# Geometry through spectral functionals.

## May 20-21, 2021 (16:00 - 19:00 CEST)

MINI-CONFERENCE

A mini-conference to review the current status of the approach on geometry through the spectral data, organized (online) at the Banach Center and Jagiellonian University.

**Organizers:**Ludwik Dąbrowski (SISSA), Andrzej Sitarz (UJ)

**Supported by:** NCN project UMO-2020/37/B/ST1/01540

**PROGRAMME:**

On action in spectral geometry
This will be a review on the notion of action in ncg with some explanations on selected technical points (pseudodifferential calculus, noncommutative integral, asymptotic expansion, ...) and few examples.

20.05, 17:00 (CEST) Thierry Masson

Geometric and algebraic structures from heat asymptotics.
It is well known that the Heat asymptotics coefficients for Laplace type operators can be written in terms of some geometrical Riemannian and Gauge invariants. Using a combinatorial procedure for the computation of these coefficients (join work with B. Iochum), we can show how these coefficients can be decomposed into some geometric structures, which depend on the operator, and some algebraic operators, which are of a more universal nature. The properties of these algebraic objects allow to perform the computation mainly on the geometrical part. I will present these structures and I will show how they are related to some other computational procedures, like the one exposed by Gilkey using pseudo-differential operators. This computation in terms of algebraic objects opens the possibility of adapting this method to non-commutative geometry.

20.05, 18:00 (CEST) Dmitri Vassilevich

Properties and applications of the heat kernel expansion
In this talk, I will review some basic properties of the heat trace asymptotics together with various applications to calculation of special values of the spectral zeta and eta functions. At the end of the talk, I will describe recent results (obtained jointly with A.Ivanov, Steklov Inst.) on the Atiyah-Patodi-Singer Theorem for domain walls.

21.05, 16:00 (CEST) Masoud Khalkhali

Curvature in Noncommutative Geometry, Spectral Aspects
This is a survey of recent progress in finding noncommutative analogues of various Riemannian curvature invariants for noncommutative manifolds.

21.05, 17:00 (CEST) Yang Liu

Modular Curvature and Morita Equivalence
I'd like to review three constructions of conformal change of metrics in the setting of spectral triples. The first two are based on papers of Connes-Moscovici and Lesch-Moscovici, concerning the modular Gaussian curvature on noncommutative two-tori. The last one comes from work of myself, in which I was trying to develop similar notion of intrinsic curvature on toric noncommutative manifolds. The main takeaway of talk is certain Morita equivalence invariant property of the modular Gaussian curvature discovered in Lesch-Moscovici's paper which does not have commutative counterpart. It would be very interesting to be able to find similar phenomenon on other examples.

21.05, 18:00 (CEST) Michał Wrochna

The spectral action on asymptotically Minkowski spacetimes
The spectral theory of the Laplace–Beltrami operator on Riemannian manifolds is known to be intimately related to geometric invariants such as the Einstein-Hilbert action. These relationships have inspired many developments in physics including the Chamseddine–Connes action principle in the non-commutative geometry programme. However, a priori they do only apply to the case of Euclidean signature. The physical setting of Lorentzian manifolds has in fact remained largely problematic: elliptic theory no longer applies and something different is needed.
In this talk I will report on joint work on this problem with Nguyen Viet Dang. We consider perturbations of Minkowski space and more general spacetimes on which the d’Alembertian P is essentially self-adjoint (thanks to recent results by Dereziński–Siemssen, Vasy and Nakamura–Taira). It is then possible to define functions of P, and we demonstrate that their Schwartz kernels have geometric content largely analogous to the Riemannian setting. In particular, we define a Lorentzian spectral zeta function and relate one of its poles to the Einstein–Hilbert action, paralleling thus a result in Euclidian signature attributed to Connes, Kastler and Kalau–Walze.
The primary consequence is that gravity can be obtained from a spectral action directly in Lorentzian signature. The proofs involve mathematical ingredients from Quantum Field Theory on curved spacetime, in particular the Feynman propagator.

**PLEASE**REGISTER HERE.

Registered participants will obtain the zoom link day before the meeting.