Michael La Croix

I am a postdoctoral associate in the Applied Mathematics department at the Massachusettes Institute of Technology, where I am currently studying random matrices under the supervision of Professor Alan Edelman. My current research is motivated by the interplay between the combinatorics of topological maps and the statistics of random matrices with independent Gaussian entries. This is a natural extension of work on enumerative combinatorics that I completed during my Doctoral studies as a student of Professor David Jackson and later also Professor Ian Goulden, in the Combinatorics and Optimization department at the University of Waterloo. Here is a recent CV.

I was previously a lecturer in the department of Statistics and Actuarial Science at the University of Waterloo, where I spent 16 months co-ordinating an introductory course in probability aimed at honours Mathematics students. The structure of the course was such that many students attended lectures via video-conferencing, or by watching recordings of lectures, and a major component of my duties consisted of adapting the course to make it more suitable for consumption in this format.

Random Matices and Combinatorics

In my PhD thesis, I introduced a combinatorial model for a map generating series defined algebraically in terms of Jack symmetric functions. Polynomials associated with this generating series also appeared in moment calculations Professor Alan Edelman performed while studying the Gaussian β-ensemble, and we have been working to expose the connection to a broader audience. Generalizations of this model can also be used to provide combinatorial interpretations to the moments of β-Hermite and β-Laguerre matrix ensembles, and I am in the process of preparing illustrated notes that explore these connections. In the particular cases of β=1 and β=2, corresponding to real matrices and complex matrices, the link between combinatorics and random matrices can be made even more explicit, and provides an analytic framework for studying questions about the center of the group algebra of the symmetric group and the double coset algebra of the hyper-octahedral group.

One of the problems motivating the creation of Jackson and Visentin's An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces is the existence of a functional equation relating the generating series of several different classes of oriented combinatorial maps (their so-called q-Conjecture). This identity can be interpreted as an identity between the distributions of singular values of two different classes of Gaussian random matrices, and this provides a common framework for explaining several different properites that nominally involve the eigenvalues of GUE matrices. The implications are explored in "The Singular Values of the GUE (Less is More)". A follow-up, "The Singular Values of the GOE" identifies analogous properties of the GOE, although the combinatorial link is less evident in this setting.

Mathematical Illustration

I believe that many mathematical concepts are best communicated with the aid of accompanying illustrations. In the process of producing my PhD thesis, I became experienced with writing high quality mathematical illustrations using the PostScript programming language. I later developed my technique as the sole illustrator of a set of course notes on linear algebra, and subsequently a text book on the same subject, Introduction to Linear Algebra for Science and Engineering (2nd edition) .

I am highly interested in how the future of illustration will be affected by the emergence of electronic books as a viable medium, and particularly want to understand the potential of animation to enhance electronic presentations of mathematics. An example using JavaScript to animate pdfs can be found here (best viewed with Adobe Acrobat reader).

Gallery

Random Matrices

Figures from a course in multi-variable calculus

Topology

Combinatorics

Maps and Polygon Glueing

Tabulations

Miscelaneous

Typesetting Math

The standard medium for typesetting math is LaTeX. If you're interested in learning LaTeX, a good place to start is The Not So Short Introduction to LaTeX 2e. Further information on math specific issues, and on using AMS-LaTeX, can be found in the Short Math Guide For LaTeX.

As an example, here is the LaTeX Source source for the main file of my Master's essay on "Approaches to the Enumerative Theory of Meanders". The main file reveals all the macros I used, and shows the general structure of a LaTeX document. Here is the finished product for comparison.

My interests in typesetting extend to a professional level. I am the author of LaTeX classes to control page layout of user submissions to an online economics journal, Review of Economic Analysis, and also designed the class used for course notes for core-courses for math majors at the University of Waterloo.

Slides from Past Talks

Talks about maps tend to be graphically intensive. Most of these presentations use animations that are controlled by inline JavaScript. So far as I know, only Acrobat reader will display most of them correctly.

The Link Between Random Matrices and Map Combinatorics

Munich 2015 MIT 2013 MIT 2012 Boston 2013 β–ensembles from polygon glueings

Random Matrices

Brandeis 2015 A handout about the GUE

Talks about Map Enumeration

Minneapolis 2014 Non-constructive bijections Boston 2012 Non-Orientability and Jack Symmetric Functions Boston 2011 What is Map Enumeration?

Talks from my PhD

Waterloo 2009 PhD Defence Waterloo 2008 Q-Conjecture Waterloo 2006 2nd Stage Exam

Miscelaneous talks

Meanders Rubik's Cube

Photography

Prior to becoming interested in digital imaging, I spent a lot of time dabbling in exotic photography techniques.

• These represent the results of a project to create double-exposure Red/Blue 3D Photographs directly on film.
• I am also facinated by the colors present in low-light settings. A selection of photographs shows the color of stars and illustrates the daylight spectrum available under the light of a full moon.