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2017 | Buch

Analysis Meets Geometry

The Mikael Passare Memorial Volume

herausgegeben von: Prof. Mats Andersson, Prof. Jan Boman, Christer Kiselman, Prof. Pavel Kurasov, Prof. Ragnar Sigurdsson

Verlag: Springer International Publishing

Buchreihe : Trends in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is dedicated to the memory of Mikael Passare, an outstanding Swedish mathematician who devoted his life to developing the theory of analytic functions in several complex variables and exploring geometric ideas first-hand. It includes several papers describing Mikael’s life as well as his contributions to mathematics, written by friends of Mikael’s who share his attitude and passion for science. A major section of the book presents original research articles that further develop Mikael’s ideas and which were written by his former students and co-authors. All these mathematicians work at the interface of analysis and geometry, and Mikael’s impact on their research cannot be underestimated. Most of the contributors were invited speakers at the conference organized at Stockholm University in his honor. This book is an attempt to express our gratitude towards this great mathematician, who left us full of energy and new creative mathematical ideas.

Inhaltsverzeichnis

Frontmatter

Memorial Contributions

Frontmatter

Mikael Passare

Curriculum Vitae

Abstract

1959-01-01. Kjell Alrik Mikael Pettersson is born in Västerås, Sweden. Mother: Britt Gunvor Emilia Pettersson, later with the family name Elfström. Father: Werner Siems. Stepfathers: Kjell Pettersson and Hans Elfström.

Galina Passare, Mats Andersson, Jan Boman, Christer Kiselman, Pavel Kurasov, Ragnar Sigurdsson

Mikael Passare (1959–2011)

Abstract

Mikael Passare was a brilliant mathematician who died much too early. In this chapter we present a sketch of his work and life.

Christer O. Kiselman

Mikael Passare

Abstract

I got to know Mikael in the eighties, when I was a PhD student at Stockholm University. I had completed a number of graduate courses and became interested in complex analysis. Mikael was a young lecturer in Stockholm, and our overlapping interest in that subject brought us into each other’s orbits.

Magnus Carlehed, Mats Andersson, Jan Boman, Christer Kiselman, Pavel Kurasov, Ragnar Sigurdsson

Research Articles

Frontmatter

Amoebas and Coamoebas of Linear Spaces

Abstract

We give a complete description of amoebas and coamoebas of k-dimensional very affine linear spaces in (ℂ*)ⁿ. This include an upper bound of their dimension, and we show that if a k-dimensional very affine linear space in (ℂ*)ⁿ is generic, then the dimension of its (co)amoeba is equal to min{2k, n}. Moreover, we prove that the volume of its coamoeba is equal to π^2k. In addition, if the space is generic and real, then the volume of its amoeba is equal to π^2k/2^k.

Mounir Nisse, Mikael Passare

One Parameter Regularizations of Products of Residue Currents

Abstract

We show that Coleff–Herrera type products of residue currents can be defined by analytic continuation of natural functions depending on one complex variable.

Mats Andersson, Håkan Samuelsson Kalm, Elizabeth Wulcan, Alain Yger

On the Effective Membership Problem for Polynomial Ideals

Abstract

We discuss the possibility of representing elements in polynomial ideals in ℂ^N with optimal degree bounds. Classical theorems due to Macaulay and Max Noether say that such a representation is possible under certain conditions on the variety of the associated homogeneous ideal. We present some variants of these results, as well as generalizations to subvarieties of ℂ^N.

Mats Andersson, Elizabeth Wulcan

On the Optimal Regularity of Weak Geodesics in the Space of Metrics on a Polarized Manifold

Abstract

Let (X,L) be a polarized compact manifold, i.e., L is an ample line bundle over X and denote by ℋ the infinite-dimensional space of all positively curved Hermitian metrics on L equipped with the Mabuchi metric. In this short note we show, using Bedford–Taylor type envelope techniques developed in the authors previous work [3], that Chen’s weak geodesic connecting any two elements in ℋ are C1,1-smooth, i.e., the real Hessian is bounded, for any fixed time t, thus improving the original bound on the Laplacians due to Chen. This also gives a partial generalization of Blocki’s refinement of Chen’s regularity result. More generally, a regularity result for complex Monge–Ampère equations over X × D, for D a pseudoconvex domain in ℂⁿ is given.

