Mathematical Physics Group

The research lines in our group pertain to the study of mathematical structures emerging from fundamental theoretical physics and interdisciplinary applications. In particular, the group members have interests in:

• General relativity and relativistic theories of gravity, including relativistic positioning; 

• Calculus  of variations on fiber bundles and related conservation laws; 

• Geometric structures on (pseudo-)Riemannian manifolds deriving from string theory and holography  (AdS/CFT);

• Mathematical methods for the exact non-perturbative study of quantum field theories (e.g. localization and integrability);

• Fluid dynamics and optical non-linear  models, their integrability and singularities, dispersive and Hamiltonian PDEs;

• Geometric-algebraic and qualitative aspects of differential equations of physical, biological and evolutionary interest.

You can learn about these topics (and others) by attending our seminars and special events that we organize regularly. 

The Department has several openings for various positions every year. Some of the opportunities to join our group are summarized here.

October 2, 2025 - Workshop

Website

Title: Extremal Volumes and Related Functionals in Physics and Geometry

Speakers: Edoardo Colombo (UniTo), Matteo Kevin Crisafio (UniTo), Elia Fusi (UniTo), James Sparks (Oxford) 

The study of extremal volumes and related functionals lies at the intersection of modern geometry and theoretical physics, with deep connections to problems in Kähler geometry, K-stability, special holonomy, calibrations and supersymmetric field theories. These objects often provide variational characterizations of structures of central importance in both mathematics and physics.

The aim of the workshop is to bring together researchers from these different perspectives to foster dialogue and interdisciplinary collaboration. The program will feature two introductory lectures, complemented by talks on more specialised topics.

This one-day event is intended as both an introduction to current developments in the field and a platform to exchange ideas across geometry and physics. 

Registration: There is no registration fee, but we kindly ask you to send an email to lorenzo.ruggeri@unito.it so we can keep track of attendance. In the evening, we will have an informal dinner: if you would like to join, just let us know in your email at your earliest convenience, as space at the restaurant is limited.

Lectures:

Elia Fusi (University of Turin -  Italy)

Title: A first introduction to K-stability

Abstract: The aim of this lecture 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.

James Sparks (University of Oxford - UK)

Title: Geometry, extremization and black hole entropy

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. 

Talks:

Matteo Kevin Crisafio (University of Turin -  Italy)

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

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. 

Edoardo Colombo (University of Turin - Italy)

Title: Equivariant volumes from localization in supergravity

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. 

Place and Time: 

Morning session, Aula Monod (1st floor), 10am - 1pm

Afternoon session, Aula S (1st floor), 2.30pm - 5.30pm 

September 16-18, 2025 - Supergravity

Website

Place: Politecnico di Torino, Via Nizza, 230, 10126 Torino TO

September 11-12, 2025 - QuanTO Lectures

Website

Title: Von Neumann Algebras and Entropy/Energy Inequalities in QFT

Lecturer: Roberto Longo (Roma 2)

Abstract: After a brief introduction to operator algebras, I will present new inequalities concerning entropy and energy; in particular, my recent relative entropy/energy ratio local bound in Quantum Field Theory. This bound is model-independent and rigorous; it follows solely from first principles in the framework of translation-covariant, local Quantum Field Theory on the Minkowski spacetime.  

Place: Palazzo Campana, UniTO  Aula Magna

The webpage of the season 2024-25 can be found here.

The webpage of the season 2023-24 can be found here.