The 26th Internet Seminar on Evolution Equations is devoted to the treatment of graphs and discrete Dirichlet spaces. A graph is a geometric structure on a set of vertices and comes with both a Dirichlet form and a Laplacian defined on the set of functions on its vertices. More precisely, given a discrete and countable set $X$ of vertices and a measure $m$ on $X$ of full support a graph on $X$ consists of an edge weight $b\colon X\times X\to [0,\infty)$ satisfying $b(x,y) = b(y,x)$, $b(x,x) = 0$ and $\sum_{y\in X} b(x,y) < \infty$ for all $x,y\in X$, and a killing term $c\colon X\to [0,\infty)$. The corresponding energy form $\mathcal{Q}$ is given by \[\mathcal{Q}(f,g):= \frac{1}{2}\sum_{x,y\in X} b(x,y) \bigl(f(x)-f(y)\bigr)\bigl(g(x)-g(y)\bigr) + \sum_{x\in X} c(x) f(x) g(x)\] for all $f,g\in C(X)$ such that $Q(f,f), Q(g,g)<\infty$. Moreover, the associated (formal) Laplacian $\mathcal{L}$ is given by \[\mathcal{L}f(x) := \frac{1}{m(x)} \sum_{y\in X} b(x,y) \bigl(f(x)-f(y)\bigr) + \frac{c(x)}{m(x)} f(x),\quad x\in X\] for all $f\in C(X)$ such that $\sum_{y\in X} b(x,y) |f(y)| <\infty$ for all $x\in X$.

We will study the interplay between the geometric structure of a graph $(b,c)$ and the spectral theory of the (or better: an) $\ell^2(X,m)$-realisation $L$ of the Laplacian $\mathcal{L}$ as well as properties of the corresponding evolution equation \begin{align*} u'(t) & = -L u(t),\quad t>0,\\ u(0) & = u_0 \in D(L). \end{align*}

We expect the participants to have a basic knowledge in functional analysis (bounded operators, uniform boundedness principle, closed graph theorem, Hahn-Banach theorem), on foundations of Hilbert spaces as well as on foundations in complex analysis of one variable.

The annual Internet Seminars introduce master, Ph.D. and postdoc students to varying subjects related to evolution equations. The course consists of three phases.

• Phase 1 (October-February): A weekly lecture will be provided via the ISem website. These lectures will be self-contained, and references for additional reading will be provided. The weekly lecture will be accompanied by exercises, and the participants are supposed to solve these problems.
• Phase 2 (March-June): The participants will form small international groups to work on diverse projects which supplement the theory of Phase 1 and provide some applications.
• Phase 3 (July 16 to July 22, 2023): Final one-week workshop at the Bundeshöhe in Wuppertal (Germany). There the project teams of Phase 2 will present their projects and additional lectures will be delivered by leading experts.

The ISem team of 2022/23 consists of

• Matthias Keller (Potsdam)
• Daniel Lenz (Jena)
• Marcel Schmidt (Leipzig)
• Christian Seifert (Hamburg)

The website of the 26th ISem is https://www.mat.tuhh.de/isem26
If you have any questions or remarks you can contact us using the e-mail address isem26@tuhh.de

A poster of the 26th ISem can be downloaded soon.

The registration for the 26th ISem opens in August. (Registration will be open until the end of October.)

The first lecture will be delivered mid October.

Organised by the European Consortium “Internet School on Evolution Equations”, the Internet Seminar is an international academic event dedicated to modern analysis. It was founded in 1997 by the functional analysis group of Tübingen (lead by Rainer Nagel). Since then, it has been held every year, organized by different groups from different countries. The aim of the course is to introduce master students, Ph.D students and post-docs to subjects related to functional analysis and evolution equations. For a nice overview on past Internet Seminars, see http://www.math.kit.edu/iana3/seite/isem/en.