2015 USRA Research Prizes!
Congratulations to the 2015 Undergraduate Research 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.
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”.
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 Liberté”.