An Overview of Schema Theory

The purpose of this paper is to give an introduction to the field of Schema Theory written by a mathematician and for mathematicians. In particular, we endeavor to to highlight areas of the field which might be of interest to a mathematician, to point out some related open problems, and to suggest some large-scale projects. Schema theory seeks to give a theoretical justification for the efficacy of the field of genetic algorithms, so readers who have studied genetic algorithms stand to gain the most from this paper. However, nothing beyond basic probability theory is assumed of the reader, and for this reason we write in a fairly informal style. Because the mathematics behind the theorems in schema theory is relatively elementary, we focus more on the motivation and philosophy. Many of these results have been proven elsewhere, so this paper is designed to serve a primarily expository role. We attempt to cast known results in a new light, which makes the suggested future directions natural. This involves devoting a substantial amount of time to the history of the field. We hope that this exposition will entice some mathematicians to do research in this area, that it will serve as a road map for researchers new to the field, and that it will help explain how schema theory developed. Furthermore, we hope that the results collected in this document will serve as a useful reference. Finally, as far as the author knows, the questions raised in the final section are new.Comment: 27 pages. Originally written in 2009 and hosted on my website, I've decided to put it on the arXiv as a more permanent home. The paper is primarily expository, so I don't really know where to submit it, but perhaps one day I will find an appropriate journa

Abbott & Costello Meet Frankenstein

In lieu of an abstract, here is the article\u27s first paragraph: Years after writing Frankenstein, Mary Shelley published her Rambles in Germany and Italy in 1840, 1842, and 1843. Early on in it she states her therapeutic intent: “Travelling will cure all: my busy, brooding thoughts will be scattered abroad; and, to use a figure of speech, my mind will, amidst novel and various scenes, renew the outworn and tattered garments in which it has long been clothed, and array itself in a vesture all gay in fresh and glossy hues, when we are beyond the Alps.” (Part I, Letter I, p.2) Even if the classic 1948 comedy Abbott and Costello Meet Frankenstein deviates from Mary Shelley’s novel too much, it is spot-on regarding her larger project of how best to navigate in the pilgrimage of life. By pilgrimage here I mean not a predetermined track, but rather just the opposite, since what is most abhorrent is to let someone else determine your proper path, instead of having a keyed-up watchfulness for the full range of possible futures

On the number of minimal surfaces with a given boundary

We generalize the following result of White: Suppose $N$ is a compact, strictly convex domain in \RR^3 with smooth boundary. Let $\Sigma$ be a compact 2-manifold with boundary. Then a generic smooth curve $\Gamma\cong \partial\Sigma$ in $\partial N$ bounds an odd or even number of embedded minimal surfaces diffeomorphic to $\Sigma$ according to whether $\Sigma$ is or is not a union of disks. First, we prove that the parity theorem holds for any compact riemannian 3-manifold $N$ such that $N$ is strictly mean convex, $N$ is homeomorphic to a ball, $\partial N$ is smooth, and $N$ contains no closed minimal surfaces. We then further relax the hypotheses by allowing $N$ to be weakly mean convex and to have piecewise smooth boundary. We extend the parity theorem yet further by showing that, under an additional hypothesis, it remains true for minimal surfaces with prescribed symmetries. The parity theorems are used in an essential way to prove the existence of embedded genus-$g$ helicoids in \SS^2\times \RR. We give a very brief outline of this application. (The full argument will appear elsewhere.)Comment: 13 pages Dedicated to Jean Pierre Bourguignon on the occasion of his 60th birthday. One tex 'newcommand' revised because arxiv version had an error. Two illustrations and one proof have been added. May 2009: Abstract, key words, MSC codes added. One typo fixed. Paper has been published in Asterisqu

The Geometry of Genus-One Helicoids

We prove: a properly embedded, genus-one minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into two connected components that lie on either side of the helicoid. We prove an analogous result for periodic helicoid-like surfaces. We also give a simple condition guaranteeing that an immersed minimal surface with finite genus and bounded curvature is asymptotic to a helicoid at infinity.Comment: 22 pages. This updated version (Apr 17, 2009) contains a much simplified statement and proof of Lemma 3.2. This version will appear in Comm. Math. Hel