# Research group Mathematical analysis

Mathematical analysis deals generally with problems that involve questions like how does a quantity develop, that is, how does it depend on variables/parameters? In our group we are mainly working in operator theory, in particular, differential operators, as well as in complex analysis.

## Group description

More precisely, recent interests include the following overlapping topics:

- Quantum Graphs are differential operators on metric graphs, which exhibit features of both ordinary and partial differential operators. Of particular interest is the relation of geometric properties of the graph with spectral properties of the operator.
- Another class of operators of interest are (linear and multilinear) Fourier integral and Pseudodifferential operators, for which in particular boundedness is investigated.
- Within singular problems we focus on both on differential operators with strongly singular potentials as well as more abstractly super singular perturbations of self-adjoint operators.
- Beside direct spectral and scattering problems also Inverse Problems are considered, in particular for quantum graphs as well as with problem related to tomography.
- Another research topic of the group is spectral theory of partial differential equations. Properties of eigenvalues and eigenfunctions of, e.g., the Laplacian or Schrödinger operators and their relation to the geometry of the underlying space are studied.
- In complex analysis we are working on different topics related to functions of several variables, including integral representations for Herglotz-Nevanlinna functions and boundary behavior and integrability questions for bounded functions on polydisks.
- The study of Banach spaces of analytic functions, such as Hardy, Bergman, and Dirichlet spaces, and their operators, also involves heavy use of complex analysis. The main interest here is to identify cyclic vectors and analyze the structure of invariant subspaces for shift operators.
- Conformal mapping models of Laplacian growth and other conformal proesses are studied using a combination of complex-analytic methods, such as Loewner's differential equation, and methods from probability theory and stochastic analysis.
- We also study black holes and the Big Bang in mathematical general relativity, which is formulated using differential geometry. A central goal is to understand wave propagation in curved spacetimes, based on geometric and microlocal analysis.
- Applications of harmonic analysis and probability theory to additive combinatorics and number theory. Analytic results, such as almost-periodicity properties of convolutions, can often say interesting things about additive problems, such as Roth's theorem on arithmetic progressions and Freiman's theorem classifying sets with small sumset. Many different tools are used to study such topics, from Fourier analysis and concentration of measure to properties of spaces of polynomials. We are particularly interested in probabilistic methods and in the study of the large spectra of certain classes of functions.

## Seminar series

The Stockholm Analysis seminar is organized jointly with the analysis group at KTH. Write an email to the group manager if you want to get a reminder each week.

Seminars are announced in the calendar at Stockholm Mathematics Centre, SMC

## Group members

### Group managers

Annemarie Luger

Professor

### Members

Nausica Aldeghi

PhD Student

Sergi Arias Garcia

Doktorand

Jan Boman

Bitr professor emeritus

Alice Brolin

PhD Student

Matthew deCourcy-Ireland

Postdoc

Christian Emmel

PhD Student

Dario Giandinoto

PhD Student

Pavel Kurasov

Professor

Oliver Lindblad Petersen

Universitetslektor

Jacob Gregory Muller

Doktorand

Jakob Reiffenstein

Postdoc

Salvador Rodriguez Lopez

Universitetslektor

Jonathan Rohleder

Universitetslektor

Timotheus Lee Schmatzler

PhD Student

Olof Sisask

Universitetslektor

Alan Sola

Universitetslektor

Andrzej Szulkin

Professor emeritus

Eleftherios Theodosiadis

PhD Student