(See also my Metamathematics research page here)
For a very long time, I have been deeply fascinated by L-functions, and in one way or another, almost all of my research projects have been attempts to gain a better understanding of these most remarkable objects. The simplest example of an L-function is the Riemann zeta function, and other L-functions can be constructed from number-theoretic gadgets like Dirichlet characters, elliptic curves, or more generally automorphic representations and motives. (They also appear out of nothing (!), in this YouTube talk).
In some poorly understood sense, L-functions are dual to prime numbers, and many of the deepest mysteries in mathematics are directly or indirectly connected to the interplay between L-functions and primes. For a general introduction to what L-functions are, see these four papers:
- Farmer, Pitale, Ryan, and Schmidt: Analytic L-functions: Definitions, Theorems, and Connections
- Iwaniec and Sarnak: Perspectives on the Analytic Theory of L-functions (from the GAFA2000 Visions in Math volume)
- Ram Murty: Selberg’s Conjectures and Artin L-functions
- Bombieri: Classical theory of Zeta and L-functions (requires SpringerLink subscription)
My PhD thesis was an attempt to reformulate the Beilinson conjectures (on special values of L-functions) in a more conceptual and unified way, via the development of a new cohomology theory for schemes over Z, called Arakelov motivic cohomology. This work overlapped with a similar thesis project carried out at the same time by Jakob Scholbach, and we ended up publishing parts of it together. See the papers below together with several other papers on Jakob’s webpage for details on this project, in particular Jakob’s PhD thesis and Habilitation thesis which contain very nice introductions to all the relevant ideas.
- Thesis: Arakelov motivic cohomology (pdf)
- Holmstrom and Scholbach: Arakelov motivic cohomology I (Journal of Algebraic Geometry, arXiv)
The idea of understanding special values of L-functions via new cohomology theories has also been pursued in more sophisticated directions by Matthias Flach and Baptiste Morin – I recommend all of the papers available on Baptiste’s webpage.
Ultimately, I felt that my own thesis was a disappointment, in that although we could make certain abstract constructions using the tools of motivic homotopy theory, and hence (in Jakob’s work) reformulate the Beilinson conjectures in a nice way, all this abstract machinery did not lead to any new results on special values or on other questions about actual L-functions. The dream I had as a graduate student was that the introduction of new homotopical foundations in number theory would lead to some significant improvement in our understanding of the world of L-functions. This certainly did not happen in my thesis, and my impression is that a decade later, even though homotopy theory and higher categories have permeated significant parts of arithmetic geometry (e.g. p-adic Hodge theory) and there are many abstract theorems about abstract constructions, it is still the case (I believe) that no-one can prove any new explicit statements about actual prime numbers or actual L-functions using homotopical ideas. On the few occasions where a real link is forged between homotopy theory and L-functions (e.g. by Hesselholt), it is done in positive characteristic and not in characteristic zero (where almost all the outstanding L-function mysteries reside).
Partly because of this disillusionment (and partly because of my limited technical ability with higher categories), I have in recent years become more interested in computational aspects of L-functions. The hope I have now is that by letting go of the idea that L-functions are constructed from schemes and motives, and listening carefully to numerical patterns in the L-functions themselves, we may discover some hints as to what new foundations of arithmetic may look like. Together with some of my students, I have a few (small but still nontrivial) observations, which we are in the process of writing up. It may or may not be the case that these observations will one day be connected to some form of homotopical foundations, along the lines suggested by Shai Haran, or in some other form yet to be discovered.