Complex Analysis Lab

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 $\|p(T)\|$.

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.

Linear matrix inequalities (LMIs) play a role in many areas of applications and the set of solutions to one is called a spectrahedron. LMIs in (dimension-free) matrix variables model most problems in linear systems engineering, and their solution sets are called free spectrahedra. These are exactly the semialgebraic matrix convex sets. This talk will discuss analytic maps between free spectrahedra and, under certain irreducibility assumptions, classify all those that are bianalytic. The foundation of such maps turns out to be a very small distinguished class of birational maps we call convexotonic. The results depend on new tools in noncommutative analysis, such as a Positivstellensatz for analytic functions whose real part is positive on a free spectrahedron, and fine detail, geometric in nature locally and algebraic in nature globally, about the boundary of free spectrahedra.

The theory of free probability was developed by Voiculescu in the 1980's as an extension of classical probability to the non-commutative setting and aims to study algebras of operators from a probabilistic point of view. The connection of free probability with random matrix theory, as well as the development of notions of entropy in the non-commutative framework, lead to significant breakthroughs regarding the structure of certain von Neumann algebras. More recently, in 2013 Voiculescu laid the foundations of bi-free probability theory, which extends the free setting and involves the simultaneous study of left and right actions of algebras on reduced free product spaces. In this talk, we will give an overview of the contributions of free probability theory to the field of operator algebras and will discuss the development of notions of entropy within the context of bi-free probability theory.

PhD thesis defense

Finite automata are a model of computation in finite memory in theoretical computer science. An automaton gets a string of symbols as an input, and either accepts or rejects it. A deterministic two-way finite automaton (2DFA) has a finite set of states and works by moving over the string back and forth, changing its state. The question studied in this talk is how long could be the shortest string accepted by an $n$-state 2DFA, as a function of $n$. The previously known lower bound is of the order $\Omega(1.626^n)$, and the upper bound is $\binom{2n}{n+1}=O(\frac{1}{\sqrt{n}}4^n)$. In my talk, I construct a family of $n$-state automata with shortest accepted strings of length $\frac{3}{4} \cdot 2^n - 1$. Also I study a special case of direction-determinate automata (those that always remember in the current state whether the last move was to the left or to the right), and determine the maximum length of the shortest accepted string precisely as $\binom{n}{\lfloor\frac{n}{2}\rfloor}-1 = \Theta(\frac{1}{\sqrt{n}} 2^n)$.

In this talk, I will present some techniques to deal with quaternionic slice regular functions in order to obtain global topological properties. These techniques come from abstract algebra and complex analysis. In particular, I will discuss the nature of quaternionic exponential and that of k-th roots as covering maps. I will try to present the topic with several explicit examples. The results are obtained in some works in collaboration with Chiara de Fabritiis and with Samuele Mongodi.

Il problema e' di sapere quali gruppi di Lie nilpotenti nascono naturalmente nell'analisi al bordo di domini complessi alla Nagel--Stein--et al, e la risposta e' tanti.

It is known that no general statements can be made about the rate of convergence in Birkhoff’s ergodic theorem. However, it is possible to obtain some results in specific settings. In this talk I will consider weighted sums and discuss their speed of convergence to the mean value. I will provide estimates for such speed of convergence in terms of Diophantine properties of the involved parameters. This talk is based on an ongoing investigation with N. Chalmoukis, L. Colzani and B. Gariboldi.

After having recalled the content of the first part, I will present a framework developed in the context of logic that allows one to measure the complexity of classification problems in mathematics. I will then explain how the "definable" algebraic invariants developed with Bergfalk and Panagiotopoulos can be used to obtain information on the complexity of classification problems mentioned in the title.

In this talk we will discuss some problems about interpolation in Hilbert spaces of analytic functions and their multiplier algebras. In particular we give a positive answer to a question raised by J. E. McCarthy and J. Agler about strong separation and simply interpolating sequences in reproducing kernel Hilbert spaces with the complete Nevanlinna Pick property. Furthermore, we use a result of A. F. Beardon from Fuchsian groups to show that, in the same class of spaces, there exist sequences with infinite associated measure which are simply interpolating.

The study of interpolating sequences for analytic functions in one or more complex variables is one of the main research areas in complex analysis. It has plenty of applications in fields such as signal theory, control theory, operator theory, etc. For many spaces, like Hardy spaces, these sequences are well understood while for others, like Dirichlet spaces, there exists a characterization which is not very easy to verify. In other circumstances, a characterisation does even not exist. In this scenario it is useful to consider a random setting, which can help us to understand when interpolation is “generic”. In particular in this talk we are going to start introducing deterministic interpolation in the Nevanlinna class, to then consider a random setting, more specifically a radial model (where points’ radii are fixed, while the arguments are uniformly distributed). In this case we will give sufficient condition for a sequence to be or to not be interpolating.

