Research Projects for Students
I welcome any expressions of interest in pursuing a Ph.D. under my supervision at DCU from prospective students who are interested in Discrete Mathematics and Algebra. Proposals of any description in areas closely related to these fields are welcome. To give some ideas, I have listed below a couple of different areas of research that could be refined into a Ph.D. proposal. If any of these interest you, or if you have another topic in mind that you would like to work on, please feel free to send me an email at ronan.egan@dcu.ie and we can discuss funding applications. The descriptions below are reasonably open ended to give you an idea of the type of project we could discuss, and can be further refined in later discussion. These projects will likely involve computation using computational algebra software such as Magma or GAP, so programming skills would be a plus, but not essential to begin with.
Existence questions and constructions of real or complex Hadamard and weighing matrices: A complex weighing matrix is a square n x n matrix such that every entry is either zero, or a complex number of modulus 1, with the property that there are exactly w non-zero entries in every row and column, and the inner product of any two distinct rows is zero (i.e., the matrix is orthogonal up to a scaling factor). The number w is the weight of the matrix. If every entry is non-zero, then it is a complex Hadamard matrix. If the non-zero entries are real, meaning either 1 or -1, then the matrix is a (real) weighing matrix, or Hadamard matrix if there are no entries equal to zero. Numerous question surround the existence of these matrices for different values of n and of different weights, particularly when the non-zero entries are restricted to being real, or being complex roots of unity. The applications of these matrices extend from encoding and encrypting, to the design of statistical experiments, to quantum information theory. This project would involve settling some of these questions of existence, and developing new constructions to find new objects not yet known. Depending on the questions asked, this will involve a combination of techniques from algebra, combinatorics, finite geometry, and number theory.
Constructions of error-correcting codes, both classical and quantum: Error-correcting codes are essential for all forms of digital communication and data storage. The typical objective is to encode information or data such as an image or text with binary vectors suitable for digital transmission, in such a way that the information can be recovered even if some errors are introduced. In Coding Theory, our goal is often to determine the best way to encode data onto vectors (binary or perhaps over another field) such that it can be done efficiently (i.e., quickly and cheaply) but the code can still be robust enough to correct errors that are likely to be introduced. The quantum analog of this applies to quantum computation, but with qubits in place of classical bits, and with added difficulties due to the restrictions of quantum mechanics. In this project you would explore different constructions of error-correcting codes of different types, and develop new constructions of codes with suitable properties. Combinatorial objects like Hadamard and weighing matrices from the project above could play a large role, and thus the techniques may be broadly similar, involving algebra, combinatorics and finite geometry.