Extrema of functional determinants on orthogonal tori

• Julie Marie Rowlett (Chalmers University of Technology)

Room: Aula Dini The search for extrema is a central theme in both mathematics and physics. In this talk, we address the following question: among all n-dimensional orthogonal tori of unit volume, which one maximizes the determinant of the Laplacian? This determinant is obtained via the zeta regularization method, originally introduced by Ray and Singer to define an analytic analogue of the topological invariant, the Reidemeister torsion. While this determinant is connected to topology, geometry, analysis, and number theory, it is also important in physics. For example, Stephen Hawking observed that the zeta-function regularization technique could be used to regularize quadratic path integrals on a curved background space-time. This talk is based on joint work with Fabio Francesconi and is intended to be broadly accessible to everyone from doctoral students to emeriti with backgrounds ranging from mathematics to physics.

The spacelike capillary problem in Minkowski spacetime

• Lorenzo Maniscalco (Università di Torino)

Room: Aula Dini In this talk we will discuss the spacelike capillary problem in a cylinder of Minkowski spacetime $\mathbb{L}^{m+1}$: given a bounded domain $\Omega\subseteq \R^m$, it asks whether there exists a spacelike graph in the cylinder over $\Omega$ with prescribed mean curvature and Lorentzian contact angle with the wall of the cylinder. From a PDE point of view, this is a quasi-linear elliptic problem with a non-linear oblique derivative boundary condition and arises as the Euler--Lagrange equation of an area-like functional. The ellipticity of the equation depends on the slope of the graph and degenerates as it approaches the light direction, making regularity issues quite subtle. A major difficulty is the possible presence of light segments in the graph of the maximizer of the functional. We will present existence and regularity results for this problem over convex domains of $\R^m$, when the prescribed mean curvature and angle are bounded and Lipschitz, respectively.

Geometry and PDEs: an unexpected journey

• Luca Benatti (Universität Wien)

Room: Aula Dini Monotonicity formulas appear in many areas of geometric analysis and have proved to be remarkably effective tools for a wide range of purposes, from geometric inequalities and rigidity phenomena to the study of regularity properties of solutions to PDEs. In this talk, I will revisit a few stages of this story, showing how formulas arising in seemingly different settings share a common underlying structure. This point of view can provide new perspectives on known results and help tackle problems where some of the usual tools are less readily available.

Analysis on surfaces with bounded integral curvature

• Batu Güneysu (Christian-Albrechts-Universität zu Kiel)

Room: Aula Dini The class of metric surfaces of bounded integral curvature (BIC) is a large class of singular surfaces including e.g. the class of polyhedral surfaces. These surfaces have just enough regularity to define the curvature as a signed Radon measure, and compact BIC surfaces admit a natural Gauss-Bonnet Theorem. Given a complete surface of bounded integral curvature (without cusps), together with its 2-dimensional Hausdorff measure, I will explain in this talk the basics of local geometric analysis on such spaces: namely, they turn out to be infinitesimally Hilbertian, intrinsic, locally doubling and locally Poincare. In particular, one obtains a heat kernel on such spaces. We also show that under a global Dynkin-Kato type condition on the negative part of the curvature measure, the above results may be globalized in the natural way. This is joint work with Sebastian Boldt and Maxime Marot.

The nodal length of random spherical harmonics

• Maurizia Rossi (Università degli Studi di Milano-Bicocca)

Room: Aula Dini In this talk we investigate the behavior of the "typical" Laplacian eigenfunction of a compact smooth Riemannian manifold. In particular, motivated by both Yau's conjecture on nodal sets and Berry's ansatz on planar random waves, we consider Gaussian eigenfunctions on the sphere and study the distribution of the length of their nodal lines in the high energy limit. This result raises several questions regarding both the distribution of other geometric functionals and the behavior of nodal statistics of random eigenfunctions on a "generic" manifold. (This talk is mainly based on a joint work with D. Marinucci and I. Wigman.)

