University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Rothschild Lecture: Higher algebra and arithmetic - Monday 8th October 2018

Rothschild Lecture: Higher algebra and arithmetic - Monday 8th October 2018

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HHH - Homotopy harnessing higher structures

This talk concerns a twenty-thousand-year old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to a cohomological interpretation of the Hasse-Weil zeta function by regularized determinants, as envisioned by Deninger.

This talk is part of the Isaac Newton Institute Seminar Series series.

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