Dalhousie University Mathematics Colloquium, 2008/09

Abstracts

Problems is low height polynomials have been investigated by both mathematians and engineers, and, until fairly recently, in almost parallel tracks. I will discuss several of these - the existence of Barker sequences, flatness, Chowla's problem, among others - and present computational approaches and results. Included is the resolution of a problem presented by Littlewood. Joint work with Peter Borwein, Tamas Erdelyi, Richard Lockhard, Joshua Knauer.

Richard Nowakowski (Dalhousie): "Play Time: a History of Combinatorial Game Theory"

Combinatorial Game Theory is 106 (or 109) years old and most of the developers are still alive. This talk will cover the development of the theory with an emphasis on who got it right and who got it wrong (and why). Little mathematical knowledge is required or assumed.

Mitja Mastnak (Saint Mary's): "Combinatorial Hopf Algebras"

A bialgebra is, roughly speaking, an algebra on which there exists a dual structure, called a coalgebra structure, such that the two structures satisfy a compatibility relation. A Hopf algebra is a bialgebra with an antipode, an endomorphism satisfying a certain condition generalizing the concept of inverse in groups. Examples of Hopf algebras include group algebras, universal envelopes of Lie algebras, algebras of representative functions on Lie groups, coordinate algebras of algebraic groups, and quantum groups. Perhaps the most striking aspect of Hopf algebras is their extraordinary ubiquity in virtually all fields of mathematics.

In the talk I will focus on the combinatorial aspects of Hopf algebras. Hopf algebras first appeared in combinatorics in 1970's, when Joni and Rota made the fundamental observation that many discrete structures give rise to natural Hopf algebras whose comultiplications encode the disassembly of those structures. Recently, there were several important developments in the area. Most notably applications in physics, including the famous Hopf algebra of Connes and Kreimer, which encodes the combinatorics of the renormalization.

Alexander Turbiner (CRM and National University of Mexico): "Solvable Schroedinger equations and representation theory"

Exact solutions of non-trivial Schroedinger equations are crucially important for applications. Almost unique source of these solutions is the Olshanetsky-Perelomov quantum Hamiltonians (rational and trigonometric) emerging in the Harish-Chandra theory. A Lie-algebraic theory of these solutions can be developed. It can be shown that all A-B-C-D Olshanetsky-Perelomov Hamiltonians (rational and trigonometric) which include celebrated Calogero-Sutherland Hamiltonians come from a single quadratic polynomial in generators of the maximal affine subalgebra of the gl(n)-algebra but unusually realized. The memory about A-B-C-D origin is kept in coefficients of the polynomial. For the case of exceptional (E) Olshanetsky-Perelomov Hamiltonians previously unknown infinite-dimensional algebras admitting finite-dimensional irreps appear. Lie-algebraic theory allows to construct the 'quasi-exactly-solvable' generalizations of the above Hamiltonians where a finite number of eigenstates is known exactly.

Renzo A. Piccinini (Dalhousie): "Conjugacy Classes of Gauge Groups"

Let G be a topological group and let B be a space with an open covering U = { U_i | i in J}. A principal G-bundle over U is a space E with a right G-action, which is the product of G and U_i over each U_i in U. Sensible physical theories of fundamental interactions can be represented by principal G-bundles; G-equivariant fibrewise automorphisms of the bundle xi representing the theory do not alter the theory, that is, they are the symmetries of the theory called gauge transformations. The gauge transformations of xi form a topological group G(xi), the gauge group of xi.

If the space B is a Riemannian manifold or a CW-complex, the gauge groups G(xi) of all principal G-bundles over B can be viewed as subgroups of a group L which depends on G and U. Hence, we can partition the set of all gauge groups G(xi) into conjugacy classes. We study these classes for certain Bs and Gs.

This is work started by C. Morgan and R. Piccinini (Manuscripta Math. 1989) and continued by M. Crabb, M. Marcolli, A. Minatta, R. Piccinini, M. Spreafico and W. Sutherland.

Daniel Klain (UMass Lowell): "If you can hide behind it, can you hide inside it?"

Suppose that K and L are compact convex subsets of Euclidean space, and suppose that, for every direction u, the shadow (that is, the orthogonal projection) of K onto the subspace normal to u can be translated inside the corresponding shadow of the body L. Does this mean that the original body K can itself be translated into L? Can we even conclude that K has smaller volume than L?

