Kempner Colloquium Abstracts

Fall 2003

One of the oldest questions in algebra is to determine the Galois group of a given rational polynomial. While this question arises naturally already in a first year algebra course, typically it is only touched upon in textbooks, mainly because the calculations involved would be horrendous to do by hand.

On a computer, however this becomes practical and provides a nice application of Group theory and Invariant theory and leads to research problems in group theory and computer algebra.

I plan to show the techniques which are used for this, and indicate the practicability.

In the early 1990's Reyer Sjamaar and I proved that singular symplectic quotients are stratified spaces. That is, singular quotients naturally decompose into a locally finite union of manifolds and these manifolds fit together in a nice way. In particular intersection cohomology of such spaces makes sense. Unfortunately, since the links of singularities of singular symplectic quotients are not symplectic, our proof ended up being indirect. More recently Chris Willett and I found a direct proof based on the fact that the links in question carry a contact structure.

Over a year ago, Manindra Agrawal, Neeraj Kayal and Nitin Saxena, from the Computer Science Department of the Indian Institute of Technology in Kanpur, India, surprised the international mathematical and computer science community with their publication "Primes is in P", in which the authors produced an algorithm, now called the AKS Algorithm, that determines whether a given number n is prime or composite and that runs in polynomial time.

The simplicity of the mathematics involved in the description of the AKS Algorithm added fascination to their remarkable achievement.

Since then, various authors in different parts of the world have worked to improve AKS, some by strengthening insights given by the authors of AKS, others by producing variants of AKS that work faster, for a large family of numbers.

In this talk we will begin by describing the AKS Algorithm, and explaining some of the ideas involved in it. We will then briefly discuss the improvement obtained by F. Voloch and those obtained by H. W. Lenstra and C. Pomerance. Next we will discuss the variant of the AKS Algorithm that I obtained, and finally we will describe the various extensions of this variant produced by Qi-Cheng, by D. Bernstein, and by P. Mihailescu and R. Avanzi.

We hope to provide the audience with a feeling for where the subject stands today, as far as we know!

Recently several new ideas were introduced of how the notion of vertex algebra can be generalized. Among these are the field algebras of Bakalov and Kac and open string vertex algebras of Huang and Kong. The common feature of these structures is that they retain the associativity property of vertex algebras, but do not necessarily require commutativity. We suggest a very general framework of defining such non-commutative generalized vertex algebras and constructing explicit examples. Our construction depends on a choice of a class of complex analytic functions; for example, the rational functions would yield the ordinary vertex algebras.

Helaman Ferguson's mathematical sculptures in stone and bronze celebrate ancient and modern mathematical discoveries, melding the universal languages of sculpture and mathematics. Using slides and video, Helaman and Claire trace Helaman's creations from initial concept, mathematical design, computer graphics, diamond cutting and final form. Their lectures have fascinated audiences worldwide, bringing together multiple disciplines and stimulating dialogue among them.

Last year's ideas of Agrawal, Kayal, and Saxena have laid a milestone in the area of deterministic primality testing. Unfortunately, their method is mainly of theoretical interest, as it is much too slow for practical applications.

In the area of random time deterministic tests, improved versions of AKS have been obtained by Berrizbeitia, Mihailescu, and Bernstein by essentially using Kummer extensions.

Via a totally different approach, H.C. Williams and collaborators have developed a test which is conjectured to prove the primality of n in time (lg n)^(3+o(1)). The main advantage of their algorithm is that it is extremely simple, as it is based on Miller-Rabin procedures. Their (plausible) conjecture concerns the distribution of pseudosquares. These are numbers which behave locally like a perfect square modulo all primes in a certain range, but are nevertheless not a perfect square.

While squares are much easier to deal with, this naturally gives rise to the question of whether the pseudosquares can be replaced by more general types of numbers.

We have succeeded in extending the theory to the cubic case. To capture pseudocubes we rely on interesting properties of elements in the ring of Eisenstein integers and suitable applications of cubic residuacity. Surprisingly, the test itself is very simple as it can be formulated in the integers only.

This is joint work with Hugh Williams and Pedro Berrizbeitia.

One of the most important subfields of Riemannian Geometry is the study of the Laplace spectrum of a compact Riemannian manifold. Another spectrum defined in an entirely different manner is the length spectrum of a manifold: the set of lengths of smoothly closed geodesics. We define a new spectrum for compact length spaces and Riemannian manifolds called the "covering spectrum" which roughly measures the size of the one dimensional holes in the space. We investigate the relationship between these spectrums. We analyze the behavior of the covering spectrum under Gromov-Hausdorff convergence and study its gap phenomenon. This is a joint work with Christina Sormani.

We consider the satisfiability of a set \Sigma of equations on a space A, i.e. the existence of continuous A-operations that satisfy the equations \Sigma. In my talk at CU on 9/14/98, I spoke about spaces A (e.g. most spheres) on which only the most trivial of equations are satisfiable. In 2003 we consider the ostensibly simpler space R and its subintervals. For example, if \Sigma is a set of axioms for group theory, then \Sigma is satisfiable on R but not on a closed interval.

The main result of the talk will be that the collection of finite \Sigma satisfiable on R is not recursive, in stark contrast to the 1998 result for spheres. I will outline a proof that is accessible to almost all mathematicians, from first-year graduate students on up. I'll invoke a functional equation for sine and cosine, which we all know. I'll mention Cauchy. I'll use an algorithm-based treatment of recursiveness (alias decidability) with which one can easily and understandably prove undecidability (by referring to known undecidability results, which I shall take as given). I will even refer to one of Hilbert's problems.