October 12-14, 2007
Pure Math Graduate Student Conference 2007
Simon Fraser University

Abstracts

Plenary Talks

Jason Bell, Simon Fraser University
Growth in algebras

An algebra A is just a ring that is also a vector space over a field k. We define and give the basic facts about Gelfand-Kirillov dimension, which is a way of studying how quickly an infinite dimensional algebra "grows". We'll give some examples and show how this new dimension allows one to obtain noncommutative analogues of well-known theorems from classical algebraic geometry.

Petr Lisonek, Simon Fraser University
Steganography with linear codes

Steganography is the science of information hiding. The sender embeds a secret message into a cover object (e.g., a multimedia file) by slightly distorting it in a way that enables the intended recipient to retrieve the hidden message; at the same time the very existence of the hidden message should be impossible to detect by any third party. The main goal of Steganography is to manage the trade-off between the amount of the communicated information and the amount of introduced distortion. We will show how linear codes can be used for this purpose. We will survey the recent results and list some open problems.

Greg Martin, University of British Columbia
Prime numbers: what we know, and what we know we think

Questions about the distribution of prime numbers, and about the existence of prime numbers of special forms, have been stymieing mathematicians for over two thousand years. It's almost necessary to study two different subjects: the theorems about prime numbers that we have been able to prove, and the (vastly more numerous) conjectures about prime numbers that we haven't yet succeeded at proving. In this talk I'll describe many of the open problems (and a few closed ones) concerning the distribution of primes, indicating when I can the techniques that have been used to attack them.


Contributed Talks

Andrew Arnold, Simon Fraser University
Calculating Cyclotomic Polynomials

This talk will detail two methods of computing cyclotomic polynomials. The first method computes cyclotomic polynomials recursively through a series of polynomial divisions. We've recently implemented a new, surprisingly fast algorithm which calculates cyclotomic polynomials as a product of sparse power series. We will also show some results we've obtained on the height and length of cyclotomic polynomials. In particular, we have found a number of cyclotomic polynomials with very large height, as well as the cyclotomic polynomial of smallest order whose height exceeds its order squared.

Matthew Ballard, University of Washington
Some homological algebra for A-infinity algebras

A-infinity algebras were originally introduced by Stasheff, but have found new life thanks in large part to the homological mirror symmetry conjecture of Kontsevich. After motivating with a brief discussion of homological mirror symmetry, I will cover the basic definitions of A-infinity algebras, modules, and morphisms. From there, I will discuss the bounded derived category of A-infinity modules over an A-infinity algebra and give sufficient conditions for the existence of a Serre functor.

Stephen Benecke, University of Victoria
Trees with Domination Subdivision Number One
Joint work with: Christina M Mynhardt

The domination subdivision number \mathrm{sd}_{\gamma}(G) of a graph G is the minimum number of edges that must be subdivided to increase the domination number of G. We present a simple characterization of trees T with \mathrm{sd}_{\gamma}(T)=1 and a fast algorithm to determine whether a tree has this property.

Robert Bradshaw, University of Washington
SAGE: Open Source Mathematics Software
Joint work with: William Stein

Open Source math software such as GAP, Pari, SciPy, NTL, and others as well as providing new functionality. It aims to be a free alternative to Magma, Maple, Mathematica, and Matlab. It now has many developers from around the world and is used in serious research by many people. I would talk about its history, where it is now, and what plans are for its future.

Karel Casteels, Simon Fraser University
Generalizations of De Bruijn Cycles
Joint work with: Brett Stevens

In 1992 Chung, Diaconis and Graham generalized de Bruijn cycles to other combinatorial families with universal cycles. Universal cycles have been investigated for permutations, partitions, k-partitions and k-subsets. In 1990 Hurlbert proved that there exists at least one Ucycle of n-1-partitions of an n-set when n is odd and conjectured that when n is even, they do not exist. Here we discuss the structure of these Ucycles which leads to a proof of Hurlbert's conjecture. In addition we can give lower bounds on the number of such Ucycles (when n is odd of course!).

Adam Clay, University of British Columbia
Densely Ordered Braid Subgroups
Joint work with: Dale Rolfsen

Dehornoy showed us in the early 90's that the braid groups admit an ordering that is invariant under left-multiplication. This ordering is discrete, in the sense that every element has a unique predecessor and successor. Upon restricting the ordering to certain normal subgroups, we find that the ordering becomes a dense order: any two elements of the subgroup have another subgroup element strictly between them. In particular, this counter-intuitive result holds for the commutator subgroup, and the kernels of the Burau representation (for those n for which it is not injective). This is joint work with Dale Rolfsen.

Michael Coons, Simon Fraser University
A density-residue theorem

We present a concrete connection between elementary and analytic number theory: that the asymptotic density of a subset A of \mathbb{N} is equal to the residue of the zeta function associated to A at s=1. This result is applied to produce a week version of the prime number theorem.

