# Analysis

Analysis is a field of Mathematics which has its roots in infinitesimal calculus, developed by Newton and Leibniz in the late 17th century. The idea of continuous motion was difficult to grasp in antiquity, as illustrated by Zeno's paradoxes. The purpose of calculus was initially to clarify the mathematical understanding of motion. Today, Analysis has expanded to cover differential equations of one or several variables, the theory of integration, geometric analysis, functional analysis, operator theory, harmonic analysis, complex analysis and dynamical systems. The Swedish school in Mathematics is strong in Analysis and the research at SMC covers many aspects of Analysis, such as random matrix theory, dynamical systems, free boundary problems in partial differential equations, variational methods, Fourier analysis, complex analysis in one and several variables, differential geometry, mathematical physics and spectral theory.

## Kristian Bjerklöv

I work in the area of smooth dynamical systems. I am especially interested in the dynamics of quasi-periodically driven systems. My research aims at understanding and rigorously show how regular systems can be forced into chaotic ones, describing the route from order to chaos, and how chaotic behavior of systems which are nonuniformly expanding can persist under small quasi-periodic perturbations. In my research I use analytic and geometric methods. Within this framework I am also interested in the ergodic and spectral theory of ODE's with quasi-periodic coefficients, in particular the Schrödinger equation.

## Mattias Dahl

My research is concerned with the interplay of geometry and analysis, in particular the study of Riemannian manifolds, curvature, and geometric elliptic operators. Specific research areas include: Spectral theory of the Dirac operator, problems concerning metrics of constant scalar curvature, the Yamabe invariant, surgery constructions, the constraint equations of General relativity.

## Tomas Ekholm

I work in spectral theory with problems from mathematical physics. My research is about Schrödinger operators and its generalizations in different setups. This includes p-Laplacian, magnetic fields, Hardy inequalities, Sobolev inequalities, higher order operators, etc. I am also interested in how the geometry affects the spectrum of operators. I have investigated geometric perturbations of a waveguide, operators on metric trees with a certain growth at infinity, Lieb-Thirring inequalities and Hardy-Lieb-Thirring inequalities.

## Björn Gustafsson

My research fields are complex analysis (mainly in one variable), potential theory, certain aspects of partial differential equations (in particular free boundary problems), together with applications of these areas in fluid mechanics and other branches of mathematical physics. Among more specific areas can be mentioned Laplacian growth, or moving boundary problems for Hele-Shaw flow, quadrature domains, the exponential transform, potential theoretic skeletons.

## Håkan Hedenmalm

My interest in Analysis covers several topics. The starting point was completeness problems in spaces of functions on semi-groups (here, the main work is with Borichev). Later, influenced by Korenblum, my focus shifted to the Bergman spaces and the Bergman kernel. This topic has connections with PDE (partial differential equations) and the geometry of abstract Riemannian surfaces. Some of this work is with Borichev, Jakobsson, Perdomo, Shimorin, and Zhu. Another line of investigation is the theory of Hardy spaces of Dirichlet series (joint with Seip et al.). Later, with Aleman and Shimorin, we got interested in conformal mapping and the still open Brennan and Kraetzer conjectures. In another direction, with Ameur and Makarov, we studied normal random matrix models, and obtained a connection with Hele-Shaw flow, obstacle problems, and with the GFF (Gaussian Free Field). Finally, in work with Montes-Rodriguez, we have investigated uniqueness sets for solutions to the Klein-Gordon equation.

## Jens Hoppe

Aspects of Classical Differential Geometry, and their Discrete Analogues: (collaborations with J.Arnlind, M.Bordemann, J.Choe, J.Fröhlich, G.Huisken) construction of non-commutative surfaces; new multi-linear formulation of geometric concepts (e.g. algebraically discretizable formulation of curvature); construction of new minimal (hyper)surfaces; discrete minimal surfaces.

Singularity Formation, Quantization, and Signs of Integrability in the Theory of Relativistic Extended Objects: Self-Similar Solutions; Non-Linear Realizations of Poincare' Invariance; Dynamical Symmetries; Infinite Dimensional Lie-Algebras; Investigation of Existence, Uniqueness, and Structure of Zero-Energy States.

