Mathematical Colloquium

The Classification of Finite Simple Groups (CFSG), also known as the enormous theorem, is a highlight of 20th-century mathematics, both with respect to its mathematical content and to the complex process of proving the result. We will give a brief mathematical introduction and then focus on historical and cultural aspects, such as: changing perceptions of what a mathematical proof is, the character and the many contexts of mathematics as an intergenerational and international collaborative enterprise, the roles that trust and consensus play within this enterprise, and financial funding.

How much does modular arithmetic (i.e., calculating modulo $p$) tell us about the integers and the rational numbers? Under which circumstances can we use insights about fields of positive characteristic (e.g., finite fields, function fields over finite fields, or power series fields over finite fields) to understand fields of characteristic 0 (and conversely)?
Classical methods to transfer results between fields of different characteristics are the Lefschetz principle and the Ax-Kochen/Ershov Theorem which states that asymptotically, the theory of the $p$-adic numbers $\mathbb{Q}_p$ and of power series fields $\mathbb{F}_p((t))$ coincide. Tilting perfectoid fields, a recent approach developed by Scholze, gives a transfer principle between certain henselian fields of mixed characteristic and their positive characteristic counterparts and vice versa. In this talk, we survey various transfer principles and present a model-theoretic approach to tilting via ultraproducts, which allows us to transfer many first-order properties between a perfectoid field and its tilt. A key ingredient in our approach is an Ax-Kochen/Ershov principle for perfectoid fields (and generalizations thereof). This is joint work with Konstantinos Kartas (Sorbonne Université).

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