In other words, suppose K can "hide behind" L from any point of view (and without rotating). Does this imply that K can "hide inside" the body L? Or, if not, do we know, at least, that K has smaller volume?

Although these questions have easily demonstrated negative answers in dimension 2 (since the projections are 1-dimensional, and convex 1-dimensional sets have very little structure), the (possibly surprising) answer to these questions continues to be "No" in Euclidean space of any finite dimension.

In this talk I will give concrete constructions for sets K and L in n-dimensional Euclidean space such that every (n-1)-dimensional shadow of K can be translated inside the corresponding shadow of L, while at the same time K has strictly greater volume than L. This construction turns out to be sufficiently straightforward that a talented person could conceivably mold 3-dimensional examples out of modeling clay.

It turns out, however, that the body L with larger (covering) shadows is guaranteed to have greater or equal volume than the set K provided that L is a cylinder, or, even more generally, provided that L can be approximated by Blaschke combinations of cylinders. This cylinder covering theorem will also be presented, along with variations, generalizations, and a number of consequential open questions regarding projections in convex geometry.

This talk is designed for a general mathematical audience, and a substantial portion of the talk should be accessible to students.

Phil Scott (Ottawa): "Reflections on a Categorical Foundations of Mathematics"

We examine the foundations of mathematics based on higher-order logic (type theory), via the notion of Lawvere's elementary topos. We shall discuss various properties of the free topos (a universe of sets acceptable to moderate intuitionists) as well as showing how, for a classical mathematician, Hilbert's formalist program is not compatible with belief in a Platonic standard model (of sets). For this, we shall examine Godel's Incompleteness Theorem both classically and intuitionistically, as well as Tarski's definition of truth. If time permits, we examine how Godel's completeness theorems can be sharpened to sheaf representation theorems (and how truth varies continuously from point to point). The talk is meant for a general audience.

Eva Knoll (Mount St. Vincent): "A Medley of Mathematics in (Contemporary) Art"

In the past few months, the Student Resource Center of the Department of Mathematics and Statistics at Dalhousie University has been moved and refurbished. This project included the commission and purchase of several pieces of contemporary mathematical art. As part of the colloquium, the artist will discuss the pieces and their inspiration, including the mathematics underlying some of their conception. This talk is accessible to a wider audience with a curiosity about mathematics.

Karl H. Hofmann (Darmstadt and Tulane): "Pro-Lie groups: a class of infinite dimensional Lie groups"

A pro-Lie group is a closed subgroup of a product of Lie groups. A connected locally compact group is a pro-Lie group. A product of infinitely many copies of the additive real group is a pro-Lie group that is not locally compact.

The class of pro-Lie groups is the smallest class of topological groups containing all Lie groups which is closed under the formation of arbitrary products and passage to closed subgroups.

The lecture is intended as a motivation for and an exposition of some of the essential pieces of the theory of pro-Lie groups and pro-Lie algebras [1], enough to provide some understanding of a recent structure theorem [2], whose proof requires an analysis of the structure of the automorphism group of an infinite product of real simple Lie groups.

Literature:

[1] K.H.Hofmann and S.A.Morris: The Lie Theory of Connected Pro-Lie Groups, EMS Publishing House, Zürich, 2007, xvii+678pp.

[2] ---: The Structure of Almost Connected Pro-Lie Groups, Preprint, TU Darmstadt 2009, 37 pp.

Mike Bennett (UBC): "Why would anyone study Diophantine equations?"

Diophantine equations are one of the oldest, frequently celebrated and most abstract objects in mathematics. In this talk, I'll attempt to show some of the roles these equations play in modern mathematics and maybe even provide some answers to the question of the title! This talk will be accessible to a general audience.

Jeff Egger (Edinburgh): "Hilbert C*-modules as dagger profunctors"

Hilbert C*-modules are a generalization of inner product spaces where the inner product is valued in a C*-algebra instead of the complex numbers. In this talk, I will investigate the theory of Hilbert C*-modules from the viewpoint of category theory.

Davide Ferrario (Milano-Bicocca): "Symmetric periodic orbits in the n-body problem"

Topological and symmetry constraints have been used extensively and successfully in the variational approach to the n-body problem. We will illustrate the problem, its colliding solutions, symmetries and how suitable constraints yield the existence and qualitative properties of periodic solutions.

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