Emilie Dufresne, Queen's University
Separating Invariants: an Introduction

The idea behind separating invariants comes from the desire to use "invariants" (ideally finitely many) to distinguish the orbits of a group action. This is not possible in general (think isomorphism classes of knots), but we can still get some level of separation: separating as much as the whole ring of invariants. In this talk I will introduce the setting in which (my kind of) invariant theory is done and define a notion of separating invariants in this setting. If time permits, I will discuss some basic results.

Javad Ebrahimi B, Simon Fraser University
On the sum of two largest eigenvalues of a graph and Gernert's conjecture
Joint work with: Bojan Mohar, Azhvan Sheikh Ahmady

D. Gernert conjectured that the sum of two largest eigenvalues of the adjacency matrix of any simple graph is at most the number of vertices of the graph. This can be proved, in particular, for all regular graphs. Examples will be given that disprove this conjecture. We will also exhibit a nontrivial upper bound for the sum of the two largest eigenvalues and derive a new upper bound for the second largest eigenvalue of the adjacency matrices of the graphs.

Dennis D.A. Epple, University of Victoria
Characterizing bipartite (2,1)-polar graphs

A graph is said to be (2,1)-polar if it can be partitioned into a complete bipartite graph and a clique. As such this is a superclass of split graphs. The talk will investigate this property and give a complete forbidden subgraph characterization for bipartite (2,1)-polar graphs.

Kseniya Garaschuk, Simon Fraser University
On binary and ternary Kloosterman sums
Joint work with: Petr Lisonek

Recently there has been a lot of work done on the divisibility of binary Kloosterman sums and the number of codewords of a certain weight of the binary Melas code (for example, a sequence of papers by Charpin, Helleseth and Zinoviev). We give full characterization of those arguments for which the Kloosterman sum is divisible by 3 as well as establish the exact spectrum of the number of coset leaders for cosets of weight 3 of the binary Melas code.

Mamad Ghebleh, Simon Fraser University
On colourings of planar graphs

We present results and open problems concerning circular colourings of planar graphs.

Seung-byong Light Go, Memorial University of Newfoundland
Combinatorial Techniques in DNA Sequence Comparison

Primary sequences of DNA, RNA, or protein of an organism can be aligned in sequences to identify the similar region. This comparison may be analyzed to study the functional and/or the structural similarities between the two sequences being compared. Sequence alignment attempts to identify the point-mutations in genes that may be responsible for cancer. If two comparing sequences are from the same species, mismatches within the alignment indicate point-mutation or insertion/deletion (indels). Coding Theory and Designs Theory in Combinatorics allow us to detect and correct errors if these sequences were to expressed numerically. We explore the mathematical models and computer algorithms presented by Levenshtein, Sankoff, and others in DNA sequence comparison.

Nathan Grieve, Queen's University
Simplicial complexes and minimal generators of lattice ideals

Briales, et al. establish an explicit relationship between the minimal free resolution, \mathcal{F}, of a lattice ideal I, over a multi-graded polynomial ring and a particular simplicial complex \Delta_\mathbf{b}. This correspondence also appears in the recent book by Miller and Sturmfels. In this talk we use these accounts to establish this correspondence. In particular, we use the Koszul Complex, \mathcal{K}(x), to establish a relationship between \mathcal{F} and \Delta_\mathbf{b}. We then use the total complex of \mathcal{K}(x)\otimes_S\mathcal{F} to explicitly describe how \Delta_\mathbf{b} translates into minimal ideal generators of degree \mathbf{b}.

Bradford Hovinen, University of Toronto
Matrix factorizations of the classical discriminant

The classical discriminant detects whether a univariate polynomial over a field has repeated roots. Classical results of Cayley, Legendre, Sylvester, and Bzout show that there exist nontrivial determinantal formulae for the classical discriminant, allowing for more efficient evaluation. However, the classical formulae are all equivalent in the sense that the cokernels of the matrices are all isomorphic. This begs the question of whether there are any nontrivial inequivalent determinantal formulae. This talk will cover recent work in classifying determinantal formulae. In particular, for the degree-4 discriminant, a moduli space of equivalence classes of formulae the cokernels of whose matrices are graded will be described. The structure of the moduli space closely relates to the geometry of the hypersurface cut out by the discriminant, and this connexion will be discussed. The talk will also touch on the construction of a specific formula --- that of the \textit{open swallowtail} --- that is inequivalent to those hitherto known.

Mahdad Khatirinejad, Simon Fraser University
Regular structures of lines in the complex spaces

I present a survey on sets of complex lines with restricted angles.

Tak Lun Koo, University of Washington
On the generation of the coefficient field of a newform by a single Hecke eigenvalue
Joint work with: William Stein, Gabor Wiese

We study the set of primes p such that the eigenvalue a_p(f) of the Hecke operator T_p acting on a fixed non-CM weight k newform f without inner twists generates the field of coefficients of f. We show that this set has density 1, and prove a natural analogue for newforms having inner twists. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation.

