Archivio 2024

In this talk I will explain how a certain stochastic pressure equation appears in modelling enhanced geothermal heating (EGS) and how we approach the existence problem. EGS consists of pushing water through crystalline crustal rock at depths of 6-8km, the heat from the rock can then be extracted. Based on empirical observations it seems that the porosity satisfies a log like correlation from at least mm to km scale, the permeability (diffusion coefficient) is assumed to be the exponential of permeability. As a “simple” model, we model the porosity using a Gaussian log-correlated field, the properly normalized exponential thus becomes the so called multiplicative chaos measure. Using Gaussian analysis, we transform the Wick renormalized stochastic problem into a family of weighted elliptic equations, and I will show how regularity of these equations imply existence for the stochastic solution. Joint work with: Benny Avelin (Uppsala), Tuomo Kuusi (Helsinki), Patrik Nummi (Aalto), Eero Saksman (Helsinki), and Lauri Viitasaari (Aalto).

We study a mean field game in continuous time over a finite horizon, T, where the state of each agent is binary and where players base their strategic decisions on two, possibly competing, factors: the willingness to align with the majority (conformism) and the aspiration of sticking with the own type (stubbornness). We also consider a quadratic cost related to the rate at which a change in the state happens: changing opinion may be a costly operation. Depending on the parameters of the model, the game may have more than one Nash equilibrium, even though the corresponding N-player game does not. Moreover, it exhibits a very rich phase diagram, where polarized/unpolarized, coherent/incoherent equilibria may coexist, except for T small, where the equilibrium is always unique. We fully describe such phase diagram in closed form and provide a detailed numerical analysis of the N-player counterpart of the mean field game. Joint work with Paolo Dai Pra (Verona) and Elena Sartori (Padova).

Intrinsic regular surfaces in Carnot groups play the same role as C^1 surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level sets or as continuously intrinsic differentiable graphs. The equivalence of these natural definitions is the problem that we are studying. Precisely our aim is to generalize some results proved by Ambrosio, Serra Cassano, Vittone valid in Heisenberg groups to the more general setting of Carnot groups

Tensors have become widely used in various domains due to their practicality. Tensor factorization techniques are used to solve computationally demanding problems, analyze large datasets, and refine descriptions of complex phenomena. This presentation outlines the development of my research on tensors, including an overview of commonly used tensor methods and their applications in various fields such as remote sensing, multilinear algebra, numerical simulation, and signal processing. Criteria for selecting the most appropriate tensor technique depending on the problem under consideration will be emphasized. The presentation aims to outline the advantages and limitations inherent in these techniques. It explores the challenges and offers insights into current research directions driven by real-world, computational, and applied problems.

In Tannakian formalism, groups and generalizations like groupoids and Hopf algebras can be reconstructed from the fiber functor which is a forgetful strict monoidal functor from its category of modules to the category of vector spaces. Can we do something similar for the actions of groups, their properties and the generalizations ? If a Hopf algebra H coacts on an algebra A by a Hopf action (that is, A becomes an H-comodule algebra) then the category of H-modules acts on the category of A-modules, this action strictly lifts the trivial action of the category of vector spaces on A-modules and also H lifts to a comonoid in H-modules inducing a comonad on the category of A-modules. The comodules over this comonad are the analogues of H-equivariant sheaves and a Galois condition can be stated in terms of affinity in the sense of Rosenberg. We propose taking these properties as defining for a general framework allowing for the definition of Galois condition/principal bundles/torsors in a number of geometric situations beyond the cases of Hopf algebras coacting on algebras. We also sketch how many other examples like coalgebra-Galois extensions and locally trivial nonaffine noncommutative torsors fit into this framework.

