Analysis seminar

The seminar will usually take place on Mondays 13:15-14:00, room 656.

Abstract:

'We consider operators of the type \(D^\alpha \Gamma_\phi : H^2(\mathcal{H}) \to H^2(\mathcal{H})\), where \(D^\alpha\) denotes a fractional differentiation operator, and \(\Gamma_\phi\) is a Hankel operator. For \(\alpha>0\), we characterize boundedness in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries. The first is that our main result does not extend to \(\alpha = 0\), i.e. Nehari-Page BMOA is not characterized by the natural anti-analytic Carleson embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of \(\Gamma_\phi\). The third is that there exists a bounded analytic function for which the associated anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of an analytic Carleson embedding does not imply that the anti-analytic ditto is bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg

Abstract: We prove that for every \(c<1/\sqrt{2}\) there exists arbitrarily large \(T\) with \(|\zeta(1/2+iT)|>\exp(c\sqrt{\log T\log\log\log T/\log\log T})\). This improves classical results by Montgomery, Balasubramanian-Ramachandra, and Soundararajan. Our proof uses Soundararajan's resonance method, multiplicative functions, and a certain large greatest common divisor sum.

Abstract: We shall discuss the influence of Omega bounds on Lebesgue measure of certain sets on oscillation of error terms appearing in various asymptotic formulas. The very general setup of this method allows some new and interesting applications.

Abstract: In this talk, we will discuss the embedding problem for \(H^p\) spaces of Dirichlet series and some connections to Hankel forms and composition operators. For even integers \(p\) we will give a simple proof of the embedding, which gives the best constant. We will also show how a bad inequality in the unit disc can be iterated to obtain a good inequality for Dirichlet series. This inequality allows us to optimally embed \(H^p\) spaces of Dirichlet series into Bergman spaces in the half-plane. If time permits, we may consider functionals and/or sharp norm estimates for composition operators.

Abstract: Wrapping it all up. Or, less succinctly, rapping about statistical properties (such as certain types of concentration of measure bounds) of a dynamical system, their consequences to the regularity of the Lyapunov exponents of a linear cocycle, their connections and applications to the spectral theory of discrete Schrodinger operators, Jacobi operators and the like.

Gunnar Taraldsen and Bo Henry Lindqvist

Abstract: Alfred Renyi formulated, in his book “Foundations of Probability”, a generalization of the axioms of Kolmogorov to allow unbounded measures in the theory. It turns out that this theory is connected with a project we started out in 1997 to clarify certain aspects of a conditional Monte Carlo method suggested by Engen and Lillegård. The method produces conditional samples given a sufficient statistic, and can hence be used to construct optimal inference procedures.

After an introduction we will present the theory of Renyi, including a proof of his structure theorem for conditional probability spaces. This theory seems – unfortunately – to be unfamiliar for many researchers in measure theory. Readers who acknowledge the need for a theoretical basis for statistical inference including unbounded measures are urged to consider the theory of Renyi. The theory gives also a possible and promising direction for research in both pure mathematics and applied statistics since Renyi left the theory unexplored to a large extent. Renyi died at the age of 48 before his book “Foundations of Probability” was published.

Abstract: The theory of Gabor frames yields convenient series representations of functions in \(L^2(R^d)\). However, the Balian-Low theorem states that such a series representation can never be an orthonormal basis for \(L^2(R^d)\). The mysterious construction of Wilson bases allows a way to circumvent this problem. In this talk we shed light and new insight on this (not well understood) corner of time-frequency analysis. The talk is based on joint work with Marcin Bownik, Jakob Lemvig, and Kasso Okoudjou.

Abstract: Nadirashvili conjectured that for any non-constant harmonic function in \(R^3\) its zero set has infinite area. This question was motivated by the Yau conjecture on zero sets of Laplace eigenfunctions. Both conjectures can be treated as an attempt to control the zero set of a solution of elliptic PDE in terms of growth of the solution. For holomorhpic functions such kind of control is possible only from one side: there is a plenty of holomorphic functions that have no zeros. While for a real-valued harmonic function on a plane the length of the zero set can be estimated (locally) from above and below by the frequency, which is a characteristic of growth of the harmonic function. We will discuss the notion of frequency, its properties and applications to zero sets in the higher dimensional case, where the understanding is far from being complete.

