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2018 Prize Holders
Samantha Zimmerman: Optimizing Nurse Schedules at a Community Health Centre
Samantha's project was to develop a method for improving nurse schedules at Community Health Centres operated by Vancouver Coastal Health (VCH). These Centres provide medical care and social support services to marginalized people in downtown Vancouver. Some of the challenges faced by VCH are wait times for patients, gaps in triage coverage, and too many short shifts arising from nurses being called in to meet demand peaks. Samantha developed a mixed integer linear programming model to find the shift schedule for registered nurses and licensed practical nurses. This model was applied to both booked and walk-in patient visits, which the Community Health Centre operates concurrently. Samantha then took the clever approach of devising a second linear programming problem that would provide the nurse-patient contact hours for a given fixed nurse schedule. In order to compute additional key performance indicators of her schedule solution, she developed a discrete event simulation model of the community health centre. Samantha has not only carried out this research, but played a significant role in working with VCH to develop the project and present the results at VCH. The work was presented as a poster at the 31st Annual Health Policy Conference at the Centre for Health Services and Policy Research and she will give a talk on her research at the 2019 INFORMS Annual Meeting in Seattle in October. A manuscript on her work is in the final stages of preparation and will be submitted to a health-care focused operations research journal.
2017 Prize Holders
Matthew King-Roskamp: Optimal Sampling Strategies for Compressive Imaging *supervisor Ben Adcock*
Matt’s research project involved the development and analysis of optimal sampling strategies for compressive imaging. Compressive imaging concerns the recovery of an image from a limited set of measurements. This problem arises in most modern imaging modalities, with applications ranging from medical (e.g. MRI, X-Ray CT) to scientific and industrial imaging. While the development of compressive imaging has been an area of significant research over the last decade, the question of how best to sample had not been fully investigated. Matt’s research was an in-depth study of this question, using the language of approximation theory; specifically, nonlinear approximation of piecewise regular functions using orthonormal wavelets. The culmination of his work is the derivation of a sampling strategy for the class of piecewise Hölder-q functions, and the theorem establishing the optimality of this approach. This is the first work that rigorously connects compressed sensing (the mathematical theory for compressive imaging) with nonlinear approximation theory. Matt has presented this work at the 2017 SIAM Pacific Northwest Biennial Conference – where he was awarded the runner-up prize in the poster competition – and most recently at the 2018 SIAM Annual Meeting, where he was invited to speak in a minisymposium on Mathematics of Signal Processing, Optimization and Inverse Problems. A journal paper based on this work is currently in the final stages of preparation.
Koen van Greevenbroek: Investigate Existence Pattern for Difference Matrices in Abelian Groups (Directed Studies Project) *supervisor Jonathan Jedwab*
Difference matrices are a type of combinatorial design related to orthogonal arrays, transversal designs, mutually orthogonal Latin squares, orthogonal orthomorphisms, and linking systems of difference sets. Koen’s Directed Studies project during Fall 2017 was to investigate the existence pattern for difference matrices in abelian groups and to try to construct larger examples than those currently known. His key discovery is that there are special examples of difference matrices in abelian groups that can be concisely encoded using a much smaller “contracted" difference matrix. This is analogous to the use of a generator matrix to represent a linear code in coding theory, and in hindsight it is entirely natural. Koen used this discovery to find four examples of difference matrices in abelian groups having twice as many rows as the best previously known, each of which gives rise to a new infinite family of examples.
Koen presented his research as an invited talk at the October 2018 Workshop on Pseudorandomness and Finite Fields in Linz, Austria, and a joint paper with his supervisor has been accepted for publication in the conference proceedings. The referee wrote: "The contracted matrices provide a neat and simple interpretation of many known constructions of difference matrices...This paper is very well written and constitutes a solid contribution to difference matrices.” Koen's supervisor also presented this work as a plenary address at the July 2018 Conference on Combinatorics and its Applications in Singapore. The resulting discussions have sparked new research questions which are likely to lead to a further publication involving Koen.
Koen showed significant initiative and independence during his research, and in particular wrote all the search programs himself. His technical writing is clear and concise. While at SFU, Koen took on several leadership positions within the Mathematics Student Union and worked as a community advisor in the student residences. He began his Master’s studies in mathematics at the University of Bonn in September 2018.