In topology, principal bundles are often assumed to be locally trivial. While the original Cartan definition of a principal bundle is inherently global, and has been successfully promoted to noncommutative geometry in the compact/unital setting, the standard concept of local triviality is hard to implement for quantum principal bundles. The goal of this talk is to review the state of the art of the concept of locality for quantum principal bundles. We begin with a guided tour concerning the topology of compact Hausdorff principal bundles. In particular, the difference between the piecewise and local triviality shall be explained. In the main part of the talk, the local triviality of compact quantum principal bundles will be compared with the piecewise triviality of principal comodule algebras. (This is a review talk based on joint works with many collaborators.)

The functional analytic setting of various variational models in Fracture Mechanics requires the use of classes of functions with set of discontinuities of codimension one. The difficulty of finding good discretization for such classes of functions makes the direct numerical simulation of those variational problems challenging and highly problematic. For this reason, numerous regularizations have been proposed, the most successful of which are phase-field functionals. These elliptic regularizations were first introduced and analyzed in the work of Ambrosio and Tortorelli for the Mumford-Shah energy in image segmentation, inspired by a now classical example in phase transition by Modica and Mortola. Ambrosio and Tortorelli type approximations have become very popular both in the communities of Calculus of Variations and of Computational Mechanics to address a number of problems in applied sciences, especially in brittle fracture. In the talk, we will comment on some of those phase-field models, starting with Ambrosio and Tortorelli's, which eventually led to a useful variant for approximating cohesive energies in Fracture Mechanics.

Sunlight constitutes an abundant and endless natural fuel, available worldwide. In a society where a substantial part of the global energy yield is being directly expended at the city scale, urban areas appear as serious candidates for the production of solar energy. Their intrinsic complexity yet makes it challenging. The morphological heterogeneity between urban geometries and intricacy of their materials optical properties especially contribute together to causing important spatiotemporal variations in the distribution of incident solar radiations. The field of irradiance received by a specific urban region (e.g. façade, building, district) may thus rapidely become the result of complex miscellaneous interactions between many degrees of freedom. Besides, Principal Component Analysis (PCA) has been widely validated as an efficient algorithm to identify the principal behavioural features, or modes of variability, of a high-dimensional phenomenon. An approach is proposed here for analysing the variations in space and time of the solar resource within an urban context by means of PCA. A parametric investigation is conducted on a set of theoretical 100×100 m² urban districts, defined as arrangements of cuboid-like buildings, with various typological indicators (Total Site Coverage, Average Building Height) and surface materials (Lambertian, highly-specular) at three different latitudes. For each configuration, the distribution of irradiance incident on the facets of the central building is modelled via backwards Monte-Carlo ray tracing over a full year and under clear sky conditions, with a 15 min timestep and 1 m spatial resolution. PCA is subsequently applied to the simulated radiative fields to extract dominant modes of variation. First results validate energy-based orthogonal decompositions like PCA as efficient tools for characterising the variability distribution of multivariate phenomena in this context, allowing for the identification of district areas subjected to important spatial and temporal variations of the solar resource. Characteristic time scales are clearly represented across successive orders of decomposition. Information about the district morphology is also obtained, with the contribution of surrounding geometries being portrayed by specific spatial modes. Similar prevalent variables are further repetitively encountered across multiple evaluated surfaces, but at different modal ranks.

