Forum for matematiske perler (og kuriositeter)

Helge Holden: Where are we going in (research) mathematics?

Sted: Lunsjrommet i 13. etg., Sentralbygg 2

Tid: Fredag 6. mars 2026 klokken 12:15–13:00

The very rapid development of artificial intelligence and machine learning will affect all aspects of science. We will look at some of the consequences for mathematics as of the winter of 2026. Several questions will be posed, and some examples will be presented.

Johanne Haugland: Quivers and Connections: Idun Reiten and her Mathematical Legacy

Sted: Lunsjrommet i 13. etg., Sentralbygg 2

Tid: Fredag 30. januar 2026 klokken 12:15–13:00

NTNU professor Idun Reiten (1942–2025) was a pioneering figure in modern algebra. As one of the founders of what is now called Auslander–Reiten theory, her contributions have fundamentally influenced the study of quivers and finite-dimensional algebras. Her work illuminates deep connections between different areas of mathematics and has left a lasting impact both within and beyond the research area known as representation theory.

After a brief biographical overview of Idun Reiten and some of her achievements, the aim of the talk is to give an introduction to Auslander–Reiten theory accessible to a broad mathematical audience. The focus will be on the intuition and conceptual beauty behind the theory, and on why it plays such a central role in understanding the structure of algebras. No prior background in representation theory will be assumed.

Harald Hanche-Olsen: Emmy Noether's theorem on invariants of variational integrals

Sted: Lunsjrommet i 13. etg., Sentralbygg 2

Tid: Fredag 28. november 2025 klokken 12:15–13:00

Slides

Emmy Noether (1882–1935) is best known for her pioneering work in algebra. However, the theorem that bears her name stems from an earlier point in her career. She was working with David Hilbert and Felix Klein in Göttingen, trying to develop the mathematics that Einstein needed for his general theory of relativity. Her theorem on invariants came about as part of this effort. The theorem is often stated in terms of classical mechanics, as describing a one-to-one correspondence between conserved quantities and one-parameter groups of automorphisms of the system under study. Stated this way, the theorem seems surrounded by a sense of mystique.

After a brief biographical sketch of Emmy Noether, I will try to explain the theorem in the special setting of classical mechanics, starting with variational calculus, then moving on first to the Lagrangian, then to the Hamiltonian formulation of mechanics and its connection with symplectic geometry, where we can state the theorem in the form which is most commonly used.

Rostislav Grigorchuk: The Banach-Tarski Paradox, amenability and growth of groups

Sted: Lunsjrommet i 13. etg., Sentralbygg 2

Tid: Fredag 31. oktober 2025 klokken 12:15–13:00

The Banach-Tarski Paradox shocked the mathematical community in the beginning of the 20th century. Analyzing this phenomenon, John von Neumann discovered in 1929 that the paradox has algebraic roots, and introduced the notion that we now call amenability. Von Neuman observed that a free group \(F_2\) on two generators is not amenable and the same holds for all groups containing \(F_2\) as a subgroup. On the other hand, Mahlon Day using the results of von Neumann introduced the class EG of elementary amenable groups. The questions whether the classes NF of groups without free subgroup \(F_2\) and EG coincide with the class AG of amenable groups are known as von Neumann-Day problems (or conjectures).

The notion of growth of finitely generated groups was introduced by A.S. Schwarz in the 50s and by John Milnor in 1968. Milnor raised the question of existence of groups of intermediate growth, that is, growth strictly between polynomial and exponential. Such groups were constructed in 1984 by the speaker and play an important role in algebra, and many other areas of mathematics. In particular, these groups separated the classes EG and AG, thus answering the von Neumann-Day question on non-elementary amenability. In the talk, we will also touch on Gromov’s outstanding classification of groups of polynomial growth.