Time: Tuesday 21:00-22:00 (Beijing Time)
Organizers: Jialong Deng; Akito Futaki
Venue: ZOOM Meeting
Schedule
Date: 13 May, 2025 (Last talk of the Spring Semester)
Speaker: Eveline Legendre (Université Claude Bernard Lyon 1)
Title: The CR-Yamabe invariants and K-semistability.
Abstract: I will present a joint work in progress with Abdellah Lahdili and Carlo Scarpa, where we explore in more detail the results of a previously established correspondence between the constant scalar curvature Kähler problem (cscK) on a polarized manifold and a family of CR-Yamabe problems on the associated circle bundle. We show, among other things, that the supremum of the associated CR-Yamabe invariants detects K-semistability.
.Date: 6 May, 2025 （Colloquium talk, in person）
Speaker: Lei Ni (University of California, San Diego)
Title： Compact Lie algebras admitting integrable complex structures
Abstract: With N. Wallach we develop a Lie algebraic approach to the celebrated results of H.-C. Wang, Tits, and Wolf-Wang-Ziller.
The new approach provides additional information and can be applied to study homogenous complex (Hermitian) manifolds.
.Date: 29 Apr., 2025
Speaker: Gerard Misiolek (University of Notre Dame)
Title: Diffeomorphism groups and information geometry
Abstract: The pioneering work of V. Arnold in the 1960s introduced spaces of diffeomorphisms into the study of hydrodynamics. Motions of an ideal fluid in a fixed domain were recognized as geodesics in the group of volume-preserving diffeomorphisms equipped with an L^2 ("kinetic energy") metric. Subsequently, the space of densities was shown to play a similar role in the geometric description of optimal mass transport with the Kantorovich-Wasserstein distance. It turns out that information geometry with its attendant Fisher-Rao metric and the Hellinger distance can be placed in the same infinite-dimensional differential geometric framework and seen as an analogue of optimal transport albeit with respect to a higher-order Sobolev metric. In the talk I will describe this framework and give an application to integrable systems.
.Date: 22 Apr., 2025
Speaker: Yoshinori Hashimoto (Osaka Metropolitan University）
Title: Coupled Kähler--Einstein metrics and coupled Ding stability
Abstract: A foundational theorem in Kähler geometry states that a Kähler--Einstein metric exists on a Fano manifold (with discrete automorphisms) if and only if it is uniformly Ding stable, which happens if and only if the stability threshold is larger than 1. In the absence of Kähler--Einstein metrics, we can seek coupled Kähler--Einstein metrics defined in terms of a decomposition of the anticanonical bundle. The main result of this talk is the equivalence between the coupled Ding stability, the coupled stability threshold larger than 1 (once they are appropriately defined), and the existence of coupled Kähler--Einstein metrics. This is a joint work with Kento Fujita.
.Date: 15 Apr., 2025
Speaker: Eleonora Di Nezza (IMJ-PRG, Sorbonne Université)
Title: Almost convexity of the Mabuchi functional in singular settings
Abstract: The Mabuchi functional M was introduced by Mabuchi in the 80's in relation to the existence of canonical metrics on a compact Kähler manifold. The critical points of M are indeed constant scalar curvature Kähler (cscK) metrics. Recently, Chen and Cheng proved that the existence of a (smooth) cscK metric is equivalent to the properness of such functional. In order to look for singular metrics, it is then natural to study the properties of the Mabuchi functional in singular settings. In this talk we prove that this functional is (almost) convex in the very general "big case". This is based on a joint work with Trapani and Trusiani.
.Date: 11 Apr., 2025 (4PM-5PM, Friday) (In person, 静斋 105)
Speaker: Wilderich Tuschmann (Karlsruhe Institute of Technology)
Title: Moduli Spaces of Metrics and their Compactifications
Abstract: Focussing on lower curvature bounds, I will present recent results and open questions about the global topological properties of moduli spaces of metrics on manifolds and RCD spaces, and also discuss metric approaches to obtain and study suitable compactifications of the former.
.Date: 18 Mar., 2025
Speaker: Yanir Rubinstein (University of Maryland)
Title: Statistics meets algebraic geometry
Abstract: This 30-minute talk will describe a serendipitous encounter of notions from statistics (e.g., quantiles, tail distributions, cdfs) to the study of thresholds and Okounkov bodies in algebraic geometry. An unexpected twist in the plot is provided by Weierstrass gap theory.
.Date: 11 Mar., 2025
Speaker: Kartick Ghosh (Tata Institute of Fundamental Research, Mumbai)
Title: Computing the α-Futaki character
Abstract: Alvarez-Consul–Garcia-Fernandez–Garcia-Prada introduced the Kahler-Yang-Mills equations (coupled equations for a Kahler metric and a metric on a holomorphic vector bundle). They also introduced the α-Futaki character, an analog of the Futaki character, as an obstruction to the existence of solutions of the Kahler-Yang-Mills equations. In this talk, we will prove a formula for the α-Futaki character for certain ample line bundles over toric manifolds. We will then compute the α-Futaki character in some cases and compare it with the known existence result of Keller–Tønnesen-Friedman.
.Date: 4 Mar., 2025 （No talks)
.Date: 25 Feb., 2025
Speaker: Marco Mazzucchelli (ENS Lyon)
Title: Closed geodesics and the first Betti number
Abstract: In this talk, based on a joint work with Gonzalo Contreras, I will present the following result: on a closed Riemannian manifold of dimension at least two with non-trivial first Betti number, the existence of a minimal closed geodesic, in the sense of Aubry-Mather theory, implies the existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic flow of a suitable -close Riemannian metric. As a corollary of this result and of existing literature, we infer that on any closed manifold of dimension at least two with non-trivial first Betti number, a generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length.