Calendar
- 22September 22, 2021
CMSA Quantum Matter in Mathematics and Physics Seminar: Symmetry types in QFT and the CRT theorem
I will discuss ideas around symmetry and Wick rotation contained in joint work with Mike Hopkins (https://arxiv.org/abs/1604.06527). This includes general symmetry types for relativistic field
theories and their Wick rotation. I will then indicate how thebasic CRT theorem works for general symmetry types, focusing on the case of the pin groups. In particular, I expand on a subtlety first flagged by Greaves-Thomas.Galois action on the pro-algebraic completion of the fundamental group
1 Oxford Street, Cambridge, MA 02138 USAGiven a variety over a number field, its geometric etale fundamental group comes equipped with an action of the Galois group. This induces a Galois action on the pro-algebraic completion of the etale fundamental group and hence the ring of functions on that pro-algebraic
completion provides a supply of Galois representations.It turns out that any finite-dimensional p-adic Galois representation contained in the ring of functions on the pro-algebraic completion of the fundamental group of a smooth variety satisfies the assumptions of the Fontaine-Mazur conjecture: it is de Rham at places above p and is a. e. unramified.
Conversely, we will show that every semi-simple representation of the Galois group of a number field coming from algebraic geometry (that is, appearing as a subquotient of the etale cohomology of an algebraic variety) can be established as a subquotient of the ring of functions on the pro-algebraic completion of the fundamental group of the projective line with 3 punctures.
Projective vs. abelian geometry
1 Oxford Street, Cambridge, MA 02138 USAProjective space and complex tori are two of the simplest types of manifolds we encounter, and in many ways they seem very different from each other. I will try to convince you however that, at least if we consider a special (but at the same time very common) class of tori called principally polarized abelian varieties, then the geometry of their subvarieties exhibits surprising, and to date mostly unexplained, similarities to the geometry of subvarieties in projective space.