# Dr. Joseph Horan

Sessional Instructor
University of Victoria
Department of Mathematics and Statistics

### Contact Info:

Office (Fall 2021): DTB A451 (David Turpin Building, formerly Social Sciences and Mathematics (SSM))
Office Phone Number: None, please e-mail.
E-mail: jahoran at uvic dot ca -- Please replace words with the appropriate symbols.

### Mathematical Research:

I am interested in, well, everything, but my research generally is on ergodic theory. My Master's and Ph.D. supervisors were Dr. Christopher Bose and Dr. Anthony Quas, listed alphabetically by last name. My Master's work focused on the Multiplicative Ergodic Theorem. My Ph.D work also focused on aspects of multiplicative ergodic theory, specifically the second-largest Lyapunov exponent and associated subspaces.

For the sake of interest (and possibly enticing others to look into the subject), ergodic theory is an area of research spanning a number of branches of analysis, including measure theory and functional analysis. The basic theory is focused on maps on measurable spaces, and the measures which are invariant under such maps... or vice-versa, that is, measures and the maps which preserve those measures. We say that a measure and map pair (m, T) is ergodic if for every T-invariant set A (that is, the preimage of A under T is A), m(A) = 0 or 1. Taking the map T to be fixed, it turns out that we can decompose any measure m into ergodic parts, but the method of combining those parts might actually be an integral! Pretty neat stuff; look up Choquet's Theorem for more information. I've spent some time going through a book by Boyarsky and Góra, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, and it's a somewhat-decent intro, although there are some errors scattered throughout the book. A very good option is Peter Walters' book, An Introduction to Ergodic Theory.

#### Commented List of Notes:

• At the end of the Fall 2013 term, for my Math 585 seminar course I gave a presentation on normal numbers; here, you may find the extended abstract for the talk, and here are the slides.
• I wrote up an interesting proof of the ergodicity of a particular map of a type considered by Furstenberg (a skew product of rotations on the torus); you may find that writeup here.
• You may also check out the presentation I gave at my Master's thesis defense, in case you missed it, or you'd like to have a look at it again, or if you want to read my Master's thesis, that's available too, though be warned: it is way too long. I suppose, however, that it's got all the proofs for which you could really ask, modulo the Birkhoff Ergodic Theorem (which you may find here) and the one result by Klaus Schmidt I used (found in Cocycles of ergodic transformation groups).
• More recently (September 2019), I gave a talk at the BIRS Retreat for Young Researchers in Probability and areas of Application; here are the notes.
• One of my favourite talks recently was in the UVic Dynamics Seminar in Spring 2020; in a single hour-long talk, I brought together dynamical systems, linear algebra, Eisenstein's criterion (okay, tangentially), and algebraic geometry! You can find the talk slides and the algebraic geometry supplement by clicking on those links.
• Of course, the culmination of my five years of Ph.D. studies is my doctoral dissertation, on the subject of certain cocycles of Perron-Frobenius operators that I show have useful spectral gap and asymptotic properties; while it is quite lengthy, it is packed full of examples and proofs, so that it should be fairly instructive should you choose to read it.

### Educational Research:

In addition to having those lovely letters beside my name, I also hold the LATHE graduate certificate, which I earned for completing a two-year for-credit program in Learning and Teaching in Higher Education. I am very interested in continually improving both my own teaching and the teaching mission of my university.

Some of my educational research interests include active learning, writing skills, TA training, student motivation, and better methods of assessment (especially since we just went through four-plus terms of emergency online teaching), all from a mathematics perspective. One SoTL (Scholarship of Teaching and Learning) project that I'd like to investigate further is based on a presentation on "Service Teaching" that I gave in one of my LATHE classes.

### Teaching:

Fall 2021: MATH 110 - Matrix Algebra for Engineers, MATH 212 - Introduction to Algebra. See Brightspace for the course pages!

#### Previous Teaching:

I've also previously run tutorials for almost every math course at UVic that has tutorials, except for Math 120, Math 248, and Stat 254. I've marked for a number of different courses at different skill levels, as well. My favourite TA work to do was tutor in the Assistance Centre; I've been a Floating Tutor multiple times, and those really were the days. It was a wonderfully positive experience. Running tutorials was also enjoyable, and hopefully one day I will become a more consistently happy marker.

### Education:

I began working towards my PhD here at UVic in September 2015, and successfully defended my dissertation in April 2020. It was a very interesting journey, throughout! Some milestones included: finishing my comprehensive exams in January 2017 (Real Analysis in January 2016, Topology in April 2016, Algebra in January 2017), and finishing my courses in December 2017 (yes, backwards).