An elementary argument (for sure well-known to the operator theory community) allows to compare the orthogonal projection of a Hilbert space H onto a given closed subspace of H, with (any) bounded non-orthogonal projection acting among the same spaces: this yields an operator identity that is valid in the Hilbert space H. This paradigm has deep implications in analysis, at least in two settings: -in the specific context where the Hilbert space consists of the square-integrable functions along the boundary of a rectifiable domain D in Euclidean space, taken with with respect to, say, induced Lebesgue measure ds (the Lebesgue space L^2(bD)), and the closed subspace is the holomorphic Hardy space H^2(D). In this context the orthogonal projection is the Szego projection, and the non-orthogonal projection is the Cauchy transform (for planar D), or a so-called Cauchy-Fantappie’ transform (for D in C^n with n¥geq 2). -in the specific context where the Hilbert space is the space of square-integrable functions on a domain D in Euclidean space taken with respect to Lebesgue measure dV, and the closed subspace is the Bergman space of functions holomorphic in D that are square-integrable on D. Here the orthogonal projection is the Bergman projection, and the non-orthogonal projection is some ``solid’’ analog of the Cauchy (or Cauchy-Fantappie’) transform. A prototypical problem in both of these settings is the so-called ``L^p-regularity problem’’ for the orthogonal projection where p¥neq 2. This is because the Szego and Bergman projections, which are trivially bounded in L^2 (by orthogonality), are also meaningful in L^p, p¥neq 2 but proving their regularity in L^p is in general a very difficult problem which is of great interest in the theory of singular integral operators (harmonic analysis). Three threads emerge from all this: (1) a link between the (geometric and/or analytic) regularity of the ambient domain and the regularity properties of these projection operators. (2) applications to the numerical solution of a number of boundary value problems on a planar domain D that model phenomena in fluid dynamics. For a few of these problems there can be no representation formula for the solution: numerical methods are all there is. (3) the effect of dimension: for planar D the projection operators are essentially two and can be studied either directly or indirectly via conformal mapping (allowing for a great variety of treatable domains); as is well known, in higher dimensional Euclidean space there is no Riemann mapping theorem: conformal mapping is no longer a useful tool. On the other hand the basic identity in L^2 (see above) is still meaningful but geometric obstructions arise (the notion of pseudoconvexity) that must be reckoned with.

We will discuss some recent results concerning weak and strong well-posedness of nonlinear stable driven SDEs with convolution interaction kernel, where the kernel belongs to a suitable Besov space. We will in particular characterize how singular the kernel can be in function of the stability index of the driving noise. In connection with some concrete models, some convergence rates for an approximating particle system will be discussed.

In this paper, we introduce a novel observation-driven model for high-dimensional correlation matrices, wherein the largest conditional eigenvalues are modelled dynamically. We impose equal correlations for any pair of assets from the same sector(s), which facilitates the use of a highly efficient alternative expression of the likelihood of a tν-distributed random vector. This alternative expression utilises the canonical form for block correlation matrices by Archakov and Hansen (2020). The dynamics of the eigenvalues is obtained from the Generalised Autoregressive Score (GAS) framework by Creal et al. (2011). We provide an empirical application by constructing Global Minimum Variance (GMV) portfolios using daily returns of 200 assets. In its simplest form, where just a single eigenvalue is updated, our model is extremely fast to estimate. It surpasses the Dynamic Equicorrelation (DECO) model model by Engle and Kelly (2012) and rivals their Block DECO (BDECO) model’s performance in achieving low variance in GMV portfolio returns. Joint work with: Stan Thijssen and Andre Lucas.

The opening segment will explore the conjectural relationships between Hodge structures and Chow groups. The Bloch-Beilinson conjecture suggests a functorial filtration on the Chow groups of smooth projective varieties, underpinned by natural axioms. We anticipate refined structures of Bloch-Beilinson filtrations, particularly within projective hyper-Kähler and Calabi-Yau manifolds, as proposed by Beauville and Voisin. Linking these to the generalized Hodge conjecture allows the formation of explicit conjectures. Verifying these for specific Calabi-Yau manifolds or projective hyper-Kähler manifolds could substantiate both the Bloch-Beilinson and generalized Hodge conjectures. **Part 2 title:** *Voisin's Conjecture and Voisin's Map* Voisin's work, which crafts a series of K-trivial varieties from cubic hype-resurfaces and self-rational maps on them, called the Voisin's map will be the focus here. Notable among these is the Fano variety of lines of a cubic fourfold, a dimension 4 hyper-Kähler manifold. The Voisin's map in this case has been extensively studied. We'll examine higher-dimensional examples, which are all Calabi-Yau manifolds. This session aims to study the geometry of these manifolds and apply their structural insights to the conjectures on algebraic cycles discussed in Part 1, utilizing Voisin's self-rational map as a pivotal analytical tool.

