I will give an introduction to the amplituhedron, a geometric object generalizing the positive Grassmannian, which was introduced by Arkani-Hamed and Trnka in the context of scattering amplitudes in N=4 super Yang Mills theory. I will focus in particular on its connections to cluster algebras, including the cluster adjacency conjecture. (Based on joint works with multiple coauthors, especially Evan-Zohar, Lakrec, Parisi, Sherman-Bennett, and Tessler.)[-]

We show how to construct 'simple' symbolic dynamical systems in terms of renormalisation schemes associated with multidimensional continued fractions. Continued fractions are used here to generate infinite words thanks to the iteration of infinite sequences of substitutions. Simple means that these symbolic systems have a linear number of factors of a given length, or that they have pure discrete spectrum, or else, that they have a low symbolic discrepancy. We also discuss the relation between these notions.[-]

Mapping classes of surfaces of finite type have been classified by Nielsen and Thurston. For surfaces of infinite type (e.g. surfaces of infinite genus), no such classification is known. I will talk about the difficulties that arise when trying to generalize the Nielsen-Thurston classification to infinite-type surfaces and present a first result in this direction, concerning maps which - loosely speaking - do not show any pseudo-Anosov behavior. Joint work with Mladen Bestvina and Jing Tao.[-]

The shear coordinate is a countable coordinate system to describe increasing self-maps of the unit circle, which is furthermore invariant under modular transformations. Characterizations of circle homeomorphism and quasisymmetric homeomorphisms were obtained by D. Šarić. We are interested in characterizing Weil-Petersson circle homeomorphisms using shears. This class of homeomorphisms arises from the Kähler geometry on the universal Teichmüller space.
For this, we introduce diamond shear which is the minimal combination of shears producing WP homeomorphisms. Diamond shears are closely related to the log-Lambda length introduced by R. Penner, which can be viewed as a renormalized length of an infinite geodesic. We obtain sharp results comparing the class of circle homeomorphisms with square summable diamond shears with the Weil-Petersson class and Hölder classes. We also express the Weil-Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.
This talk is based on joint work with Dragomir Šarić and Catherine Wolfram. See https://arxiv.org/abs/2211.11497.[-]

Statistical fairness seeks to ensure an equitable distribution of predictions or algorithmic decisions across different sensitive groups. Among the fairness criteria under consideration, demographic parity is arguably the most conceptually straightforward: it simply requires that the distribution of outcomes is identical across all sensitive groups. In this talk, we explore the relationship between classification and regression problems under this constraint.
We provide several fundamental characterizations of the optimal classification function under the demographic parity constraint. In the awareness framework, analogous to the classical unconstrained classification scenario, we demonstrate that maximizing accuracy under this fairness constraint is equivalent to solving a fair regression problem followed by thresholding at level 1/2. We extend this result to linear-fractional classification measures (e.g., 𝐹-score, AM measure, balanced accuracy, etc.), emphasizing the pivotal role played by regression in this framework. Our findings leverage the recently developed connection between the demographic parity constraint and the multi-marginal optimal transport formulation. Informally, our result shows that the transition between the unconstrained problem and the fair one is achieved by replacing the conditional expectation of the label by the solution of the fair regression problem. Leveraging our analysis, we also demonstrate an equivalence between the awareness and the unawareness setups for two sensitive groups.[-]

Let $\Omega \subset \mathbb{R}^3$ be a sheared waveguide, i.e., $\Omega$ is built by translating a cross-section (an arbitrary bounded connected open set of $\mathbb{R}^2$ ) in a constant direction along an unbounded spatial curve. Consider $-\Delta_{\Omega}^D$ the Dirichlet Laplacian operator in $\Omega$. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of $-\Delta_{\Omega}^D$. After that, we state sufficient conditions that give rise to a non-empty discrete spectrum for $-\Delta_{\Omega}^D$. Finally, in case the cross section translates along a broken line in $\mathbb{R}^3$, we prove that the discrete spectrum of $-\Delta_{\Omega}^D$ is finite, furthermore, we show a particular geometry for $\Omega$ which implies that the total multiplicity of the discrete spectrum is equals 1.[-]

In this talk, we start by studying a particular model for opinion dynamics where the influence weights of agents evolve in time via an equation which is coupled with the opinions' evolution. We explore the natural question of the large population limit with two approaches: the now classical mean-field limit and the more recent graph limit. After establishing the existence and uniqueness of solutions to the models that we will consider, we provide a rigorous mathematical justification for taking the graph limit in a general context. Then, establishing the key notion of indistinguishability, which is a necessary framework to consider the mean-field limit, we prove the subordination of the mean-field limit to the graph one in that context. We finish with the study of interacting particle systems posed on weighted random graphs. In that aim, we introduce a general framework for the construction of weighted random graphs. We prove that as the number of particles tends to infinity, the finite-dimensional particle system converges in probability to the solution of a deterministic graph-limit equation in which the graphon prescribing the interaction is given by the first moment of the weighted random graph law.[-]