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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 $\mathcal{Q}(f,f), \mathcal{Q}(g,g)<\infty$. 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 is now closed. If you still want to register/participate, please send us an e-mail.

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.

## Partners

The workshop is supported by the DFG Priority Programme SPP 2026 Geometry at Infinity (https://www.spp2026.de/).

The workshop is co-organized by the COST Action CA18232 Mathematical models for interacting dynamics on networks ( mat-dyn-net) from COST (European Cooperation in Science and Technology).

COST (European Cooperation in Science and Technology) is a funding agency for research and innovation networks. Our Actions help connect research initiatives across Europe and enable scientists to grow their ideas by sharing them with their peers. This boosts their research, career and innovation.

The main aim and objective of the COST Action CA18232 is to bring together leading groups in Europe working on analytical and numerical approaches to a range of issues connected with modelling and analysing mathematical models for dynamical systems on networks (DSN), in order to be able to address its research challenges at a European level.

## Anouncement of NANT

The Networking in Applied Network Theory event will be held in October 2022 to February 2023. It focuses on applications of dynamical systems on networks.