C.M.Ringel (Bielefeld): New Activities

A new age has started!

(9.10.2026)

It is said that AI has exhibited examples which show that all the classical homological conjectures are wrong. Whether these assertions are correct, has still to be verified (like any claim of a mathematician). But at least someone has started to take representation theory serious! It was always the mantra of Happel: We do not know enough examples.

Let me stress that in mathematics there is nothing what may be called a "counter example". The concept of a counter-example belongs to History of Mathematics, not to Mathematics. It just asserts that someone (usually considered to be famous) had wrong impressions about reality.

Now finally one may hope that the next ICRA workshop will be devoted to present examples so that mathematicians can start to develop a representation theory based on knowledge, not on wishes.

Seminar BIREP

Vorträge - Lectures

The Cape Cod Lectures

Here will be the slides of my Cape Cod lectures, April 2026,
as part of the Maurice Auslander Centenary Lectures
(WHOI, Woods Hole, Ma, USA, April 22-27, 2026), only slightly revised.

Tilting with Maurice. II. (April 25, 2026)
- My lecture Tilting with Maurice, at the Auslander Memorial Conference 1995
- The old and the new view of tilting (and Auslander's visions)
- Finitely generated torsion classes, and the torsional brick chain filtrations.
- Normal tau-rigid modules (and functorially finite torsion classes).
- The lattice of torsion classes
- Example: The tau-rigid modules for one tubular algebra of rank 3.

Homological Theory of Representations. (April 26, 2026)
The file is presently not available, since some corrections have to be made.

Keszthely 1971, Carleton 1072, Bonn 1973
Homological invariants and the classical homological conjectures.
The agemo quiver.
The search for semi-Gorenstein projective modules which are not torsionless.
Parity features of homology
The Nakayama algebras with large finite global dimension (Madsen-Marczinzik-Zaimi, Sen)

The abstract:

Abstract. At the memorial meeting in 1995, I gave a lecture with this title: "Tilting with Maurice". It was devoted to fights about the relevance of concepts in representation theory. Some participants complained afterwards that they had expected (say as a final pun) a discussion about Auslander's contribution to tilting theory. At that time I felt that tilting theory was my business, not his. But I was wrong. Auslanders's ideas (quite different from mine) have turned out to be essential for the further developments: for example, tau-tilting is strongly based on his visions.

The second lecture will focus the attention on Auslander's aim of providing a homological theory of representations, as seen already in his early work in representation theory: the Paris lectures and the Queen Mary lectures. Surprisingly, some of his suggestions have yet been neglected, for example his proposal to look at what he called the t-torsion-free modules. Graphical visualisation (the agemo quiver, the dominance quiver) may stimulate further interest. Maurice himself never relied on visual means - it seems that this has to be my task.

Both lectures will start with historical recollections. This then will be followed by a report on recent investigations of various mathematicians which correspond to Auslander's ideas. The first lecture will concentrate on the role of bricks in module categories, the second on parity features of homology groups, always with emphasis on suitable examples.

Previous lectures

Discrete Structures in Mathematics - the SFB 343 (1989-2000) (Bielefeld, 26.09.2019)
Luise Unger: Algebra und Kombinatorik (Hagen, 30.09.2021)
Bernd Fischer at Bielefeld - some recollections
(29.9.2023. Groups 2023 - Northern Group Theory conference in honour of Bernd Fischer)
My pre-dinner Presentation at the BNU Conference on Representation Theory of Alge\ bras, May 31, 2025.

German Fascination with Chinese Culture in the 18th century: The "Causa Wolffiana".

The basic lattices

Given a Dynkin diagram Δ, let Φ₊(Δ) be its root poset. Given a vertex x of Δ and a natural number t, let Φ_t(Δ,x) be the subposet of Φ₊(Δ) given by all roots a with a_x = t.
The posets Φ₁(Δ,x) with Δ simply laced and x a leaf of Δ are called the basic lattices. They are the lattices of the form

Φ₁(Δ,x) = J^r([p-1]⊕[q-1]) = J^r-1([p]×[q])
where Δ = T_pqr is the star with arms of rank p, q, r ≥ 1 and x is the leaf on the arm of rank r. For example, T₃₃₂ = E₆ and T₃₂₁ = A₅.
(Here, [n] is the chain of cardinality n, ⊕ is the disjoint union, × the product of posets, and J(P) is the poset of ideals of a poset P.)
Here are the Hasse diagrams for the basic lattices.
(1)

All basic lattices have a symmetric chain decomposition.
Thus, they are Peck posets (graded posets which are rank symmetric, rank unimodal and strongly Sperner). Of course, the basic lattices are distributive (since they are ideal lattices of posets).
(2) Let Δ be a simply laced Dynkin diagram, x a vertex of Δ and t a natural number:

The poset Φ_t(Δ,x) is a product of basic lattices.
Here is an example.
(3) If ω is an orientation of a simply laced Dynkin diagram Δ, let H(Δ,ω,x) be the hammock for the Dynkin quiver (Δ,ω) given by all indecomposable representations with at least one composition factor S(x).
Let H_t(Δ,ω,x) be the subset of all M with precisely t composition factors S(x); this is a poset with respect to x-full maps and

H_t(Δ,ω,x) ≃ Φ_t(Δ,x)

Xiamen: Mathematics and Puzzles
Announcement
The collection
Puzzles and Mathematics (pdf-file, March 15, 2019)

Conferences and Lectures
- Permutationen - Das Fotoautomaten-Paradox und weitere Überraschungen.
  Mathematikdidaktisches Kolloquium, Bielefeld, 19.12.2017
- Conferences and lectures, 2003 - 2016
Themes / Themen

Preprints, Notes from Lectures
- Das Fotoautomaten-Paradox
- Preprints and Manuscripts
- Book Project: Claus Michael Ringel, Jan Schröer: Introduction to Representation Theory of Algebras (in preparation)
- General texts (For example on puzzles)
- Fully documented lectures and list of further lectures
Some older presentations
- Gießen: Mathematikum, 15.01.2010
  Unmöglich - oder doch möglich? Denkspiele und Puzzles, die die Köpfe rauchen lassen!
  Präsentation
- Eleganz in der Mathematik.
  Kunsthalle Bielefeld, 30. April 2010.
  Präsentation
- Eleganz in der Mathematik: Verkettete Kreise im Raum
  Ankündigung
  Präsentation
- The Impossible. Some Puzzles (version June 17, 2011)
  Englische Fassung (leicht erweitert) von Teil (3) der Reihe: Denkspiele aus aller Welt
Corrections

Recommendation

Jones, Korol, Szamboti, Walter: 15 years later: On the physics of high-rise buildings collapses. in: Europhysics News, Volume 47, Number 4, July-August 2016 (Europhysics News is published by EPS, the European Physical Society).
Text

Fakultät für Mathematik, Universität Bielefeld
Verantwortlich: C.M.Ringel
E-Mail: ringel@math.uni-bielefeld.de
Last modified: Oktober 13, 2016

A new age has started!

The Cape Cod Lectures

Previous lectures

German Fascination with Chinese Culture in the 18th century: The "Causa Wolffiana".