The Yamabe invariant is a real-valued diffeomorphism invariant coming from Riemannian geometry. Using Seiberg-Witten theory, LeBrun showed that the sign of the Yamabe invariant of a Kähler surface is determined by its Kodaira dimension. We consider the extent to which this remains true when the Kähler hypothesis is removed. This is partly based on joint work with Claude LeBrun.

To understand the energy of a dilute gas of interacting bosons has been a fundamental challenge in mathematical physics for many decades. This is one of the most simple, yet still a very difficult, quantum mechanical many-particle systems. Recently several breakthroughs have been made on this problem. In the talk I will review the classical, non-rigorous Bogoliubov calculation, which gives a good guide to the modern approaches, and point out where new ideas have to be added in order to reach a mathematical proof. The focus will be on the first correction term to the energy, the so-called Lee-Huang-Yang term.

Based on joint work with Jan Philip Solovej (Copenhagen).

Kähler-Ricci solitons are a natural generalization of the concept of a Kähler-Einstein metric which arise in the study of the Kähler-Ricci flow. In particular, shrinking gradient Kähler-Ricci solitons on non-compact manifolds model the singularity development of the Kähler-Ricci flow. In this talk, I will present some of my thesis work on the uniqueness of shrinking gradient Kähler-Ricci solitons on non-compact toric manifolds. In particular, the standard product of the Fubini-Study metric on $\mathbb{CP}^1$ (the round metric on $S^2$) and the Euclidean metric on $\mathbb{C}$ is the only shrinking gradient Kähler-Ricci soliton on $\mathbb{CP}^1 \times \mathbb{C}$ with bounded scalar curvature.

In this talk, we discuss the stability issue for the inverse problem of determining the electric potential appearing in a Schrödinger equation defined on an infinite cylindrical waveguide. We consider both results of stability from full and partial boundary measurements associated with the so-called Dirichlet-to-Neumann map. In the presence of the magnetic potential, a second problem is considered for which we prove that the electric potential and the magnetic field depend stably on the global and partial Dirichlet-to-Neumann maps. Our approach combines construction of complex geometric optics solutions and Carleman estimates suitably designed for our stability results stated in an unbounded domain.

On commencera dans un premier temps par une introduction aux problèmes de modules formels dérivés d’une part, et aux structures algébriques paramétrées par les props d’autre part. On verra ensuite comment décrire dans ce contexte les problèmes de déformation et leurs algèbres de Lie en termes de préchamps classifiants d’algèbres. Les résultats qui en découlent apportent une explication conceptuelle de diverses variantes de complexes de déformations apparaissant dans la littérature tout en en proposant une vaste généralisation. On en tirera quelques applications à la résolution de conjectures de Kontsevich en quantification par déformation, ainsi que des liens avec la géométrie symplectique et de Poisson dérivée si le temps le permet.

We introduce a family of real-valued homology cobordism invariants of homology 3-spheres. The invariants are derived from filtered instanton Floer homology, and those values are critical values of the SU(2)-Chern-Simons functionals. As its application, we produce infinitely many homology 3-spheres that cannot bound either a positive or negative definite 4-manifold. As another application, we show that if the 1-surgery of a knot has the Froyshov invariant negative, then the 1/n-surgeries (n>0) of the knot are linearly independent in the homology cobordism group. This is joint work with Yuta Nozaki and Masaki Taniguchi.

We study the classical problem of recovering a multidimensional source process from observations of nonlinear mixtures of this process. Assuming statistical independence of the coordinate processes of the source, we show that this recovery is possible for many popular models of stochastic processes (up to order and monotone scaling of their coordinates) if the mixture is given by a sufficiently differentiable, invertible function. Key to our approach is the combination of tools from stochastic analysis and recent contrastive learning approaches to nonlinear ICA. This yields a scalable method with widely applicable theoretical guarantees for which our experiments indicate good performance.

In this talk, I will introduce the notion of n-morphisms between two A-infinity algebras. These higher morphisms are such that 0-morphisms correspond to standard A-infinity morphisms and 1-morphisms correspond to A-infinity homotopies. The set of higher morphisms between two A-infinity-algebras then defines a simplicial set which has the property of being an infinity-groupoid. The combinatorics of n-morphisms are moreover encoded by new families of polytopes, which I call the n-multiplihedra and which generalize the standard multiplihedra.