Archivio 2023 278 seminari

We study the Cauchy problem \begin{equation} \label{CP} \begin{cases} P(t,x,D_t,D_x)u(t,x) =f(t,x) \\ u(0,x)=g(x) \end{cases}, \qquad (t,x) \in [0,T] \times \mathbb{R}, \end{equation} for $p$-evolution operators of the form $$P(t,x,D_t,D_x)= D_t + a_p(t) D_x^p + \sum_{j=1}^{p-1} a_j(t,x)D_x^j, \qquad (t,x) \in [0,T]\times \mathbb{R},$$ where $a_p \in C([0,T], \mathbb{R})$ and $a_j \in C([0,T], C^\infty(\mathbb{R}; \mathbb{C})), j=0,\ldots,p-1,$ in the Gevrey functional setting. When the coefficients $a_j(t,x), j=0,\ldots,p-1,$ of the lower order terms are complex-valued, it is possible to obtain well-posedness results in Gevrey spaces under suitable decay assumptions on $a_j$ for $|x| \to \infty.$ In the first part of the talk, we present a well-posedness result for $3$-evolution equations obtained in [1]. In the second part we discuss necessary conditions for Gevrey well-posedness in the case of $p$-evolution equations for an arbitrary positive integer $p$, see [2]. The results presented in the talk are obtained in collaboration with Alexandre Arias Junior (Universit\`{a} di Torino) and Alessia Ascanelli (Universit\`{a} di Ferrara). References: [1] A. Arias Junior, A. Ascanelli, M. Cappiello, \textit{Gevrey well-posedness for $3$-evolution equatons with variable coefficients}, 2022. To appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci. DOI: 10.2422/2036-2145.202202\_011, https://arxiv.org/abs/2106.09511; [2]A. Arias Junior, A. Ascanelli, M. Cappiello, {\it On the Cauchy problem for $p$-evolution equations with variable coefficients: a necessary condition for Gevrey well-posedness}. Preprint (2023), https://arxiv.org/abs/2309.05571

We illustrate the asymptotic behaviour of the eigenvalue counting function for self-adjoint, positive, elliptic linear operators, defined through classical weighted symbols of order (1,1), on an asymptotically Euclidean manifold X. We first prove a two term Weyl formula, improving previously known remainder estimates. Subsequently, we show that, under a geometric assumption on the Hamiltonian flow at infinity, there is a refined Weyl asymptotics with three terms. This is joint work with Moritz Doll.

In 1978 Metivier showed that a differential operator with real-analytic coefficients is elliptic if and only if any non-analytic Gevrey vector is a Gevrey function of the same order. In this talk we generalize Metivier's Theorem to Denjoy-Carleman classes given by weight sequences. In particular we show that if $\mathcal{E}^{\{\mathbf{M}\}}$ is a Denjoy-Carleman class such that the associated Borel map is surjective, then there is a vector $u$ of class $\{\mathbf{M}\} $ for any non-elliptic differential operator with real-analytic coefficients, which is not an element of $\mathcal{E}^{\{\mathbf{M}\}}$. This is joint work with Gerhard Schindl.

Magic angles are a topic of current interest in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. In this talk, we shall discuss a simple operator describing the chiral limit of twisted bilayer graphene, whose spectral properties are thought to determine which angles are magical. It comes from a 2019 PR Letter by Tarnopolsky--Kruchkov--Vishwanath. By adapting analytic hypoellipticity results of Kashiwara, Trepreau, Sjöstrand, and Himonas, we show that the corresponding eigenfunctions decay exponentially in suitable geometrically determined regions, as the angle of twisting decreases. This is joint work with Maciej Zworski.

It is well known that the De Rham cohomology of a compact Lie group is isomorphic to the Chevalley-Eilenberg complex. While the former is a topological invariant of the Lie group, the latter can be computed by using simple linear algebra methods. In this talk, we discuss how to obtain an injective homomorphism between the cohomology spaces associated with left-invariant involutive structures and the cohomology of a generalized Chevalley-Eilenberg complex. We discuss some cases in which the homomorphism is surjective, such as the Dolbeault cohomology and certain elliptic and CR structures. The results provide new insights regarding the general theory of involutive structures as, for example, they reveal algebraic obstructions for solvability for the associated differential complexes.