Abstract: L. A. Bunimovich and B. Z. Webb developed a theory for transforming a finite weighted graph while preserving its spectrum, referred as isospectral reduction theory. A key concept in this theory is that of "structural set", corresponding to subsets of the vertex set to which a graph may be reduced.

In this seminar we discuss an extension of this theory to a class of Markov type operators on Banach spaces, applicable to infinite countable weighted graphs with a finite structural set.

Joint work with M. Joana Torres.

The seminar will usually take place on Mondays, 14:15-15:00, room 656 SB2.

Abstract: "We will discuss interrelations between approximations by polynomials in the closed domain of the complex plane and in the segment."

Abstract: "Microlocal analysis exploits mathematical manifestations of the classical/quantum (particle/wave) correspondence and has been a very successful tool in spectral theory and partial differential equations. We can say that these last two fields lie on the "quantum/wave side".

Recently, microlocal methods have been applied to the study of classical dynamical problems, in particular of chaotic (Anosov) flows. I will explain how it works in the context of Ruelle resonances, decay of correlations and meromorphy of dynamical zeta functions.

The talk, based on recent works of Datchev, Dyatlov, Faure, Guillarmou, Giulietti, Liverani, Nonnenmacher, Sjöstrand, Paternain, Pollicott, Salo, Tsujii, Uhlmann and the speaker, will be non-technical and is intended as an introduction to both microlocal analysis and to chaotic dynamics."

Abstract: "The genetic code is connection between 64 codons and 20 amino acids with 1 stop signal. Codons are the building blocks of genes and amino acids are the building blocks of proteins. There is a huge number of possibilities to map 64 elements onto set of 21 element, but in living cells there is 1 possibility (one genetic code) with some slight variations. Using p-adic distance between codons we get ultrametric structure of the codon space which life employed in construction of the genetic code: if 5-adic and 2-adic distance between two codons is the shortest, then they code the same amino acid. p-Adic distance can be also extended to investigation of similarity (nearness) between strings of DNA, RNA and amino acids. In this talk a review of p-adic aspects of the genetic code will be presented."

Abstract: "One possible setup for studying Gabor frames is to study modules over certain convolution algebras. More precisely, a function defined on the real line can be acted upon by time-shifts (which are unitary operators), and multiplication operators by periodic functions (which are linear combinations of frequency-shifts). The algebra generated by these multiplication operators and unitaries is a convolution algebra over a well-studies dynamical system: a rotation on the circle.

The abstract theory of equivalence between dynamical systems (and groupoids) allows to obtain some concrete results for the frame operator associated with a given family of time-frequency shifts.

[joint work with Franz Luef]"

Abstract:"Describing the behavior of the orbits of a dynamical system can be a challenging task, especially for systems that have a complicated topological and geometrical structure. A very useful way to obtain features of such systems is via invariant probability measures. In the case that the system admits more than one invariant probability measure, an efficient way to chose an interesting one is to select those that have regular Jacobians. These types of measures are called equilibrium states. Their study is a field in ergodic theory called thermodynamical formalism, which was introduced by Sinai in the 70s and it is being developed ever since.

In this talk I will discuss the existence and uniqueness of equilibrium states for a family of partially hyperbolic systems with respect to Holder continuous potentials with small variation. Moreover, I will also describe some statistical properties for the unique equilibrium measure."

Abstract: "I will present a joint work with Jordi Marzo. We construct an orthonormal system of homogeneous polynomials in the sphere, uniformly bounded, where the cardinality is almost the dimension of the space of polynomials. In this construction the Fekete points in the sphere and the reproducing kernels of the space of polynomials play a key role."

Abstract: "The famous Carleson theorem ensures that, given any \(f\in L^2(\mathbb T)\), its Fourier series \((S_n(f)(x))\) converges for almost every \(x\in\mathbb T\). We are interested in the set of points of divergence. By the Cauchy-Schwarz inequality, \(|S_nf(x)|\leq C\dot n^{1/2}\). This leads us to introduce, for \(\beta\in [0,1/2]\), \(E^-(\beta)=\{x\in\mathbb T;\ |S_n(f)(x)|\geq C\cdot n^\beta\textrm{ infinitely often}\}\) and \(E^+(\beta)=\{x\in\mathbb T;\ |S_n(f)(x)|\geq C\cdot n^{\beta}\textrm{ for all }n\}\). We investigate the size of these sets using the notions of Hausdorff and packing dimension. We then show that there exist multifractal functions, in the sense that all the sets \(E^-(\beta)\) or \(E^+(\beta)\) can be as large as possible. Similar considerations are done for the Hardy space of Dirichlet series."

