## 2015 Prize Winners

**Gregory Maxedon**

Supervised by Karen Yeats

Paolo Aluffi asked when, given a pair of a graph and an edge of this graph, is the Kirchhoff polynomial of the graph with the edge contracted in the ideal of partial derivatives of the Kirchhoff polynomial of the original graph. Gregory looked for a graph theoretic understanding of when this happens. After quickly learning all the algebraic background for the project, he sorted out the situation for multiple edges, showing that more than double edges can always be reduced to double edges, and determined exactly which edges of wheel graphs have the property. He also wrote some software to investigate the problem experimentally. Together with Avi Kulkarni, who became involved after seeing Gregory's poster on his work at the 2015 SFU Symposium on Mathematics and Computation, Gregory and his supervisor wrote the paper arXiv:1602.00356. Three of the core sections of the paper are essentially entirely Gregory's work both in results and presentation.

**Alexandre (Sasha) Zotine**

Supervised by Nathan Ilten

Sasha studied linear subspaces contained in projective toric varieties, which are special varieties describable in terms of lattice polytopes. Sasha's main result was to prove that a projective toric variety contains infinitely many codimension-one linear spaces if and only if its corresponding lattice polytope is contained in a prism over an empty simplex. This result was motivated by his supervisor’s study of lines on toric surfaces, and Sasha’s methods and result provided considerable insight into a complete understanding of all linear subspaces contained in a projective toric variety. Sasha turned his research into a very good undergraduate honors thesis. It has also been incorporated into a preprint co-authored with his supervisor. The contents of this preprint have been used by David Cox as the basis of an expository lecture at the renowned algebraic geometry summer school “Geometrie Algebrique en Liberte”.

**Charles Turo**

Supervised by Nathan Ilten

Charles studied a connection between a certain family of algebro-geometric objects and a special family of simplicial complexes. In particular, he showed that the natural set of generators of a special family of ideals forms a Gröbner basis with regard to a particular term order. His research generalizes work of Sturmfels connecting the Grassmannian parametrizing 2-planes in a vector space to the polytope known as the associahedron. Additionally, it provides useful tools for studying so-called Fano varieties, as well as for the study of Aluffi algebras. Charles incorporated his results into a paper co-authored with his supervisor which will appear in the *Journal of Pure and Applied Algebra*. This paper garnered high praise from the referee: “a beautiful paper, fun to read, and thought provoking”.

## 2014 Prize Winners

**Joshua Horacsek**

Nominated by Marni Mishna

Joshua’s project involved combinatorial modelling in comparative genomics. The goal was to build a mathematical framework to evaluate different models of gene evolution. Joshua designed and coded software which allows the user to specify a particular topology, and returns a quantitative evaluation of the underlying model together with a visual representation. His software was applied to data on mammalian genomes and yeast genomes. He presented his work at the 2014 Canadian Undergraduate Mathematics Conference.

Joshua’s project is an excellent demonstration of how nontrivial combinatorics can arise in bioinformatics. His poster for the 2014 Symposium on Mathematics and Computation has been used in SFU recruitment activities. His USRA work led an eight month internship at the BC Centre for Excellence in HIV/AIDS. He subsequently won an NSERC Master's scholarship to study computing science at the University of Calgary.

## 2013 Prize Winners

**Matthew Gibson**

Nominated by Michael Monagan

Matthew worked on the problem of computing the Greatest Common Divisor of multivariate polynomials. He implemented Brown's algorithm in C and several optimizations. His code is typically 10 times faster than the code in Magma. Matthew subsequently parallelized the algorithm. His Cilk implementation achieves a speedup of a factor of 12 on a 16 core computer.

**Layla Trummer**

Nominated by Petr Lisonek

Layla studied the algorithm for determining the minimum distance of a linear error correcting code, originally proposed by Brouwer and later modified by Zimmermann. Since computing the minimum distance is an NP-hard problem, the algorithm has long running times even on codes of modest size. The algorithm offers a lot of flexibility in its initial setup phase, and the research plan was to take advantage of this flexibility to accelerate the algorithm.

