Complex Analysis Lab

Seminari passati

2022
04 Marzo
Michelangelo Cavina
A stochastic view of Caffarelli-Silvestre theorem
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica, probabilità

Caffarelli and Silvestre gave a celebrated interpretation of the fractional Laplacian in terms of a Dirichlet problem for an elliptic operator. In this introductory and expository seminar we show how this can be viewed in terms of stochastic processes.

2022
28 Gennaio
Isidoros Iakovidis
SZEMEREDI’S THEOREM II
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario interdisciplinare

In this series of expository talks we introduce and discuss tools of ergodic theory such as recurrence theorems in order to give the proof of Szemeredi’s theorem.

2022
21 Gennaio
Isidoros Iakovidis
Szemeredi’s theorem I
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario interdisciplinare

In this series of expository talks we introduce and discuss tools of ergodic theory such as recurrence theorems in order to give the proof of Szemeredi’s theorem.

2021
06 Dicembre
Alessandro Monguzzi, Università di Bergamo
Proper holomorphic mappings and L^p boundedness of the Bergman projection in C^2
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

Reprising from my last seminar I will present some results recently obtained with G. M. Dall'Ara on the L^p boundedness of the Bergman projection of domains that can be covered by the unit ball in C^n. These results were inspired by recent works by Chen, Krantz and Yuan and by Bender, Chakrabarti, Edholm and Mainkar and these latter works were in turn motivated by the L^p regularity of the Bergman projection on generalized Hartogs triangles.

2021
22 Novembre
Gianluca Giacchi
From the time-frequency analysis of Gabor frames to the complex analysis of Fock spaces
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

This seminar introduces the time-frequency analysis of Gabor frames and the sampling problem for the Bargmann-Fock space, which is still an open question in the multivariate situation. It is theoretically possible to reconstruct a signal (a square-integrable function) and its Fourier transform, as integral superpositions of time-frequency shifts operators. In practical applications, Gabor theory provides a discrete version of this reconstruction formula, up to a proper choice of a window function and a discrete subset of the phase-space. More precisely, these two objects have to define a frame of L 2 . For several reasons, Gaussian windows are the best suited for the analysis of signals. It turns out that Gaussian frames are characterized by discrete subsets of the phase-space that define sampling sequences for the Bargmann-Fock space, a problem which falls within the realm of complex analysis.

2021
15 Novembre
Alessandro Monguzzi
An invitation to holomorphic function spaces on Hartogs triangles
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

In 1906 F. Hartogs introduced the domain which is now known as the Hartogs triangle. Such domain is a peculiar non-smooth pseudoconvex domain which often happens to be a good test domain for several conjectures. L. Edholm and J. McNeal recently introduced (2016) some generalizations of the Hartogs triangle in order to study the mapping properties of their associated Bergman projection. Since then several mathematicians studied such generalized Hartogs triangles and this investigation led to new and deep results in more general settings as well. In this expository talk I will review and describe some of these results pointing out possible directions for future research.

2021
18 Ottobre
Javier Gonzalez
Halmos' Invariant Subspace Problem and Local Spectral Theory
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

Abstract: Compact and Normal Operators are two of the best understood classes of operators in terms of existence of invariant subspaces. Nevertheless, it is still open the Halmos Problem: Does every compact perturbation of a normal operator have a non-trivial invariant subspace? In this talk, we will motivate the interest of this particular problem and will focus on the 'baby' case: rank-one perturbations of diagonal operators. We will also introduce some basic aspects from Local Spectral Theory and will show the relation of this theory with invariant subspaces, that will play a prominent role on > the study of these particular operators.

2021
08 Ottobre
Javier Gonzalez
A Brief Introduction to the Invariant Subspace Problem
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

The Invariant Subspace Problem asks if every bounded, linear operator on a complex Hilbert space has a non-trivial invariant subspace. In spite of its simple statement, this is one of the most famous unsolved problems on Operator Theory. We will present and skecth some of the most important techniques that have been developed to try to answer this question. In particular, we will discuss methods to provide non-trivial invariant subspaces for some classes of operators (Compact and Normal Operators) and methods to reduce the problem to the study of lattices of non-trivial invariant subspaces for concrete operators defined on spaces of analytic functions.

