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).
Registration is open and free for everyone: Click here for the necessary info.
SCHEDULE - Avscon 2025
- Prof Dr Marcus Waurick - Avscon the Schur topology - 2.4.2025/9.4.2025/16.04.2025
- Dr Nora Doll - Avscon the spectral flow for selfadjoint Fredholm operators - 30.4.2025/07.05.2025/14.05.2025
- Prof Dr Dohnal - Avscon Bifurcations from the Essential Spectrum - 28.05.2025/04.06.2025/11.06.2025
- Prof Dr Bernstein - Avscon Dirac Operators and Clifford Analysis - 25.06.2025/02.07.2025/09.07.2025
TENTATIVE SPEAKERS - Avscon 2026
- Prof Dr Ralph Chill (TU Dresden)
- Prof Dr Uta Freiberg (TU Chemnitz)
- Prof Dr Philipp Reiter (TU Chemnitz)
- Prof Dr Friedrich Martin Schneider (TU Bergakademie Freiberg)
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.
... Quantum Systems to Clifford Algebras
Lecturer: Prof Swanhild Bernstein (TU Freiberg)
Abstract: When we discuss quantum theory or quantum systems, we enter a realm of complexity and mystery. Physicists developed these fundamental physical systems in the last century to explain the observed results in the realm of el- ementary particles. Quanta, the building blocks of quantum theory, can be challenging to comprehend, yet they are essential in the realm of quantum computing. This technology is expanding the boundaries of our understand- ing. Or, as Richard Feynman emphasized, “I think it is safe to say that no one understands quantum mechanics.”
Many people are familiar with Heisenberg’s uncertainty principle; how- ever, it is less well-known that quantum theory has also influenced modern mathematics, leading to the creation of new mathematical concepts and the- ories, as well as the specific application of existing ones. Examples of this include partial differential equations, group theory, and representation the- ory. The mathematician Hermann Weyl realized early on that quantum me- chanics had similar structures to those found in mathematical representation theory. Physicists were not necessarily happy about this, as S. Sternberg put it: "Die ’Gruppenpest’ would last for three decades" (E. Wigner). It is less about mathematical rigor than about examples that demonstrate the close connection between mathematical structures and physics and are intended to encourage people to delve deeper into the subject.
In four lectures, we will embrace this complexity and present various phenomena in a somewhat heuristic manner, pointing out relevant references along the way.
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.