p-adic Numbers and the Hasse Principle

Hasse’s local-global principle is the idea that one can find an integer solution to anequation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p. The aim of this course is to give firstly an introduction into p-adic numbers and analysis (different constructions of p-adic integers) and secondly to apply them to study Diophantine equations. In particular we will sketch the proof of the local-global principles for quadratic forms, i.e., of the Hasse-Minkowski theorem, and discuss a counter example for cubic forms. Finally, we want to point out that - while the above results are classical - p-adic methods still play a crucial role in modern arithmetic geometry, i.e., in areas like p- adic Hodge theory, p-adic representation theory/p-adic local Langlands or Iwasawa theory

Wikipedia as a tool for contemporary history of science: A case study on CRISPR.

Rapid developments and methodological divides hinder the study of how scientific knowledge accumulates, consolidates and transfers to the public sphere. Our work proposes using Wikipedia, the online encyclopedia, as a historiographical source for contemporary science. We chose the high-profile field of gene editing as our test case, performing a historical analysis of the English-language Wikipedia articles on CRISPR. Using a mixed-method approach, we qualitatively and quantitatively analyzed the CRISPR article's text, sections and references, alongside 50 affiliated articles. These, we found, documented the CRISPR field's maturation from a fundamental scientific discovery to a biotechnological revolution with vast social and cultural implications. We developed automated tools to support such research and demonstrated its applicability to two other scientific fields-coronavirus and circadian clocks. Our method utilizes Wikipedia as a digital and free archive, showing it can document the incremental growth of knowledge and the manner scientific research accumulates and translates into public discourse. Using Wikipedia in this manner compliments and overcomes some issues with contemporary histories and can also augment existing bibliometric research