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Moduli Spaces, Manifolds and Arithmetic

Periodic Reporting for period 3 - MSMA (Moduli Spaces, Manifolds and Arithmetic)

Reporting period: 2019-12-01 to 2021-05-31

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.
Important parts of the work on manifolds has been performed in collaboration with Randal-Williams. For manifolds of dimension 6, 8, 10, ... most 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.
"Hermitian K-theory" was introduced by Karoubi in the 1970s, and is by now somewhat well understood. Ongoing joint work with Venkatesh and Feng aims at using this understanding for number theoretic purposes: to begin with we expect to give geometric and explicit constructions of certain Galois representations, known to exist only by abstract means.

Ongoing work with Chan and Payne aims to use "graph complexes", introduced in the early 1990's by Kontsevich inspired by ideas from physics, for a better understanding of certain object in algebraic geometry. Interestingly, it seems likely that this work will also have implications for moduli spaces of manifolds.

Continuing the joint work with Kupers and Randal-Williams on "cellular E_2 algebras", we expect to apply the methods to general linear groups. Further down the road we hope to give new applications to algebraic K-theory.
Half Dehn twist