Undergraduate Summer Research Projects

^{Attention Students wanting first-hand experience in mathematical research this Spring/Summer in the field of Orthogonal Matrices*}

(*Warning:  exposure to research in this field has been demonstrated to be highly addictive!)

I hope to take on students as RAs (research assistants) this summer (2017) but that will be dependent on grant funding, which will be pending until about April 2017.  In the meantime students wanting to work should find a professor with an NSERC grant and apply for an NSERC USRA award -- the deadline is in December; watch for announcements and/or speak to the Faculty of Science office.  Note you must have a grantholder (therefore not me this year) sign your application.  You are not bound to work for that grant holder, but there must be a credible project you can do for them.  If your intention is to work with me or anyone other than the grant holder you should discuss this with them so there are no misunderstandings.

There are a few awards at UM which do not depend on grant funding. You should also apply for those, and I would be able to take on a student with such an award regardless of whether I have a grant in 2017.

Seminar Series: Introduction to orthogonal matrices
If I will be taking on students I will announce a once-weekly seminar beginning in the first week of April, open to all interested students. We will cover the background to several problems to be explored by undergrad RA’s this term.  Watch this space and my office door for announcements.

Those who end up employed as RAs will be reimbursed for their time attending and doing the (required for them) weekly reading, in preparation for the summer. Other students may participate as they see fit.

If you do not receive an award I still may consider hiring you part time if you have the right background.  For this I will want to see:

• your most recent grade report (unofficial/photocopy ok) 
• a list of your current courses (if they don’t appear on the report)
• a 1-2 page letter explaining why you are interested in mathematical research and listing any prior experience, if any.
• mention anything that will affect your availability and any additional employment you will have during the summer.

If either the seminars or the RA positions interest you, let me know your schedule by Tuesday March 14 and it will be taken into account in planning the seminars. Students interested in the seminar and not sure whether to apply for the RA position should attend; applications will be accepted until positions are filled.

Some recent projects (some of which are ongoing):

• Circulant partial Hadamard matrices.  This subject began over 10 years ago arising out of a question about stream-cipher cryptography.  As it turns out the objects we introduced are relevant to some key questions concerning Hadamard matrices.  Students working for me over several subjects have produced some lovely new approaches and many computational results, and we have developed the theory for these interesting objects.  An r-H(kxn) is a kxn matrix obtained from a row of 1s and -1s of length n, with subsequent rows obtained by circulating entries of prior rows to the right by one position.  The key property we want is that any pair of rows are orthogonal, which means they match in exactly half the positions.  r is the sum of the entries along that row.  In the cryptographic setting we always insisted that r=0, whereas in the general setting we allow all values, and understanding the relationship between r, k and n is critical.  If we can refine this relationship sufficiently we have a good chance of solving one of the big unsolved problems in design theory, the circulant Hadamard matrix conjecture (sometimes also called "Ryser's Conjecture").
• Negacyclic weighing matrices.  This topic grew as a side-project in 2013 when a team of four students were working on Circulant partial Hadamard matrices.  We kept happening upon questions about these little-studied objects.  Hampered by a too-thin literature on the subject we embarked on our own investigation into the existence of these objects, and two of the students produced a significant body of new information on the subject.  Their work is summarized on a poster on the 5th floor.
• 2x2 block decomposition conjecture for Hadamard matrices. A decade ago a summer RA working for me showed that, in order 24, there exists a Hadamard matrix which cannot be decomposed entirely into 2x2 submatrices which are all rank 1.  This was the first instance of a class of objects of which we now have a number of examples.  A number of mathematicians have been putting effort into the 2x2 block decompositions of Hadamard matrices to learn something of their hidden structure.  A related conjecture -- which remains unsolved -- asks whether every Hadamard matrix can be decomposed into 2x2 submatrices which are all rank 2.  Most people in the field appear to believe this is true, but nobody has proven it.  This is a lovely project for student exploration.
• Sequences with zero autocorrelation. A number of sets of sequences with a propery we call "zero autocorrelation" are useful for building powerful constructions for important combinatorial objects.  About 15 years ago we pushed a class called ternary complementary pairs past what was known, and an RA working for me found the very first case having a weight of 29.  We have some idea where to look for a weight-37 case (the current smallest unknown weight) but this will require some very clever computer-based searching.  A good project for those interested in the computational side of combinatorics!
• Synthetic Orthogonality Theory.  This is a new and completely different project, which pushes inroads into an unexplored field -- what exactly is "orthogonality"?  That is, what do we mean in combinatorics when we speak of this?  It has been a concept that has grown without direction as we consider discrete variants on the notion, which are consistent with certain kinds of configurations that, in some abstract sense, embody "orthogonality" but do not necessarily bear much resemblance to the usual geometric notion of perpendicularity as in the usual linear algebra.  A few years ago associates of mine in Ireland (Dr. Dane Flannery) and California (Dr. Warwick de Launey) wrote a book, Algebraic Design Theory, introducing an axiomatic notion of "pairwise orthogonality" that embraces the gamut of these meanings ... but they leave the elementary theory largely undeveloped ... and most of the sweeping terrain thus called into existence simply unexplored.  Their groundbreaking book opened up an entirely new field, but left another field, which I call Synthetic Orthogonality Theory, more-or-less virgin territory.  I plan, over the next few years, to map it out with some aspiring new explorers.
• Lots more ... I'll add some when I have time.

Further info:

R. Craigen
MH 523
474-7489
craigenr@umanitoba.ca

server.maths.umanitoba/~craigen/PWP/