TBA

• Simone Vincini (Universität Wien)

Room: Aula Dini

Towards stability of the Neumann spectral gap via needle decomposition

• Sara Farinelli (Università di Genova)

Room: Aula Dini The Payne–Weinberger inequality gives a sharp lower bound for the first non-zero Neumann eigenvalue of a convex domain in terms of its diameter. A recent result by Amato, Bucur and Fragalà provides a quantitative stability result in the Euclidean setting, based on a suitable convex partition argument. In this talk, I will present an approach to the problem in which convex partitions are replaced by needle decompositions in Euclidean space. This provides a natural framework for addressing the same question in non-flat settings, such the one of Riemannian manifolds with non-negative Ricci curvature, which will be also discussed. This is based on ongoing joint work with N. De Ponti and I. Y. Violo.

Extracting subcomplexes on Carnot groups

• Francesca Tripaldi (University of Leeds)

Room: Aula Dini Differential complexes are fundamental tools in geometry and analysis, encoding geometric structures through differential operators and cohomological invariants. In subRiemannian geometry, the classical de Rham complex is often replaced by more intrinsic subcomplexes, such as the Rumin complex, which better reflect the underlying graded structure. In this talk, I will discuss the problem of extracting differential subcomplexes adapted to Carnot groups and introduce a new family of complexes, called spectral complexes, arising from spectral sequence methods. I will present the construction of these complexes and outline some applications.

TBA

• Pieralberto Sicbaldi (Universidad de Granada)

Room: Aula Dini

Stable hypersurfaces: rigidity and topology at infinity.

• Claudia Pontuale (Università degli Studi dell’Aquila)

Room: Aula Dini We discuss rigidity phenomena for complete, noncompact, stable hypersurfaces with constant mean curvature, with particular emphasis on questions related to Do Carmo’s problem. We will review some classical results and then consider extensions to the anisotropic setting, focusing on how stability can constrain the global geometry and topology at infinity. Some recent results and open questions will also be discussed.

Magnetic Laplacian and the conformal class of a surface

• Alessandro Savo (Sapienza Università di Roma)

Room: Aula Dini On a closed Riemannian surface we study the magnetic Laplacian with magnetic potential given by a harmonic 1-form A. Its lowest eigenvalue (magnetic ground state energy) is positive, unless A represents an integral cohomology class. We isolate a countable set of ground state energies which we call ''ground state spectrum'' of the metric. Our main result is to show that the ground state spectrum determines the volume and the conformal class of the metric. In particular, hyperbolic metrics are distinguished by their ground state spectrum (which is not true for the classical, non-magnetic Laplace spectrum). Joint work with Bruno Colbois and Luigi Provenzano.

Leibenson's equation on Riemannian manifolds

• Alexander Grigor’yan (Universität Bielefeld)

Room: Aula Dini

TBA

• Davide Carazzato (Universität Wien)

Room: Aula Dini

A solution operator for the linearized constant scalar curvature equation at the hyperbolic space

• Albachiara Cogo (Scuola Normale Superiore)

Room: Aula Dini We discuss a new approach to constructing a solution operator to the linearized scalar curvature equation at the hyperbolic metric, which has negative constant scalar curvature. Such an operator is of Bogovskii-type with good support propagation properties and gains two derivatives relative to standard norms; it extends the corresponding construction for the linearized scalar curvature operator at the Euclidean metric, which has vanishing scalar curvature. As an application, this provides a tool for gluing solutions of the Einstein Constraint Equations in the time-symmetric setting and hence for constructing new initial data for the Vacuum Einstein Equations of General Relativity: such gluing methods rely on suitable iterative schemes, in which one finds a sequence of solutions to the corresponding linearised problems. This is based on joint work with P. Chuściel and A. Nützi.

Integral and spectral curvature condition in dimension 3

• Gilles Carron (Nantes Université)

Room: Aula Dini I will discuss of various curvature or spectral constraints implying that a complete Riemannian manifold of dimension 3 is diffeomorphic to the Euclidean space. I will also provide some recent topological finiteness result obtained in collaboration with Dorian Martino (ETH Zürich).