## Kurt Johansson

The main focus of my research is Random Matrix Theory and related problems in statistical mechanics and probability theory. The statistical properties of the spectra of random matrices give rise to natural probability distributions and processes that occur also in many other contexts. For example the Tracy-Widom or largest eigenvalue distribution turn up in certain planar random growth models and also in random tilings of planar regions. This has been proved rigorously only in certain special cases, but it is expected to be true in much greater generality. This type of of problem is often referred to as a universality problem for the random matrix distributions. Another type of universality problem is proving that the spectral properties of large dimensional random matrices is the same within large classes of probability measures on random matrices. Many distributions from random matrix theory can be found also, at least in simulations, in number theory and asymptotics of spectra in quantum systems. These type of universality problems can be compared with the problem of understanding the fact that the normal distribution occur in so many contexts. In the case of random matrix distributions the understanding of the universality is much less developed and getting a better understanding of the random matrix distributions and processes and their universality are central problems in my research.

## Torbjörn Kolsrud

In a joint project with K. Iwata, Hiroshima, we study stochastic elliptic functions with only simple poles and their moments (n-point functions). We have proved that in the continuum limit, under reasonable conditions, one obtains the free Gaussian field on the 2-torus. We also also study moments for finite constellations, and their interior relations, a kind of Ward identities. Under a certain condition on independence, the moments become modular forms. I also study homogeneous functions solving all equations in the stationary NLS and KdV hierarchies on the line.The limit as the hierarchy number goes to infinity can be studied and expressed by known analytic functions. The corresponding periodic problems are interesting but certainly more difficult.

## Pavel Kurasov

My main area of research is operator theory, in particular spectral theory and its applications to mathematical physics. My studies have roots in mathematical analysis, especially in the theory of analytic functions, but requires deep knowledge of modern physics.

For example spectral properties of Schrödinger operators can effectively be described by Nevanlinna class analytic functions. Exploring this connection in the case of singular perturbations of self-adjoint operators we have written monograph devoted to Solvable Schrödinger type operators widely used to create quantum mechanical models. Consideration of supersingular perturbations naturally leads to triplet extensions and generalized Nevanlinna functions usually connected with operators in spaces with indefinite metrics. This is a promising area for future studies with numerous open mathematical problems.

In recent years I became interested in quantum graphs - ordinary differential equations on metric graphs, which are important e.g. in applications to nano-electronics. To solve the corresponding inverse problem means to reconstruct the metric graph, the differential operator on the edges and the matching conditions at the vertices. Here the theory of differential equations, geometry and matrix theory are interacting in a rather sophisticated way. These three subproblems are entirely connected and their interplay is not fully understood even for trees - graphs without cycles. Thus the first two subproblems possess unique solutions (for certain classes of matching conditions), but we constructed counterexamples showing that the general inverse problem may not be uniquely solvable. For quantum graphs with cycles I established formula connecting their spectra to the Euler characteristics of geometric graphs. Complete solution of the inverse problem remain open, also considering of the magnetic field may help to determine a unique solution.

## Pär Kurlberg

I am a number theorist with particular interests in applications to arithmetic dynamics (especially quantum chaos), as well as various statistical properties of arithmetic objects, such as curves and polynomial actions over finite fields. The main question in Quantum Chaos is how chaos in classical dynamical systems manifests itself quantum mechanically, and the study of this frequently gives rise to problems that can be investigated using techniques from number theory (for example, the study of eigenfunctions of the Laplacian on tori are closely related to integers represented as sums of two squares.) Recently I have studied the distribution of the number of points (over finite fields) of curves in certain families. In many cases it turns out that the (normalized) distribution is Gaussian; in a sense a law of large number for these point counts.

## Sara Maad Sasane

I work in nonlinear PDEs using tools from functional analysis and dynamical systems. Recently I have been studying problems in spectral theory involving embedded eigenvalues. I have a background in critical point theory and the calculus of variations.

## Kirsti Mattila

My research interests include: Functional analysis, Spectral properties of operators on Banach spaces and Operator semigroups.

## Hans Ringström

I study mathematical problems arising in the general theory of relativity, in particular in cosmology. In standard cosmology, the universe is taken to be spatially homogeneous and isotropic. Even though there is support for this assumption, it is clear that it is not exactly fulfilled. As a consequence, several questions arise. Of particular interest is the question of stability: do small perturbations of the initial data corresponding to the standard models give rise to solutions that are similar globally to the future? Another question of interest is related to the global topology (shape) of the universe: what are the restrictions imposed by the assumption that all observers consider the universe to be close to a standard model? These are the problems I currently work on.