Abstract: TBA

Abstract: "Almost every mathematician encounters hypergeometric functions in their lives, most often without realizing it. For example \(e^x = \sum_{n\geq 0} x^n/n!\) and the well-known combinatorial identity \(\sum_{k=0}^n \binom{n}{k} =2^n\) are examples of hypergeometric series. Indeed, the topic of hypergeometric functions and their basic hypergeometric generalizations is very old, mathematicians like Euler and Gau\ss\ have already made major contributions. Therefore it is surprising that only 20 years ago a further generalization, the elliptic hypergeometric series appeared; in a physics paper no less. In this seminar I will discuss what this generalization entails, and why we study it. The talk should be accessible for anyone with a basic knowledge of complex analysis, in particular no prior knowledge of hypergeometric series is required. "

Abstract: "First we will discuss some of the ideas behind non-Archimedean physics. Then we will show how non-Archimedean stochastics can be used to approximate physical models over local fields by finite models."

Abstract: "Lately there have been a number of results regarding Fatou components of polynomial skew product. These polynomial maps in two complex variables leave invariant the family of vertical lines, and therefore can often be studied using one-dimensional techniques. At this time polynomial skew products are the only rational maps for which the existence of wandering Fatou components is known. The construction takes place in a neighborhood of a parabolic invariant fibre.

On the other hand, in recent joint work with Iris Smit we showed that near an attracting parabolic fibre (that satisfies some extra conditions), there exist no wandering Fatou components. In October 2015 at the PhD defense of Smit, Berit Stensones asked whether such results also hold for small perturbations of polynomial skew products. Such perturbation could be equal to a skew product up to some high order, but do not leave invariant the family of vertical lines. It would be quite interesting to see whether the same proofs still hold, or whether different, two-dimensional techniques are required for these maps.

In this talk I will suggest a large class of maps to consider, and make a few preliminary observations."

Abstract: "In this talk I shall briefly discuss local holomorphic dynamics in dimension one, focusing on the the normalization and the linearization problems for germs of biholomorphism, and on parabolic bifurcation. Then I will discuss the local dynamics for germs of biholomorphism in several complex variables with an isolated fixed point and in particular I will focus on the dynamics of polynomial skew-products. If time allows I will show how the techniques of parabolic bifurcation can be used to deduce the existence of wandering Fatou components in dimension 2 as done in the joint work with M. Astorg, X. Buff, R. Dujardin and H. Peters."

Abstract: "Avelin, Hed, and Persson recently extended an old result of Fornaess and myself to the effect that plurisubharmonic functions on a domain D that extend continuously to its boundary, can be approximated uniformly with functions that are plurisubharmonic in (shrinking) neighborhoods of the closure of D, if D has $C^0$ boundary. We extend this result to domains with more general boundary, in particular we show that such approximation is possible on the Reinhardt triangle. We also give an example of a fat hyperconvex domain where this approximation result does not hold. This is work in progress jointly with Håkan Persson (Uppsala)"

Abstract: "The study of the Neumann-Poincare (NP) operator (or the boundary double layer potential) of a domain dates back to Poincare and Carleman's doctoral dissertation, at the time serving as a prominent example in the abstract spectral theories proposed by Hilbert, Fredholm, and F. Riesz. Later, the NP operator was central in (quasi)conformal mapping and in the development of the theory of singular integral operators. Very recently, the theory of new materials has revived the interest in the spectral properties of the NP operator, acting on the energy space of the domain. We use a classical similarity equivalence between the NP operator and the Ahlfors-Beurling transform of the domain to characterize the spectrum on a wedge in two variables. A localization argument combined with distortion estimates from conformal mapping leads to a complete description of the essential spectrum of the Neumann-Poincare operator on planar domains with corners. Joint work with Mihai Putinar."