In this talk, we will explore the uniqueness of solutions to variable coefficient Schrödinger equations. We will show that, given appropriate decay assumptions on the coefficients and on the solution at two different times, the corresponding solution must be identically zero. Based on a joint work with Zongyuan Li and Xueying Yu.

CholeskyQR is a popular algorithm for QR factorization in both academia and industry. In order to have good orthogonality, CholeskyQR2 is developed by repeating CholeskyQR twice. Shifted CholeskyQR3 introduces a shifted item in order to deal with ill-conditioned matrices with good orthogonality. This talk primarily focuses on deterministric methods. We define a new matrix norm and make improvements to the shifted item and error estimations in CholeskyQR algorithms. We use such a technique and provide an analysis to some sparse matrices in the industry for CholeskyQR. Moreover, we combine CholeskyQR and our new matrix norm with randomized models for probabilistic error analysis and make amelioration to CholeskyQR. A new 3-step algorithm without CholeskyQR2 is also developed with good orthogonality.

In any context of life, human beings aim to achieve the best possible result with minimal effort. In this talk, we discuss how to implement this general principle to the numerical approximation of partial differential equations, where the aim is to obtain accurate approximations at low computational costs. Using the approximation of the Poisson equation by standard finite element methods as a prototypical example, we show how adaptive algorithms based on rigorous a posteriori error estimation lead to approximations that are, in a certain sense, optimal.

The classical von Neumann inequality shows that for any contraction T on a Hilbert space, the operator norm of $p(T)$ satisfies \[ \|p(T)\| \le \sup_{|z| \le 1} |p(z)|. \] Whereas Ando extended this inequality to pairs of commuting contractions, the corresponding statement for triples of commuting contractions is false. However, it is still not known whether von Neumann's inequality for triples of commuting contractions holds up to a constant. I will talk about this question and about function theoretic upper bounds for $\|p(T)\|$.

The Gauss-Green and integration by parts formulas are of significant relevance in many areas of mathematical analysis and physics, and such applications motivated several investigations to extend these formulas to less regular integration domains and vector fields. These endeavours naturally led to the definition of the divergence-measure fields, which are L^p-summable vector fields whose divergence is a Radon measure. By applying a Leibniz rule between functions of bounded variation and essentially bounded divergence-measure fields, we will prove Gauss--Green formulas for these fields on sets with finite perimeter. It is also of interest to consider as integration domains sets with possibly fractal boundary, such as sets with finite fractional perimeter. To this purpose, we will present a distributional approach to fractional Sobolev spaces and fractional variation, which exploits the notions of fractional Riesz gradient and divergence. This will allow us introduce the fractional divergence-measure fields, which, in perfect analogy with the integer case, are L^p-summable vector fields whose fractional divergence is a Radon measure. Finally, we will provide Leibniz rules involving such fields and suitably regular scalar functions, leading to the fractional version of the Gauss-Green formula. The talk is mainly based on joint works with Kevin R. Payne and Giorgio Stefani.

We present some recent results about a way of defining suitable fractional powers of the sub-Laplacian on an arbitrary Carnot group through an analytic continuation approach introduced by Landkof in Euclidean spaces. Furthermore, we present a stronger outcome in the setting of the Heisenberg group, which is the simplest non-commutative stratified group. Eventually, in this context we propose a geometrical application of our result: we compute the value of suitable momenta with respect to the heat kernel. This is joint work with Fausto Ferrari.

