Seminari BAD

While density compensation has been introduced artificially in iterative SENSE reconstructions as a preconditioner for accelerating the convergence of the conjugate gradient descent algorithm, it turns out that this density compensation appears naturally if the problem is reformulated with appropriate Euclidean products in the image space and the data space. This is also true for compressed sensing reconstruction, which can be seen as solving the same least-squares problem as iterative SENSE, but with L1 regularization. Modifying the Euclidean products also changes the adjoint operator of the linear model, which has important consequences for implementations. The purpose of this talk is to illustrate how conjugate gradient descent can be modified to take different inner products into account.

Compact Hyperkähler manifolds (also called Irreducible Holmorphic Symplectic) are of great interest because of the famous classification theorem of complex manifold with vanishing first Chern class and for the very rich structure on the second cohomology. Most of the geometrical information is in fact recovered by the data of the Hodge decomposition in degree two and its relation with the lattice structure on the integral second cohomology. Due to their rigidity, very few examples have been found and it is very not known if indeed there are others. A family of Hyperkähler fourfolds arise as double covers of EPW sextics associated with a certain Lagrangian space. In a joint work with T. Wawak we construct two quite explicit examples of such fourfolds with a symplectic action of the alternating group A7. These are polarized manifolds with polarization 2 and from a classification of the possible automorphisms only two such examples are expected, but no information on the Picard lattice is available so we are not able to distinguish them yet.

Since their first introduction in 1935, matroids have always been considered a great combinatorial tool for branches of applied mathematics like Coding Theory and Optimization. However, in the last years, a deep interplay between Matroid Theory, Algebraic Topology and Algebraic Geometry has been found: the Combinatorics of matroids allows us to fully describe the cohomology of some interesting varieties (the complement of an arrangement of hyperplanes, the wonderful model of De Concini-Procesi, the reciprocal plane); their Geometry led the way to solve long-standing combinatorial conjectures like the Heron-Rota-Welsh Conjecture and the Top-Heavy conjecture. In this talk, we will present Kazhdan-Lusztig polynomials for matroids, outlining the geometric setting in which they were first introduced in 2016. Many of their properties are still only conjectured, since their definition is given through an intricate recursion. We will show how one can partially solve these conjectures for wide classes of matroids using just combinatorial tools. This is a joint work with L. Ferroni and G.D. Nasr.

A score contains all the information needed to reproduce a music signal in time: the octaves representing different frequency intervals. Time-frequency analysis is a branch of harmonic analysis that studies the relations between tempered distributions and their Fourier transforms that, in this context, precisely represent the amplitude of a signal in time and the amplitude of its frequencies. This seminar aims to provide a brief introduction on several aspects of time-frequency analysis: from uncertainty principles, mathematical results providing hindrances to the exact localization of signals, to pseudodifferential operators, whose study fits surprisingly and naturally into a time-frequency analysis setting.

For a domain in M, the relative heat content is defined as the total amount of heat contained in the domain at time t, allowing the heat to flow outside the domain. We study the small-time asymptotics of the relative heat content associated with smooth non-characteristic domains of a general rank-varying sub-Riemannian structure, equipped with an arbitrary smooth measure. By adapting to the sub-Riemannian case a technique due to Savo, we establish the existence of an asymptotic series, up to order 4. Significant difficulties emerges, as the boundary behavior of the temperature function is not known: we use an “asymptotic” symmetry argument of the heat diffusion to obtain information on the small-time behavior of temperature at the boundary of the domain. This is a joint work with Andrei Agrachev and Luca Rizzi.

\emph{Simplicial volume} is a homotopy invariant of compact manifolds introduced by Gromov in the early '80s. It measures the complexity of manifolds in terms of (real) singular chains. Despite its topological meaning, simplicial volume has many applications in geometry. For instance it provides useful information about the Riemannian volume of negatively curved manifolds. However, as soon as we consider non-compact manifolds its geometric meaning is much more mysterious. Indeed, one may extend the notion of simplicial volume to non-compact manifolds by considering locally finite homology, but its behaviour is not yet well understood. Among the key ingredients for studying the simplicial volume of (non-)compact manifolds, \emph{amenable groups} play a fundamental role. Recall that amenable groups are groups carrying invariant means. The aim of this talk is to investigate the relation between simplicial volume and amenable groups. More precisely, after having introduced the notion of \emph{amenable covering} of compact manifolds, we will discuss a classical vanishing result for the simplicial volume. Later we will construct special amenable coverings of non-compact manifolds. This will allow us to obtain the corresponding vanishing result in this setting. If there will be enough time, we will discuss a striking application of these results: the simplicial volume of the product of at least three non-compact manifolds always vanishes. Some results presented in this talk are part of a joint work with Roberto Frigerio.

