Prossimi seminari del Dipartimento di Matematica

We introduce the concept of Ramsey pairs, and show how they can be prove several infinitary results in combinatorics.

In this talk we present an explicit area formula to compute the spherical Hausdorff measure of an intrinsic regular graph in an arbitrary homogeneous group. We assume the intrinsic graph to be intrinsically differentiable at any point with continuous intrinsic differential. The key aspect of the result lies in the introduction of a suitable notion of intrinsic Jacobian and in the computation of an explicit expression for this object. Eventually, we present recent results about the symmetries of some homogeneous distances for which the area formula takes a simplified expression. This is a joint work with V. Magnani (Unipi). Attività di ricerca supportata dal progetto INDAM-GNAMPA-2024:"Free boundary problems in noncommutative structures and degenerate operators" CUP E53C23001670001

The simplicial volume is a homotopy invariant of manifolds; this talk is about the simplicial volume of a Davis' manifolds, obtained from the so-called reflection group trick, which is a powerful method for constructing aspherical manifolds. I will describe an approach based on the study of triangulations of spheres and simplicial maps between them. This approach also presents connections with the theory of graph minors. No knowledge about simplicial volume or Davis' reflection group trick is expected from the audience.

Given a (nice) group G, we are interested in how fast the length of a group element can grow when we apply powers of a given outer automorphism of G. If the group G is free or the fundamental group of a closed surface, classical train-track techniques give a complete and precise picture. This can be extended to automorphisms of all negatively curved (a.k.a. Gromov hyperbolic) groups G, using Rips-Sela theory and the canonical JSJ decomposition. Very little seems to be known beyond this setting. We study this problem for a broad class of non-positively curved groups: "special" groups in the Haglund-Wise sense. In this setting, we prove that: (1) the top exponential growth rate of any automorphism is an algebraic integer; (2) if the automorphism is untwisted, then it admits only finitely many growth rates, and each of these is polynomial-times-exponential.

The classical Plateau problem asks which surface in three-dimensional space spans the least area among all the surfaces with boundary given by an assigned curve S. This problem has many variants and generalizations, along with (partial) answers, and has inspired numerous new ideas and techniques. In this talk, we will briefly introduce the problem in both its classical and modern contexts, and then we will focus on a specific vectorial (planar) type of the Plateau problem. - Given a curve S in the plane, we can ask which diffeomorphism T of the disk D maps the boundary of D to S and spans the least area, computed as the integral of the Jacobian of T, among competitors with the same boundary condition. For simply connected curves, the answer is provided by the Riemann map, and the minimal area achieved is the Lebesgue measure of the region enclosed by S. For more complex curves, possibly self-intersecting, new analysis is required. I will present a recent result in this sense, obtained in collaboration with Prof. Riccardo Scala from the University of Siena, where the value of the minimum area is computed with an explicit formula that depends on the topology of S.

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Transmission problems describe physical phenomena where the behavior of a system changes across a fixed interface, resulting in different PDEs on each side. These PDEs are coupled through a transmission condition, typically of the Neumann type. In this talk, we will discuss some recent results concerning transmission problems governed by fully nonlinear operators which degenerate near the transmission interface. We obtain Holder differentiability of solutions up to the interface, with optimal exponent, which depends pointwise on the rate of degeneracy. Research activity supported by PRIN 2022 7HX33Z - CUP J53D23003610006, Pattern formation in nonlinear phenomena

I will discuss the existence of a solution to the logarithmic Schrödinger equation in a bounded convex domain, with the property that log u is concave. Since the reaction is sign-changing and non-monotone, the classical techniques to attack the problem fail. We instead rely on a continuity argument for the approximating Lane-Emden problems based on the heuristic argument. We will discuss the optimality of the result, exhibiting for any α > 0 a solution such that u^a is not concave. This is a joint work with M. Squassina and M. Gallo.

Given a realcompact space X, we denote by Exp(X) the smallest infinite cardinal κ such that X is homeomorphic to a closed subspace of Rκ. In this talk, we analyze the realcompactness number of countable spaces. We will show that, for every cardinal κ, there exists a countable crowded space X such that Exp(X) = κ if and only if p ≤ κ ≤ c. On the other hand, we show that a scattered space of weight κ has pseudocharacter at most κ in any compactification. This will allow us to calculate Exp(X) for an arbitrary (that is, not necessarily crowded) countable space. This is a joint work with Andrea Medini and Lyubomyr Zdomskyy.

In the first part of the seminar, I will discuss the concept of Birational Geometry, introduce some birational iInvariants and the moduli spaces of varieties of general type, along with a few examples that I consider significant. In the second part, I will present some recent results, obtained in collaboration with S. Coughlan, Y. Hu, and T. Zhang, on certain moduli spaces of three-dimensional varieties of general type.

Given a measured groupoid G, together with Filippo Sarti, we defined a cohomology theory which generalizes the measurable bounded cohomology of a locally compact group. In the particular case of a groupoid associated to a measure preserving action, our cohomology boils down to the usual bounded cohomology of the group with twisted coefficients. We will discuss the possible applications of this result to orbit equivalence. 

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