Avscon -- A very short course on is an online mathematical convention to obtain glimpses into various mathematical theories. Every summer semester, up to 4 mini courses on a particular mathematical aspect are presented. The lecturers may change from mini course to mini course. The courses are not interdependent so that every new course is the start of something new. Lectures are presented online via the online learning platform OPAL (see below for more information).

Lectures start on the 2nd of April 2025!

 

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 - 25.06.2025/02.07.2025/09.07.2025/16.07.2025

TENTATIVE SPEAKERS  - Avscon 2026

  • Prof Dr Uta Freiberg (TU Chemnitz)
  • Prof Dr Philipp Reiter (TU Chemnitz)
  • Prof Dr Friedrich Martin Schneider (TU Bergakademie Freiberg)
  • Prof Dr Ralph Chill (TU Dresden)

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.

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 soon as it will be publicly available.