This talk will be an overview of some relations between the geometry and analysis of CR structures in three dimensions. The subject began with Poincare's observation that in 2-dimensional complex space there are more real hypersurfaces through a given point than there are local biholomorphisms leaving that point fixed. Elie Cartan then computed the geometric invariants that solve the local equivalency problem. Hans Lewy studied Cartan's work, in particular the partial differential operator on the hyperquatric and its relation to the underlying geometry. This led to his famous counterexample - a linear partial differential equation with no solution. The talk will conclude with the realization problem and a false analogy.

Given a function $f\in L^2(\mathbb{R})$, we consider means and variances associated to $f$ and its Fourier transform $\hat{f}$, and explore their relations with the Wigner transform $W(f)$, obtaining, as particular cases, a simple new proof of Shapiro's mean-dispersion principle, as well as a stronger result due to Jaming and Powell. Uncertainty principles for orthonormal sequences in $L^2(\mathbb{R})$ involving linear partial differential operators with polynomial coefficients and the Wigner distribution, or different Cohen class representations, are obtained, and an extension to the case of Riesz bases is studied. This is a joint work with Chiara Boiti (Università degli Studi di Ferrara) and Alessandro Oliaro (Università di Torino)

Time-frequency localization operators (with Gaussian window) were introduced by Daubechies in 1988. Since then they have been studied intensively, in particular regarding boundedness, compactness, Schatten properties and estimates for the eigenvalues. However, sharp estimates for the norm of these operators are still few. In this talk I will present a classical result by Lieb and two new result that give sharp estimates for the norm of localization operators under the assumption that the weight function belongs to one or more $L^p$ spaces.

We study classes of ultradifferentiable functions defined in terms of small weight sequences violating standard growth and regularity requirements. First, we show that such classes can be viewed as weighted spaces of entire functions for which the crucial weight is given by the associated weight function of the so-called conjugate weight sequence. Moreover, we generalize results from M. Markin from the so-called small Gevrey-setting to arbitrary convenient families of (small) sequences and show how the corresponding ultradifferentiable function classes can be used to detect boundedness of normal linear operators on Hilbert spaces (associated to an evolution equation problem). Finally, we study the connection between small sequences and the recent notion of dual sequences introduced in the PhD-thesis of Javier Jiménez-Garrido. This is joint work with David Nicolas Nenning from the University of Vienna.

Starting out from a new description of a class of parameter-dependent pseudodifferential operators with finite regularity number due to G. Grubb, we introduce a calculus of parameter-dependent, poly-homogeneous symbols whose homogeneous components have a particular type of point-singularity in the covariable-parameter space. Such symbols admit intrinsically a second kind of expansion which is closely related to the expansion in the Grubb-Seeley calculus and permits to recover the resolvent-trace expansion for elliptic pseudodifferential oerators originally proved by Grubb-Seeley. Another application is the invertibility of parameter-dependent operators of Toeplitz type, i.e., operators acting in subspaces determined by zero-order pseudodifferential idempotents.

Minicourse for PhD and master students. Title: Deformations of Fano and Calabi-Yau varieties. Abstract: In the classification of algebraic varieties, we parametrize varieties with similarities, thus it is natural to consider families/deformations of varieties. In good cases, we can parametrize some varieties over a smooth base space and this makes the description of the moduli space of those varieties reasonable. Fano varieties and Calabi-Yau varieties are fundamental objects in the classification. In the talks, I'll explain that, when varieties are Fano or Calabi-Yau, we have such smooth parameter spaces. In most parts, I'll concentrate on smooth Fano/Calabi-Yau varieties. If time permits, I'll also talk about more general cases.