Abstract: "In this talk we will give an overview of the theory of modular forms. We will focus on the interplay of the theory of modular forms and Fourier analysis and their applications to discrete geometry. In particular, we will report on recent results on sphere packing problem."

Abstract: "The setting of this talk is the weighted Fock space \(F^2_m\) of entire functons on \(\mathbb{C}\) which are square integrable with respect to the measure

\(d\mu_m(z)=e^{-|z|^{2m}}dz, m>0\),

where \(dz\) is the normalized Lebesgue measure.

In the context of the Segal-Bargmann space (\(m=1\)), Cho, Park and Zhu studied the boundedness of the product of Toeplitz operators

\(T_u T_{\overline{v}}\) on \(F^2_1\).

We extend their work to the case of general \(m\geq 1\), and give necessary and sufficient conditions on \(u, v \in F^2_m\) for the product

\(T_u T_{\overline{v}}\)

to be bounded on \(F^2_m\). In particular, we relate the boundedness of \(T_u T_{\overline{v}}\) with the boundedness of product of the Berezin transforms of \(|u|^2\) and \(|v|^2\) (Sarason's conjecture)."

Abstract: "After reviewing some basic facts about Hardy spaces of Dirichlet series, we discuss the important (open) problem of whether they can be embedded in certain classical Hardy spaces in the half-plane Re(s)>1/2. Through Bohr's point of view, we demonstrate that the problem is more intimately connected to composition operators than what was previously known."

The seminar will usually take place on Mondays, 14:15-15:00, room 656 SB2.

Abstract: "For a function f we consider the operation S_f(A) = Uf(|A|) where A is an operator on a Hilbert space and A = U|A| its polar decomposition (U is a partial isometry). For finite-dimensional matrices the polar decomposition is usually called the singular value decomposition, and it is written A = USV*, U, V unitary, S diagonal with the singular values of A as entries. Then S_f(A) = Uf(S)V*. In this talk we consider when S_f is Lipschitz in the Hilbert-Schmidt norm (usually called the Frobenius norm in the finite case). I will review some of the history of this problem and the corresponding problem for the usual functional calculus for normal operators. I will then give a proof that S_f is Operator-Lipschitz when f is ordinary Lipschitz, with optimal constants, based on the Birkhoff-von Neumann theorem that permutation matrices are the extreme points of the convex set of doubly stochastic matrices.

Based on joint work with Fredrik Andersson and Marcus Carlsson."

Abstract: gcd_talk.pdf

Abstract: "Consider the fundamental solution to a discrete quasi-periodic Schrödinger-like equation (or more generally, consider a linear quasi-periodic cocycle). Assuming that the potential function is an analytic function of several variables, the growth rate of the fundamental solution may be related to the mean of certain pluri-subharmonic functions. In this talk I will describe some recent results on Lyapunov exponents of quasi-periodic cocycles and emphasize the role played in their proofs by certain types of estimates on pluri-subharmonic functions. [Joint work with Pedro Duarte from University of Lisbon.]"

Abstract: "In time-frequency analysis one is, in particular, interested in lattice Gabor systems: a family of time-frequency shifts of a generator along a lattice in the time-frequency plane. Concerning lattice Gabor systems there are important, so-called density, structure and duality results. They are characterizations, sufficient conditions, and observations about the behaviour of Gabor systems and the possibility of such a system to give rise to certain series expansions of elements in a Hilbert space like L^2(R^d), also called frames. In this talk we will see that the classical results of density, structure and duality theorems such as the Wexler-Raz conditions, Janssen representation and the duality principle have a generalization to Gabor systems on locally compact abelian groups where the time-frequency shifts are along closed subgroups - not necessarily lattices. Moreover, the proofs of these statements stay inside time-frequency analysis and rely essentially on properties of the Fourier transform and frame theory! Even in the familiar setting of R^n the results improve the known theory. Moreover, the setting includes the so-called non-separable Gabor systems, which up until now had to be approached via Von Neumann algebra-techniques and pseudo differential operators."

Abstract: "Let $P_n$ be a polynomial of degree at most $n$ of one complex variable. We prove that the norm of $P_n$ in the space $L^1(T)$ on the unit circle $T$ is comparable to the atomic norm of $P_n$ with respect to the equidistributed measure on $T$ supported at $n+1$ point. Some generalizations and consequences of this result will be also discussed."

