Nash equilibrium in finite-population games, and we will share some preliminary results on the convergence rate as the number of players approaches infinity.
This is a joint work with Nabil Kazi-Tani. 

Title : Patterns in Mallows permutations

Abstract: The Mallows distribution is a one-parameter model of non-uniform random permutations. It originated in data ranking and is now widely used, both in theory due to its relations with several other mathematical objects, and in applications due to its simplicity and tractability. Many properties of Mallows permutations have been studied extensively in recent years, such as their longest monotone subsequences or their cycle structure. Another statistic of interest is the number of occurrences of given patterns, which is used for example in the context of property testing and quasirandomness.
In this talk, we will present some results on pattern counts in Mallows random permutations. Their asymptotics highly depend on the parameter of the model: we will focus on three different regimes, and the tools available within them. In the weakly biased regime, coupling techniques can be used to transfer results from uniform permutations to Mallows permutations. In the strongly biased regime, the study of Mallows random permutations relies on a fundamental regenerative property. When the bias is "moderately strong'", this regenerative property can be recovered after transforming our permutation.

Title : Exploring the topological characteristics of basketball substitution and measuring the versatility of players

Abstract: The outcomes of basketball games are significantly influenced by player substitution strategies, a factor pivotal to a team’s success. While numerous studies have endeavored to optimize substitution decisions and lineup configurations under certain circumstances, the broader implications of substitution practices on a team's long-term performance remain underexplored. These practices, often reflective of a team's overarching roster management and coaching philosophy, can markedly impact collective performance over a season. For example, rigorous schedule may lead to the overutilization of core players, consequently limiting the involvement and development of the broader team roster, which could result in inconsistent team performance.
To investigate the relationship between long-term substitution dynamics and team standings, we need to model teams’ substitution behaviors. The application of complex network analysis, a methodology that has been widely used in sports performance analysis, offers an appropriate framework to model the relationships between team members. Analogous to constructing pass networks in football to ascertain player significance and team pass patterns, or region networks in tennis and badminton, complex network analysis can provide valuable insights into basketball substitution strategies.
Therefore, our works are integrated into the following three parts:
1.     Player substitution network: narrow the gap of substitution importance between players.
2.     Lineup substitution network: make more lineups connected with each other.
3.     Versatile players: how different players perform in different lineups.
Through the construction of player and lineup substitution networks of the first two parts, we delve into the substitution patterns of high-performing teams, then measure the versatility of players based on their performance difference among lineups. 

Title: Limit theorems for p-domain functionals of stationary Gaussian fields

Abstract: By means of the Malliavin-Stein approach, I will discuss central and non-central limit theorems for p-domain functionals of stationary Gaussian fields. Based on a joint work with Nikolai Leonenko, Leonardo Maini and Francesca Pistolato.

Title : Differential topology for dynamical random fields

Abstract: Joint work with Michele Stecconi

As an illustrative first example, consider the zero set of a smooth function on the sphere. If the function is close to the constant 1, then the zero locus is empty; likewise if it is close to the constant -1. However, by the intermediate value theorem, if we deform the first function into the second, then at some point the zero set is going to be non-empty. Under reasonable assumptions, it will first be a point, then a loop, then it will grow in size, then possibly the topology will become richer (more connected components), then it will start to shrink, until it is just a loop, shrinking to a point, and disappearing.
We will introduce this model and similar one, such as the rubber band moving on a table, which can go over itself but never break. The fundamental questions we will ask are what kind of topological phases can these objects present, and how and when do they transition between those.