Speakers

Elia Fusi (University of Turin -  Italy)

Lecture Title: A First Introduction to K-Stability

Lecture Abstract: The aim of this talk is to describe the main ideas and ingredients behind K-stability. In order to achieve that,  in the first part, I will discuss some famous results in GIT, such as the Hilbert-Mumford criterion and the Kempf-Ness' Theorem. With them serving as guidelines, I will move to discuss the moment map interpretation of the scalar curvature in Kähler geometry, due to Donaldson and Fujiki.  Finally,  I will focus on introducing the concept of K-stability, ultimately, stating the Yau-Tian-Donaldson conjecture and the known results concerning its validity.

Matteo Kevin Crisafio (University of Turin -  Italy)

Seminar Title: K-stability beyond Kähler–Einstein: functionals and obstructions for Sasaki and conformally Kähler Einstein–Maxwell metrics 

Seminar Abstract: After Chen, Sun and Donaldson resolved the long-standing Yau–Tian–Donaldson conjecture, establishing the equivalence between K-polystability and the existence of Kähler–Einstein metrics on smooth Fano varieties, interest in extending the notion of K-stability to broader geometric settings has surged. In this seminar, I will focus on recent results by Futaki and Ono concerning a volume functional whose extremisation yields conformally Kähler Einstein–Maxwell metrics together with a Futaki-like invariant acting as an obstruction to their existence; I will also speculate on possible connections between this functional and the on-shell action of 4d supergravity solutions. Finally, I will mention K-stability results for Sasaki geometry, as developed in Collins’s PhD thesis, which, among other things, provide a more fundamental geometric interpretation of the celebrated results of Gauntlett, Martelli, Sparks and Yau about volume extremisation and the existence of Sasaki–Einstein metrics. 

James Sparks (University of Oxford - UK)

Lecture Title: Geometry, extremization and black hole entropy

Lecture Abstract: I will give an overview of various geometric techniques that allow one to compute observables in gravity via extremal problems, focusing on black hole entropy functions as the main example. Topics covered will include GK geometry and equivariant localization, together with relations between them. I will also explain how this leads to gravitational blocks, and a new viewpoint on the attractor equations for black holes. 

Edoardo Colombo (University of Turin - Italy)

Seminar Title: Equivariant volumes from localization in supergravity

Seminar Abstract: In this talk, I will review how the equivariant volume functional can be computed with localization techniques, such as the Duistermaat–Heckman formula and the theorem of Goertsches–Nozawa–Toeben, and I will explain how this volume functional enters in various extremization problems in supergravity that are relevant for holography. I will then focus on the localization of the on-shell action of D=5 gauged supergravity, and showcase how the equivariant volume appears from the computation.

Organizers

• Dario Martelli

• Fabrizio Nieri

• Antonio Pittelli

• Lorenzo Ruggeri