Abstract: "By a result due to F. Forstnerič, injective holomorphic maps defined and close to the identity on certain subsets of C^n admit a compositional splitting by injective holomorphic maps on certain larger domains, which are close to the identity as well. For recent application it has been important to derive a parameter version of said result, i.e. to show that the maps obtained from such a splitting can be chosen to depend continuously on a parameter, given that the original domains and maps do. In this talk I will explain the original result and talk about how continuous dependence on a parameter can be ensured along the way. "

Abstract: "We'll recall some known results on Reinhardt domains and circular domains, and then present some recent work generalizing these results in the setting of group actions."

Abstract: "In this elementary talk, we will discuss the relationship between the abscissas of simple, uniform and absolute convergence for ordinary Dirichlet series with multiplicative coefficients. Our main tool will be the so-called Bohr lift, which connects Dirichlet series to function theory in polydiscs.

Based on a joint work with Winston Heap."

Abstract: "The talk discusses a link between the theorem of Balian and Low on the non-existence of well-localized Gabor-Rieszbases and a constant curvature connection on projective modules over noncommutative tori. "

Abstract: "A function f on the real line is said to tile by translates along a discrete set Λ if the sum of all the functions f(x-λ), λ in Λ, is identically equal to 1. Which functions can tile by translates, and what can be said about the translation set Λ? I will survey the subject and discuss some recent results joint with Mihail Kolountzakis."

Abstract: "Given a sequence of holomorphic automorphisms of C^2 with uniformly attracting fixed point, we can study its attracting basin. This basin is definitely diffeomorphic to R^4. But under what conditions will it be biholomorphic to C^2? In this talk, I will give an overview of old and new results and counterexamples for this question. Based on joint work with Han Peters."

Abstract: "In this talk, we'll give an overview on sharp L^2 extensions and multiplier ideal sheaves, mainly focusing on our own recent results published in Ann. of Math., Invent. Math., and China Science Math."

Abstract: "The dynamics of billiard maps has been extensively studied in ergodic theory. Recently a contractive modification on the billiard reflexion law has been suggested which imprints a dissipative and hyperbolic character to its dynamics. We will talk about the ergodic theory of the class of polygonal billiard maps with a contractive reflexion law. This is a joint work with G. Del Magno, J. L. Dias and J. P. Gaivao."

Abstract: "We use Soundararajan's resonance method and estimates for certain large GCD sums to obtain a new lower bound for the maximum of $|\zeta(1/2+it)|$ on $[0,T]$."

This is joint work with Andriy Bondarenko.

The seminar will take place on Mondays, 13:15-14:00, room 734 SB2.

Abstract: For K a product of discs D_i, with i running over some index set I, we present a characterization of uniform limits on K of polynomials, due to Aron, Falco, Maestre and Nestoridis. The index set I may be of arbitrary cardinality.

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Abstract: This talk is about motion planning, motion representation and steps in orbital stabilization of motions of mechanical systems, which might be redundant or have one or several passive degrees of freedom. Given a motion, we suggest to search for its representation without explicit time dependence: an evolution of one of degrees of freedom is defined by certain differential equation (a motion generator); while other degrees of freedom are found through relations valid between coordinates on the motion. Such representation of a motion becomes compact and, as shown, it is often useful in analysis of dynamics in vicinity of its orbit and controller design. In particular, we show steps in a feedback control design that are based on construction of a transverse linearization. Roughly speaking, the transverse linearization is a linear system of dimension one less than the nonlinear system such that stabilization of this system is in certain sense equivalent to exponential orbital stabilization of a desired (periodic) motion of the original nonlinear system. The proposed approach is illustrated on popular research benchmark set-ups (the Furuta pendulum, the Acrobot, a pendulum on a cart, a spherical pendulum on a puck) and applications (design stable gaits for bipeds, quadrupeds; analysis of recorded motions of humans). Remarkably, for mechanical systems the transverse linearization of any feasible (forced) orbit, which in general is related to defining moving Poincaré sections, can be introduced analytically. This fact opens a broad range of opportunities.

