Dalhousie University Mathematics Colloquium, 2010/11

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Abstracts

Integrable Killing tensors are intimately linked to the question in which coordinates the Hamilton-Jacobi equation can be solved via a separation of variables. We use an isomorphism between the space of valence two Killing tensors on an n-dimensional constant sectional curvature manifold and the irreducible GL(n+1)-representation space of algebraic curvature tensors in order to translate the Tonolo-Nijenhuis-Schouten integrability conditions for a Killing tensor into purely algebraic integrability conditions for the corresponding algebraic curvature tensor. This results in two simple algebraic equations of degree two and three. As an application we (i) construct a new family of integrable Killing tensors and (ii) solve the integrability conditions for the 3-sphere.

Ian Anderson (Utah State University): "A Symbolic Approach to Classification Problems in Differential Geometry and General Relativity"

The classification of finite groups of low order and the classification of 2-dimensional surfaces are important examples of classification problems that mathematicians usually encounter in their first course in algebra and topology. In this talk I'll begin with some simple examples of classification problems in differential geometry and show how Maple can be used to create a user-friend implementation of the solution to these classification problems. We shall then build upon these ideas and discuss current efforts to solve the much more difficult problem of the classification of 4-dimensional space-times and solutions of the Einstein field equations. The talk will be accessible to graduate students and upper-level undergraduates; all are welcome to attend.

Andre Joyal (UQAM): "Higher topos theory, higher categories and brave new algebra"

The notion of a higher topos was introduced by Rezk and it is now the basic framework behind homotopical algebraic geometry developed by Lurie, Toen and Vezzosi. I will explain the difference between a higher topos, a topos, and an ordinary topological space. I will discuss the observation of Bierdemann that Goodwillie calculus is attaching a sequence of Grothendieck topologies to every object of a higher topos. I will discuss the higher topos of parametrized spectra.

Alex Hoffnung (U. Ottawa): "Incidence Geometries, Buildings and Hecke Algebras"

We will introduce one notion of a Hecke algebra — the deformation of the group algebra of a Coxeter group. We will motivate the definition from the very simple perspective offered by category theory and incidence geometries. We will look at the particular example of the Hecke algebra associated to the A_n Dynkin diagram for n=2.

John Cosgrave (Dublin): "What is a Gauss factorial?"

This talk is aimed at a non-specialist audience with no previous knowledge of Number Theory.

Our definition of a Gauss factorial is motivated by Gauss' generalisation of Wilson's theorem. This entirely elementary object opens up a whole new world of interest, revealing many new results in Number Theory, especially in connection with classic, beautiful binomial coefficient congruences of Gauss and Jacobi. In particular it has enabled us to greatly extend work of Chowla, Dwork and Evans (1986), who settled a difficult (1983) conjecture of Beukers relating to Gauss' congruence. I shall briefly outline those works.

Principally I will focus on one very special kind of Gauss factorial (those generated by prime powers), and I shall present some recent (2010) work of ours, which increases a mystery surrounding two particular primes: 29789 and 76543, primes so exceptional that it has been suggested we name them "lonely" primes.

All of this will be done using easily explained concepts. (Joint work with Karl Dilcher).

Karen Yeats (SFU): "Patterns in Feynman graph denominators"

If we write a Feynman integral in parametric form and then integrate one variable at a time, the resulting denominators have nice combinatorial interpretations in terms of spanning forest polynomials. We can use this to find identities of these denominators. Without expecting any background in these areas I'll explain how this picture comes about, some nice identities, and how this is suggestive of knots and matroids.

Russ Woodroofe (Washington University in St. Louis): "Minimax chains and topology of the subgroup lattice"

Kohler proved in the 60's that a finite solvable group has a maximal chain of subgroups having the same length as its chief series. We discuss a converse to this result: that every maximal chain of subgroups of any finite group is at least as long as the chief series, and is strictly longer unless G is solvable. As a consequence we obtain a characterization of finite solvable groups in terms of topological (or commutative algebraic) properties of the order complex of the associated subgroup lattice. This is joint work with John Shareshian.

Peter Paule (RISC, Johannes Kepler University, Linz): "Symbolic Computation in Combinatorics: Recent Progress achieved at RISC"

The first part of the talk is a report on the "partition analysis project" with George E. Andrews (Penn State) and Axel Riese (RISC). In his pioneering book "Combinatory Analysis" MacMahon described a generating function method to solve systems of linear Diophantine inequalities, resp. equations, over nonnegative integers. After more than hundred years, Andrews initiated a revitalization of MacMahon's partition analysis with computer algebra. Applications range from (plane) partitions and compositions of numbers, combinatorial objects like magic squares, to number theoretical congruences. The second part of the talk is devoted to a novel computer algebra method for proving special function inequalities. This method bases on Collins' cylindrical algebraic decomposition (CAD) and has been put into new action by Stefan Gerhold and Manuel Kauers (both RISC). Illustrating examples include a new proof of a classical inequality due to Wallis, as well as a proof a Victor Moll's log-concavity conjecture for coefficients of associated Legendre polynomials. The second part is joint work with Manuel Kauers and Veronika Pillwein.

Hester Graves (Queen's University): "Lagrange's Four-Square Theorem and its Generalizations"

Lagrange proved that every natural number can be written as a sum of four squares. This simple theorem inspired both Waring's problem and the Fifteen theorem. In this talk I will give a basic overview of the directions of study it motivated, and, if there is time, some recent related results.

Karl Hofmann (Darmstadt): "On some types of subgroups of compact groups"

At a conference in Barcelona last summer, called "Algebra meets Topology", Salvador Hernández asked me the following questions:

(1) Does every compact group contain a metric subgroup?
(2) Does every compact group contain a nonmeasurable subgroup?

In the situation of the first question one thinks of large compact groups and infinite metric subgroups. Regarding the second, the concept of measurability refers to Haar measure on a compact groups (which is unique when it is normalized to giving the whole group measure 1.) We shall explain terms and circumstances, give an ample answer to question one, and explain some unsolved issues regarding the second question. (Joint project with Salvador Hernández, and Sidney A. Morris.)

Gert Almkvist (University of Lund, Sweden): "Ramanujan-like formulas for 1/pi^2 and String Theory"

This is joint work with Jesus Guillera, Zaragoza. Using the Gromov-Witten potential from String Theory we design a Maple programme to find formulas for 1/pi^2 . One of the new formulas can be used to compute an arbitrary decimal digit of 1/pi^2 without computing the earlier digits.

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