Walter Talbot's thesis

Walter Richard Talbot was the fourth African American to earn a PhD in Mathematics. His doctoral degree is from the University of Pittsburgh in 1934 in geometric group theory. A contemporary research program was the determination of fundamental domains of finite group actions on complex vector spaces. His thesis is not widely available, and this note gives a brief synopsis of the main results of his thesis, expressed using modern mathematical methods and language, and placed in general context

The Reinhardt Conjecture as an Optimal Control Problem

In 1934, Reinhardt conjectured that the shape of the centrally symmetric convex body in the plane whose densest lattice packing has the smallest density is a smoothed octagon. This conjecture is still open. We formulate the Reinhardt Conjecture as a problem in optimal control theory. The smoothed octagon is a Pontryagin extremal trajectory with bang-bang control. More generally, the smoothed regular $6k+2$-gon is a Pontryagin extremal with bang-bang control. The smoothed octagon is a strict (micro) local minimum to the optimal control problem. The optimal solution to the Reinhardt problem is a trajectory without singular arcs. The extremal trajectories that do not meet the singular locus have bang-bang controls with finitely many switching times. Finally, we reduce the Reinhardt problem to an optimization problem on a five-dimensional manifold. (Each point on the manifold is an initial condition for a potential Pontryagin extremal lifted trajectory.) We suggest that the Reinhardt conjecture might eventually be fully resolved through optimal control theory. Some proofs are computer-assisted using a computer algebra system.Comment: 41 page

Mathematics in the Age of the Turing Machine

The article gives a survey of mathematical proofs that rely on computer calculations and formal proofs.Comment: 45 pages. This article will appear in "Turing's Legacy," ASL Lecture Notes in Logic, editor Rodney G. Downe

On the Reinhardt Conjecture

In 1934, Reinhardt asked for the centrally symmetric convex domain in the plane whose best lattice packing has the lowest density. He conjectured that the unique solution up to an affine transformation is the smoothed octagon (an octagon rounded at corners by arcs of hyperbolas). This article offers a detailed strategy of proof. In particular, we show that the problem is an instance of the classical problem of Bolza in the calculus of variations. A minimizing solution is known to exist. The boundary of every minimizer is a differentiable curve with Lipschitz continuous derivative. If a minimizer is piecewise analytic, then it is a smoothed polygon (a polygon rounded at corners by arcs of hyperbolas). To complete the proof of the Reinhardt conjecture, the assumption of piecewise analyticity must be removed, and the conclusion of smoothed polygon must be strengthened to smoothed octagon

Developments in Formal Proofs

This report describes three particular technological advances in formal proofs. The HOL Light proof assistant will be used to illustrate the design of a highly reliable system. Today, proof assistants can verify large bodies of advanced mathematics; and as an example, we turn to the formal proof in Coq of the Feit-Thompson Odd Order theorem in group theory. Finally, we discuss advances in the automation of formal proofs, as implemented in proof assistants such as Mizar, Coq, Isabelle, and HOL Light.Comment: Bourbaki seminar report 1086, June 201

Some algorithms arising in the proof of the Kepler conjecture

By any account, the 1998 proof of the Kepler conjecture is complex. The thesis underlying this article is that the proof is complex because it is highly under-automated. Throughout that proof, manual procedures are used where automated ones would have been better suited. This article gives a series of nonlinear optimization algorithms and shows how a systematic application of these algorithms would bring substantial simplifications to the original proof. This article includes a discussion of quantifier elimination, linear assembly problems, automated inequality proving, and plane graph generation in the context of discrete geometry.Comment: 14 pages, 5 figure

What is Motivic Measure?

These notes give an exposition of the theory of arithmetic motivic integration, as developed by J. Denef and F. Loeser. An appendix by M. Fried gives some historical comments on Galois stratifications.Comment: 20 pages, 2 figures. These are notes for the AMS Special Session on Current Events, Jan 9, 2004. This work is licensed under the Creative Commons Attribution Licens

The fundamental lemma and the Hitchin fibration [after Ngo Bao Chau]

This article is a Bourbaki seminar report on Ngo Bao Chau's proof of the fundamental lemma. About thirty years ago, R. P. Langlands conjectured a collection of identities to hold among integrals over conjugacy classes in reductive groups. Ngo Bao Chau has proved these identities (collectively called the fundamental lemma) by interpreting the integrals in terms of the cohomology of the fibers of the Hitchin fibration. The fundamental lemma has profound consequences for the theory of automorphic representations. Significant recent theorems in number theory use the fundamental lemma as an ingredient in their proofs.Comment: Bourbaki seminar 1035, April 201

Orbital Integrals are Motivic

This article shows that under general conditions, p-adic orbital integrals of definable functions are represented by virtual Chow motives. This gives an explicit example of the philosophy of Denef and Loeser, which predicts that all naturally occurring p-adic integrals are motivic.Comment: 10 pages, no figures. This paper is granted to the public domai

Can p-adic integrals be computed?

This article gives an introduction to arithmetic motivic integration in the context of p-adic integrals that arise in representation theory. A special case of the fundamental lemma is interpreted as an identity of Chow motives.Comment: 11 pages, 2 figures, based on a lecture at IAS on April 6, 2001 (also at http://www.math.ias.edu/amf/), to appear in a volume "Contributions to Automorphic Forms, Geometry and Arithmetic" dedicated to J. Shalika and published by Johns Hopkins University Press. This article has been granted to the public domain. No rights are reserved by the autho