An Allen-Cahn approach to the Plateau problem

Monday 11:00 h - 12:00 h

Abstract: Modelling non-separating surfaces as level sets of solutions to an elliptic equation, requires thinking of these as sections of a line bundle rather than as usual functions. This perspective introduces an arena full of interesting new problems for elliptic functionals. In this talk I would like to present a few of these problems, show you how they arise naturally from geometric settings, and discuss my recent work with Marco Badran and Manuel Del Pino on Plateau’s problem.

First Integrals of the Geodesic Flow on Nilpotent Lie Groups

Monday 16:30 h - 17:30 h

Abstract: A symmetric Killing tensor on a Riemannian manifold is a symmetric tensor such that the complete symmetrization of its covariant derivative vanishes. This concept generalizes the notion of a Killing field, since such tensors define first integrals of the equations of motion and are therefore conserved along geodesics. On a Riemannian manifold, parallel symmetric tensors and symmetric products of Killing fields, along with their linear combinations, give rise to symmetric Killing tensors known as decomposable. However, determining whether a manifold admits indecomposable symmetric Killing tensors is generally not straightforward. In this talk, we first introduce the geometry of Riemannian 2-step nilpotent Lie groups modeled on Euclidean space and develop intuition about geodesic flow. Using the algebraic structure of these spaces, we present recent results on the existence and characterization of left-invariant symmetric Killing tensors in Lie groups with invariant metrics, with a focus on 2-step nilpotent and almost abelian groups. We provide explicit examples of indecomposable tensors and discuss their relationship to the underlying Lie algebra structure.

Extracting subcomplexes on Carnot groups

Tuesday 11:00 h - 12:00 h

Abstract: 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.

On stable extremal domains for the first Dirichlet eigenvalue of the Laplacian operator

Tuesday 15:00 h - 16:00 h 

Abstract: In this talk, we discuss the concept of stable extremal domains for the first Dirichlet eigenvalue of the Laplacian. We classify stable extremal domains in the 2-sphere and in higher-dimensional spheres under the assumption that either the boundary is minimal or the eigenvalue equals the dimension of the sphere. Additionally, we establish topological bounds for stable domains in general compact Riemannian surfaces, assuming either nonnegative total Gaussian curvature or sufficiently small volume. This is joint work with Marcos P. Cavalcante (UFAL, Brazil).

Geometry of critical constant mean curvature surfaces in hyperbolic 3-Spaces

Tuesday 16:30 h - 17:30 h

Abstract: I will describe old and new results on the geometry of surfaces whose mean curvature takes a critical value: minimal surfaces in \mathbb{R}^3, mean curvature 1 surfaces in \mathbb{H}^3, minimal surfaces in Nil, and mean curvature 1/2 surfaces in \mathbb{H}^2\times\mathbb{R} .

From Gaussian to Q-Curvature: A Conformal Viewpoint

Wednesday 11:00 h - 12:00 h

Abstract: Gaussian curvature plays a central role in the study of two-dimensional surfaces, from characterizing their local shape to determining global topological properties. Its transformation law under conformal changes connects this quantity to both complex analysis and partial differential equations, and has motivated significant advancements in both fields. A natural question is whether an analogous theory exists in higher dimensions. In this talk, we would like to discuss problems in conformal geometry concerning how curvature quantities transform under conformal changes, concluding with the Q-curvature. Originally introduced as a fourth-order generalization of Gaussian curvature to dimension 4, the Q-curvature can also be defined in higher dimensions and has come to play a central role in modern conformal geometry. We present results on the associated Q-curvature equation in both compact and noncompact settings, along with some open problems and promising directions.

Min-max widths associated with distance functions

Wednesday 15:00 h - 16:00 h

Abstract: The min-max theory for the area functional is a Morse theory on the space of hypersurfaces contained in a Riemannian manifold. The theory experienced remarkable developments and found deep applications in differential geometry. The min-max widths are invariants that naturally emerge from this theory as special critical values of the area. It is very interesting to compare these numbers to other geometric quantities, such as the volume and curvature bounds of the ambient manifold. In this talk, I will discuss a new notion of min-max width associated with the distance function, together with several comparison results. More precisely, we develop a Morse–Lusternik–Schnirelmann theory for the distance between two points on a smoothly embedded circle in a complete Riemannian manifold. This framework leads naturally to a definition of width that extends the classical notion of width for plane curves. We further investigate curves that may be regarded as Riemannian analogues of plane curves of constant width, establishing characterization results and geometric properties. Finally, we present inequalities relating this new width to other geometric invariants, as well as rigidity results characterizing the cases of equality.

Solitons for the mean curvature flow in H2xR: rotators and asymptotic geometry of vertical translators

Wednesday 17:00 h - 18:00 h

Abstract: Solitons for the mean curvature flow are special solutions that evolve by isometries of the ambient space. In this talk, I will discuss solitons in the product space H2xR, with emphasis on vertical translators and rotators. The talk will start with a brief overview of symmetric vertical translators in H2xR, including rotational, parabolic, and hyperbolic examples. This will be followed by recent contributions on rotator-translators, obtained in joint work with R. F. de Lima and A. K. Ramos, where helicoidal symmetry is used to construct a family of rotators and to relate them to vertical translators. Finally, I will discuss the asymptotic geometry of vertical translators in H2xR, based on joint work with G. Pipoli and G. Tinaglia. In particular, I will describe characterization results for the components of the asymptotic boundary, taking into account both the vertical and horizontal parts of the boundary at infinity.

Tensile minimal surfaces

Thursday 11:00 h - 12:00 h

Abstract: In architecture, minimal surfaces belong to the structural family of tensile surfaces. Tensile surfaces are lightweight, form-active structural membranes held in tension by cables, masts, or air pressure, creating durable and efficient shapes. The study of these surfaces saw significant progress in the 19th century, mainly thanks to the work of Frei Otro and his team. In this talk we will introduce a method to produce new minimal surfaces bounded by asymptotic lines with constant curvature that provide tensile structures very efficient in architecture. These surfaces are also the solution to certain symmetric thread problems.

Alexandrov theorem for crystalline bubbles

Thursday 15:00 h - 16:00 h

Abstract: In this talk, I discuss some recent progress concerning bubbling phenomena for almost critical points of anisotropic surface energies. In particular, given a sequence of uniformly convex norms $ \phi_h $ on $ \R^{n+1} $ converging to an arbitrary norm $ \phi $, I discuss the rigidity of $ L^1 $-accumulation points of sequences of sets (of finite perimeter) $ E_h \subseteq \R^{n+1} $, that are volume-constrained almost-critical points of the anisotropic surface energy associated with the norm $ \phi_h $.   Such limits are finite union of disjointed and possibly mutually tangent $ \phi $-Wulff shapes.

Scalar curvature stability of hyperbolic 3-manifolds

Friday 11:30 h - 12:30 h

Abstract: One of the consequences of Perelman's remarkable work on Ricci flow is the following comparison+rigidity result: let (M,h) be a closed hyperbolic 3-manifold; if g is a Riemannian metric on M with scalar curvature at least -6, then the volume of (M,g) is greater thab or equal the volume of (M,h); moreover in the case of equality there is an isometry between the two metrics. This answered in the afirmative the three dimensional version of a conjecture of Schoen. In this talk, we are going to discuss quantitative versions of the rigidity statement. The main tool is the theory of Ricci flows whose initial condition are singular spaces. This is joint work with Ben Lowe (University of Chicago).