Periodic Reporting for period 4 - GRANT (Groups, Representations and Analysis in Number Theory)
Reporting period: 2020-03-01 to 2021-08-31
There are two related families of objectives on which I have worked within framework of the project. I will leave aside some subdivisions of the project that have resulted in a couple of publications, so as to focus on the main tasks.
a) Growth in groups. Consider a finite subset A of a group, that is, a set whose elements can be multiplied. Now consider the set B of all things that can be obtained as the product of, say, elements of A. Generally, B will have more elements than A, but how many more?
This question - growth - is closely related to that of diameter. In the case of permutation groups, the concept of diameter can be explained very simply. Rubik's cube, and other puzzles like it, can be solved by means of some combination of moves; they give good examples of bpermutation groups, in that we permute pieces in some allowable ways. What is the smallest number d such that every reachable position in a puzzle can be reached in at most d moves? This number d is called the diameter (of the Cayley graph of the group, which describes all possible positions and how to move between them). In fact, the diameter can be defined in the same way for other finite groups, including matrix groups. (Matrix groups give us all the infinite families of finite simple groups other than the alternating groups, which are permutation groups).
There is a third, closely related question, particularly important for applications - the notion of expansion. For a graph, being an expander is a stronger property than having small diameter (that is, an expander necessarily has small diameter, but not viceversa); roughly speaking, an expander graph is one in which a random walk will lead to an approximately equal distribution quickly. While the study of expander graphs was initially motivated (1967-1970s) by the problem of building robust, efficient telecommunication networks, their current applications lie mainly in theoretical computer science, group theory and, increasingly, number theory (see objective b below). The fact that a concept has applications other than the ones that motivated it demonstrates that it is natural and robust -- the "right" concept, so to speak.
My work before the start of the project consisted in proving growth in groups for some families of matrix groups and for permutation groups. My results had already been generalized by other researchers to other families of matrix groups and applied (mainly by Bourgain and Gamburd) to give expansion results. (Growth in groups implies small diameter immediately).
b) Expansion in a divisibility graph.
One of the central issues of analytic number theory is the parity problem: namely, the great difficulty in distinguishing between numbers with n prime factors and n+1 prime factors, or between numbers with an odd or even number of prime factors. Many standard tools break down at that level. We are speaking about the key lacking step in natural approaches to a plethora of hard problems. There has been remarkable progress on this issue in the last 15 years, in several ways. One direction has been that followed by Matomäki, Radziwiłł, Tao et al.; Matomäki and Radziwiłł initiated it by proving that numbers with an odd or even number of prime factors are about equally common in most short intervals. It became clear that much further progress could be made if certain graphs (joining integers n to integers n+p, for any prime p dividing n) were shown to be expanders. My objective has been to prove such expansion by means of a study of random walks in those graphs.
For objectives and conclusions, see below.
a) Growth in groups. Consider a finite subset A of a group, that is, a set whose elements can be multiplied. Now consider the set B of all things that can be obtained as the product of, say, elements of A. Generally, B will have more elements than A, but how many more?
This question - growth - is closely related to that of diameter. In the case of permutation groups, the concept of diameter can be explained very simply. Rubik's cube, and other puzzles like it, can be solved by means of some combination of moves; they give good examples of bpermutation groups, in that we permute pieces in some allowable ways. What is the smallest number d such that every reachable position in a puzzle can be reached in at most d moves? This number d is called the diameter (of the Cayley graph of the group, which describes all possible positions and how to move between them). In fact, the diameter can be defined in the same way for other finite groups, including matrix groups. (Matrix groups give us all the infinite families of finite simple groups other than the alternating groups, which are permutation groups).
There is a third, closely related question, particularly important for applications - the notion of expansion. For a graph, being an expander is a stronger property than having small diameter (that is, an expander necessarily has small diameter, but not viceversa); roughly speaking, an expander graph is one in which a random walk will lead to an approximately equal distribution quickly. While the study of expander graphs was initially motivated (1967-1970s) by the problem of building robust, efficient telecommunication networks, their current applications lie mainly in theoretical computer science, group theory and, increasingly, number theory (see objective b below). The fact that a concept has applications other than the ones that motivated it demonstrates that it is natural and robust -- the "right" concept, so to speak.
My work before the start of the project consisted in proving growth in groups for some families of matrix groups and for permutation groups. My results had already been generalized by other researchers to other families of matrix groups and applied (mainly by Bourgain and Gamburd) to give expansion results. (Growth in groups implies small diameter immediately).
b) Expansion in a divisibility graph.
One of the central issues of analytic number theory is the parity problem: namely, the great difficulty in distinguishing between numbers with n prime factors and n+1 prime factors, or between numbers with an odd or even number of prime factors. Many standard tools break down at that level. We are speaking about the key lacking step in natural approaches to a plethora of hard problems. There has been remarkable progress on this issue in the last 15 years, in several ways. One direction has been that followed by Matomäki, Radziwiłł, Tao et al.; Matomäki and Radziwiłł initiated it by proving that numbers with an odd or even number of prime factors are about equally common in most short intervals. It became clear that much further progress could be made if certain graphs (joining integers n to integers n+p, for any prime p dividing n) were shown to be expanders. My objective has been to prove such expansion by means of a study of random walks in those graphs.
For objectives and conclusions, see below.
Subproject a.- Two of my junior collaborators (Daniele Dona and Jitendra Bajpai) and I have succeeded in proving bounds on growth in matrix groups that are much stronger than those proved before; in particular, the dependence of the speed of growth on the rank (roughly:
how bounds degrade when we consider matrices of larger and larger side) has been improved vastly. Grosso modo, the dependence is as good as it can be (namely, it is now polynomial on the rank, rather than being given by an exponential tower of variable length).
I have also been able to give a second proof of my result with Seress (Ann. of Math., 2014) on growth in permutation groups, bringing matters closer to the strategy used for matrix groups. (See: Proceedings of Groups St. Andrews 2017.) I have worked on some further ideas in that direction with H. Bradford, another junior member of my team.
Subproject b.-
I am glad to report that M. Radziwiłł have succeeded in showing that our prime divisibility graphs (described above) are strong local expanders almost everywhere. There are already several strong consequences in analytic number theory, and there are likely to be more. We obtain a much stronger version of Tao's celebrated "weak Chowla in degree 2" result; we also obtain finer estimates. We are looking forward to see what other researchers (and ourselves) will be able to prove using our result.
The main results of subproject a and b have been written up and submitted for publication.
how bounds degrade when we consider matrices of larger and larger side) has been improved vastly. Grosso modo, the dependence is as good as it can be (namely, it is now polynomial on the rank, rather than being given by an exponential tower of variable length).
I have also been able to give a second proof of my result with Seress (Ann. of Math., 2014) on growth in permutation groups, bringing matters closer to the strategy used for matrix groups. (See: Proceedings of Groups St. Andrews 2017.) I have worked on some further ideas in that direction with H. Bradford, another junior member of my team.
Subproject b.-
I am glad to report that M. Radziwiłł have succeeded in showing that our prime divisibility graphs (described above) are strong local expanders almost everywhere. There are already several strong consequences in analytic number theory, and there are likely to be more. We obtain a much stronger version of Tao's celebrated "weak Chowla in degree 2" result; we also obtain finer estimates. We are looking forward to see what other researchers (and ourselves) will be able to prove using our result.
The main results of subproject a and b have been written up and submitted for publication.