I tend to advise projects in topics related to algebra and combinatorics, but have wide-ranging interests. Students with a project in mind should contact me directly to discuss.
Masters Theses Advised
1. Yu Cheng, The Transfer: Coverings, Homology and Group Theory. (2019-21)
The transfer homomorphism in group theory is rather mysterious when encountered in a group theory textbook (e.g. Isaacs 'Algebra'). The motivation comes from the theory of fundamental groups and covering spaces in algebraic topology. The existence of Eilenberg-MacLane spaces allows the topological version of the transfer to be applied to groups, recovering the usual definition. Yu is working through the details of this translation (teaching me a lot in the process!). Marshall Hall's discussion of the transfer is in terms of monomial representations -- Yu plans to also fit this into her framework.
2. Nikolaos Kalampalikis, Weighing matrices and the Sensitivity Conjecture. (2019-21)
Jenny Seberry's theory of weighing matrices is one of the highlights of algebraic design theory, used in establishing strong existence results for real Hadamard matrices. In 2018, Huang resolved an old and famous conjecture in computer science through the use of a weighing matrix. Neither Huang nor subsequent authors have mentioned the connection to weighing matrices. In his thesis, Nikolaos gives an exposition of the theory of weighing matrices and background combinatorial results connected to the sensitivity conjecture. The general theory of weighing matrices may well give further insight into problems of the type considered by Huang.
Undergraduate Theses Advised
1. Theo Marks, Symmetric and diagonalisable matrices in positive characteristic. (2020-21)
Everyone knows that symmetric matrices are diagonalisable. In fact, the proof of this result requires an inner product, and so a field of characteristic 0.
Theo is investigating the failure of this result in positive characteristic: what are the obstructions to diagonalisability of a symmetric matrix over a finite field?
The techniques involved are linear algebra, theory of rings and fields and some computation in MAGMA.
2. Kyle Dituro, Packings and coverings in finite groups. (2020-21)
A packing in a finite group is a subset P for which all products of pairs of elements of P are distinct. A covering is a subset C such that the set of products of pairs from C contains every element of G. Optimal packings and coverings are closely related to planar difference sets, and have been studied quite extensively in cyclic groups but the literature is rather scattered. Recently Banakh and Gavrylkiv gave recursive constructions for packings in abelian and dihedral groups. Kyle is assembling the known results for various classes of groups and providing complete and comprehensive proofs, as well as investigating more general non-abelian groups
3. Kwabena Adweteba-Badu: Digraphs and nilpotent algebras. (2019-21)
A famous old theorem of Schur gives the maximal dimension of a nilpotent matrix algebra of nilpotency class 2. The generators of such an algebra correspond to edges in directed complete bipartite graph with no directed paths of length 2. Similarly, algebras of nilpotentcy class k are obtained from d.a.g.s with no directed paths of length k. Dense examples of such graphs are given by the Gallai-Hasse-Roy-Vitaver theorem. Kwabena is investigating whether these graphs have maximal dimension among the k-step nilpotent subalgefbras of the n x n matrices.
4. Phillip Heikoop, Dimensions of Semi-simple matrix algebras. (2018-19)
The simple matrix algebras over an algebraically closed field are just the matrix algebras, and so have dimension the square of an integer. So an integer L between 1 and n^2 is the dimension of a semi-simple subalgebra of the n x n matrices if and only if there exists a partition of k < n such that the sum of the squares of the parts is equal to L (this is less complicated than it sounds). As n becomes large, what proportion of integers in [1.. n^2] are the dimension of a semi-simple algebra? By adapting a recursive argument of Savitt and Stanley, Phillip proved that all but at most c*n^(3/2) integers in this interval are dimensions of semi-simple algebras. For sufficiently large n, we can choose c < 9/2. This work was published in the Pi Mu Epsilon journal.
5. Daniel Kim & Jack Pugmire, Metrics on graphs (2018-19)
Motivated originally by the idea of performing threat detection on Twitter via graph theoretic methods, this project began with an investigation of measures of similarity on the vertices of a graph. Daniel and Jack investigated various generalizations of the small-world property for a graph and developed classifier algorithms based realistic structural assumptions about graphs. Their new classifiers were able to distinguish between users in real world graphs that they analysed. This project was jointly supervised by Prof. Robert Walls, in CS.