In cooperation with her supervisor, Layla identified a condition that guarantees the fastest termination of the algorithm for a fairly large class of codes. Her paper (co-authored with Petr Lisonek) was accepted by the 4th International Castle Meeting on Coding Theory and Applications in Palmela, Portugal. A journal version of the paper was published in Advances in Mathematics of Communications in 2016.

## 2012 Prize Winners

**Graham Banero**

Nominated by Tom Brown & Veselin Jungic

Graham worked on an open problem in double arithmetic progressions. He also considered variations of this problem and its relation with the problem of the avoidability of additive squares.Graham presented his work at the Prairie Discrete Mathematics Workshop and the Canadian Undergraduate Mathematics Conference. His research was published as “On additive complexity of infinite works" in the Journal of Integer Sequences in 2013. He continued research in this area during Summer 2013.

**Mark Strange**

Nominated by Jonathan Jedwab

Mark studied a problem of multislit spectrometer design, originally proposed by Golay in 1951 but subsequently forgotten. Mark showed that Golay's formulation of the problem is unduly restrictive, and by relaxing the restrictions was able to obtain infinitely many spectrometer designs satisfying all the original physical criteria.

Mark won first prize in the undergraduate category of the Computational Mathematics Day 2012 poster competition for this work. He published his research in 2013 in the IEEE Transactions on Information Theory (jointly with his supervisor).

**Andrew Butler**

Nominated by Paul Tupper

Andrew came up with a definition of a type of space, which he called a conformity, that has the same relation to diversities as uniform spaces have to metric spaces. Andrew demonstrated that this analogy holds by proving the analogues of two classical results from the theory of uniform spaces. He also mapped out some further interesting relationships between topologies, uniform spaces, and conformities.

He published his findings in the Journal of Function Spaces in 2013.

## 2011 Prize Winners

**Stephen Melczer**

Nominated by Marni Mishna

Stephen considered lattice path enumeration problems from both a complex analytic and a computational point of view. He studied some lattice models that had previously resisted powerful strategies by performing an intricate analysis of the behaviour of the singularities of related generating functions. In the course of his research he found new modular computational methods for generating coefficients of the generating function.

He presented his work in a national meeting in Victoria, and he won the undergraduate category in the 2011 Poster competition of SFU Computational Math Day.

On the basis of this work, he was invited for a prestigious 3 month internship at the French Computer Science research institute Inria, where he considered related problems for lattice paths in three dimensions. He was invited to give lectures in Paris, Marseilles and Austria. He continued this work in an MSc supervised by Marni Mishna and Alin Bostan (Inria).

**Lee Safranek**

Nominated by Adam Oberman

Lee worked on solving nonlinear Partial Differential Equations for Convex Envelopes. Convex Envelopes are related to convex hulls, which are the smallest convex object which contains a given shape. Computing the convex envelope requires the solution of a nonlinear partial differential equation by finite difference methods.

This involves discretizations of directional derivatives in many directions. The number of these directions increase dramatically with the dimensions. Lee generalized the solver from two to three and four dimensions. He also built a fast Newton solver for the resulting nonlinear systems.

The reason for the four dimensional solver is that generalized (quasi-) convex envelopes occur in Material Science, and lead to the computation of layered materials. Lee is continuing his research on generalized convex envelopes.

**Julian Sahasrabudhe**

Nominated by Tom Brown & Veselin Jungic

Julian was a full participant in work that led to introduction of the notion of additive complexity for infinite sequences over finite sets of integers. This new approach led to the discovery of a large family of infinite sequences that must contain additive k-powers.

Julian was accepted in a graduate program at the University of Cambridge and was awarded the Canadian Scholarship Trust Fund Wright-Pigott-Lloyd-Neale-Carter Graduate Scholarship Award for 2012 and the NSERC Alexander Graham Bell Graduate Scholarship for 2012.

** **

## 2010 Prize Winners

**Yuanxun (Bill) Bao**

Nominated by Dave Muraki

As an NSERC USRA summer researcher, Yuanxun (Bill) Bao, studied the fluid dynamical equations that describe a process by which an atmospheric gravity wave can go unstable. This mechanism, known as a parametric resonance, is one of the leading suspects for the generation of clear air turbulence (CAT) -- the aviation phenomenon of in-flight disturbances in the absence of adverse weather conditions. Bill is currently continuing this research as his thesis project for his Masters Degree in Applied Mathematics. Bill won first prize in the undergraduate category of the Computational Mathematics Day 2010 poster competition for this work.