In recent years there has been a growing interest in Ramsey theory and related problems in combinatorics of numbers. Historically, the earliest results in this field are Schur's Theorem ("In every finite coloring of the naturals there exists a monochromatic triple a, b, a+b") and van der Waerden's Theorem ("In every finite coloring of the naturals there exist monochromatic arithmetic progressions of arbitrary length"). A peculiar aspect of this area of research is the wide variety of methods used: in addition to the tools of elementary combinatorics, also methods of discrete Fourier analysis, ergodic theory, and ultrafilter space algebra have been successfully applied. Recently, a further line of research has been undertaken, in which combinatorial properties of sets of integers are studied by methods of nonstandard analysis. In this seminar I will discuss these methods and present some examples of their applications.

Monads and algebraic theories are two categorical approach to universal algebra. In his book Higher Algebra, Jacob Lurie established a relatively comprehensive theory of monads on infinity categories. However, his approach can be difficult in practice to use due to its highly technical nature. In this talk, we will describe a version of generalized algebraic theories in the $\infty$-categorical setting, and show that it is recovers Lurie's theory for nice monads. As an application, we will prove several structural results about monads in the $\infty$-categorical setting. We will also use our result to describe the algebraic theories of E_1, E_2, and E_\infty algebras.

We discuss the characterization of gauge-symmetric functions in the Heisenberg group via various geometric prescriptions on their level sets. In this talk we mainly focus on a family of overdetermined boundary value problems of Serrin type which exhibit both similarities and differences with respect to the classical symmetry result for the torsion function. We show uniqueness results for gauge balls under suitable partial symmetry assumptions for the class of competitor sets. The main technical tool is a new Bochner-type identity for functions with toric/cylindrical invariances. This is a joint project with V. Martino.

In the context of finite horizon mean field games with continuous time dynamics driven by additive Wiener noise, we introduce a notion of coarse correlated equilibrium in open-loop strategies. For non-cooperative many-player games, a coarse correlated equilibrium can be seen as a lottery on strategy profiles run according to a publicly known mechanism by a moderator who uses the (non-public) lottery outcomes to tell players in private which strategy to play. Players have to decide in advance whether to pre-commit to the mediator's recommendations or to play without seeing them. We justify our definition by showing that any coarse correlated solution of the mean field game induces approximate coarse correlated equilibria for the underlying N-player games. An existence result for coarse correlated mean field game solutions, not relying on the existence of classical solutions, will be given; an explicitly solvable example will be discussed as well. Joint work with Luciano Campi and Federico Cannerozzi (University of Milan "La Statale").

Modeling traffic dynamics has highlighted some universal properties of emergent phenomena, like the stop and go congestion when the vehicle density overcomes a certain threshold. The congestion formation on a urban road network is one of the main issues for the development of a sustainable mobility in the future smart cities and different models have been proposed. The quantification of the congestion degree for a city has been considered by various authors and data driven models have been develpoed using the large data sets on individual mobility provided by the Information Communication Technologies. However the simulation results suggest the existence of universal features for the transition to global congested states on a road network. We cope with the question if simple transport models on graph can reproduce universal features of congestion formation and the existence of control parameters is still an open problem. We propose a reductionist approach to this problem studying a simple transport model on a homogeneous road network by means of a random process on a graph. Each node represents a location and the links connect the different locations. We assume that each node has a finite transport capacity and it can contain a finite number of particles (vehicles). The dynamics is realized by a random walk on graphs where each node has a finite flow and move particles toward the connected nodes according to given transition rates (link weights). Each displacement is possible if the number of particles in the destination nodes is smaller than their maximal capacity. The graph structure can be very simple, like a uniform grid, but we have also considered random graphs with maximum in and out degree, to simulate more realistic transport networks. We study the properties the stationary distributions of the particles on the graph and the possibility of the applying the entropy concept of Statistical Mechanics to characterize the stationary distributions and to understand the congestion formation.

TBA

Starting from the pioneering works of Beilinson and Kapranov, derived categories of coherent sheaves on homogeneous varieties G/P have attracted a lot of attention over the last decades. We’ll begin by an introduction into this area and then discuss more recent developments related to Lefschetz exceptional collections, quantum cohomology and homological mirror symmetry.

