The 29th Internet seminar consists of the following three phases:

• Lecture phase: October 2025 - February 2026
  During the lecture phase, the "virtual lecturers" upload a new chapter of lectures 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 where they study one specific topic in debth - usually by reading a research article about this topic and discussing it online.
  Each group prepares a presentation about their topic that will be held on-site at the workshop in the third phase of the ISEM. Thus, participating in the project phase implies a commitment to participate in person in the final workshop.
  Each group is coordinated by one or two more experienced mathematicians who can offer support.
  
  There will be enough slots for approximately 50-60 students to participate in the project phase.
   
• Workshop: 8th - 12th June 2026 in Wuppertal
  The workshop takes place at the meeting center "Bundeshöhe" in Wuppertal, Germany, where accomodations 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 a couple of 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. Participants should have studied the following prerequisites: 

• 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 workhshop phase will travel to Wuppertal in Germany for the workshop.

The virtual lecturers will try to organize funding for the accommodation in Wuppertal for a large fraction of the participants. Financial support for the travel to Germany 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 expenses.

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 seminars is available here.

See here for the website of our immediate predecessor, the 28th Internet Seminar on "Ergodic Structure Theory and Applications".