29th Internet Seminar: Eventual Positivity
Registration:
- Registration for participants will open soon and will be linked here.
Virtual lecturers:
- Sahiba Arora (Leibniz Universität Hannover)
- Jochen Glück (Bergische Universität Wuppertal)
- Jonathan Mui (Bergische Universität Wuppertal)
Three phases:
- Lecture phase: 13th October 2025 - February 2026
- Project phase: February 2026 - June 2026
- Workshop: 8th - 12th June 2026 in Wuppertal
What is the 29th Internet Seminar?
The "Internet Seminar on Evolution Equations" is an international mathematical event that is organized annually by different mathematicians. Each year's organizers choose a topic - usually closely related to evolution equations and/or operator theory - and design a course that consists of three phases: an online lecture phase, an online project phase and an on-site workshop. Students from all over the world participate in the event.
The 29th iteration of the Internet Seminar has the topic "Eventual Positivity." See below for a preliminary table of contents.
The 29th Internet seminar consists of the following three phases:
- Lecture phase: 13th October 2025 - February 2026
During the lecture phase, the "virtual lecturers" upload a new chapter of lecture notes on this webpage each week. Each chapter consists of approximately 10 pages plus a collection of exercise problems.
The participants read the lecture notes and work on the exercise problems. While doing so, they discuss and collaborate in two different ways:- Via the official course chat platform. We will use the chat platform Zulip (which has LaTeX support) for this purpose. All participants of the ISEM will receive an invitation to the chat at the beginning of the lecture phase.
- Local groups, seminars or courses at the participants' universities. At many universities, the participating students or lecturers organize a local ISEM group that meets regularly to discuss the lectures and the exercise problems.
Each week, the virtual lecturers will ask the local group of one university to provide the solutions to the week's exercises written in LaTeX, such that they can be uploaded to the ISEM webpage.
- Project phase: February 2026 - June 2026
Students who are interested to get even more engaged after the lecture phase can apply to participate in the ISEM's project phase. The participants of the project phase will be split into international groups of 3-5 students coordinated by experienced mathematicians. Each group studies one specific topic in depth - usually by reading a research article about the topic and discussing it online. Each group prepares a presentation for the on-site workshop in the third and final phase of the ISEM. Thus, participating in the project phase implies a commitment to participate in person in the final workshop.
There will be enough slots for approximately 50-60 students to participate in the project phase. - Workshop: 8th - 12th June 2026 in Wuppertal, Germany
The workshop takes place at the meeting center Bundeshöhe in Wuppertal, where accomodation for all participants and a lecture room for the workshop will be available. Accomodation for participating students (PhD, Master and Bachelor students) usually consists of double rooms.
Many of the project coordinators will also participate in the workshop. In addition, there will be talks by invited speakers who are established experts in their fields.
The 29th Internet Seminar is designed for PhD students, Master students and advanced Bachelor students. We expect that participants should have the following prerequisite knowledge:
- Calculus/analysis in one and several variables.
- Linear algebra
- An introduction to real analysis/measure and intregration theory (in particular L^p-spaces)
- An introduction to functional analysis (Banach spaces, Hilbert spaces, bounded linear operators)
- Basic theory of ordinary differential equations (ODEs) would be great, but is not strictly required.
Further important topics that are needed but cannot be covered in-depth in the course (e.g. Sobolev spaces) will be provided in appendices to the lecture notes for those participants that are not already familiar with them.
Students (PhD, Master, and Bachelor) who participate in the project phase and thus in the workshop phase will travel to Wuppertal in Germany for the workshop.
Funding for accommodation and board in Wuppertal will likely be available for a large fraction of the participants. Financial support for the travel to Wuppertal might be available for a small number of participants but cannot be guaranteed at the moment. Most of the participating students will be required to organize their own funding for their travel to Wuppertal.
The Internet Seminar was initiated in 1997 by the functional analysis group of Tübingen, lead by Rainer Nagel. Since then it has been held with participants from more and more universities from all around the world. A list of all 28 previous courses is available here.
See here for the website of our immediate predecessor, the 28th Internet Seminar on "Ergodic Structure Theory and Applications".
The topic: Eventual positivity
Positive solutions to evolution equations
Linear evolution equations are differential equations where one looks for a time-dependent function with values in a Banach space, whose time derivative is connected to the current value of the function by a linear operator. Typical examples of such evolution equations include:
- The heat equation: it describes how the temperature distribution of a material evolves over time.
- Transport and kinetic equations: they describe how the density of particles evolves due to their kinetic motion.
- The differential equation that governs the time development of a continuous-time Markov process.
Those three examples are all differential equations with values in a funtion space, and all three of them have the following property in common: If the initial value of the equation is positive, then so is the solution for all subsequent times. This is called a positivity preserving property and it can be studied by using the - nowadays well-developed - theory of positive operator semigroups.
Eventual positivity
A more surprising phenomena occurs for the following evolution equations:
- Linear ordinary differential equations for which the matrix on the right hand side has negative entries "at the wrong positions".
- A differential equation with fourth- instead of second-order spatial derivatives.
- A heat equation with non-local boundary conditions that connect different parts of the boundary.
In each of these cases it can happen that, for positive initial values, the solution of the equations first has a sign change, but then again becomes and stays positive for large times. This phenomenon is called eventual positivity and it can be studied by using the theory of eventually positive semigroups. The development of this theory in infinite dimensions started about ten years ago.
What makes eventually positive semigroups intriguing is that they are notoriously difficult to characterize and that they show a much more subtle behaviour than their positive counterparts, for instance when it comes to perturbation theory. Large parts of the theory of eventually positive semigroups thus have a different focus and a different flavour than the theory of positive semigroups.
What is in the course?
The course will roughly follow this outline (see further below for a preliminary list of chapters):
- In the first two weeks we will, as a starter, give an introdution to positivity and eventual positivity on finite-dimensional spaces.
- Distributed over several lectures there will be an introduction to unbounded linear operator, their spectral theory, and the relation to partial differential equations (PDEs).
- Positivity and eventual positivity of evolution equations can be studied for PDEs in a variety of function spaces. To be able to treat them all at once, we give an introduction to Banach lattices, which can be thought of as an abstract generalization of classical function spaces and their pointwise (almost everywhere) order.
- We discuss and compare how positivity for solutions of PDEs can be shown by different methods, in particular by the classical maximum principle and by a variational approach.
- We show how maximum principles are related to eventual positivity of so-called resolvents of linear operators.
- There will be an introduction to C_0-semigroups, which are used to describe the solutions of linear evolution equations. We will also briefly discuss when such semigroups are positive.
- Then we focus on eventually positive C_0-semigroups. We prove currently known characterizations of eventual positivity and use them discuss several examples. Typical semigroup topics such as spectral properties, convergence as time tends to infinity, and perturbation theory will be studied and used in this latter part of the course.
1. Positive matrices and matrix semigroups
2. Eventual positivity in finite dimensions
3. Unbounded operators and their spectral theory
4. Ordered function spaces and Banach lattices
5. Positive solutions to PDEs
6. Eventual positivity of resolvents
7. Uniformly eventually positive resolvents: a characterization
8. An introduction to (positive) C_0-semigroups
9. Eventually positive semigroups
10. Convergence to equilibrium
11. Characterizations of eventually positive semigroups
12. Perturbation theory
13. Irreducibility
14. Local eventual positivity
Poster
Poster design: Vivek Kumar
Design idea: Amanda Glück
