David Pengelley

Self-dual orientable embeddings of Kn

Abstract Whenever Eulers Formula does not exclude a self-dual embedding of Kn in an orientable 2-manifold, we construct one. This completes a problem partially solved by Lothar Heffter in 1898 and Arthur White in 1973. The method employs a more general type of current graph than that used to construct triangular embeddings. Self-duality does not follow directly from the index one nature of the constructed embeddings.

A global structure theorem for the mod 2 Dickson algebras, and unstable cyclic modules over the Steenrod and Kudo-Araki-May algebras

The Dickson algebra W n +1 of invariants in a polynomial algebra over [ ] 2 is an unstable algebra over the mod 2 Steenrod algebra [Ascr ], or equivalently, over the Kudo–Araki–May algebra [Kscr ] of ‘lower’ operations. We prove that W n +1 is a free unstable algebra on a certain cyclic module, modulo just one additional relation. To achieve this, we analyse the interplay of actions over [Ascr ] and [Kscr ] to characterize unstable cyclic modules with trivial action by the subalgebra [Ascr ] n −2 on a fundamental class in degree 2 n – a . This involves a new family of left ideals [Iscr ] a in [Kscr ], which play the role filled by the ideals [Ascr ][Ascr ] n−2 in the Steenrod algebra.

The homology of MSpin

The mod two homology of MSpin, the Spin-cobordism Thom spectrum, has a rich algebraic structure. We will describe it explicitly as a comodule algebra, and give some applications to the ring structure of the Spin-cobordism ring.

Global structure of the mod two symmetric algebra, H*(BO;F2), over the Steenrod algebra

The algebra S of symmetric invariants over the eld with two elements is an unstable algebra over the Steenrod algebraA, and is isomor- phic to the mod two cohomology of BO, the classifying space for vector bundles. We provide a minimal presentation for S in the category of un- stable A-algebras, i.e., minimal generators and minimal relations. From this we produce minimal presentations for various unstableA-algebras associated with the cohomology of related spaces, such as the BO(2 m 1) that classify nite dimensional vector bundles, and the connected covers of BO. The presentations then show that certain of these unstableA-algebras coalesce to produce the Dickson algebras of general linear group invariants, and we speculate about possible related topological realizability. Our methods also produce a related simple minimal A-module presenta- tion of the cohomology of innite dimensional real projective space, with ltered quotients the unstable modulesF (2 p 1)=AAp 2 , as described in

“Voici ce que j’ai trouvé:” Sophie Germain’s grand plan to prove Fermat’s Last Theorem

Abstract A study of Sophie Germain’s extensive manuscripts on Fermat’s Last Theorem calls for a reassessment of her work in number theory. There is much in these manuscripts beyond the single theorem for Case 1 for which she is known from a published footnote by Legendre. Germain had a full-fledged, highly developed, sophisticated plan of attack on Fermat’s Last Theorem. The supporting algorithms she invented for this plan are based on ideas and results discovered independently only much later by others, and her methods are quite different from any of Legendre’s. In addition to her program for proving Fermat’s Last Theorem in its entirety, Germain also made major efforts at proofs for particular families of exponents. The isolation Germain worked in, due in substantial part to her difficult position as a woman, was perhaps sufficient that much of this extensive and impressive work may never have been studied and understood by anyone.

Did Euclid Need the Euclidean Algorithm to Prove Unique Factorization

Euclid’s lemma can be derived from the algebraic gcd property, but it is not at all apparent that Euclid himself does this. We would be quite surprised if he didn’t use this property because he points it out early on and because we expect him to make use of the Euclidean algorithm in some significant way. In this paper, we explore the question of just how the algebraic gcd property enters into Euclid’s proof, if indeed it does. Central to Euclid’s development is the idea of four numbers being proportional: a is to b as c is to d. Euclid gives two different definitions of proportionality, one in Book VII for numbers (“Pythagorean proportionality”) and one in Book V for general magnitudes (“Eudoxean proportionality”). We will discover that it is essential to keep in mind the difference between these two definitions and that many authorities, possibly including Euclid himself, have fallen into the trap of believing that Eudoxean proportionality for numbers is easily seen to be the same as Pythagorean proportionality. Finally, we will suggest a way to make Euclid’s proof good after 2300 years.

Index four orientable embeddings and case zero of the Heawood conjecture

Abstract By imposing a special symmetry, we are able to construct index four triangular embeddings of graphs in compact orientable 2-manifolds. Because of the complexity of the current graphs required, such embeddings have heretofore been unattainable, but the imposed symmetry reduces the problem to constructing a special kind of index two current graph. We illustrate the method with a solution for case zero of the Heawood conjecture, using an abelian group, thus completing a constructive proof of the Heawood map color theorem, and eliminating the need for Galois field theory and nonabelian groups in its solution. The method has also been used in the determination of the genus of K n , n , n , n .

The odd-primary Kudo-Araki-May algebra of algebraic Steenrod operations and invariant theory

We describe bialgebras of lower-indexed algebraic Steenrod operations over the field with p elements, p an odd prime. These go beyond the operations that can act nontrivially in topology, and their duals are closely related to algebras of polynomial invariants under subgroups of the general linear groups that contain the unipotent upper triangular groups. There are significant differences between these algebras and the analogous one for pD 2, in particular in the nature and consequences of the defining Adem relations.

Teaching and Learning Mathematics from Primary Historical Sources.

Abstract Why would anyone think of teaching and learning mathematics directly from primary historical sources? We aim to answer this question while sharing our own experiences, and those of our students across several decades. We will first describe the evolution of our motivation for teaching with primary sources, and our current view of the advantages and challenges of a pedagogy based on teaching with primary sources. We then present three lower-division case studies based on our classroom experience of teaching discrete mathematics courses with student projects based on primary sources, and comment on how these could be adapted for use with other lower-division audiences.

Sparseness for the symmetric hit problem at all primes

The hit problem for a module over the Steenrod algebra