## Maria Saprykina

Maria Saprykina works in the field of smooth dynamical systems and ergodic theory. Recently, Maria's main interest has been in Hamiltonian dynamics. Hamiltonian systems appear in mathematical modeling of many physical processes, from a simple pendulum to the movement of the planets. Easiest examples of Hamiltonian systems are so-called integrable systems, that have a rather regular dynamics. Many interesting systems can be seen as close to integrable. A system of weakly coupled mathematical pendula is an example. The description of solutions to systems close to integrable consists of a wide range of interesting problems. Maria's research focuses on the rigorous investigation of near-integrable systems with the help of analytic and geometric methods.

## Henrik Shahgholian

My interest is in applied and numerical problems. I work in the field of partial differential equations and free boundary problems. Most of my recent works concern regularity properties of free boundaries and geometric aspects.

## Per Sjölin

I work on maximal operators of Schrödinger type. In particular one obtains estimates for solutions to the time-dependent Schrödinger equation. I have also studied multiparameter maximal operators of Schrödinger type and maximal operators of Schrödinger type with a complex parameter. In particular I have obtained estimates for maximal operators of Schrödinger type applied to radial functions. I also study localization of Schrödinger means for functions in Sobolev spaces.

I also work on Heisenberg uniqueness pairs in the plane and have used a theorem of Beurling and Malliavin to obtain a uniqueness result.

## Alan Sola

My primary research interests are in complex and harmonic analysis, several complex variables, and probability theory and stochastic processes. In recent years, I have studied cyclic vectors for shift operators acting on spaces of analytic functions, singularities of rational functions in several complex variables, and scaling limits in conformal mapping models of Laplacian growth. Much of my work is collaborative, primarily with researchers based in the US and the UK.

## Jan-Olov Strömberg

My research is along the following main lines: A commercialisation project supported by Vinnova; a project in applied mathematics concerning analysis of large and high dimensional data sets; and finally two problems in pure mathematics: one concerning new identities for inverse Radon transforms and one concerning estimates of maximal function in very high dimensions.

Vinnovas is supporting a project for commercialisation of a patented video compression method based on wavelets. This method is especially effective for coding surveillance videos. The purpose for this projects is to find potential industrial partners.

Diffusion geometry is a rather new method for handling high dimensional data. In a Ph.D. project for Joel Andersson we have applied this data on images from micrograms of cell kernels, data obtained from the company Vironova. Using these images are developing methods for identification and classification of virus types.

The is a still increasing need for mathematical tool for handling massive high dimensional data sets. Diffusion geometry seems to be a plausible tool for this. We are also beginning to look into the area of compressive sensing. We have several both KTH internal and external intressents for using such methods. This year we have the post.doc Zhijie Wen working with us.

In a more mathematics project we a considering a simple identity the might give new high-light to the inversion formula for the inverse Radon transforms. This is a joint project the the Ph.D. student Joel Andersson.

In a joint project with the Ph.D. student Alexander Iakovlev we consider the finding a lower estimation of constant for the weak type (1.1) estimate of the Hardy Littlewood maximal function on centred cubes in high dimension. The question was raised for about 30 years ago by S. and E.M. Stein whether this constant goes to infinity as the as the dimension increases. The affirmative answer to this was found a few years ago by Aldas. We a now giving a more quantitative answer to this question.

## Andrzej Szulkin

The field of my research is Nonlinear Functional Analysis, in particular, applications of topological and variational methods in functional analysis to nonlinear differential equations. Solutions of certain differential equations correspond to critical points of a functional (usually - though not always - of Euler-Lagrange type). One wants to find criteria for solvability and estimate the minimal number of solutions that a given problem (or class of problems) must have. Since the minimal number of critical points depends very much on the geometry of the functional and not on the particular equation, results obtained by variational methods often apply to a rather broad class of such equations. Among problems to which these methods have been successfully employed I can mention the existence and the number of solutions to semilinear elliptic boundary value problems, the existence and the number of time-periodic solutions to the semilinear wave equation of vibrating string type, the existence and the number of periodic, homoclinic and heteroclinic orbits for Hamiltonian systems.