Minicourse for master and PhD students. Title: Hilbert scheme of points on a surface and its degeneration. Abstract: Hilbert scheme of zero dimensional subschemes on a (quasi-)projectuve surface gives an interesting construction of higher dimensional smooth (quasi-)projective varieties of even dimension. For example, if the surface is a K3 surface, the Hilbert scheme gives an example of higher dimensional irreducible symplectic projective manifold. In the first part of this mini course, I explain the basic properties of the Hilbert scheme of points only assuming Hartshorne (i.e. a basic knowledge of modern algebraic geometry). In the second part, I put emphasis on the degeneration of Hilbert schemes. The motivation comes from the study of the boundary behavior of the period map of irreducibel symplectic Kähler manifolds. I also explain an explicit construction of the degeneration of Hilbert schemes.

Title: Moduli spaces of bundles in low genus and degeneracy loci. Abstract: Coble hypersurfaces enjoy very special properties related to abelian varieties and moduli of semistable bundles of rank r with trivial determinant on a curve C of genus g, notably when (g,r) equal to (2,3) or (3,2). Using orbital degeneracy loci arising from Vinberg theta-groups and Hecke cycles, we describe moduli of semistable bundles with fixed odd determinant as subvarieties of Grassmannians, again when (g,r) equals (2,3) or (3,2). The geometry of these loci and of their singularities parallels that of Coble hypersurfaces and is related to projective models of K3 surfaces of genus 13 and 19. Joint work with Vladimiro Benedetti, Michele Bolognesi, Laurent Manivel.

Title: Gauss-Prym maps on Enriques surfaces. Abstract: Let C be a complex projective algebraic curve and let L and M be two line bundles on C. One can associate L and M with some natural maps between spaces of global sections of certain sheaves on C. These are called Gaussian-Wahl maps. These maps have been classically studied in connection with extendability questions of curves on surfaces. In this talk I will focus on the case of Enriques surfaces, presenting some natural questions that arise in this situation.

Title: The Fano variety of lines on a cyclic cubic fourfold. Abstract: Among Fano varieties of K3 type (or FK3 for short) the cubic fourfold stands out for its historical significance. Indeed, long before the terminology FK3 was born there were already examples of a relation between this famous cubic hypersurface and irreducible holomorphic symplectic (or IHS for short) manifolds. One method to associate a smooth cubic fourfold with an IHS manifold involves the Fano variety of lines on it. This is, as proven by Beauville-Donagi, an IHS manifold of type K3[2]. This relation becomes even more intriguing when considering mildly singular cubic fourfolds, e.g. cubic fourfolds Y that are triple covering of P4 branched over a singular cubic threefold. In this case we have that F(Y), the Fano variety of lines on Y, is birational to an IHS manifold of type K3[2]. This fact has been used by Boissière-Camere-Sarti and by me to study some compactification of the moduli spaces of irreducible holomorphic symplectic manifolds with an order three non-symplectic automorphism. In order to achieve this result the authors do not consider the rich geometry of F(Y). I will present recent results obtained in collaboration with Samuel Boissière and Paola Comparin that explain how the geometry of F(Y) gives us a better understanding of the deep relation between cyclic cubic fourfolds and IHS manifolds of type K3[2] with a non-symplectic automorphism of order three.

Title: On the (uni)rationality problem for quadric bundles and hypersurfaces. Abstract: A variety X over a field is unirational if there is a dominant rational map from a projective space to X. We will discuss the unirationality problem for quartic hypersurfaces and quadric bundles over a arbitrary field in the the perspective of the relation between unirationality and rational connectedness. We will prove unirationality of quadric bundles under certain positivity assumptions on their anti-canonical divisor. As a consequence we will get the unirationality of any smooth 4-fold quadric bundle over the projective plane, over an algebraically closed field, and with discriminant of degree at most 12.

Title: Symplectic action of groups of order four on K3^[2]-type manifolds. Abstract: When a group of order four G (either Z/4Z or (Z/2Z)^2) acts symplectically on a K3^[2]-type manifold X, then its action is always standard, meaning that we can deform the pair (X,G) to a pair (S^[2],G), where S^[2] is the Hilbert square of a K3 surface with a symplectic action of G, and the action of G on S^[2] is naturally induced. I will describe these two actions and construct for each one the general member of a projective family that admits it, starting from a family of K3 surfaces with a mixed (symplectic and non-symplectic) action of a group of order 4. Time permitting, I will also talk about the induced involutions on the Nikulin orbifold which is obtained by partial resolution of the quotient X/i, where i is a symplectic involution normal in G.

