Workshop 'Operators, Spectra and Applications'
at AIMS Senegal, 20.06.2016 - 24.06.2016
organized by Mouhamed Moustapha Fall (AIMS Senegal) and Peter Stollmann (TU Chemnitz)
The workshop will feature extended lectures by C. Helmberg, T. Kalmes and P. Stollmann and give the possibility for participants to present their own research.
The following courses were delivered:
'The Laplacian in Spectral Graph Theory and its Applications' by C. Helmberg, TU Chemnitz
Abstract: Spectral Graph Theory studies properties of eigenvalues and
eigenvectors of matrix representations of graphs. For a graph on
nodes 1 to n the classical graph Laplacian is a symmetric matrix of
order n with diagonal element i carrying the degree of node i and
offdiagonal element ij having value -1 if i and j are adjacent and 0
otherwise. It is frequently interpreted as the discrete analogon to
the Laplace operator and its spectral properties show tight
connections to the graphs connectivity properties and random walks in
the graph.
In this course we will introduce these notions in detail and start
with some basic spectral properties. We then discuss the relation to
connectivity questions in graphs and the use of the Laplacian in graph
partitioning applications, in particular in combination with
semidefinite relaxations. Finally we present some possibilities to
study characteristic properties of eigenvectors to extremal
eigenvalues by making use of semidefinite eigenvalue optimization
techniques and highlight some connections to embedding problems,
random walks and expansion properties.
'General solvability of linear partial differential equations and the problem of parameter dependence' by T. Kalmes, TU Chemnitz
Abstract: A fundamental question in mathematical analysis, motivated by various problems from different areas of applications, is the solvability of a constant coefficient linear partial differential equation P(∂)u=f in an open subset of ℝd. Depending on the properties of the given right hand side f, this problem leads to the question of surjectivity of the linear differential operator P(∂) on/between various spaces of functions and distributions. Although surjectivity of P(∂) on the space of smooth functions, on local Sobolev spaces, and on the space of distributions has been characterized by Malgrange and Hörmander in the late 1950s and 1960s, respectively, even today there are only few concrete situations when these characterizations can be evaluated in a satisfactory manner.
For a surjective differential operaor P(∂), in view of the impossibility to determine real life data with arbitrary precision, it is moreover important to know whether for arbitrary right hand sides fλ depending "nicely" on a parameter (like, e.g. continuously, holomorphically, etc.) it is always possible to find solutions uλ of P(∂)uλ=fλ depending in the same way on λ.
In our lecture series we will take the point of view of abstract functional analysis in order to discuss conditions for surjectivity and we will present satisfactory evaluations thereof for some classes of differential operators. Moreover, we present a recent sufficient condition due to Bonet and Domanski from 2006 concerning the parameter dependence problem together with a series of interesting, concrete examples.
'Unique continuation, uncertainty principles and sampling theorems.' by P. Stollmann, TU Chemnitz
Abstract: It is a phenomenon of general interest that subsolutions of elliptic partial differential equations are extended in some sense. Several closely related concepts deal with that fact. One of them is unique continuation. In its original form, unique continuation means that such subsolutions cannot vanish to infinite order. Due to its fundamental importance in partial differential equations, this problem has been treated in various levels of generality since Aronszajn's fundamental work in the 1950's.
There was renewed interest in quantitative versions due to the importance of such results for random Schrödinger operators, as seen in
the pioneering paper by Bourgain and Kenig in 2005 that led to quite a list of recent contributions. The main focus here is sometimes indicated by the notion quantitative uncertainty principle and refers to the fact that eigenfunctions of the corresponding partial differential operators are spread out, quite reminiscent of the famous uncertainty principle from quantum mechanics and sampling theorems from harmonic analysis.
In our lecture series we will introduce all these concepts and make connection to a spectral uncertainty principle that has the advantage that it also applies in discrete and non-local situations for which, e.g. unique continuation in its original form is known to fail.
The workshop is supported by the
-- Gefördert vom DAAD aus Mitteln des Bundesministeriums für Bildung und Forschung (BMBF) --