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Abstract: Truncated correlation and convolution operators is a general operator-class containing popular operators such as Toeplitz (Wiener-Hopf), Hankel and finite interval convolution operators as well as small and big Hankel operators in several variables. We completely characterize the symbols for which such operators have finite rank, and develop methods for determining the rank in concrete cases. Such results are well known for the one-dimensional objects, the first discovered by L. Kronecker during the 19th century. We show that the results for the multidimensional case differ in various key aspects

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Abstract:We show that a recent result of Demailly and Pham Hoang Hiep implies a description of plurisubharmonic functions with given Monge-Ampere mass and smallest possible log canonical threshold. It also gives a new proof of a result of Qi'an Guan and Xiangyu Zhou on plurisubharmonic functions with given log canonical threshold and smallest possible Lelong number.

Abstract: Demailly's strong openness conjecture gives a new openness property of multiplier ideal sheaves. I'll give a survey on multiplier ideal sheaves, the concept and basic properties; then explain our solution of the strong openness conjecture.

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Abstract: In 1977 P. Yang asked whether there exist bounded complete immersed k-dimensional complex submanifolds of C^N. A positive answer is known for holomorphic curves (k=1) and partial answers are known for the case when k>1. In the talk we will describe how to construct a holomorphic function on the open unit ball B of C^N, N>1, whose real part is unbounded on every path in B of finite length that ends on bB. This implies the existence of a complete closed complex hypersurface in B and thus gives a positive answer to Yang's question in all dimensions k, N, 0<k<N.

Abstract: Let a solution of the above-mentioned equation decays (sufficiently fast) at two distinct moments of time. Then it vanishes identically.

Based on joint work with Philippe Jamming, Eugenia Malinnikova, Karl-Mikael Perfekt.

Abstract: Large deviation principles describe the asymptotic probabilities of rare events in probability theory and dynamical systems. The continuity of the Lyapunov exponents of linear co-cycles can be reduced to deriving certain types of large deviation estimates. We explain how these estimates are obtained, via a spectral approach due to S. V. Nagaev, for irreducible co-cycles over mixing Markov systems. This talk's subject is a joint work with Slivius Klein.

Abstract: will present main results on sharp L^2 extensions, sketch the main idea of the proof, and give applications.

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Abstract: Sullivan's famous non-wandering domains theorem states that the Fatou components of a rational function on the Riemann sphere are all periodic or pre-periodic. It was shown in 2014 by M. Astorg, X. Buff, R. Dujardin, H. Peters and J. Raissy that wandering domain can exist for polynomial maps in C^2. Their examples are parabolic skew products. In current work-in-progress with Han Peters, we investigate the existence of wandering domains for attracting skew products.

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Continuation of talk on 16.02

Abstract: We shall discuss how one can use intgral kernals to get supnormestimates for the solution of the d-bar equation in real analytisc pseudoconvex domains in C^3

Abstract. Set capacities on Trees and in more classical contexts (Euclidean space and, more generally, Ahlfors-regular metric spaces) can be compared. This is sometimes useful.

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ABSTRACT

Abstract: The classical Heisenberg uncertainty inequality shows that a function in the Sobolev space W^{1,2} can be localized near some some point only if the ratio of the norm of its derivative and the norm of the function is large. In 1980s Strichartz obtained a number of uncertainty inequalities in Euclidean spaces and his starting point was a sampling inequality for functions in W^{1,2} over a discrete sequence of points. We follow the idea of Strichartz and obtain some uncertainty inequalities that connect the concentration of the function on some suitably uniformly distributed subset to the norm of the function in some Besov space the results are closely connected to the classical trace theorems for Besov spaces. We also rewrite this result as sampling inequalities for functions in the corresponding Besov spaces.

This is a joint work with Ph. Jaming

Abstract: In the 80s, the Neumann-Poincare operator K (or the boundary double layer potential) was used to solve the Dirichlet and Neumann problems for the Laplacian with L2-data on Lipschitz boundaries. The method depends on showing that the (largest) Fredholm eigenvalue of K is less than 1. This talk will suggest the study of the spectrum of K on the Sobolev space of order 1/2 on the boundary, the space of charges giving rise to potentials of finite energy. For smooth domains, the spectrum is the same as in the L2-situation, but for curvilinear polygons, in which we are particularly interested to characterize the spectrum, the situation is completely different. Even for square in R^2, the spectral picture is unclear. For 2D domains, the Neumann-Poincare operator turns out be similar to a Beurling-Ahlfors transform on a Bergman space, which together with results from quasiconformal mapping theory gives partial results for certain polygonal domains.

