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is a Scientia Fellow in the School of Mathematics and Statistics. In this fascinating interview, he reveals��what first drew him towards��maths, some of the applications of his research, his hope that his��forthcoming book will shake up calculus teaching, his passion and talent for music, and his love for his miniature dachshunds - and much more!��

What first triggered your interest in mathematics, and when was that?

I always was keen on mathematics as a child, and it was clear from an early age that I had an excellent mind for numbers, but the moment I first thought about devoting my life to mathematics was in middle school (the seventh grade to be exact). Although my father was a neonatologist by training, he was tremendously gifted mathematically, and in his library at our house he had many books on mathematics.

Two of these were The Beat of a Different Drum, which is a biography of the American physicist Richard Feynman, and Chaos: Making a New Science, which explains the evolution of the mathematical sub-discipline of chaos theory and dynamical systems. I read these two books at the age of 11, and I recall being awestruck at the mathematics mentioned in these books, even if I couldn’t fully understand all of it at the time. But I knew then that mathematics was something I wanted to play an important role in my future.

When did you first consider pursuing mathematics as an academic career, and what do you like most about your job?

It wasn’t until after I had finished my undergraduate studies. As an undergraduate I studied mathematics, theology, and secondary education (yes, an unusual combination, I know). I then taught high school mathematics, with my intention being to go to medical school after a couple years of teaching and thus to follow in my father’s footsteps. But the more I taught mathematics, the more I began to consider a career as a mathematician, and, well, here I am. I do wonder occasionally how my life would have turned out had I become a physician instead, but, at the same time, I have no regrets about devoting my life to academic mathematics.

Speaking of not regretting getting into academic mathematics, teaching has been a very satisfying part of my work. Thinking about how to explain mathematics in an appealing and engaging way is a satisfying challenge, and it’s one that I’ve always enjoyed.

I also have to say that one of the other gratifying aspects of having an academic career in mathematics has been the international network of research collaborators it has allowed me to create. In no particular order I’ve worked with mathematics in the US, Italy, Germany, Spain, Portugal, China, India, Australia, and Saudi Arabia, and I’ve visited several of those countries, many of which I likely would never had visited were I not an academic mathematician. It’s allowed me to have a good and full life. And for that I am grateful.

Your research often centres on difference equations, a discrete version of differential equations. You have also worked on, and��, the approach of time scales which tries to connect difference to differential equations in a unified way. What appeals to you more: discrete time or continuous time problems? Which do you think is a better model to describe the world around us and the phenomena we observe?

I really don’t have a preference between difference calculus (discrete time problems) and differential equations (continuous time problems). There are challenges in studying each of these, and I enjoy them equally, truth be told.

My personal opinion is that continuous time problems are seen as somehow more “important”. For example, it is generally much easier to get a paper on differential equations in a “high impact” journal than it is a paper on difference equations. Sadly, discrete problems are seen as somehow easier and less interesting than their continuous counterparts. This is a very ignorant view, if I may dare say so, because discrete time problems can be very challenging - far more so than sometimes meets the eye. But it does seem to be pervasive in the mathematics community. (Just as, it seems to me, applied mathematics, in general, is seen as less worthy than more abstract mathematics.)

As for modelling I suppose that it depends on the type of model. For example, if you want to calculate the sequence of payments required to pay off a mortgage (what is called amortisation), then a discrete time model is what you want. I actually have taught courses in financial mathematics before, and most of the models we considered were discrete time models because frequently you have a sequence of payments occurring at discrete time points with interest calculated on a monthly basis. On the other hand, if we want to model a process that evolves continuously (say, the temperature of a cake as it cools), then while one could use a discrete time model as an approximation (a type of maths called numerical analysis), a continuous time model such as a differential equation might be more appropriate. So, each type of model has its place in the scheme of things.

One of my PhD advisors, Al Peterson, is an expert in what you mentioned: the so-called time scales calculus, which��unifies both viewpoints. I haven’t published very much in the area of time scales. But it is highly interesting in that it allows one to see (and, more to the point, understand) similarities and dissimilarities between the discrete and continuous approaches. So, in my opinion, there is much value in it.