Title: Recent results on Fano varieties. Abstract: In this talk we present some recent results on complex smooth Fano varieties. To this end, we first recall an invariant introduced by Casagrande, called Lefschetz defect. We review the literature to deduce that all Fano manifolds with Lefschetz defect greater than three are well known. Then we focus on the case in which the Lefschetz defect is equal to three, by discussing a structure theorem for such varieties. As an application, we use this result to classify all Fano 4-folds with Lefschetz defect equal to three: there are 19 families, among which 14 are toric. This is a joint work with C. Casagrande and S. Secci.

Title: On the classification of fine compactified Jacobians of nodal curves. Abstract: If a smooth curve degenerates to a nodal curve, what are the possible modular degenerations of the Jacobian? I will give a complete answer to this question, using some recent results of Pagani-Tommasi.

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In recent years there have been an increasing number of results where quantum physics has been combined with machine learning for different reasons. On the one hand, quantum computers promise to significantly speed up some of the computational techniques used in machine learning and, on the other hand, “classical” machine learning methods can help us with the verification and classification of complex quantum systems. Moreover, the rich mathematical structure of quantum mechanics can help define new models and learning paradigms. In this talk, we will introduce quantum machine learning in all of these flavors, and then discuss how to bound the accuracy and generalization errors via entropic quantities. These bounds establish a link between the compression of information into quantum states and the ability to learn, and allow us to understand how difficult it is, namely how many samples are needed in the worst case scenario, to learn a quantum classification problem from examples. Different applications will be considered, such as the classification of complex phases of matter, entanglement classification, and the optimization of quantum embeddings of classical data.

We will first work through the construction of moduli spaces of quiver representations as GIT quotients, and collect basic geometric properties. We will then work out classes of examples where these spaces can be described explicitly. We will describe geometric techniques for studying moduli spaces, for example coordinates, vector bundles, torus localization, Hilbert scheme.

In these lectures I will present the calculation of the title in the case of the star-shaped quivers related to character varieties based on my joint work with E. Letellier and T. Hausel. The starting point will be a formula of Hua for a general quiver. The basic tool used is the combinatorics of symmetric functions and generating functions, which I will discuss from scratch.

Semi-invariants of quivers and their applications.

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In 2021 Fang and Reineke described the support of linear degenerations of flag varieties in terms of Motzkin paths, by using Knight-Zelevinsky multi-segment duality. In a joint project with Esposito and Marietti (IMRN 2023, arXiv 2206.10281) we give a new characterization of supports in representation-theoretic terms by what we call excessive multi-segments. To do so we consider an algebraic structure on the set of Motzkin paths that we call Motzkin monoid. By using a universal property of the Motzkin monoid, we show that excessive multi segments are parametrized in a natural way by Motzkin paths. Moreover, we show that this parametrization coincides exactly with the Fang-Reineke parametrization. As a byproduct we have an elementary combinatorial criterion to decide if a multisegment is a support. We have an inductive procedure to describe the inverse of the Fang-Reineke map. In this term there is a very beautiful (as yet conjectural) formula for the coefficients.

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Symmetric quivers and symmetric varieties In this talk I will report on ongoing joint work with Ryan Kinser and Jenna Rajchgot on varieties of symmetric quiver representations. These varieties are acted upon by a reductive group via change of basis, and it is natural to ask for a parametrisation of the orbits, for the closure inclusion relation among them, for information about the singularities arising in orbit closures. Since the Eigthies, same (and further) questions about representation varieties for type A quivers have been attached by relating such varieties to Schubert varieties in type A flag varieties (Zelevinsky, Bobinski-Zwara, ...). I will explain that in the symmetric setting it is possible to interpret the above questions in terms of certain symmetric varieties. More precisely, we show that singularities of an orbit closure of a symmetric quiver representation variety are smoothly equivalent to singularities of an appropriate Borel orbit closure in a symmetric variety.