2021
01 Ottobre
Nicola Arcozzi
Why do engineers care about the Hardy space?
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

The Hardy space and signal processing

2021
30 Giugno
Alberto Dayan, Norwegian University of Science and Technology in Trondheim
Hausdorff Measures, Dyadic Approximation and Dobinski Set
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

We study the size of the Dobinski set, a subset of the unit interval defined in terms of dyadic approximation. This can be done by using two distinct approaches: first, we give an analogous of Jarnik’s Theorem in Diophantine approximation that fits our dyadic setting, and we use such a result ( together with some reference to quasi-independence) to prove that the Dobinski set has logarithmic Hausdorff dimension equal to 1, with associated logarithmic Hausdorff measure equal to infinity. We will also see how the same result can be proven by using technical estimates on the Hausdorff dimension for willow Cantor sets constructed inside the Dobinski set. This is a joint work with José Luís Fernández and Maria Jose González.

2021
14 Aprile
Nikolaos Chalmoukis
A generalization of a theorem of Sarason for elliptic semigroups of analytic functions
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

We will discuss the maximal subspace of strong continuity of a semigroup of composition operators acting on the space of analytic functions of bounded mean oscillation in the unit disc. The minimality of this space is related to a well known theorem of Sarason about the space of analytic functions of vanishing mean oscillation. In the case of elliptic semigroups we give a complete characterization in terms of the Koenigs function of the semigroups that can replace rotations in Sarason's Theorem. This improves previous results of Blasco et al. Similar results are also obtained for the Bloch space. This is a joint work with V. Daskalogiannis.

2021
12 Aprile
Alexander Bufetov
POINT PROCESSES AND INTERPOLATION
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario interdisciplinare

The Kotelnikov theorem recovers a Paley-Wiener function from its restriction onto an arithmetic progression. A Paley-Wiener function can also be recovered from its restriction onto a realization of the sine-process with one particle removed. If no particles are removed, then the possibility of such interpolation for the sine-process is due to Ghosh, for general determinantal point processes governed by orthogonal projections, to Qiu, Shamov and the speaker. If two particles are removed, then there exists a nonzero Paley-Wiener function vanishing at all the remaining particles. How explicitly to interpolate a function belonging to Hilbert space that admits a reproducing kernel, given the restriction of our function onto a realization of the determinantal pont process governed by the kernel? For the sine-process, the Ginibre process, the determinantal point process with the Bessel kernel and the determinantal point process with the Airy kernel, A.A. Borichev, A.V. Klimenko and the speaker proved that if the function decays as a sufficiently high negative power of the distance to the origin, then the answer is given by the Lagrange interpolation formula.

2021
31 Marzo
Nicola Arcozzi
The Siegel half-space: a toolbox II
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

2021
24 Marzo
Nicola Arcozzi
The Siegel upper-half space: a toolbox I
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

The Siegel domain plays in several complex variables the role of the upper half plane in the one-dimensional theory. One of its advantages over the unit ball is that the boundary behavior of the Bergman metric is rather transparent in Siegel coordinates; another one is that its boundary has a (noncommutative) group structure and (noncommutative) Fourier analysis can be used; another one is that the derivative with respect to the "vertical" direction encodes much information concerning classical holomorphic spaces. In these seminars we will give an overview of the rich toolbox available when studying function theory over such domain. The first one will be devoted to the Bergman metric and the automorphism group.

2021
17 Marzo
Matteo Fiacchi
Gromov hyperbolic spaces: general theory and applications to complex geometry
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario interdisciplinare

Gromov hyperbolic spaces are a generalization of negative curved spaces in the general context of metric spaces. We present different (but equivalent) definitions and main results (Shadowing lemma, Gromov compactification and homeomorphic extension of quasi-isometries). Finally, we apply the general theory to complex domains endowed with the Kobayashi distance.

2021
10 Marzo
Matteo Fiacchi
An introduction to pseudoconvexity in several complex variables
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario interdisciplinare

We give a introduction, assuming no prerequisite, to the notion of pseudoconvexity in several complex variables.