**Michael Fry **

Nominated by Paul Tupper

Michael Fry proved the correctness of BrbrNet, a neural network model of syllabification in the language Berber. The model had been proposed by the cognitive scientist Paul Smolensky as a neurally plausible implementation of a well-known linguistic phenomenon. In the original publication, there was no proof that the model would generate the correct syllabification for all inputs. Fry together with his supervisor, Paul Tupper, were able to identify a set of parameters for BrbNet such that correct convergence for all inputs could be proven.

**Gordon Hiscott**

Nominated by Nilima Nigam

Gordon's USRA project was to develop and implement numerical approximation strategies for a mathematical model of cyclical neutropenia, a blood disorder. He studied this model, and used techniques from the mathematical analysis of PDE to understand some of the theoretical properties of the model. None of the standard numerical strategies applied in this instance, and Gordon was able to prove this fact. Gordon also realized that many problems in mathematical biology lead to similar systems of age-structured population models, and than a robust, accurate and efficient method for our problem would prove useful in other settings as well. He is presently continuing this work in the SFU MSc program under the supervision of Nilima Nigam.

**Aleksandar Vlasev**

Nominated by Karen Yeats

The classical Dodgson identity of determinants applied to the Laplacian matrix of a graph can be interpreted as a quadratic identity of spanning forest polynomials of the graph, where the spanning forests appearing in each polynomial are defined by how three marked vertices are divided among the component trees. Aleks Vlasev found a new identity of spanning forest polynomials which is the analogous identity with four marked vertices. He presented a poster jointly with Konrad Duch at Computational Math day 2010, and recently submitted a paper on the result jointly with Karen Yeats.

## 2009 Prize Winners

**Ryan Coghlan**

Supervisors: Cedric Chauve and Tamon Stephen

Ryan’s project found combinatorial properties of matrices that have the gapped consecutive ones property, with application in computational paleogenomics. He wrote software and applied it to real vertebrate genomics data for a poster in CECM Day.

He implemented an exponential time algorithm for a slightly more general problem to decide the bandwidth of a graph. This implementation is very challenging as the algorithm is based on the enumeration of an exponential number of equivalence classes of permutations, that can require a very large memory space. Ryan did study several non-trivial data structures and approaches to manage this problem, with finally some success on a smaller scale dataset of fungi genomes. The work of Ryan was presented in a poster at the RECOMB Comparative Genomics conference in Ottawa in 2010.

**Stephen Melzcer **

Supervisor: Adam Oberman

Steve developed original fast solution methods for nonlinear Partial Differential equations in two contexts.

The first task was building generalized fast solvers for Hamilton Jacobi nonlinear Partial Differential equations.

Steve went on to work on fast solvers for second order equations. Steve implemented Interior point solvers from a book in Linear Programming in Matlab, and then implemented his own solver, and compared it to the state of the art off the shelf solver: MOSEK. For medium sized problems his solver beat the additional overhead of MOSEK, and was superior, but, as expected, for the largest problems MOSEK was superior. Stephen presented this work at the Canadian Undergraduate Mathematics Conference (Waterloo) in 2010.

**Asif Zaman**

Supervisors: Peter Borwein and Sandy Rutherford

Asif Zaman's project was a queue network model for booked elective admissions to hospitals. The models arising out of this program will be used by the Ministry to project the number of hospital beds required to maintain the desired access levels for each hospital in BC.

Asif presented his work at the Canadian Undergraduate Mathematics Conference at Carleton University. His work also forms an important part of acute care planning models being developed for the Ministry of Health Services. His results were among those reported on by S. Rutherford in a talk at Operational Research in Health Services 2009. A paper based in part on Asif's work is currently being written.