This talk will be an introduction to the field of topological data analysis, emphasizing the role played in it by quiver representation theory. Specifically, I will describe how the supports of the indecomposables can be used as descriptors for data, with stability guarantees under suitable choices of metrics on the representation categories of the quivers under consideration. I will also explain how one proceeds when those quivers are of wild representation type.

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This talk is devoted to the validity and the failure of the strong maximum principle for equations involving the k-th fractional truncated Laplacian or the k-th fractional eigenvalue, which are fully nonlinear integral operators whose nonlocality is somehow k-dimensional. We give in particular geometric characterizations of the sets of minima for nonnegative supersolutions. Based on joint works with I. Birindelli (Sapienza University), H. Ishii (Tsuda University) and D. Schiera (University of Lisbon).

In this talk, I will show what problems arise when trying to obtain Gaffney-type inequalities in subRiemannian geometry, since one cannot simply apply the classical Riemannian tools to this particular setting. I will then present some of the tools that are currently available to tackle this problem, and how they can be applied to obtain the desired results in the case of contact manifolds.

I will discuss a class of quasilinear elliptic equations involving the p-Laplace operator and nonlinearities of Sobolev-critical growth, focusing on existence, non-existence, and compactness issues related to their variational formulation.

We study existence and qualitative properties of solutions to subelliptic problems with Hardy potential and critical nonlinearities on stratified groups. We investigate both the semilinear and the quasilinear case. First, we determine the existence, Lorentz regularity and asymptotic behavior of entire solutions. By convenient transformations, we are naturally lead to study the equation satisfied by the extremal functions to some weighted Sobolev-type inequalities on groups, whose analytic expression is not known. As a byproduct, we derive existence results for the associated Brezis-Nirenberg type problem, depending on the involved parameters. We also obtain non-existence Pohozaev-type results.

In this seminar we present some existence results, in the spirit of the celebrated paper by Brezis and Nirenberg (CPAM, 1983), for a critical problem driven by a mixed local and nonlocal linear operator. More precisely, given a bounded open set in R^n (with n ≥ 4), we consider a perturbed critical problem and we develop an existence theory, both in the case of linear (that is, p = 1) and superlinear (that is, p > 1) perturbations. In the particular case p = 1, we also investigate the mixed Sobolev inequality associated with (P), detecting the optimal constant, which we show that is never achieved. The results discussed in this talk are obtained in collaboration with S. Dipierro, E. Valdinoci and E. Vecchi.

The general themes of the talk are Dirichlet forms, fractional operators and associated Sobolev type spaces on groups of homogeneous type. Our results lead to explicit integral formulas of the infinitesimal generators, which are the studied fractional operators, and embedding theorems between the relevant spaces. The considered groups are not assumed to be Carnot groups or to satisfy a Hörmander type conditions. Finally, we will describe a result on sharp asymptotic decay of solutions to non-linear equations modeled on the fractional Yamabe equation.

We consider the problem of existence of extremal functions for second order Adams' inequalities with Navier boundary conditions on balls in R^n in any dimension n\geq 4. We also discuss some sharp weighted versions of Adams' inequality on the same spaces. The weights that we consider determine a supercritical exponential growth, except in the origin, and the corresponding inequalities hold for spherically symmetric functions only.

Given a complete, connected Riemannian manifold with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium equation with respect to the 2-Wasserstein distance. We produce stability estimates under negative curvature bounds, which to some extent generalize well-known results by Sturm and Otto-Westdickenberg.

We consider the classical geometric problem of prescribing scalar and boundary mean curvature via conformal deformation of the metric on a n-dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature and positive boundary mean curvature. It is known that if n=3 all the blow-up points are isolated and simple. In this work we prove that this is not true anymore in low dimensions (that is n=4, 5, 6, 7). In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and minimizes the norm of the trace-free second fundamental form of the boundary.