2021
03 Marzo
Nicola Arcozzi
The Carleson measures for the Dirichlet space II (Zoom: 842 3922 3003)
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

A characterization of the Carleson measures for the Dirichlet space is given.

2021
24 Febbraio
Nicola Arcozzi
The Carleson measures for the Dirichlet space (Zoom: 837 9129 8028)
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

The Carleson measures for the Dirichlet space: their use and their characterizations. Codice Zoom: 837 9129 8028

2021
19 Febbraio
Nicola Arcozzi, Cristiana De Filippis, Pavel Mozolyako, Nikolaos Chalmoukis
Microworkshop on Wolff's inequality and potential
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

The goal of this very informal, online workshop is bringing together some researchers who use, or who are interested in, Wolff's inequality and potentials in different sub-areas of mathematics, to exchange their viewpoints on the topic. id Zoom: 842 2115 6698 Nicola Arcozzi (UniBo): Wolff's inequality and dyadicization 2.00pm Cristiana De Filippis (U. Torino): Calderón-Zygmund estimates and non-uniformly elliptic operators 2.45PM Pavel Mozolyako (St. Petersburg State University): A probabilistic proof of Wolff's inequality 3.30pm Nikolaos Chalmoukis (UniBo): TBA 4.15pm

2021
17 Febbraio
Nicola Arcozzi
The Holomorphic Dirichlet space: a crash course
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

The Dirichlet space is the only conformally invariant, holomorphic Hilbert space on the unit disc. In this elementary introduction (no prerequisites) we see it from the viewpoint of hyperbolic geometry, Reproducing Kernel Hilbert Spaces, fractional derivatives, and dyadic analysis. (In presenza: Seminario VIII piano)

2018
16 Novembre
Pavel Mozolyako
THE DIRICHLET SPACE ON THE BI-DISC V
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

On work in collaboration with N. Arcozzi, M. Perfekt, G. Sarfatti,

2018
14 Novembre
Artur Nicolau (Universitat Autònoma de Barcelona)
The Corona Theorem in Nevanlinna quotient algebras and interpolating sequences.
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

Let H be the algebra of bounded analytic functions in the unit disc and let I be an inner function. In 2007, Gorkin, Mortini and Nikolski studied the Corona problem in the quotient algebra H/IH and proved that there is no corona if and only if I satisfies the so called weak embedding property. We discuss an analogue problem for quotients of the Nevanlinna class and show that in contrast with the previous case, a complete answer can be given in terms of interpolating sequences. This is joint work with Xavier Massaneda and Pascal Thomas.

2018
09 Novembre
Nicola Arcozzi
The Dirichlet space on the bidisc IV
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

On work in collaboration with P. Mozolyako, M. Perfekt, G. Sarfatti,

2018
05 Novembre
Nikolaos Chalmoukis
Onto Interpolation for the Dirichlet space and for Sobolev W1,2(D)
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

We will discuss a characterization of onto interpolating sequences with finite associated measure for the Dirichlet space in terms of capacity of some condensers. The same condition in fact characterizes all onto interpolating sequences for W1,2(D) even if the associated measure is infinite.

2018
31 Ottobre
Matteo Levi
Distortion of sets under inner functions
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

Fernandéz and Pestana in the 90’s proved that the Hausdorff content of a set E in the unit circle is bounded by the content of its preimage under any inner function of the unit disc fixing the origin. We extend this result in to inner functions whose fixed points lie on the boundary of the disc. To this purpose, we introduce a specific measure, which happens to be of some interest by its own. Time permitting, we will also present some applications. The talk is based on a joint work with Artur Nicolau and Odí Soler, from Universitat Autònoma de Barcelona.

2018
26 Ottobre
Nicola Arcozzi
THE DIRICHLET SPACE ON THE BI-DISC III
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

Having left the holomorphic sea, we go into combinatorial arguments on the bi-tree.