Robert J. Berman

A Comparison Principle for Bergman Kernels

Abstract

We give a version of the comparison principle from pluripotential theory where the Monge–Ampère measure is replaced by the Bergman kernel and use it to derive a maximum principle.

Bo Berndtsson

Suita Conjecture from the One-dimensional Viewpoint

Abstract

The Suita conjecture predicted the optimal lower bound for the Bergman kernel of a domain on the plane in terms of logarithmic capacity. It was recently proved as a special case of the optimal version of the Ohsawa–Takegoshi extension theorem. We present here a purely one-dimensional approach that should be suited to readers not interested in several complex variables.

Zbigniew Błocki

Siciak’s Theorem on Separate Analyticity

Abstract

We give a simple proof of an important special case of the famous theorem of Jósef Siciak on separate analyticity.

Jan Boman

Mikael Passare, a Jaunt in Approximation Theory

Abstract

A personal and informal presentation of the contributions of Mikael Passare to the field of approximation theory.

Jean-Paul Calvi

Amoebas and their Tropicalizations – a Survey

Abstract

Let V(f) be the complex hypersurface of a Laurent polynomial f. The amoeba A(f) is the projection of V(f) under the Log-absolute map. Amoebas have countless applications and, in particular, they form a key connection between “classical” algebraic geometry and tropical geometry. There exist multiple different tropical hypersurfaces related to amoebas. In this survey, we introduce the most important of these tropical hypersurfaces and compare their relations to amoebas. Moreover, we discuss related open problems in amoeba theory.

As a new contribution we provide an example of an amoeba in ℝ² which has a component in the complement with an order not contained in the support of the defining polynomial. As a consequence, we conclude that an amoeba and its corresponding complement induced tropical hypersurface are not homotopy equivalent in general. Similarly, we prove that Archimedean amoebas and non-Archimedean amoebas are not homotopy equivalent in general.

Timo de Wolff

Coamoebas of Polynomials Supported on Circuits

Abstract

We study coamoebas of polynomials supported on circuits. Our results include an explicit description of the space of coamoebas, a relation between connected components of the coamoeba complement and critical points of the polynomial, an upper bound on the area of a planar coamoeba, and a recovered bound on the number of positive solutions of a fewnomial system.

Jens Forsgård

Limit of Green Functions and Ideals, the Case of Four Poles

Abstract

We study the limits of pluricomplex Green functions with four poles tending to the origin in a hyperconvex domain, and the (related) limits of the ideals of holomorphic functions vanishing on those points. Taking subsequences, we always assume that the directions defined by pairs of points stabilize as they tend to 0. We prove that in a generic case, the limit of the Green functions is always the same, while the limits of ideals are distinct (in contrast to the three point case). We also study some exceptional cases, where only the limits of ideals are determined. In order to do this, we establish a useful result linking the length of the upper or lower limits of a family of ideals, and its convergence.

Duong Quang Hai, Pascal J. Thomas

Geodesics on Ellipsoids

Abstract

Various ways of describing geodesic motion on Ellipsoids are presented (intrinsic and constrained formulations) including Jacobi’s solution, Weierstrass’ solution, and level set Liouville integrability.

Jens Hoppe

Welschinger Invariants Revisited

Abstract

We establish the enumerativity of (original and modified) Welschinger invariants for every real divisor on any real algebraic del Pezzo surface and give an algebro-geometric proof of the invariance of that count both up to variation of the point constraints on a given surface and variation of the complex structure of the surface itself.

Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin

Some Results on Amoebas and Coamoebas of Affine Spaces

Abstract

We give some topological characteristics of the coamoeba of a generic k-dimensional affine space and two stronger versions, specific for the affine case, of a result by Nisse, Sottile and the author. We also give topological and partly algebraical characterizations of the amoeba and coamoeba in some special cases: k = n − 1, k = 1 and, when n is even, k = n/2, in the last case with a certain emphasis on the example n = 4.

Petter Johansson

Convexity of Marginal Functions in the Discrete Case

Abstract

We define, using difference operators, classes of functions defined on the set of points with integer coordinates which are preserved under the formation of marginal functions. The duality between classes of functions with certain convexity properties and families of second-order difference operators plays an important role and is explained using notions from mathematical morphology.

Christer O. Kiselman, Shiva Samieinia

Modules of Square Integrable Holomorphic Germs

Abstract

This paper was inspired by Guan and Zhou’s recent proof of the socalled strong openness conjecture for plurisubharmonic functions. We give a proof shorter than theirs and extend the result to possibly singular Hermitian metrics on vector bundles.

László Lempert

An Effective Uniform Artin–Rees Lemma

Abstract

We prove a global uniform Artin–Rees lemma type theorem for sections of ample line bundles over smooth projective varieties. This result is used to prove an Artin–Rees lemma for the polynomial ring with uniform degree bounds. The proof is based on multidimensional residue calculus.

Johannes Lundqvist

Amoebas of Half-dimensional Varieties

Abstract

An n-dimensional algebraic variety in (ℂ^×)²ⁿ covers its amoeba as well as its coamoeba generically finite-to-one. We provide an upper bound for the volume of these amoebas as well as for the number of points in the inverse images under the amoeba and coamoeba maps.

Grigory Mikhalkin

A log Canonical Threshold Test

Abstract

In terms of log canonical threshold, we characterize plurisubharmonic functions with logarithmic asymptotical behaviour.

Alexander Rashkovskii

Root-counting Measures of Jacobi Polynomials and Topological Types and Critical Geodesics of Related Quadratic Differentials

Abstract

Two main topics of this paper are asymptotic distributions of zeros of Jacobi polynomials and topology of critical trajectories of related quadratic differentials. First, we will discuss recent developments and some new results concerning the limit of the root-counting measures of these polynomials. In particular, we will show that the support of the limit measure sits on the critical trajectories of a quadratic differential of the form \( Q(z) \, dz^{2} = \frac{az^{2}+bz+c} {(z^{2}-1)^{2}} \, dz^{2} \) Then we will give a complete classification, in terms of complex parameters a, b, and c, of possible topological types of critical geodesics for the quadratic differential of this type.

Boris Shapiro, Alexander Solynin

Interior Eigenvalue Density of Jordan Matrices with Random Perturbations

Abstract

We study the eigenvalue distribution of a large Jordan block subject to a small random Gaussian perturbation. A result by E. B. Davies and M. Hager shows that as the dimension of the matrix gets large, with probability close to 1, most of the eigenvalues are close to a circle.

We study the expected eigenvalue density of the perturbed Jordan block in the interior of that circle and give a precise asymptotic description.

Résumé. Nous étudions la distribution de valeurs propres d’un grand bloc de Jordan soumis à une petite perturbation gaussienne aléatoire. Un résultat de E. B. Davies et M. Hager montre que quand la dimension de la matrice devient grande, alors avec probabilité proche de 1, la plupart des valeurs propres sont proches d’un cercle.

Nous étudions la répartitions moyenne des valeurs propres à l’intérieur de ce cercle et nous en donnons une description asymptotique précise.

Johannes Sjöstrand, Martin Vogel

Titel: Analysis Meets Geometry
herausgegeben von: Prof. Mats Andersson
Prof. Jan Boman
Christer Kiselman
Prof. Pavel Kurasov
Prof. Ragnar Sigurdsson
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-52471-9
Print ISBN: 978-3-319-52469-6
DOI: https://doi.org/10.1007/978-3-319-52471-9