# The Geometry, Algebra and Analysis of Algebraic Numbers (15w5054)

Arriving in Banff, Alberta Sunday, October 4 and departing Friday October 9, 2015

## Organizers

Francesco Amoroso (Université de Caen, Laboratoire LMNO)

(Oklahoma State University)

(University of Edinburgh)

Jeffrey Vaaler (University of Texas at Austin)

## Objectives

The meeting will present a tremendous opportunity to bring experts in many different areas together, and enable them to learn from and build on each others' specialized knowledge. Thus, on the one hand the topic of the workshop is clearly focused, while on the other hand the diversity of participants' interests will give the workshop great potential for cross-fertilisation between different areas of mathematics.
Mahler measure and heights. We plan to summarize recent progress and stimulate further development in this area, where the central problem is the $80$-year old Lehmer conjecture [36],[51], on the smallest Mahler measure of a nonzero noncyclotomic algebraic integer. Thus, while this fundamental problem is old and difficult, there has been steady progress towards its positive solution in recent years. Some important advances include its proofs for polynomials having a bounded number of monomials [23], with odd coefficients [16] and for algebraic numbers that generate Galois extensions of the rationals [9], generalizations of Dobrowolski-type bounds for multiplicatively independent algebraic numbers [9] and for multivariate polynomials [8], and an absolute lower bound for the height in abelian extensions [10]. The polynomials of smallest Mahler measure coming from integer symmetric matrices have also been found [24],[39]. There are many rapidly developing related directions such as computing explicit values of the Mahler measure [41], its connection with algebraic geometry and dynamics [50], [42], counting results [37], infinite fields with the Bogomolov [15] and the Northcott property [27],[20], etc. The aim of the proposed conference is to preserve and enhance this positive momentum, and to create a strong foundation for further progress of this rich area. We believe that the topic of the workshop is quite appropriate, and we hope that it will lead to solutions of important problems.The legacy of Schur and Siegel. One of the aims of the meeting will be to highlight the inheritance of Schur and Siegel in this area. (Their achievements in other areas are so substantial that these have perhaps not received enough attention up to now.) One important topic is the Schur-Siegel trace problem on the smallest limit point for arithmetic means for totally real and positive algebraic integers [48], [49] ,[52],[3] . Another is the application of the Schur theory of Hardy functions to the study of Pisot numbers [19],[54], and, more generally, to the theory of nonreciprocal polynomials [5],[6]. The distribution of sets of conjugates algebraic numbers in the complex plane, whose study began with Kronecker [35], and was continued by Schur [48] and Fekete [30], is still far from being well understood. A notable exception is the area of asymptotic equidistribution for algebraic numbers of small height (or small Mahler measure), which received substantial attention in recent years (see, e.g., [13],[53],[29],[55]). The ideas of equidistribution were successfully applied in many problems, including the Schur-Siegel trace problem, cf. [46],[47].Relations between conjugate algebraic numbers. The problem of how conjugate algebraic numbers may be connected algebraically is poorly understood -- the difficulty of Lehmer's problem being a consequence of this. A related problem is to find strong lower bounds for discriminants of algebraic numbers, current known exponential lower bounds all coming from consideration of field discriminants. A breakthrough on this would have consequences for Lehmer's problem.Other connections between conjugates have been studied in [11],[12] ,[25].itembf Integer Chebyshev problem and other extremal problems for polynomials with integer coefficients in various norms. The problems of minimizing norms by polynomials with integer coefficients date back to at least the work of Hilbert of 1894 [32]. They were developed by Fekete and many others [7],[17],[18],[44], but the integer Chebyshev problem remains open even in its classical setting. It has intimate connections with the distribution of conjugate algebraic integers, and with the Schur-Siegel trace problem. In the workshop, we plan to consider a range of extremal problems that have many applications in number theory.Applications of functional analysis. Recent work has led to new theorems about the Weil height of algebraic numbers using techniques from functional analysis. In some cases the results can be stated in the classical language of algebraic number theory, but the proofs use a special Schauder basis for the Banach spaces identified in [4]. At present proofs using classical methods of algebraic number theory are not known. We expect that these results and techniques will be described at the workshop and become familiar to a wider audience of researchers.Related conferences. There was a meeting in Paris in 1988: Cinquante ans des Polynômes [1], a meeting in Banff in 2003: The many aspects of Mahler's measure, and a meeting in Bristol in 2006: Number Theory and Polynomials} [2] , but no others we know of. Since significant progress is intermittent and breakthroughs sparse, there is no need for frequent conferences. But now the time has come for another one. Thus the proposed meeting in 2015 is both timely and appropriate.
In addition to regular talks, we plan to have discussion groups and problem sessions. There are a number of promising graduate students and recent PhDs in the area from Europe and North America. We intend to particularly target this group with the intention of fostering long term collaborations.

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