2018
19 Ottobre
Marco Peloso
A brief introduction to de Branges spaces in one and several variables
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

The de Branges spaces constitute a fundamental class of Hilbert spaces of entire functions. The theory of such spaces is well developed but well known only to a small group of experts. We intend to point out a few fundamental facts about these spaces and discuss a non-obvious but very natural generalisation to several variables.

2018
18 Ottobre
Nicola Arcozzi
THE DIRICHLET SPACE ON THE BI-DISC II
nell'ambito della serie: COMPLEX ANALYSIS LAB
This is the second of a series of seminars based on a preprint by N. Arcozzi, P. Mozolyako, K.M. Perfekt, G. Sarfatti. We consider "soft" arguments from Functional Analysis and a discretization scheme.

2018
12 Ottobre
Nicola Arcozzi
The Dirichlet Space on the bi-disc
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

This is the first of a series of seminars based on a preprint by N. Arcozzi, P. Mozolyako, K.M. Perfekt, G. Sarfatti. Many questions and problems will be mentioned in the course of the talks. One of the goals is to have, at the end of the series, some ideas on how to develop a Multiparameter Potential Theory.

2018
11 Settembre
Charles Curry
Sub-Riemannian Brownian motion and numerics for SDEs
nell'ambito della serie: COMPLEX ANALYSIS LAB
We place the numerical method of Cruzeiro, Malliavin and Thalmeier for simulation of elliptic diffusions in the context of Riemannian geometry and discuss possible extensions to the hypoelliptic case.

2018
08 Giugno
Pavel Mozolyako
The Dirichlet Space: a discrete approach
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

We consider the problem of characterizing the Carleson measures for the Dirichlet space on the bidisc and reduce it to a problem concerning a bilinear Hardy operator on the direct product of two trees, which can be solved. After giving a brief introduction to the Dirichlet and Hardy spaces of analytic functions, we introduce the basics of (logarithmic) potential theory on the bitree and investigate some naturally arising capacitary-type inequalities. Possible further inquiries and related problems are discussed. Work in collaboration with Nicola Arcozzi, Karl-Mikael Perfekt, and Giulia Sarfatti.

2018
09 Marzo
Chiara De Fabritiis
s-Regular Functions which Preserve a Complex Slice
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

Regular functions on the skew-field of quaternions were introduced by Gentili and Struppa some 10 years ago in order to give an analogue of holomorphic functions in a non commutative setting. After a (short) introduction, I will give a formula which allows us to simplify the understanding of the *-product, which corresponds to the pointwise product of holomorphic functions. The peculiar structure of quaternions, foliated in copies of complex plane, drives naturally to consider the classes of functions which preserve either one or all complex slices. The main part of the talk will be devoted to characterize the functions whose sum, *-product or conjugate preserve a slice. At the end, I will address to the case of *-powers which shows an unexpected connection with a problem of algebraic geometry studied by Causa and Re. (Joint work with A. Altavilla)

2018
16 Febbraio
Berardo Ruffini
Some results on an isoperimetric model for charged liquid drops
nell'ambito della serie: COMPLEX ANALYSIS LAB
We consider an isoperimetric model, originally proposed by Lord Rayleigh, aimed to describe the (lack of) equilibria of a liquid conducting drop in presence of a charge on its surface. The resulting functional contains an attracting term, usually modeled by the perimeter of the drop, and a repulsive term depending on the amount of charge considered and the electric capacity of the drop. We show that, quite surprisingly, the resulting variational problem is ill posed. We then consider several modification of it and we investigate existence, uniqueness and stability issues about those problem. The talk is based on works in collaborations with M. Goldman, C. Muratov and M. Novaga.

2018
26 Gennaio
Giulia Sarfatti
Shift invariant subspaces of the quaternionic space of slice L^2 functions
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

In this seminar I will introduce the space of slice L^2 functions over the quaternions and I will present a characterization result for (simply and doubly) invariant subspaces for the shift operator, recently obtained in collaboration with Alessandro Monguzzi. Besides its own interest, our result gives a different proof for the quaternionic analog of the classical Beurling Theorem and allows us to obtain an inner-outer factorization for functions in the quaternionic Hardy space.