Based on work together with Johan Helsing and Mihai Putinar

Abstract: The purpose of this talk is to describe the use of certain large deviation type estimates for dynamical systems in the study of continuity properties of the Lyapunov exponents of linear cocycles. Most relevant concepts will be defined, and some analogies with ‘classical’ settings will be made. The emphasis will be on the use of large deviations in the study of certain types of dynamical systems. A (possible) future talk will be concerned with the proof of such large deviation estimates.

[Based on joint work with Pedro Duarte.]

The seminar will take place on Mondays, 14:15-15:00, room F3.

ABSTRACT

ABSTRACT: We study the 'localization of zeroes' phenomenon in spaces of Cauchy transforms of discrete measures. We also discuss its relation to other topics in analysis (weighted polynomial approximation, canonical systems of differential equations, de Branges' theory). Joint work with E.Abakumov and A. Baranov.

ABSTRACT

ABSTRACT: I will present a class of functions (introduced by Kellendonk) associated with aperiodic tilings or point-sets: pattern-equivariant functions. They can be seen as an analogue of almost-periodic functions. As for almost-periodic functions, these pattern-equivariant functions can be seen as continuous functions on an appropriate compactification of R^n. This compactification turns out to be a foliated space. I will discuss a few applications, namely de Rham cohomology and deformations of aperiodic point-sets.

ABSTRACT

ABSTRACT: A function is said to be almost time (resp. band) limited if almost all the L^2-mass of that function (resp. its Fourier transform) is contained in a given interval. As functions that are both time and band limited do not exist, in many applications it is natural to assume that the functions considered are almost time and band limited. The aim of this talk is to show that such functions are nicely approximated by (the truncation of) their expensions in some orthonormal bases like the Hermite and Legendre bases.

ABSTRACT: This talk will be an introduction to aperiodic tilings and point-sets. Such point-sets can be used to model quasicrystals (material with long-range order but no periodicity). I will describe a few methods to build such objects, as well as some of the the tools which are used to study them. I will try to draw a few links between the study of aperiodic order and other topics in mathematics such as topology and number theory.

ABSTRACT: Franz Luef's work demonstrates how projective modules over noncommutative tori provide structures which tie together many results on lattice Gabor frames. We shall show that a similar situation occurs in the case of Gabor frames coming from quasicrystals, where there are corresponding operator algebras and projective modules. In particular, there always exist multiwindow Gabor frames for a quasicrystal. The dimensions of the modules end up being equal to the frame measure as defined by Balan, Casazza, Heil, and Landau. The structure of the commutants also suggest that there are generalizations of Janssen's representation to quasicrystalline Gabor frames.

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The seminar will take place on Mondays, 14:15-15:00, room F4.

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ABSTRACT: In one complex dimension, a famous theorem by Sullivan proves that there are no wandering Fatou components for polynomial maps. In more than one dimension the non-existence is still an open conjecture. We give examples of polynomial maps in C^2 that have 1 complex dimensional discs with 'wandering' behavior. Joint work with Han Peters.

• 10:15 - 11:00 Chris Hughes, University of York, TBA
• 11:15 - 12:00 Ole Fredrik Brevig, NTNU, The Sidon constant for Dirichlet polynomials
• 12:00 - 13:30 Lunch
• 13:30 - 14:15 NN, TBA
• 14:30 - 15:15 Michel Weber, Université Louis-Pasteur et C.N.R.S. (Strasbourg), Dirichlet Polynomials and the Riemann zeta function: some old and recent results
• 19:00 Dinner

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KEYWORDS: Levinson loglog theorem, Boundary Harnack Principle.

ABSTRACT: The equidistribution theorem for irrational rotation on the circle may be stated by saying that the discrepancy N(S,n) - n mes(S) = o(n), where S is any set whose boundary has measure zero, and N(S,n) is the number of points falling into S among the first n points in the orbit. It is known that for certain special sets S, the discrepancy is in fact bounded as n tends to infinity. Hecke and Kesten characterized the intervals with this property, called "bounded remainder intervals". In this talk we discuss the Hecke-Kesten phenomenon in a multi-dimensional setting. This is joint work with Nir Lev.