Hilbert geometries have been introduced as a generalization of hyperbolic geometry, and provide a family of metric spaces where the Euclidean straight lines are geodesics. A Hilbert geometry is said to be divisible if it admits a group of isometries that acts cocompactly on the space. The aim of this talk is to introduce the class of divisible Hilbert geometries and to look at a characterization of hyperbolicity in this class.

The fractional p-Laplacian is a nonlinear, nonlocal operator with fractional order and homogeneity exponent p>1, arising in game theory and extending (in some sense) both the classical p-Laplacian and the linear fractional Laplacian. While behaving similarly to its local counterpart from the point of view of variational and topological methods, this operator requires an "ad hoc" approach in regularity theory. We will give an account on some regularity results for elliptic equations driven the fractional p-Laplacian, either free or coupled with nonlocal Dirichlet conditions: in particular we will discuss interior and boundary Hölder continuity, a special form of weighted Hölder regularity, and a recent local clustering lemma. Finally, we will rapidly hint at some applications such as comparison principles, Hopf type lemmas, Harnack inequalities, and an equivalence principle between Sobolev and Hölder minimizers of the associated energy functional. The talk is mainly based on some very recent collaborations with F.G. Düzgün, S. Mosconi, and V. Vespri.

An Enriques surface is the quotient of a K3 surface by a fixed point-free involution. Klemm and Marino conjectured a formula expressing the Gromov–Witten invariants of the local Enriques surface in terms of automorphic forms. In particular, the generating series of elliptic curve counts on the Enriques should be the Fourier expansion of (a certain power of) Borcherds famous automorphic form on the moduli space of Enriques surfaces. In this talk I will explain a proof of this conjecture.

A Polish group is TSI if it admits a two-side invariant metric. It is CLI if it admits complete and left-invariant metric. The class of CLI groups contains every TSI group but there are many CLI groups that fail to be TSI. In this talk we will introduce the class of α-balanced Polish groups where α ranges over all countable ordinals. We will show that these classes completely stratify the space between TSI and CLI. We will also introduce "generic α-unbalancedness": a turbulence-like obstruction to classification by actions of α-balanced Polish groups. Finally, for each α we will provide an action of an α-balanced Polish group whose orbit equivalence relation is not classifiable by actions of any β-balanced Polish group with β<α. This is joint work with Shaun Allison.

TBA

Transcendental Henon maps are a class of automorphisms of $C^2$ with rich dynamical behavior, yet in some sense, tame enough for general theorems to be proven. In this talk we explore some features of the dynamics of such maps, connecting them with some function theoretical properties of entire transcendental functions in one variable.

Let d be a finite tuple of commuting derivations on a field K. A classical result allows us to associate a numerical polynomial to d (the Kolchin polynomial), measuring the "growth rate" of d. We show that we can abstract from the setting of fields with derivations, and consider instead a matroid with a tuple d of commuting (quasi)-endomorphisms. In this setting too there exists a polynomial measuring the growth rate of d. Joint work with E. Kaplan

This lecture starts from two famous discrete-time dynamic models in economics, namely the Cobweb model to describe price dynamics and the Cournot duopoly model to describe competition between two firms producing homogeneous goods, and shows how their study has stimulated new fruitful streams of literature rooted in the field of qualitative analysis of nonlinear discrete dynamical systems. In the case of the Cobweb model, starting from the standard one-dimensional dynamic model the introduction of new kinds of expectations and learning mechanisms open new mathematical research about two-dimensional maps with a vanishing denominator, leading to the study of new kinds of singularities called focal points and prefocal curves. Analogously, in the case of the two-dimensional Cournot duopoly model, some recent developments are described concerning the introduction of nonlinearities leading to multistability, i.e. the coexistence of several stable equilibria, with the related problem of the delimitation of basins of attraction, which requires a global dynamical analysis based on the method of critical curves. Moreover, in the particular case of identical players, some recent results about chaos synchronization and related bifurcations (such as riddling or blowout bifurcation) are illustrated, with extensive reference to the rich and flourishing recent stream of literature.

