Prossimi seminari del Dipartimento di Matematica

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A classical tool in the study of real closed fields are the fields K((G)) of generalized power series (i.e., formal sums with well-ordered support) with coefficients in a field K of characteristic 0 and exponents in an ordered abelian group G. We generalize previous results about irreducible elements and unique factorization in the subring K((G≤0)).

This seminar presents two recent works focused on sparse signal recovery and inverse problems. The first part introduces the truncated Huber penalty, a non-convex penalty function designed for robust signal recovery. We explore its application in constrained and unconstrained models, proving theoretical properties of the optimal solutions. An efficient algorithm based on the block coordinate descent method is also discussed, along with applications in signal and image processing. The second part covers a generalized Tikhonov regularization framework with spatially varying weights estimated via a neural network. This end-to-end approach integrates adaptive parameter estimation, improving detail preservation.

A valued difference field is a field equipped with a valuation and an automorphism which preserves the valuation ring setwise. In this talk I will discuss various results on these objects, concerning the existence of a section of the valuation compatible with the automorphism and of extensions of the valuation to difference fields extension, the solvability of amalgamation problems of valued difference fields, and the classification of the theory of valued difference fields in the sense of positive model theory. This is joint work with Jan Dobrowolski and Rosario Mennuni.

In the classification of first-order theories, many "dividing lines" have been defined in order to understand the complexity and the behavior of some classes of theories. These classes are usually defined by forbidding some specific configurations of definable subsets. Patterns (of consistency and inconsistency) are essentially descriptions of abstract configurations of sets and hence can give a general framework to study dividing lines. In this talk, we describe this general framework and we introduce a notion of maximal complexity, first defined by Shelah, by requesting the presence of all the exhibitable patterns of definable sets. Weakening this notion, we define Positive Maximality and the PM_k hierarchy, and describe some results about them.

I will talk about the relationship between computability theory, the complexity of sets of reals, and set-theoretical axioms. Using all these, I will prove a ZFC consistency result in fractal geometry, which shows that the Hausdorff dimension of “simple" sets can behave badly under projections.

Inspired by a recent characterisation of coherent topoi as a class of Kan injectives, we provide a tentative definition of fragment of geometric logic. We treat them as mathematical objects, and study them from the point of view of Lindstrom-type theorems.

Group operations on a fixed countably infinite universe form a Polish space G. Thus we can view group properties as isomorphism-invariant subsets of G, and it makes sense to ask: what properties are generic (in the sense of Baire category)? In my talk, I will address this question and if time permits, I may also say a few words about generic properties of compact groups.

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

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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.

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.

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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