## 2008 Prize Winners

**Aaron Chan**

Supervisor: Jonathan Jedwab

Aaron’s project led to the joint paper "On the non-existence of a projective (75,4,12,5) set in PG(3,7)" (also with Jim Davis), which was accepted by the *Journal of Geometry* in 2009, and a presentation at the 2^{nd} Annual Pure and Applied Mathematics Graduate Student Conference in 2008.

Aaron used a combination of theoretical argument and large-scale computer search to show that if a specific projective set code exists then it must have a non-trivial automorphism group. This corresponds to the smallest open case of a coding problem posed in 1998. The verification of the search was carried out on a PS3 gaming machine.

**Jamie Lutley**

Supervisor: Jason Bell

Jamie’s Summer project led to the joint paper "An automaton-theoretic approach to the representation theory of quantum algebras" (also with Stephane Launois), which appeared in *Advances in Mathematics* in 2010.

The project involves a new approach to the representation theory of quantized coordinate algebras supporting a torus action, by combining methods from the theory of finite-state automata and algebraic combinatorics. Jamie used techniques from representation theory and automaton theory to establish a key linear recurrence relation.

## 2007 Prize Winner

**Amy Wiebe**

Supervisor: Jonathan Jedwab

Amy’s project led to the joint paper "A new source of seed pairs for Golay sequences of length 2* ^{m}*" (also with Frank Fiedler), which appeared in

*Journal of Combinatorial Theory (Series A)*in 2010.

Amy studied examples of new 6-phase Golay complementary sequences that had been found by computer search, and succeeded in identifying previously unrecognized structure in these examples. This led to a complete and concise explanation of the origin of the new sequences.

## 2006 Prize Winners

**Denis Dmitriev**

Supervisor: Jonathan Jedwab

Denis’ project led to the joint paper "Bounds on the growth rate of the peak sidelobe level of binary sequences", which appeared in *Advances in Mathematics of Communications* in 2007.

Denis found a novel algorithm to reduce the complexity of peak sidelobe level calculations for m-sequences. This allowed the range of exhaustive calculation for these sequences to be greatly increased, revealing the first clear numerical evidence in support of unproven claims from the 1960s.

**Al Erickson**

Supervisor: Michael Monagan

Al worked on two projects. The first was to write a Maple implementation of a multivariate polynomial factorization algorithm, using a new representation for multivariate polynomial data. The second was to study algorithms for generating random regular graphs, required as part of a graph theory package for a MITACS industrial partner. Al presented the second project as a poster at the 2006 Maple conference and the 2006 CECM Day.

**Nhan Nguyen**

Supervisor: Jason Bell

Nhan’s project led to the joint paper "Dimension and enumeration of primitive ideals in quantum algebras" (also with Stephane Launois), which appeared in the *Journal of Algebraic Combinatorics* in 2009.

The project connects three different areas of mathematics: enumerative combinatorics, symplectic manifolds, and quantum algebra. Nhan’s computer enumeration results led to conjectures that the authors later succeeded in proving.

## 2005 Prize Winners

**Moe Ebrahimi**

Supervisor: Michael Monagan

Moe’s considered two projects. The first was visualizing systems of first order ordinary differential equations in the plane using animation, and led to the paper "New Options to Visualize Systems of Differential Equations in Maple" which appeared in the Proceedings of 2005 Maple Conference. The second was on a spring model for drawing graphs in two and three dimensions, which was presented as a poster at the 2005 Maple conference and at the 2005 CECM Day.

**Simon Lo**

Supervisor: Michael Monagan

Simon’s project led to the paper "A Modular Algorithm for Computing the Characteristic Polynomial of an Integer Matrix in Maple", which appeared in the Proceedings of the 2005 Maple Conference.

Simon’s research looked at how to compute the characteristic polynomial of large integer matrices, and was motivated by a combinatorial problem involving arrangements of Lego blocks in three dimensions.

**Kayo Yoshida**

Supervisor: Jonathan Jedwab

Kayo’s project led to the joint paper "The peak sidelobe level of families of binary sequences", which appeared in the *IEEE Transactions on Information Theory *in 2006.

Kayo investigated claims in the radar literature dating from the 1960s, concerning the rate of growth of the peak sidelobe level of m-sequences. She showed that these claims, although possibly correct, were unsupported numerically, historically, and theoretically.