Další seminář z algebry se koná 30.4.2020 od 13.00 online na platformě ZOOM. Informace pro připojení a další program semináře je zde.

Christian Espindola

A topos theoretic proof of Shelah’s eventual categoricity conjecture

Abstrakt:
Assuming the Generalized Continuum Hypothesis (GCH), we give a topos-theoretic proof of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC) with amalgamation. The proof is based on infinitary generalizations of Deligne and Joyal's completeness theorems for certain infinitary intuitionistic logics, by means of which it is possible to attack the question of categoricity in a topos-theoretic way. Using recent results on the Scott adjunction developed by Henry and Di Liberti, we prove under GCH that an AEC with amalgamation which is categorical in a pair of cardinals is also categorical in all cardinals in between. Under some extra natural assumptions on the AEC, we also deduce a new downward categoricity transfer. We also explain how these methods can be adapted to shed some light on the categoricity spectrum of more general accessible categories.

Zrušeno

Bohužel se přednášející, Hans Zanna Munthe-Kaas, dostal v  minulých dnech do karantény kvůli novému koronaviru a musel svoji cestu zrušit.

Pro jeho přednášku najdeme společně nahradní termín později, tuto středu se neuskuteční.

Seminář se koná 4.3.2020 od 16:30 v Mendelově muzeu.

Hans Munthe-Kaas

Symmetry: From Conway’s Magic Theorem to Archimedes’ Labyrinth and Beyond

Abstract:
Symmetry is a topic which has inspired artists and mathematicians from ancient to modern times. A fundamental problem is the classification of discrete groups of isometries, such as the 17 planar wallpaper groups, which have been used in mosaics since medieval ages and were classified by Fedorov in 1891 in a complicated proof.

Conway’s Magic Formula can be used to classify discrete symmetries for spherical, plane and hyperbolic surfaces and yields the 17 wallpaper groups, the 7 frieze patterns and all discrete spherical symmetries as special cases. The formula and its proof is so simple that it is accessible to advanced high school students.

Recently, Munthe-Kaas was involved in the design of a mathematical maze in Bergen Botanical garden. Inspired by Conway, he ended up with a highly symmetric design. Under some reasonable assumptions, only one of the 17 wallpaper groups fulfils his original design criteria. The labyrinth, called Archimedes’ labyrint consists of 1234 yews (Taxus baccata, Tis červený) in 2m height and covers an area of about 800 m^2. It was presented in Science Magazine, October 2018.

In the last part of this talk we move beyond Conway, and discuss the problem of multivariate polynomial interpolation. Based on kaleidoscopic symmetry groups (Coxeter groups), we find interpolation points with remarkable properties. We show that for any d and k, there exists a unisolvent set of interpolation points for d-variate polynomial interpolation of order k. These points have optimal Lebesque constants and allow fast computation by symmetric fast Fourier transforms.