2017
15 Dicembre
Pavel Mozolyako
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

In this talk we will present the basics of harmonic measure (with focus on the probabilistic interpretation). We will give a (very brief) overview of its history in analysis, properties and discuss some related results (mostly about the geometrical properties of its support).

2017
01 Dicembre
Davide Cordella
Discrete extremal length on graphs and square tilings
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

The notions of extremal metric and extremal length can be develop on the discrete context of finite graphs. In particular, following an article by Oded Schramm (1993), they are the main tool to build a correspondence between the 1-skeleton of triangulations of a quadrilateral and square tilings: the squares are associated to the vertices in a combinatorial fashion to fill a rectangle with no overlaps. The extremal metric expresses the length of the edge of the squares. Furthermore, extremal length can be considered on trees. It is in some sense the reciprocal of the notion of capacity from potential theory.

2017
24 Novembre
Nicola Arcozzi
Interpolating sequences for the Dirichlet Space I
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

Interpolating sequences for the holomorphic Dirichlet space and its multiplier space were characterized in two preprints of 1994: Marshall and Sundberg, and Chris Bishop. Both articles never were published, but they were very influential; especially the second one, which is linked below. The plan is going through the details of the proof, following a third route opened a few years later by B. Boe. The exposition will be rather technical (the generalities about interpolation by sequences were already covered in previous seminars), but some interesting open problems will be mentioned.

2017
17 Novembre
The layer potential approach to the spectra of transmission problems on domains with singularities II
nell'ambito della serie: COMPLEX ANALYSIS LAB
The Neumann-Poincare (NP) operator (or the double layer potential) has classically been used as a tool to solve the Dirichlet and Neumann problems of a domain. However, it also serves as a prominent example in non-self adjoint spectral theory, due to its unexpected behaviour for domains with singularities. Recently, questions from materials science have revived interest in the spectral properties of the NP operator on domains with corners, edges, and conical points. This surge in attention is owed to the connection with resonances of transmission/scattering problems used to model surface plasmons in nanoparticles. I aim to give an overview of recent developments, with particular focus on the NP operator’s action on the energy space of the domain. I will also present recent work for domains in 3D with conical points featuring rotational symmetry. In this situation, we have been able to describe the spectrum both for boundary data in L^2 and for data in the energy space. In the former case, the essential spectrum consists of the union of countably many self-intersecting curves in the plane, and outside of this set the index may be computed as the winding number with respect to the essential spectrum. In the latter case the essential spectrum consists of a real interval. Based on joint work with Johan Helsing and Mihai Putinar.

2017
14 Novembre
The layer potential approach to the spectra of transmission problems on domains with singularities I
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

The Neumann-Poincare (NP) operator (or the double layer potential) has classically been used as a tool to solve the Dirichlet and Neumann problems of a domain. However, it also serves as a prominent example in non-self adjoint spectral theory, due to its unexpected behaviour for domains with singularities. Recently, questions from materials science have revived interest in the spectral properties of the NP operator on domains with corners, edges, and conical points. This surge in attention is owed to the connection with resonances of transmission/scattering problems used to model surface plasmons in nanoparticles. I aim to give an overview of recent developments, with particular focus on the NP operator’s action on the energy space of the domain. I will also present recent work for domains in 3D with conical points featuring rotational symmetry. In this situation, we have been able to describe the spectrum both for boundary data in L^2 and for data in the energy space. In the former case, the essential spectrum consists of the union of countably many self-intersecting curves in the plane, and outside of this set the index may be computed as the winding number with respect to the essential spectrum. In the latter case the essential spectrum consists of a real interval. Based on joint work with Johan Helsing and Mihai Putinar

2017
10 Novembre
Nikolaos Chalmoukis (unibo)
Generalized integration operators on Hardy spaces
nell'ambito della serie: COMPLEX ANALYSIS LAB
I'll start with presenting some known results about the boundedness and compactness properties of the generalized Cesaro operator, Tg, on Hardy spaces in the unit disc, as well as some of its applications. In the second part we introduce a variant of this operator which depends on an analytic symbol g, and we prove the analogous results for this operator. As an application we generalize a theorem of J. Rattya about complex linear differential equations, and we prove a result about factorization of derivatives of Hardy functions.