We prove a second-order smooth-fit principle for a class of infinite-dimensional singular stochastic control problems. These can be thought of as spatial monotone-follower problems and find applications in spatial models of production and climate transition. Let (D, M, μ) be a finite measure space and consider the Hilbert space H := L^2(D, M, μ; R). Let then X be a H-valued stochastic process on a suitable complete probability space, whose evolution is determined through an SPDE driven by a linear operator A and affected by a cylindrical Brownian motion. The evolution of X is controlled linearly via a vector-valued control consisting of the direction and the intensity of action, a real-valued nondecreasing right-continuous stochastic process, adapted to the underlying filtration. The goal is to minimize an infinite time-horizon, discounted convex cost-functional. By combining properties of semiconcave functions and techniques from viscosity theory, we first show that the value function of the problem V is a C^{1,Lip}(H)-viscosity solution to the corresponding dynamic programming equation, which here takes the form of a variational inequality with gradient constraint. Then, allowing the decision maker to choose only the intensity of the control, and requiring that the given direction of control n is an eigenvector of the linear operator A, we establish that the directional derivative V_n is of class C^1(H), hence a second-order smooth-fit principle in the controlled direction holds for V . This result is obtained by exploiting a connection to optimal stopping and combining results and techniques from convex analysis and viscosity theory.

Il volume simpliciale è un invariante per varietà compatte introdotto da Gromov nel 1982. Pur essendo definito solo utilizzando l'omologia singolare, è strettamente correlato alle strutture geometriche che una varietà può supportare: ad esempio, si annulla su varietà che ammettano metriche con curvatura di Ricci non negativa, ed è positivo per varietà di curvatura negativa. In questo seminario confronteremo il volume simpliciale con alcuni invarianti ad esso correlati, come il minimo numero di simplessi in una triangolazione, o il minimo numero di simplessi singolari in un rappresentante della classe fondamentale a coefficienti interi. A tale scopo, introdurremo un nuovo invariante, chiamato "Filling volume", definito sul mapping class group di varietà. Lavoro in collaborazione con Federica Bertolotti.

Bounded cohomology of groups is a variant of group cohomology that, given a group and a Banach coefficient module over such group, gives graded semi-normed vector spaces. A major role in the theory of bounded cohomology is played by amenable groups and amenable actions as they provide vanishing conditions for bounded cohomology. The goal of this talk is to introduce bounded cohomology of groups and look into its realtion with amenability.

As generative AI technologies are revolutionizing industries and our daily lives, what is going to happen to the role of the mathematician? In this talk, I will highlight recent breakthroughs in deep learning and AI and explore how current and future advancements might alter the way we do mathematics.

In the first part we will review some notions of positivity and base loci for line bundles and how to generalise them to the case of higher rank vector bundles. In the second part we will discuss some geometric interpretations of semiampleness of the cotangent bundle, and some characterizations of abelian verieties and compact complex parallelizable manifolds.

Given a Riemann surface X, character stacks and varieties are geometric objects which parametrize certain representations of the fundamental group of X or, equivalently, local systems with prescribed local monodromies. These objects have a rich geometry and are related, for instance, to the moduli spaces of Higgs bundles through non abelian Hodge correspondence. The cohomology of character stacks and varieties is almost completely understood in the case of a generic choice of monodromies, thanks to the work of Hausel, Letellier and Rodriguez-Villegas and Mellit. In the non-generic case, the geometry of these objects becomes considerably more complicated and their cohomology has not been studied much until recently. In the first part of the talk, I will introduce and define character stacks and varieties and review the known results about the generic case. In the second part, I will focus on the non-generic case and give a sketch of the proof of a formula for the E-series of non-generic character stacks, which is the main result of my PhD thesis.