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# 26th Internet Seminar "Graphs and Discrete Dirichlet Spaces"

## Description of the Course

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.

## Structure of the Internet Seminar

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

## Registration

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.