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What drew you toward this particular area of research, and what are some interesting ways that it can be applied?

Ironically, it wasn’t at all what I was planning to do. I entered the PhD program with the express intent of writing the dissertation in mathematical biology - specifically, epidemic modelling. I even contemplated doing a dual PhD in biology and mathematics. But during my first year in the PhD program I took a course on difference equations from Al Peterson, who is an amazing teacher and researcher (and just a genuinely decent person, I must say); in fact, Al become one my doctoral advisors. I was so enamoured with Al’s research focus that I abandoned my original plan and decided to devote my studies to the abstract analysis of differential and difference equations.

I also took, as a second-year graduate student, a year-long sequence in differential equations and fixed point theory (the latter being methods that can be used to detect when a given differential equation has a solution). The instructor for those courses was Lynn Erbe. Lynn is a fabulous mathematician (fun fact: he has one of the most cited mathematics papers of all time in the journal Proceedings of the American Mathematical Society). So, under Lynn’s tutelage I became even more interested in differential equations and, especially, the application of topological fixed point theory to the study of boundary value problems, which was Lynn’s special expertise. And the rest is history, as they say!

Nonetheless, and perhaps fittingly, there are many interesting applications of the mathematics I study. For example, one mathematical object that I study are called “discrete fractional operators”, and these have recently been used in modelling tumour growth. I also study what are called “boundary value problems”, and these have many applications ranging from beam deflection (in engineering) to the transfer of heat through a material (in thermodynamics) to modelling the motion of an elastic body (in kinematics). So, while the mathematics I study is rather divorced from direct application, it is very much related to highly applicable mathematics.

I hear that you’ve just worked on a massive soon-to-be-released first year calculus book. What differentiates your book from other calculus books on the market? What do you hope to convey to students from within its pages?

The idea behind the book is to unify the treatment of the integral and differential calculus. In every book of which I’m aware many pages, indeed hundreds of pages, are devoted to derivatives with absolutely nothing mentioned about integration. This I have always found very odd. On the one hand it’s deeply “wrong” historically, so far as I understand, seeing as the integral calculus was originally invented by Archimedes some 2000 years before the differential calculus. But, more importantly, it compartmentalises mathematical ideas (such as definite integration and differentiation) in ways that are frankly bizarre, unnatural, and pedagogically unhelpful, in my opinion.

So, the goal of the book is to develop both branches of calculus simultaneously. My hope is that this shows students that the ideas are naturally intertwined and, furthermore, follow from very simple physical ideas. For example, in most books (perhaps nearly all) the fundamental theorem of calculus is presented as though it is a shocking result. Yet when viewed from simple physical principles it is a very natural affirmation of, dare I say, straightforward intuition. These are the types of connections my book will emphasise, and I hope that this will lead to a deeper understanding of the calculus amongst students learning from the text - and also give instructors some new ways of teaching single variable calculus.

As a consequence of the aforementioned goals, the book upends the usual sequence of topics. For example, definite integrals are covered within the first few pages, as is differentiation. And Taylor polynomials are also introduced much earlier than usual. All in all, writing this nearly 1000-page book has been a labor of love since July of 2015 when I first sketched out the idea for it. I hope it will be well received and, ideally, shake up the way calculus is taught.

A cursory glance at your research profile reveals a long list of publications, and you now have a couple of books under your belt. Do you have any tips for achieving such a prolific workload (quick thinking, speedy writing, highly organised, insomnia...?!).

Interesting question. Well, let me first affirm the role that luck plays. This is true in many different ways. For example, I was exceptionally fortunate to have advisors (another shout out to Al and Lynn mentioned��earlier) who gave me a good, interesting, and tractable dissertation problem. It led to many nice results even before I had completed graduate school. I can’t overemphasise what a gift this was. I’m quite certain there is a parallel universe in which I’m not a tenth of the mathematician that I’ve become in this universe simply because in said parallel universe I didn’t have the good fortune of enjoying a satisfactory “initial condition”.