ABSTRACT: In this talk I want to approach rational Gabor frames from the perspective of noncommutative geometry, which turn out to be vector bundles over rational noncommutative tori. The Connes-Chern classes of these vector bundles provide a way to understand properties of rational Gabor frames in terms of topological notions.

In particular, I want to explain some of the geometric and topological properties of the Zak transform and of the representations of the rational Gabor frame operator due to Zibulski-Zeevi and Lyubarskii-Nes.

ABSTRACT:We define a class of Banach spaces coming in pairs (M_0, M) of a small space and a big space. Prototypical examples of such pairs are given by (c_0, l^infty), (VMO, BMO), weighted spaces, and Möbius invariant spaces. We establish that the bidual of M_0 is M, and that the canonical projection from M* to M_0* induces an L^1-decomposition of M^* (i.e. M_0 is an M-ideal in M). This has many strong consequences for the Banach space structure of M_0. In particular we mention that M_0 has Pelczynskis properties (u) and (V).

Abstract In this talk we will discuss the interrelation between classical optimization problems on spheres S^d such as minimal equal-weight quadratures (spherical designs) and minimal energy problems. In a joint work with A. Bondarenko and D. Radchenko we have proved the existence of certain configurations in S^d which are spherical t-designs with asymptotically minimal number of points and which simultaneously have asymptotically the best separation property. These configurations also provide approximate solutions for a wide class of minimal problems.

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Abstract: I will talk about strongly regular graphs and some of their connections to group theory, combinatorial set theory, and spherical designs.

Abstract: This talk will be concerned with describing recent results on continuity of the Lyapunov exponents for analytic, quasi-periodic co-cycles with singularities. Some of the main technical tools in obtaining such results depend upon proving certain uniform estimates on unbounded pluri-subharmonic functions defined in a neighborhood of the higher dimensional torus. I will dedicate the better part of the talk to these analytic problems, and keep the dynamics motivation brief. [Joint work with Pedro Duarte.]

Abstract: Following a recent work of Dan Mangoubi we consider harmonic functions that have the same set of zeros. We show that the ratio of this functions is always well-defined and is a real-analytic function, it satisfies the maximum and minimum principles. In dimension three we also obtain the Harnack inequality for the ratios of harmonic functions and generalize the gradient estimates obtained by Mangoubi in dimension two. This is a joint work with Alexander Logunov (St.Petersburg)

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Abstract: In this talk we discuss the maximal order of multiplicative functions. For example, the largest behaviour of the divisor function \(d(n)\) which counts the number of divisors of \(n\) is, roughly, \(2^\frac{\log n}{\log\log n}\).

We consider a general case where the function is given at the prime powers by \(f(p^k) = \exp\{ h(k)l(p)\}\) where \(h(.)\) and \(l(.)\) are increasing and decreasing functions respectively with \(l(p)\) regularly varying of index \(-\alpha\) (\(0\le \alpha<1\)). For example, we show that under appropriate conditions \[ \max_{n\le N} \log f(n) \sim \biggl(\sum_{n=1}^{\infty} \Delta h(n)^{1/\alpha}\biggr)^{\alpha}L(\log N)\] where \(L(x) = \sum_{p\le x} l(p)\) and \(\Delta h(n) = h(n)-h(n-1)\).

Abstract: We discuss Hilbert spaces of polyanalytic functions on the complex plane.The norm on these spaces is given by integration against a weight \(e^{-mQ(z)}\) where \(Q\) is a strictly subharmonic function and \(m\) a large positive scaling parameter. We obtain a near-diagonal asymptotic expansion for the reproducing kernels as \(m\) tends to infinity. In the setting of one complex variable, this generalizes the work of Tian-Yau-Catlin-Zelditch from analytic functions to polyanalytic functions.

We also study reproducing kernels of corresponding polynomials spaces. These are spanned by functions \(\bar z^r z^j\) where \(0 \leq r \leq q-1\) and \(0 \leq j \leq n-1\). The inner product is induced by the same weight as before. Keeping \(q\) fixed and letting \(n\) and \(m\) go to infinity, we obtain scaling limits for the kernels in so called bulk regime. In the model case \(Q(z)=|z|^2\), this investigation has applications in statistical quantum mechanics.

Some of the results are joint work with Håkan Hedenmalm.