Harmonic analysis was created in the early 19th century as a tool to solve many partial differential equations coming from physics. It is also called Fourier analysis, since the initiator was the Frenchman J. Fourier. The theory has since developed into a central, huge field of mathematics. The basic idea is to describe various phenomena by decomposing them into waves or oscillations of different frequencies. Obvious examples are the splitting of light into a spectrum of colours, or that of sound into different tunes. But the first idea, due to Fourier, came up in connection with the temperature evolution inside a cooling body, where waves are by no means natural. Still, the method turned out to be very successful here, and so it is in many other contexts of mathematics, physics and other sciences.
The research of the group consists both in developing this theory further and in applying it to partial differential equations, Lie group representations and other fields.
Below we give descriptions of the settings and problems that members of our group are working with.
Spectral theory and PDE
Grigori Rozenblioum investigates spectral properties of different operators arising in Mathematical Physics, especially, in Quantum Physics, in particular, the Schr¨odinger, Dirac and Pauli operators with magnetic field as well as semi-discrete models and quantum graphs. The main results consist in finding estimates and asymptotics of the eigenvalues of these operators as depending on physical parameters. This type of problems is closely related to another topic, spectral properties of Toeplitz operators in Bergman type spaces. To study these properties, one needs to develop new methods in complex analysis. Another direction in complex analysis is needed to study the zero energy eigenstates for quantum-mechanical operators.
Grigori Rozenblioum also works in global analysis of partial differential and pseudodifferential operators on singular manifolds, first of all with index theorems of Atiyah-Singer type. In this direction advanced methods of algebraic topology, K-theory and C* -algebras are applied. This field is strongly represented by his former PhD student Magnus Goffeng, who obtained recently very impressive results in global analysis, especially, regularized formulas for the degree of a mapping of manifolds.
Harmonic analysis and orthogonal expansions
Peter Sjögren’s field consists of singular integrals, maximal operators and similar basic objects of harmonic analysis, considered in settings given by various orthogonal expansions. Often the expansions are those given by the classical Hermite, Laguerre or Jacobi polynomials or some associated functions. Many of these are significant in quantum physics. Roughly speaking, this means expressing functions as superpositions not of regular, sine-shaped waves as in standard Fourier analysis, but of waves which are adapted to some particular geometry. For instance, some parts of the region considered may be given more weight than other parts.
For each expansion, there is a Laplacian, a heat operator and other notions which form part of a version of harmonic analysis, together with an underlying measure. This allows the introduction of Poisson and heat kernels and singular integral operators like Riesz transforms. The question is then what parts of classical harmonic analysis that will remain valid for these operators. In particular, this regards their Lp boundedness and convergence properties, and maximal functions inevitably come in here. The underlying space is usually of several dimensions, and some questions are meaningful, and difficult, even in infinite dimension.
As an example, take the Hermite polynomial setting. There the heat maximal operator possesses all the expected boundedness properties, at least in finite dimension. But among the Riesz transforms, only those of order at most two turn out to be bounded.
Hardy spaces also exist in the settings considered, but the various classical definitions no longer always lead to the same space. A particular problem is posed by the heat kernel for the Jacobi case, which still today is largely unknown.
Operator theory, operator algebras and harmonic analysis
Lyudmila Turowska’s main research interests lie within functional analysis, representation theory and harmonic analysis whose interaction naturally takes place in the framework of Banach and operator algebras, and operator spaces. In particular, she studies Banach algebras associated to locally compact groups such as the Fourier algebra, the group C*-algebras and the von Neumann group algebras. Representation theory of groups lies in the base of this study: e.g. the Fourier algebra of a group is the algebra of all matrix coefficients of the left regular representation of the group. Some particular contributions to the area concern questions on spectral synthesis (a reconstruction of elements of certain spaces from their parts – ”pure frequencies”, ”spectrum” and ”support”), ideal theory and multipliers of the Fourier/Beurling-Fourier algebra. Her approach relies on a new technique based on the theory of operators and operator algebras.
One of the most important developments in Analysis in recent years has been ”quantisation” of functional analysis starting with the advent of the theory of operator spaces and completely bounded maps. It provides a powerful tool to study important questions in various areas. To investigate the link between classical and quantised analysis is another direction of Turowska’s research. The study of so called Schur multipliers (certain transformations on the space of bounded operators on a Hilbert space) and their unbounded and non-commutative generalisations has been one of the subjects of her recent work.
Representations of Lie groups. Harmonic and complex analysis on homogeneous domains
Genkai Zhang studies unitary representations of Lie groups, in particular semisimple Lie groups, and their applications in harmonic and complex analysis on homogeneous domains. His Ph.D student Oskar Hamlet is studying tight embeddings of Hermitian symmetric spaces, which is related to finite-dimensional representations of Lie groups.
A central problem in representation theory is to find the unitary dual of a semisimple Lie group. There remain many important problems, even though tremendous progress has been made. We are interested in particular in the branching and tensor product decompositions of unitary small representations. Here some natural analytic questions arise, whose answers would have applications to several topics, such as Berezin and Radon transforms, Hua operators and Rankin-Cohen quantization.
Many unitary representations can be realized on nilpotent groups. We then study related sub-Riemannian geometry, Kohn Laplacians and heat kernel estimates. Among homogeneous spaces, the bounded symmetric domains are of particular interest. We study Toeplitz operators, tight embeddings and rigidity problems on those domains.
Nonlinear partial differential equations
Philip Brenner and Peter Kumlin study problems related to nonlinear dispersive partial differential equations. In particular questions on existence and regularity of solutions as well as of solution and scattering operators to nonlinear wave and Klein-Gordon equations are treated. Questions on ill and wellposedness for these problems involve the study of mapping properties for nonlinear mappings between Sobolev/Besov spaces and nonlinear interpolation.