John McCleary

On the modp Betti numbers of loop spaces

The method of proof is an elegant application of the Leray-Serre spectral sequence to the path-loop fibration. This result is one of the first spectacular applications of homotopy theory in differential geometry. In [Gr], M. Gromov improved the classical theorem of Morse above in the direction of a quantitative statement: let N(p,q;2) denote the number of geodesics in M joining p and q of length less than 2. If M is a compact, connected Riemannian manifold with finite fundamental group, and p and q are non-conjugate points in M, then there is a positive constant C, depending on M, such that

How not to prove Fermat's last theorem

The history of the search for a proof of this theorem is filled with many partial successes and many complete failures. In this article, I report on my own attempt and failure because I feel that the method is of considerable interest in spite of its lack of success. The central idea revolves around the question of how well one can approximate a given real number, a, with rational numbers. The goodness of the approximation can be used to determine whether or not a is a rational. This method has been used recently for a number of interesting problems in number theory, and with great success (see [1] for a survey of the work of Apery, Alan Baker, the Chudnovskys and Beukers). In applying this idea to Fermats Last Theorem, we reduce the problem to the study of the rationality of nth roots of certain rational numbers. A tried and true method of approximation, Newtons method, leads to a sequence of rational approximants that we try to show converge too quickly to be giving a rational number. It is at this final stage that the approach fails. In section 2 we reformulate Fermats Last Theorem suitably. In section 3 criteria are presented for recognizing when a real number is irrational; one criterion treats the rate of approximation by rational numbers, and the other concerns the continued fraction representation of the number. In section 4 Newtons Method is introduced and the Main Idea stated, followed by an example where the Main Idea is shown to work by playing the Newton approximation off against the continued fractions approximation. In section 5 we review the facts about the convergence of Newtons Method as it applies to the problem at hand. This leads to an inequality that is crucial to the success or failure of the Main Idea. In section 6 the explicit form of the Newton approximants is considered along with the growth of their denominators. Another inequality results that, together with the previous inequality, leads us to the main theorem of the paper, and the extent to which the Main Idea can be made to work.

What mathematics isn’t

ConclusionsThe examples presented above reveal the features of our notion of reorientation in some of the major turning points of the history of mathematics. An accepted view of what constitutes mathematics is called into question by a new result, or new pressures show that mathematics simplyisn’t what was previously believed; a new conception of mathematics follows and produces new ideas and results even as it incorporates old results of the previous schools. This pattern is discernible in many other instances in the history of mathematics. Some examples that we did not discuss include the introduction of the group concept, the discovery of non-Euclidean geometries, Hilbert’s Finite Basis theorem, the introduction of differential methods into the study of the foundations of geometry, the rise of functional analysis, and, from recent developments, the use of the categorical viewpoint. The reader can easily find a similar development in his or her speciality. Though this approach to the history of mathematics does not immediately arrive at a philosophy of mathematics, it does allow one to organize the stuff out of which a philosophy of mathematics is shaped, that is, the mathematicians’ answers to ‘what is mathematics?’. We have outlined a possible mechanics for the evolution of the questions of interest to the philosophers of mathematics. We leave the determination of the underlying causes to them.

String homology, and closed geodesics on manifolds which are elliptic spaces

Let

Airborne weapons accuracy: topologists and the Applied Mathematics Panel

M

Geometry from a Differentiable Viewpoint

be a closed simply connected smooth manifold. Let

Homotopy theory and closed geodesics

\F_p

Closed geodesics on stiefel manifolds

be the finite field with

CHAPTER 23 – A History of Spectral Sequences: Origins to 1953

p

Spectral sequences in combinatorial geometry: Cheeses, inscribed sets, and Borsuk–Ulam type theorems

elements where