Laurel Miller-Sims, McMaster University
Variants of Hilbert's 17th Problem in Valued Fields

Hilbert's 17th problem asks for a characterization of those rational functions f\in\mathbb{R}[X]=\mathbb{R}[X_1,...,X_n] that take only positive values. While Artin's positivstellensatz resolved Hilbert 17 in 1927, valued fields provide natural structures in which to formulate analogues of this problem as we may replace the notion of being positive with that of having positive valuation. I will discuss some recent variants of Hilbert 17 in model-complete theories of valued fields.

DeAnne Morris, Washington State University
The Peripheral Spectrum of a Nonnegative Matrix
Joint work with: Judith McDonald

Necessary and sufficient conditions for a set of Jordan blocks to correspond to the peripheral spectrum of a nonnegative matrix will be discussed. For each eigenvalue, \lambda, we will define the \lambda-level characteristic (with respect to the spectral radius). The necessary and sufficient conditions include a requirement that the \lambda-level characteristic is majorized by the \lambda-height characteristic. An algorithm which has been implemented in MATLAB is given to determine when a multiset of Jordan blocks corresponds to the peripheral spectrum of a nonnegative matrix. The algorithm is based on the necessary and sufficient conditions discussed.

Christopher Phan, University of Oregon
Generalized Koszul properties for filtered algebras

Under certain conditions, a filtration on an augmented algebra A admits a related filtration on the Yoneda algebra Y(A) := \mathrm{Ext}_A(K, K). We show that there exists a bigraded algebra monomorphism \mathrm{gr} Y(A) \hookrightarrow Y(\mathrm{gr} A). This monomorphism can be used to verify that A is Koszul or has various generalized Koszul properties, such as the \mathcal{K}_2 property recently introduced by Cassidy and Shelton.

Beth Powell, University of Alberta
R-Universal Central Extensions

In this talk I will introduce a generalized concept of universal central extensions of groups, and explain how the classical theory can be seen as one instance of this larger homological theory. I will identify when a group G has an ``R-universal central extension" and recognition criteria for when a central extension of G is R-universal. Also I will briefly mention the idea of an R-universal central extension of a module and will discuss one application of this.

Jie Sun, University of Alberta
Descent constructions for central extensions of infinite dimensional Lie algebras
Joint work with: Arturo Pianzola, Daniel Prelat

We use descent theory to construct central extensions of twisted forms of split simple Lie algebras over rings. These types of algebras arise naturally in the construction of Extended Affine Lie Algebras. The construction also gives information about the structure of the group of automorphisms of such algebras.

Maryam Verdian-Rizi, Simon Fraser University
On Coloring and Structure of Fisk Triangulation

Fisk Triangulation is a triangulation with exactly two odd vertices, which are also adjacent. We want to show that their dual is 3-edge colorable. We resolve the case for toroidal graphs, and also their structure is given. For other orientable surfaces, the claim is proven to be true for a constructive class of Fisk triangulation.

Liam Watson, Universite du Quebec a Montreal
Aspects of Khovanov homology

his talk will introduce the algebraic and combinatorial aspets of a homological knot invariant due to Khovanov. We will construct this theory from scratch; no backrgound in knot theory will be assumed (in particular, the term knot will be defined). Time permitting, some particular examples will be presented.

Oznur Yasar, Memorial University of Newfoundland
Fast Searching
Joint work with: Danny Dyer, Boting Yang

Edge searching is a graph problem that corresponds to cleaning a contaminated graph using the minimum number of searchers. We define fast searching as a variant of this widely studied problem. Fast searching corresponds to an internal monotone search in which no edge is traversed more than once and searchers are not allowed to jump. We give a polynomial time algorithm to compute the fast search number of any tree, and examine the fast search number of some families of graphs.

Amy Yielding, Washington State University
A family of Spectrally Arbitrary Zero-Nonzero
Joint work with: Judith McDonald

Spectrally arbitrary patterns have become of interest in the past decade. Work has been accomplished in classifying 4x4 and 5x5 spectrally arbitrary zero-nonzero patterns as well as families of nxn minimally spectrally arbitrary patterns. The most common method of proof is implementing the Nilpotent Jacobian method developed by Britz, McDonald, Olesky, and Van Den Driessche in Minimal Spectrally Arbitrary Sign Patterns. In this talk we will use the Nilpotent Jacobian method to establish the necessary conditions needed for a nxn irreducible zero-nonzero pattern with \frac{n(n+1)}{2}+1 nonzero entries, at least two of which lie on the diagonal, and at least two transversals to be spectrally arbitrary.

Masrour Zoghi, University of Toronto
A Generalisation of the Universal Enveloping Algebra

TBA