Beril Zhang: Long Time Behaviour of Aggregation Equations *supervisors Razvan Fetecau and Weiran Sun*
Beril’s project was to set up and investigate a mathematical model for self-collective/swarming behaviour on surfaces and manifolds. Despite the extensive research on aggregation models in Euclidean spaces in recent years, there has been very little done for models posed on arbitrary surfaces or manifolds.
In her work Beril demonstrated swarming on the sphere in R^3 and on the hyperbolic plane (regarded as a hyperboloid in R^3 endowed with the Minkowski metric). The model had to be set up on manifolds, and also interaction potentials had to be chosen suitably. This work was the first demonstration that models in the class we considered can lead to self-collective behaviour on manifolds. Consequently, it opens new research directions and new perspectives on such models (e.g., applications in robotics).
Beril worked very independently on this project, leading her own explorations with an excellent intuition. It should also be noted that Beril studied hyperbolic geometry on her own. Results of this research have turned into a paper that is currently submitted for publication in a top-tier journal.
2016 Prize Holders
Charlotte Trainor: Classification of Almost-Toric Fano Manifolds
Charlotte carried out the research project "Classification of Fano divisorial polytopes" during the summer of 2016 as part of an NSERC USRA. She has turned the results of this research into an excellent honours thesis, as well as a forthcoming research article coauthored with Professors Ilten and Mishna.
The fundamental objects Charlotte investigated were "divisorial polytopes,” a quasi-combinatorial object generalizing the notion of lattice polytopes. These objects have relevance in algebraic geometry, as they correspond to an extremely useful class of geometric objects known as complexity-one T-varieties. It is an important open problem to understand all complexity-one T-varieties with the special property of being “Fano,” that is, having positive Ricci curvature. Charlotte's research sheds light on this problem: she establishes some important and previously unknown bounds on the structure that such Fano complexity-one T-varieties possess. Further, her work establishes a foundation for an approach towards effectively classifying all such varieties. As such, it is an important contribution at the interface of algebraic geometry and combinatorics.
Throughout the research and writing process, Charlotte demonstrated the ability to work independently, and generate interesting ideas of her own. Her excellence in writing and presentation is reflected in her receipt of first prize in the 2016 poster competition at SFU's Symposium on Mathematics and Computation. Likewise, her penchant for learning and hard work led her to win a Governor General's Silver Medal in 2017, an award given annually to only two undergraduates at SFU.
William Yolland: Combinatorial Problems Arising in Digital Communication
William’s project was to determine whether or not there is a difference set in a specific nonabelian group of order 256. This was the final piece of information needed to complete a five-year international collaboration whose goal was to settle the existence question for all 56,092 groups of order 256. William knew at the outset that the existence question for the final group could in principle be resolved by computer, but the search size of 264 was computationally infeasible. William devised a series of ingenious numerical experiments to constrain the search space, and thereby succeeded in producing an example of a difference set in the required group. He then developed a novel recursive construction that completely explains the existence of a difference set in this final group, using sequences with special correlation properties. As a direct result of his work, an international workshop will take place in 2018 to discuss the completion of the collaboration and its implications. The leaders of the collaboration are also preparing a monograph in which William’s new method will play a crucial role.
Anya (Casie) Bao: Solving PDE in 100 Dimensions
Casie’s research project involved the development and analysis of a novel compressed sensing algorithm for correcting for corrupted measurements in the field of Uncertainty Quantification. This is a well-known, but challenging issue that had not been addressed previously by the community. Beside the development of the algorithm, a significant part of Casie’s work was devoted to its theoretical analysis. She was able to show that in some cases, a constant fraction of the measurements can be corrupted, but that the algorithm can account for this with surprisingly little deterioration in recovery error. Casie presented this work at the 2017 SIAM Computational Science & Engineering conference, and, in collaboration with her supervisor and researchers from the University of Utah and Sandia National Labs, she developed it into a journal paper which is currently under review. The core mathematical content of the paper is predominantly Casie’s work.
2015 Prize Holders
Gregory Maxedon: A Graph Theoretic Understanding of an Algebro-Geometric Condition
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.
Charles Turo: Degenerations to Stanley-Reisner Schemes
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.”
Alexandre (Sasha) Zotine: Determinantal Complexity of Cubic Polynomials
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 an 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.”
2014 Prize Holders
Joshua Horacsek: Combinatorial Modelling in Comparative Genomics
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.