Avscon - A very short course on ...
Avscon—a series of short online courses designed to provide accessible and engaging mathematical content. These courses are available online and open to everyone, with new offerings each summer semester.
Each mini-course consists of three lectures delivered by experts from around the world, presenting carefully selected material that is accessible to anyone with a B.Sc. in mathematics or a comparable background.
Lectures are presented online via the online learning platform OPAL (see below for more information).
Lectures start on the 2nd of April 2025!
Registration is open and free for everyone: Click here for the necessary info.
CONFIRMED SPEAKERS - Avscon 2025
- Prof Dr Swanhild Bernstein (TU Bergakademie Freiberg)
- Prof Dr Tomaš Dohnal (MLU Halle-Wittenberg)
- Dr Nora Doll (MLU Halle-Wittenberg)
- Prof Dr Marcus Waurick (TU Bergakademie Freiberg)
SCHEDULE - Avscon 2025
- Waurick - Avscon the Schur topology - 2.4.2025/9.4.2025/16.04.2025
- Doll - Avscon the spectral flow for selfadjoint Fredholm operators - 30.4.2025/07.05.2025/14.05.2025
- Dohnal - Avscon Bifurcations from the Essential Spectrum - 28.05.2025/04.06.2025/11.06.2025
- Bernstein - Avscon Dirac Operators and Clifford Analysis - 25.06.2025/02.07.2025/09.07.2025
TENTATIVE SPEAKERS - Avscon 2026
- Prof Dr Uta Freiberg (TU Chemnitz)
- Prof Dr Philipp Reiter (TU Chemnitz)
- Prof Dr Friedrich Martin Schneider (TU Bergakademie Freiberg)
- <tba>
Title and Abstracts
A very short course on ...
... the Schur topology
Lecturer: Prof Marcus Waurick (TU Freiberg)
Abstract: The aim of the course is to understand homogenisation processes in an operator-theoretic sense. In fact, we will identify an operator topology on the level of coefficients fully capturing the convergence involved in the context of homogenisation: We use solely operator-theoretic means and do not refer to the particular form of the coefficients. The upshot of this perspective will be homogenisation results for time-dependent partial differential equations (almost) for free.
... the spectral flow for selfadjoint Fredholm operators
Lecturer: Dr Nora Doll (MLU Halle-Wittenberg)
Abstract: This course provides an analytic study of the spectral flow for paths of selfadjoint Fredholm operators. Applications of the spectral flow to homotopy and index theory are discussed in detail.
... Bifurcations from the Essential Spectrum
Lecturer: Prof Tomaš Dohnal (MLU Halle-Wittenberg)
Abstract:We consider nonlinear elliptic PDEs in R^n and study bifurcations of localized (L^2(R^n)) solutions from end points of the essential spectrum. Besides the existence of the bifurcation we want to provide an asymptotic approximation of the bifurcating profile when the bifurcation parameter approaches the spectral end point. This problem arises, for instance, in the study of time harmonic solutions of certain nonlinear PDEs. Here the temproal frequency plays the role of the bifurcation parameter. Gaps in the essential spectrum arise naturally in problems with spatially periodic coefficients, which we concentrate on. The approach is then based on the Bloch decomposition, a suitable ansatz and on Banach fixed point arguments to estimate correction terms. We do not use the more traditional variational techniques for bifurcations as these do not provide asymptotic approximations of the solution branch.
... Dirac Operators and Clifford Analysis
Lecturer: Prof Swanhild Bernstein (TU Freiberg)
Abstract: This mini-series explores Clifford algebras, Clifford analysis, and Dirac operators. Clifford analysis emerged from function theory as a higher-dimensional analogue of complex function theory. Like complex function theory, it serves as a refinement of harmonic analysis, is built on an algebraic structure known as Clifford algebras, and requires an equivalent to the Cauchy-Riemann operator found in complex function theory.
The mini-series consists of three parts:
(1) Clifford Algebras:
How are these algebras defined? What are their fundamental properties? How can they be constructed? This section will also include examples that illustrate the concepts of Clifford algebras.
(2) Clifford Analysis:
Utilizing Clifford algebras and a first-order differential operator known as the Dirac operator, we define monogenic functions. These functions are the zero solutions of the Dirac operator. This leads to developing a higher-dimensional function theory and provides an alternative description of differential forms through multi-vector functions.
(3) Further Considerations:
The foundational concepts of Clifford analysis can be extended. We will provide two examples: Hermitian Clifford analysis and q-Clifford analysis. These examples will help clarify and generalize the underlying structures within a new context.
Registration for Avscon
The scripts of the individual courses will be posted to this website. If you want to take part in the discussion concerning the contents of the course, or if you want to access lecture video material, or if you want to obtain credits (ECTS) or a Microcredential degree (from TU Freiberg), please register via OPAL using these instructions and enrol in the OPAL-course related to Avscon as an Avscon participant.