2017
03 Novembre
Alessandro Monguzzi (Università di Milano)
Beurling and Lax Theorems on invariant subspaces
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

The study of invariant subspaces of Hilbert space operators is a classical problem in analysis. It is an open question whether every operator on an Hilbert space has an invariant subspace other than the trivial ones, the zero subspace and the whole space. A. Beurling completely characterized the invariant subspaces of the unilateral shift $(a_0,a_1,\ldots)\mapsto(0,a_0,a_1,\ldots)$ on $\ell^2(\mathbb{N})$ modeling the shift operator on $\ell^2(\mathbb{N})$ with the multiplication by $z$ on the Hardy space of the unit disc. Few years later P. Lax proved an analogous result, that is, he characterized the translation invariant subspace of $L^2(0,\infty)$. In this seminar I will illustrate Beurling and Lax's result. Time permitting, I will also present an analogous of Beurling's result in the setting of the quaternionic Hardy space of the unit ball. This result was recently obtained in a joint work with G. Sarfatti.

2017
27 Ottobre
Nicola Arcozzi
An application of analysis on trees: some trace inqualities for holomorphic functions
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

Why doing analysis on trees, besides the intrinsic interest? We show as application the characterization of the Carleson measures (or "trace measures") for the Dirichlet space. The seminar requires virtually no prerequisite on holomorphic functions.

2017
27 Ottobre
Nicola Arcozzi
Interpolation in Reproducing Kernel Hilbert Spaces I
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

We discuss interpolating sequences for RKHS, with special emphasis on the Dirichlet space. The general plan is to cover in a few hours: - interpolation in RKHS in general; - the case of RKHS with the complete Nevanlinna-Pick property; - the Dirichlet space; -the problem of "onto interpolation" on the Dirichlet space; - the weighted Dirichlet spaces (with the construction of Peter Jones). It would be nice to finish with the recent result of Alexandru Aleman, Michael Hartz, John E. McCarthy, Stefan Richter (https://arxiv.org/abs/1701.04885) characterizing the interpolating sequences for the spaces having the complete Nevanlinna-Pick property and their multiplier spaces (volounteers are welcome!).

2017
20 Ottobre
Matteo Levi
Equilibrium measures on trees
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

On a metric space (X,d) one can define a set function called capacity which has motivations coming from Physics and which plays a deep role in potential theory and geometric measure theory. It is well known that to any compact subset E of X can be associated a probability measure m on X, called equilibrium measure for E, such that m(E)=cap(E). These measures at present are not well understood. We will present a characterization of equilibrium measures when X is a locally finite tree of infinite depth.

2017
13 Ottobre
Artur Nicolau (Universitat Autònoma de Barcelona)
Finitely Generated Ideals in the Nevanlinna Class
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

Interpolating sequences for the Nevanlinna class will be used to discuss a natural problem on finitely generated ideals in the class

2017
29 Settembre
Pavel Mozolyako
Some problems in potential theory on the polytrees
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

Moving a continuous problem to a discrete setting is a popular and everpresent approach, that (from time to time) allows to single out the geometric and combinatorial issues of the question ad hand. A particular case we are aiming to investigate is the representation of the unit disc (polydisc) by a dyadic tree (cartesian product of dyadic trees), and its connection to Dirichlet type spaces. Following Arcozzi, Rochberg, Sawyer and Wick we give a (very brief) introduction to the potential theory on a polytree, and then present a very incomplete list of related problems.

2017
22 Settembre
Nicola Arcozzi
Wolff's proof of Wolff's inequality (in the dyadic setting)
nell'ambito della serie: COMPLEX ANALYSIS LAB

seminario di analisi matematica

In 1983 Tom Wolff proved a surprising inequality which proved to be a pivotal tool in Nonlinear Potential Theory. I will go through Wolff's proof, but in the dyadic setting. Hedberg, L. I.; Wolff, Th. H. Thin sets in nonlinear potential theory. Ann. Inst. Fourier (Grenoble) 33 (1983), no. 4, 161–187.