In this talk we will don't speak about Joseph-Louis Lagrange (1736-1813) but about Lagrange's reception at the nineteenth Century. "Who read Lagrange at this Times?", "Why and How?", "What does it mean being a mathematician or doing mathematics at this Century" are some of the questions of our conference. We will give some elements of answers and the case Lagrange will be a pretext in order to explain what are doing historians of mathematics: searching archives and – thanks to a methodology – trying to understand, read and write the Past.
Lagrange - mathematical press - complete works - bibliographic index of mathematical sciences (1894-1912) - Liouville - Boussinesq - Terquem[-]

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.[-]

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.[-]

We consider the assignment (or bipartite matching) problem between $n$ source points and $n$ target points on the real line, where the assignment cost is a concave power of the distance, i.e. |x − y|p, for 0 < p < 1. It is known that, differently from the convex case (p > 1) where the solution is rigid, i.e. it does not depend on p, in the concave case it may varies with p and exhibit interesting long-range connections, making it more appropriate to model realistic situations, e.g. in economics and biology. In the random version of the problem, the points are samples of i.i.d. random variables, and one is interested in typical properties as the sample size n grows. Barthe and Bordenave in 2013 proved asymptotic upper and lower bounds in the range 0 < p < 1/2, which they conjectured to be sharp. Bobkov and Ledoux, in 2020, using optimal transport and Fourier-analytic tools, determined explicit upper bounds for the average assignment cost in the full range 0 < p < 1, naturally yielding to the conjecture that a “phase transition” occurs at p = 1/2. We settle affirmatively both conjectures. The novel mathematical tool that we develop, and may be of independent interest, is a formulation of Kantorovich problem based on Young integration theory, where the difference between two measures is replaced by the weak derivative of a function with finite q-variation.
Joint work with M. Goldman (arXiv:2305.09234).[-]

We propose a PDE framework for modeling the distribution shift of a strategic population interacting with a learning algorithm. We consider two particular settings one, where the objective of the algorithm and population are aligned, and two, where the algorithm and population have opposite goals. We present convergence analysis for both settings, including three different timescales for the opposing-goal objective dynamics. We illustrate how our framework can accurately model real-world data and show via synthetic examples how it captures sophisticated distribution changes which cannot be modeled with simpler methods.[-]

Motivated by the task of sampling measures in high dimensions we will discuss a number of gradient flows in the spaces of measures, including the Wasserstein gradient flows of Maximum Mean Discrepancy and Hellinger gradient flows of relative entropy, the Stein Variational Gradient Descent and a new projected dynamic gradient flows. For all the flows we will consider their deterministic interacting-particle approximations. The talk is highlight some of the properties of the flows and indicate their differences. In particular we will discuss how well can the interacting particles approximate the target measures.The talk is based on joint works wit Anna Korba, Lantian Xu, Sangmin Park, Yulong Lu, and Lihan Wang.[-]

In this talk, we shall consider equivariant subproduct system of Hilbert spaces and their Toeplitz and Cuntz–Pimsner algebras. We will provide results about their topological invariants through K(K)-theory. More specifically, we will show that the Toeplitz algebra of the subproduct system of an SU(2)-representation is equivariantly KKequivalent to the algebra of complex numbers so that the (K)K- theory groups of the Cuntz–Pimsner algebra can be effectively computed using a Gysin exact sequence involving an analogue of the Euler class of a sphere bundle. Finally, we will discuss why and how C*-algebras in this class satisfy KK-theoretic Poincaré duality.[-]

The "positivity phenomenon" for Bessel sequences, frames and Riesz bases $\left(u_k\right)$ are studied in $L^2$ spaces over the compacts of homogeneous (Coifman-Weiss) type $\Omega=(\Omega, \rho, \mu)$. Under some relations between three basic metric-measure dimensions of $\Omega$, we obtain asymptotics for the mass moving norms $\left\|u_k\right\|_{K R}$ (Kantorovich-Rubinstein), as well as for singular numbers of the Lipschitz and Hajlasz-Sobolev embeddings. Our main observation shows that, quantitatively, the rate of the convergence $\left\|u_k\right\|_{K R} \longrightarrow 0$ depends on an interplay between geometric doubling and measure doubling/halving exponents. The "more homogeneous" is the space, the sharper are the results.[-]

In computational anatomy and, more generally, shape analysis, the Large Deformation Diffeomorphic Metric Mapping framework models shape variations as diffeomorphic deformations. An important shape space within this framework is the space consisting of shapes characterised by $n \geq 2$ distinct landmark points in $\mathbb{R}^d$. In diffeomorphic landmark matching, two landmark configurations are compared by solving an optimization problem which minimizes a suitable energy functional associated with flows of compactly supported diffeomorphisms transforming one landmark configuration into the other one. The landmark manifold $Q$ of $n$ distinct landmark points in $\mathbb{R}^d$ can be endowed with a Riemannian metric $g$ such that the above optimization problem is equivalent to the geodesic boundary value problem for $g$ on $Q$. Despite its importance for modeling stochastic shape evolutions, no general result concerning long-time existence of Brownian motion on the Riemannian manifold $(Q, g)$ is known. I will present joint work with Philipp Harms and Stefan Sommer on first progress in this direction which provides a full characterization of long-time existence of Brownian motion for configurations of exactly two landmarks, governed by a radial kernel.[-]