As the name declares, the theory of random walks in groups is somehow in the middle of group theory and probability theory. The central question is “how to generate a group efficiently?” In this seminar I will present the notions of diameter, mixing time and expander graphs, as well as some explicit constructions. The second part will be dedicated to the Bourgain-Gamburd Machine, a recent technique to show expansion in quasirandom groups.

Let G be a simple Lie group over C and let Lie(G) be its Lie algebra. The exterior algebra of Lie(G) is extensively studied in literature for its link with the geometry of G and the combinatorics of the Weyl group W(G). In this talk I will present an overview of some classical results about the exterior algebra in a "representation theory"-flavour, with a particular attention to two open conjectures due to Kostant and Reeder.

The biodiversity that we observe on Earth is ultimately the product of evolutionary forces: natural selection, mutations, recombination, and random genetic drift jointly shaped the genome of all living organisms. By employing Next Generation Sequencing techniques, geneticists are now able to observe patterns of DNA sequence variation at different evolutionary scale and with an unprecedented level of details. Coalescent theory represent arguably the most robust mathematical framework describing the connection between those changes in the genome and the demographic history of a species. In this seminar, I will introduce the standard coalescent model, how this arises from Poisson processes and its connection with a simple population genetic model (the Wright-Fisher model). I will then discuss how can we build more sophisticated models within the coalescent framework and how we can apply them to reconstruct the evolutionary history of a species and redefine the species concept itself.

The history of philosophy is full of deep and unanswered questions. Who are we? What is our purpose in this world? How many QQ-Fano threefolds exist in high codimension? Prophets Mori and Mukai in the late 80s provided a classification of smooth Fano threefolds of index one, and their 17+88 families are an irreplaceable tool in every birational geometer working kit. More important than the list itself was the strategy (the 'vector bundle method') and the mantra: Fano threefolds arise as linear section of "good" varieties coming from representation theory. The same results sadly does not extends to the QQ-Fano threefolds case (the natural playground for MMPers). Tom and Jerry models provides a good level of understanding up to codimension three, but the garden is still wild and untamed. In this talk we will provide an overview of the subject, and we will report on some recent progress in the classification.

General topology plays an important role in the study of Banach spaces. In this talk, after giving a glance to the main relations between general topology and Banach spaces, the classes of Corson, Valdivia and non-commutative Valdivia compact spaces, will be presented. Special care will be given to the class of set-theoretical compact trees. This class will be used to address open questions in this field.

The simplicial volume (or Gromov norm) is a homotopy invariant of compact manifolds introduced in 1982 by Gromov in his pioneering paper "Volume and Bounded Cohomology". Roughly speaking, the simplicial volume measures how it is difficult to describe the manifold in question in terms of real singular chains. Despite its topological meaning, the simplicial volume turns out to be a fundamental tool for understanding rigidity phenomena, i.e. to establish obstructions to the existence of some geometric structures in terms of topological invariants. More precisely, working with negatively curved manifolds, simplicial volume provides useful information about their Riemannian volume. The aim of this talk is to give an accessible overview about the notion of simplicial volume and to discuss the main techniques involved in this context (some key words are: degree of maps, amenability and negative curvature). If there will be enough time, I will discuss some classical applications like Mostow rigidity and the study of the variation of the volume during a hyperbolic Dehn filling.

In this talk we will discuss questions about the boundedness and continuity of classical and fractional maximal operators acting on Sobolev spaces and spaces of functions of bounded variation.

The aim of Geometric Group Theory is to get insights on the structure of groups looking at their actions on suitable topological spaces. A significant type of these spaces are classifying spaces for families of subgroups. Classifying spaces for families have been widely studied in the case of the families of finite subgroups and virtually cyclic subgroups, due to their connections with the celebrated Baum-Connes Conjecture and Farrell-Jones Conjecture, respectively. But the definitions are stated for all families of subgroups. The purpose of this seminar is to convince that even off the main road of the two standard families of finite and virtually cyclic subgroups there is fascinating mathematics, that still waits to be explored. In particular, this talk is intended to be an accessible invitation to the investigation of "exotic" classifying spaces in the realm of knot theory. We show how to build a classifying space for a significant family of subgroups of a prime knot group and we investigate its cohomological properties. While playing a little with it, interesting analogies with algebraic number theory will pop out.

Some years ago M.G.Kuhn and T.Steger introduced a class of unitary representations of free groups which they called multiplicative. This class is large enough to include many examples of unitary representations constructed henceforth. Moreover, multiplicative representations have the very interesting property of being tempered, i.e. weakly contained in the regular representation. After recalling the basic notions needed to enter into the field, we outline some algebraic and analytic properties of these representations and we advance some ideas for extending such a study to surface groups and hyperbolic groups.