I began my Masters degree program here at UVic in Fall 2013, and defended my thesis in April 2015. It certainly was interesting! I learned lots, and even managed to present my work to others without embarassing myself too many times. Or something. Maybe.

From Fall 2009 to Winter 2013, I studied at the University of Waterloo, culminating in a BMath Honours Degree, Double Major in Pure and Applied Mathematics, graduating on the Dean's Honours List With Distinction. It was - how does one put it lightly while still being correct - fun! Please ask me to regale you with stories of my Functional Analysis or Lie Groups/Lie Algebras courses; I'm sure I won't mind reminiscing. Also, not-humble-at-all brag incoming: I am the proud owner of a non-zero score on the 2012 William Lowell Putnam Mathematical Competition! In particular, I got a 2! Looking at this site and obtaining some mildly clever bounds on the number of participants with a given score, it is possible to give the median for the 2012 Putnam, or in general a bound for the median. I did this here.

### Curriculum Vitae

A much more formal summary of my academic life, including a list of my publications, may be found in my CV.

### Personal Interests:

I dabble in just about everything, so here comes a not-so-brief overview of my interests. Feel free to engage me regarding any of these! I love finding people who do some of the same activities as I do. I guess one might say that the set of my interests is dense in the set of all interests, but I think there's probably a counterexample somewhere. Actually, I don't know what sort of topology we'd put on such a set. Oh well.

• Sports: I bowl 5-pin, golf, do track and field, and play pick-up/intramural X, where X is almost any team sport, except for a set of measure zero, of course. I follow most of the North American sports leagues, and root for the Canucks, Blue Jays, BC Lions, and Raptors. I'm also a certified Level 1 BC Athletics official.
• Music: I've played the electric bass for seventeen-plus years, and I fake playing the upright bass and various percussion instruments; I also sing, most of the time in tune. I listen to essentially all genres of music except for what is currently "popular" (see: Top 40), and my favourite bands/artists include Streetlight Manifesto, Dream Theater, Stan Rogers, Dala, Lin-Manuel Miranda, Phoebe Bridgers... amongst too many to name. I also have done a bit of recording work, like this song which I co-wrote-by-mail with a friend from undergrad.
• Everything Else: Musical theatre? Check. Video and board games? Check. Baking? Check-a-roo!

Speaking of musical theatre, I'm quite fond of song parodies, and I'm also quite fond of the history of mathematics. So naturally, it was only a matter of time (ahem) until I produced a parody of the title track "Alexander Hamilton", from Lin-Manuel Miranda's musical Hamilton: The Frenchman, Évariste Galois. Feel free to share! It's almost entirely historically accurate, and the inaccuracies are so vague that they could be correct. I say this because much of the legend of Galois has been exaggerated and, at times, falsified. Check out Tony Rothman's paper, Genius and Biographers: The Fictionalization of Évariste Galois, for more details. For an annotated and complete collection and translation of Galois's mathematical writings, read Peter M. Newman's book, The mathematical writings of Évariste Galois.

### Tuesday Tea:

In the before times, every Tuesday we would do tea and snacks in the lounge, and the snacks tended to be homemade. Since the pandemic began in March 2020, Tuesday Tea fizzled out, in online form. Perhaps one day it will return to its former glory.

Once upon a time, I intended to aggregate the recipes used for Tuesday Tea goodies, so that if there was anything which you thought was amazing, you'd be able to look back and see how it was made, and then make it yourself! Unfortunately, this didn't exactly 'pan' out (you know... like a baking pan). It was worth talking about, anyways. I do, however, have a semi-complete compilation of Tea-Mail puzzles and poetry and various other literature, spanning my time at UVic. You may find it here. I make no further guarantees that it will be updated reasonably often. But, hopefully it will accurately preserve content from previous Tea Chancellors, so that even if you delete your e-mails, you'll still be able to revisit the glory and triumph that is Tea-Mail. For the record, all of the songs are spoiled (almost) immediately below the lyrics, so if you are trying to guess the song from the lyrics (which is a great game), you shouldn't scroll too far.

On June 23, 2015, we had a pie-off in the department! Joey Niezen and I made pies competitively. You may find a document recapping some of the information here, and below are some photos illustrating the event. You can left-click these to embiggen (I hope I've implemented that correctly), and you can right-click to download them.

### Daily ε of Math Calendar 2019

In 2019, the Math & Stats Assistance Centre put up a calendar, which had the following property: most days had a mathematics problem, the numerical answer to which was the day of the month. The question, then, is how one might be able to obtain the answer. For fun, I started writing solutions, or at least, outlines of solutions. Over three years later, I finished. Here are the results (sorry for the blurriness of some photos):