Periodic Reporting for period 4 - MSMA (Moduli Spaces, Manifolds and Arithmetic)
Periodo di rendicontazione: 2021-06-01 al 2022-05-31
Number theory is the study of numbers and their properties. For example, some numbers are prime numbers and while others decompose as products of prime numbers (e.g. 6 = 2 times 3). This topic may not seem especially geometric at first glance, but nevertheless modern number theory has relied on more and more tools and methods from topology. In particular the notion of cohomology, originally developed in order to distinguish manifolds with different properties, is now an essential tool in algebraic number theory. One of the goals of this project is to introduce and study new applications of tools from topology to number theory.
For the number theoretic aspects, much of what was outlined in the proposal has been achieved in joint work with Venkatesh. In a joint paper which is now published, we use methods from topology (homotopy theory) to study deformations of so-called Galois representations. Several further projects of algebraic or number theoretic flavor have been carried out by the PI and collaborators during the support of the grant. In a paper joint with Feng and Venkatesh, we defined an action of the absolute Galois group of Q on a certain kind of higher Grothendieck--Witt theory, using the moduli space of principally polarized abelian varieties and its connection to integral symplectic groups. We then proceeded to determine the induced action on homotopy groups as a certain universal extension.
Another major finding, not explicitly foreseen in the application but very much in the spirit of this project, was joint with Chan and Payne and has been published in the Journal of the American Mathematical Society. We studied the moduli of curves, and in particular the weight filtration on its rational cohomology. Our main result is a new connection between the top weight cohomology and the cohomology of a graph complex defined by Kontsevich in the 1990s. Combining with a result of Willwacher, we deduce an exponentially growing lower bound on the rank of homology of M_g in degree 4g-5. It was known beforehand that the rational homology vanishes in degrees 4g-4 and higher, so our classes live in the highest possible degree.
Work of Kupers and Randal-Williams, supported by this grant and now published in Forum of Math Pi, led to an understanding of the rational cohomology of Torelli groups for high-dimensional manifolds.
Work of Mikala Jansen under this grant, recently accepted for publication in IMRN, determines the stratified homotopy type of reductive Borel--Serre compactifications.