## Periodic Reporting for period 4 - MSMA (Moduli Spaces, Manifolds and Arithmetic)

Reporting period: 2021-06-01 to 2022-05-31

This project concerns questions in topology and number theory, two classical areas in pure mathematics. Topology is concerned with the shape of space, with particular emphasis on qualitative properties. "Space" is an abstract concept. This project has focused on so-called manifolds, a particularly interesting type of space, ubiquitous throughout mathematics and other sciences (for example, according to Einstein's general relativity the universe is a four-dimensional manifold). Manifolds of dimension 1 and 2 are easily visualized, whereas manifolds of higher dimension must be treated as abstract mathematical objects. A primary focus in this project has been to study the collection of all manifolds as a topological object in its own right, the so-called moduli spaces of manifolds. In low dimension, the moduli space of 2-dimensional manifolds has been studied for several decades with great success; one of the goals of this project is to extend this knowledge to moduli spaces of manifolds of higher dimension.

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.

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.

The proposed work on manifolds has been performed in collaboration with Randal-Williams. For manifolds of dimension 6, 8, 10, ... essentially all of the goals outlined in the proposal have been achieved, and have been published in three long papers in the highest-ranking journals. The understanding in high dimension is now almost on par with the understanding in dimension 2. Building in part on this work, other important results have been obtained by Kupers, Krannich, and others. More recently, joint work with Kupers and Randal-Williams has introduced a new method (cellular E_2 algebras) to the study of moduli spaces. In dimension 2 we discovered a new pattern (secondary stability) in the cohomology.

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.

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.

All the results by the PI mentioned in the above section go beyond the state of the art. The work with Randal-Williams on high-dimensional manifolds could be said to have reinvigorated the subject, and has sparked many further developments. Spectacular breakthroughs in the 1960s largely studied manifolds one at a time, but many recent developments have aimed at understanding families of manifolds, parametrized by a base space. Such questions can be encoded in a so-called moduli space, whose homotopy type encodes how manifolds can vary in families. The work with Randal-Williams gave a new tool for studying such moduli spaces, and subsequent developments have shown that this tool is very useful. In particular, work by various combinations of Weiss, Kupers, Krannich, and Randal-Williams have used our work to establish breakthrough new results on some of the basic building blocks of geometric topology. Kupers and Krannich have both been supported earlier on this grant.