Forum for matematiske perler (og kuriositeter)

Alexander Schmeding: Fair cake-cutting

Sted: Lunsjrommet i 13. etg., Sentralbygg 2

Tid: Fredag 21. februar 2025 klokken 12:15–13:00

Fair cake-cutting asks how one can divide a resource such as cake fairly (= for some measure of fairness) between a number of players. For two, the well-known “(you-)cut-and-(I-)choose” protocol solves the problem. Preferably, one would solve the problem for any (finite) number of players by an easy, discrete algorithm. In addition, the cake is to be divided in amenable time, say before the heat death of the universe (come to the talk in case you wonder why this awfully specific timeframe was mentioned).

Several old and some quite new ways to divide a cake will be presented and explained. In a remarkable show of bad timing, we will discuss solutions to the problem after you have already divided the cake to be served at the seminar.

Ulrich Bauer: Geometry and Topology of Data

Sted: Lunsjrommet i 13. etg., Sentralbygg 2

Tid: Fredag 6. desember 2024 klokken 12:15–13:00

I will give an illustrated introduction to the geometric and topological concepts, tool, and properties used in topological data analysis (TDA). The main geometric building block is the construction of spaces (simplicial complexes) from data points, guided by a geometric scale and possibly other parameters. The main tool from algebraic topology is persistent homology, which studies the changes in connectivity across the range of parameters.

The use of these constructions for data analysis and beyond will be motivated by concrete examples and applications, such as:

• reconstruction of a shape and its homology from a point cloud,
• faithful simplification of contours of a real-valued function,
• existence of unstable minimal surfaces, and
• identification of recurrent mutations in the evolution of COVID-19.