My work seeks to understand the world of symmetry by using a classical tool from number theory: zeta functions. It involves methods from a wide spectrum of mathematics: p-adic Lie groups, model theory, algebraic geometry, analytic methods and more.

## Zeta Functions of Groups

I began my doctorate with the following quote from Borges which for me sums up the challenge of navigating the mathematical universe.

Animals are divided into:

(a) belonging to the Emperor

(b) embalmed

(c) tame

(d) sucking pigs

(e) sirens

(f) fabulous

(g) stray dogs

(h) included in the present classification

(i) frenzied

(j) innumerable

(k) drawn with a camel hair brush

(l) et cetera

(m) having just broken the water pitcher

(n) that from a long way off look like flies.

*Borges quoting from “a certain Chinese encyclopaedia”.*

Classification is one of the big themes of science, not least in Mathematics. Like the Chinese encyclopaedia of Borges, mathematicians are constantly trying to divide their world up into pieces that they can tame and list. My exploration of this mathematical world lies at the border between group theory and number theory. Group theory is the mathematics of symmetry.

The properties of symmetry pervade the whole of science and an understanding of what sorts of symmetries can exist and how they relate can provide answers to areas as diverse as cryptography, crystallography and the prediction of the existence of fundamental particles.

The last two centuries have seen two of the great classifications of mathematics. In the late nineteenth century a list was made of the simple complex Lie groups which lie at the heart of much of modern physics. Whilst this century we have seen a list, as bizarre as Borges’ menagerie of animals, of finite simple groups, symmetrical objects with finite numbers of symmetries which can’t be built out of smaller objects.

These represent for mathematicians something as fundamental as the Periodic table of elements is for Chemists.

Number theory on the other hand seeks to understand the numbers we use which, despite one’s first impressions, still remain deeply mysterious. For example, the prime numbers are the building blocks of all numbers, yet how they are distributed is a problem that lies at the heart of the Holy Grail of mathematics, The Riemann Hypothesis.

This hypothesis or conjecture uses something called a zeta function to try to understand how prime numbers behave. Although we know the building blocks of symmetry, we know little about how to put them together. My research looks at trying to understand two wild classes of groups built from some of the simplest of the simple groups: finite p-groups and infinite nilpotent groups.

Riemann demonstrated how a zeta function could provide a powerful way to understand a wild list like prime numbers. My research extends this idea to develop a theory of zeta functions to try to tame these wild classes of groups. For example the theory allows one to say something about the number of objects that exist in nature with a fixed number of symmetries, rescuing them from categories (i) and (j) of Borges’s encyclopaedia.

The tools that have been developed to understand these zeta functions are very varied, ranging from analytic methods, geometric techniques, even to the extent of using an understanding of the very logic of the language in which we write our mathematics.

The research illustrates how interconnected mathematics truly is with tunnels connecting the most unexpected corners of the mathematical world.

## Selection of my most important Academic Publications

[1] Finitely generated groups, p-adic analytic groups and Poincaré series. Annals of Math. 137 (1993), 639-670. This paper combines the theory of p-adic analytic groups and results from model theory to prove a very general rationality result for p-adic pro-p and infinite groups. It is regarded as a fundamental paper in the theory of zeta functions of groups.

[2] Functional equations and uniformity for local zeta functions of nilpotent groups, with A. Lubotzky, Amer. J. of Math. 118 (1996), 39-90. This paper established the first functional equations for zeta functions of groups by exploiting the language of buildings attached to reductive algebraic groups.

[3] Analytic pro-p groups. Second Enlarged Edition, with J.D. Dixon, A. Mann and D. Segal. Cambridge Studies in Advanced Mathematics 61, CUP (1999). This book revolutionized the understanding of analytic pro-p groups by presenting in a new light the work of Lazard. It established the subject as a major theme in modern group theory. The principal writing of the book was undertaken by du Sautoy and Dan Segal.

[4] Analytic properties of zeta functions and subgroup growth, with Fritz Grunewald, Annals of Math, 152, no 3, 793-833 (2000). This paper established deep and subtle connections between zeta functions of groups and mainstream arithmetic geometry. It has changed the view of the subject by revealing that the zeta functions of Weil provide the best model for these non-commutative zeta functions. The paper answered central questions in the subject of subgroup growth.

[5] Counting p-groups and nilpotent groups. Inst. Hautes Études Scientifiques, Publ. Math. 92, 63-112 (2000). This paper explains a new approach to the theory of finite p-groups. Using zeta functions it is possible to show regularities emerging from this wild class of groups. The paper proves Conjecture P of Newman and O’Brien and makes a new contribution to Higman’s PORC Conjecture which has remained open for 40 years.

[6] Counting subgroups in nilpotent groups and points on elliptic curves, J. Reine Angew. Math. 549 (2002) 1-21. This paper marks a turning point in the theory of nilpotent groups constructing the first examples of nilpotent groups whose zeta functions are not uniform across primes but reflect the arithmetic of elliptic curves. It disproved a major conjecture in the area and opened up a completely unexpected new dialogue between group theory and arithmetic geometry.

[7] Motivic zeta functions of infinite-dimensional Lie algebras, with Francois Loeser. Selecta Math. 10 (2004), no 2, 253-303. The language of motivic integration is used to prove uniformity results for zeta functions of groups and to define new zeta functions for the important class of infinite dimensional Lie algebras.

[8] Zeta functions of groups and rings. International Congress of Mathematicians Vol II, 131-149. Eur. Math. Soc. Zurich, 2006. An invitation to present your work at the ICM is one of the highest accolades in mathematics. This paper addresses the current state of the art and looks forward to the new challenges ahead setting the agenda for future research.

[9] Zeta functions of groups and rings, with Luke Woodward. Lecture Notes in Mathematics, 1925, Springer-Verlag 2008. 208pp This is a significant publication bringing together for the first time many calculations of zeta functions of groups applying the powerful technique of resolution of singularities from algebraic geometry. The book also proves new results on functional equations and analytic behaviour of these zeta functions.

[10] Non-PORC behaviour of a class of descendant p-groups. With Michael Vaughan-Lee Journal of Algebra 361 (2012) 287–312 The first example of a finite p-group whose descendants cannot be counted by polynomials but depend on the behaviour of the number points on elliptic curves. It is the most serious attack yet on Higman’s PORC conjecture.

## Downloads

- » Finitely Generated Groups, p-Adic Analytic Groups and Poincare Series The Annals of Mathematics, Second Series, Vol. 137, No. 3 (May, 1993)
- » Functional equations and uniformity for local zeta functions of nilpotent groups American Journal of Mathematics, Volume 118, Number 1, February 1996
- » Analytic Pro-p groups Cambridge University Press
- » Analytic properties of zeta functions and subgroup growth Annals of Mathematics, 152 (2000)
- » Counting p-Groups and Nilpotent Extrait des Publications Mathématiques, N. 92 (2000)
- » Counting subgroups in nilpotent groups and points on elliptic curves Journal fu¨ r die reine und angewandte Mathematik, 2002
- » Motivic zeta functions of infinite-dimensional Lie algebras Selecta Mathematica, New Series, 2004
- » Zeta functions of groups and rings by Marcus du Sautoy and Fritz Grunewald
- » Zeta Functions of Groups and Rings Lecture Notes in Mathematics
- » Non-PORC behaviour of a class of descendant p-groups Journal of Algebra 361 (2012)