Forum for matematiske perler (og kuriositeter)

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.