The Bergman projection is a fundamental operator in complex analysis in one and several complex variables. Consequently, its regularity properties on L^p and other function spaces have been extensively studied. In this talk, we discuss some recent results in this direction on strongly pseudoconvex domains with near minimal boundary smoothness. In particular, weighted L^p estimates are obtained on strongly pseudoconvex domains of class C^4, where the weight belongs to a suitable generalization of the Bekolle-Bonami class. For such domains, precise estimates on the Bergman kernel function are unavailable. Consequently, we use a kernel free, operator-theoretic technique that goes back to Kerzman, Stein, and Ligocka, and was subsequently refined by Lanzani and Stein to prove (unweighted) L^p regularity. We will also discuss the fundamental obstruction to improving this result to domains of class C^2 (minimal smoothness). This talk is based on joint work with Brett Wick.

We will present a new Reproducing Kernel Hilbert Space of functions on the real line, having the complete Pick Property. We then characterize its Carleson measures, multipliers and interpolating sequences. Work in collaboration with Nikolaos Chalmoukis.

A classical theorem of Fatou [1] says that a function in the Hardy space of the disc has non tangential limits at almost every boundary point. Conversely for every compact set of Lebesgue measure zero in the boundary of the unit disc there exists a function in the Hardy space which fails to have non tangential limits exactly on this set. So we say that the exceptional sets are exactly the sets of Lebesgue measure zero. In the unit ball of C^n there are two (different) spaces which play the role of the Hardy space in the unit disc. One is the Hardy space of the ball and the other one is the Drury Arveson space. For the first one, exceptional sets in the sense of Fatou have been characterized by Korányi [2]. In this talk we will give a characterization of the exceptional sets of the Drury Arveson space by introducing a new potential theory on the boundary of the unit ball in C^n or equivalently in the Heisenberg group. [1] P. Fatou, (1906) Séries trigonométriques et séries de Taylor, Acta Mathematica, Acta Math. 30, 335–400. [2] A. Korányi, (1969). Harmonic Functions on Hermitian Hyperbolic Space. Transactions of the American Mathematical Society, 135, 507–516.

Let A be a discrete, unbounded, infinite set in R. Can we find a "large" measurable set E in R which does not contain any affine copy x + tA of A (for any in R, t > 0)? If a(n) is a real, nonnegative sequence that does not increase ex- ponentially, then, for any 0 <= p < 1, we construct a Lebesgue measurable set which has measure at least p in any unit interval and which contains no affine copy of the given sequence. We gen- eralize this to higher dimensions and also for some "non-linear" copies of the sequence. Our method is probabilistic. Joint work with M. Kolountzakis (Univ. of Crete). Current address: Department of Mathematics and Applied Mathematics, University of Crete, Heraklion, Greece

I will recall the notion of continuous trace C*-algebra, and present classical classification results for such C*-algebras and their automorphisms in terms of the Cech cohomology of the spectrum due to Dixmier--Douady and Phillips--Raeburn. I will then explain how the framework of Borel-definable homological algebra recently developed with Jeffrey Bergfalk and Aristotelis Panagiotopoulos allows one to refine the analysis and obtain more precise classification results.

The reproducing kernel introduced in the first seminar, and the corresponding Hilbert space structure, are studied using tools from Fourier theory on finite abelian groups.

In 2015 the consistency of the random forest algorothm was linked to a particular reproducing kernel. In the first of some expository talks, I introduce the algorithm and the kernel in question.

5th and last. The horocycles in the hyperbolic disc carry the structure of a commutative group, hence a (crucial) notion of Fourier transform. In the complex ball, the corresponding (Heisenberg) group is noncommutative, hence its Fourier theory is more twisted, although not less crucial. We just give an overview of the subject.

4th in this series of expository seminars. We leave the complex world and discuss some geometric aspects of the Heisenberg group (Riemannian and sub-Riemannian) per se: geodesics, metric spheres, horizontal curves, consequences of the two-step structure of the group...

[3rd in this expository series] By a change of coordinates, the unit ball is mapped onto the Siegel domain, where it is easy to spot the (Heisenberg) Lie group structure of the horocycles and the Riemannian metric obtained by restricting the Bergman-Kobayashi metric to them. Assuming the viewpoint of the visual metric (just a rescaling), the boundary of the Siegel domain inherits a (Heisenberg) sub-Riemannian structure.

In this second (expository) seminar we see that the unit ball in several complex variables carries a Riemannian metric which is invariant under automorphisms. Its restrictions to horocycles converge, after rescaling, to a sub-Riemannian metric on the boundary, which has a natural Lie group (Heisenberg) structure. These same metrics also arise from the study of the Drury-Arveson space on the unit ball (a Hilbert space modelling contractive tuples of communitng operators on Hilbert spaces), as well as from problems in control theory.

In the first of series of expository seminars we review the Poincaré metric, and its connetion to function theory in the unit disc.

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.

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.

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.

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.

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.

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.

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.

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.

The Hardy space and signal processing

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.

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.

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.

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.

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.

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

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

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

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

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)

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

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.

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

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.

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.

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

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.

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.

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.

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.

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.

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)

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.

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.

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).

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.

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.

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.

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

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.

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.

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.

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!).

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.

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

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.

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.