Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

Abstract

Atiyah and Bott used equivariant Morse theory applied to the Yang-Mills functional to calculate the Betti numbers of moduli spaces of vector bundles over a Riemann surface, rederiving inductive formulae obtained from an arithmetic approach which involved the Tamagawa number of SLn. This article attempts to survey and extend our understanding of this link between Yang-Mills theory and Tamagawa numbers, and to explain how methods used over the last three decades to study the singular cohomology of moduli spaces of bundles on a smooth projective curve over C can be adapted to the setting of A 1 -homotopy theory to study the motivic cohomology of these moduli spaces over an algebraically closed field.

The Topological Field Theory of Data: a program towards a novel strategy for data mining through data language

“…In this way, not only we fully recover through the construction proposed the whole data space topology (for example the set of Betti numbers) the space of data, but we are able to continue construct an autonomous, self-consistent Topological Data Field Theory (TDFT) on the space of data: once more the fascination of unexpected links in mathematics [40].…”

A note on the Bloch-Tamagawa space and Selmer groups

“…The Tamagawa number τ (H) is the volume of X H relative to a canonical (Tamagawa) measure [15]. The Tamagawa number theorem [8,1] (which was formerly a conjecture) states The above formulation (5) of the Tamagawa number theorem is due to T. Ono [12,17] whose study of the behavior of τ under an isogeny explains the presence of Pic(H), and reduces the semisimple case to the simply connected case. The original form of the theorem (due to A. Weil) is that τ (H) = 1 for split simply connected H. The Tamagawa number theorem (5) is valid, more generally, for any connected linear algebraic group H over F .…”

Niceness theorems

“…Further the functor of the big Witt vectors is lambda ring valued. However, this paper is not about Symm and its extraordinarily rich structure 18 , but about niceness results. That includes 'nice proofs'.…”