As another example of luck, when it comes to one of my greatest insights as a mathematician (if I can call it that), was being in the right place at the right time. Specifically, I was working on a particular problem in August of 2013, and I was having difficulty obtaining a result that I thought should be provable. The process of determining why I could not prove my desired result led to me a sudden insight about mathematical objects called “discrete fractional operators”. I can still remember that afternoon like it was yesterday, sitting in my house, with my two dachshunds napping at my side.

In short, I realised that what was preventing my desired result from being true was the fact that these operators have some very peculiar properties as concerns their ability to detect when a function is either increasing or decreasing. No one had noticed this before (or, so far as I can ascertain, even considered the possibility), and so, I along with my friend and colleague Raj Dahal published the first results in this direction. Since then, the research in this area has really blossomed and there are now many dozens of papers on the subject. Moreover, as this line of research has continued it’s been observed that these operators behave in even more complex ways than either Raj or I realised when our first results were published in early 2014.

I call it my “Sir Alexander Fleming moment,” not that I would dare compare myself to the discoverer of penicillin! But still, it’s amazing the role luck, happenstance, and fortune plays in one’s life. And this is certainly true in the life of a researcher - perhaps even more acutely so in some ways. Had I not been working on a totally unrelated problem, I would never have thought to consider the question that I ultimately did. So, much as Fleming discovered penicillin’s antibacterial properties accidentally, so I accidentally opened up an entire new area of mathematics because ultimately asking the wrong questions led to asking the right ones!

But above and beyond luck I believe that perseverance and a calm mind are important tools. Many will find this deeply bizarre, but many of my ideas have come sitting with my two dachshunds, relaxing whilst watching a movie or walking in the backyard. When my mind is in this relaxed state I find that I can allow my creativity to percolate. Of course, lots of silly ideas arise from this. But now and then a good one arises, too.

And, yes, it probably helps to sleep less than one ought to sleep! The notion of burning a candle at both ends comes to mind, for better or for worse.

I read on your Scientia Fellowship profile that you enjoy composing music and playing the piano. Can you tell us a bit more about your passion for music and its origins?

Both my parents played the piano, and my father composed music. So, even as a very young child I was exposed to music frequently. As a toddler I used to like to “conduct” the famous orchestral piece The Rite of Spring; it was my very favourite piece of music as a young child, and I listened to it incessantly (likely, this drove my parents crazy). Then when I was about seven I watched the move Coma, the musical score for which was composed by Jerry Goldsmith. I loved the score so much that I would try to repeat it from memory on my parents’ piano - also likely driving my parents mad. So, at that point my parents decided that perhaps I should actually receive some formal training, and so, I began studying the piano. Later, as an undergraduate student, I took orchestration, composition, and conducting lessons. I even briefly studied the organ.

For a time during my high school and undergraduate days I was practicing the piano or composing music for about 30 hours a week. I really was passionate about it. A few of my compositions won some minor awards in state and national composition competitions. My best composition, Sonata in an Uncertain Key, won runner-up nationally; it was written in a heavily polytonal style, somewhat of a homage to the styles of Bartok and Stravinsky, my two favourite twentieth century composers. I’d like to think it was the level I could have attained if I had decided to go to music conservatory and study composition full time. But, alas, it is very difficult to make a living as a composer. The daughter of one of my father’s friends studied music composition at UCLA, which has a fabulous school of music, and yet she just wasn’t able to get a break. I took that as a cautionary tale.

Still, I don’t regret one bit the time I put into my music studies as a child and adolescent. It was great fun. I’ve even written a few compositions for my two dachshunds! Whether they like them is, I am afraid, debatable. They are tough critics those two!

Where have some of your students ended up working or studying?

One of my former students is a professor at Stanford University (he ultimately completed a PhD in computer science, though he has an undergraduate degree in mathematics). Another just finished a PhD in physics at Rice University. And I have another former student who is a PhD student in mathematics (specialising in numerical analysis) at UC-Berkeley. I’ve had the good fortune of having taught a number of very talented and motivated students.

Who is the most famous person in your maths genealogy?

Well, I hit the jackpot on this one. I had two PhD advisors. The first, Lynn Erbe, has as one of his ancestors Gauss, who, in my humble opinion, was the greatest mathematician of all time. The second, Al Peterson, has as one of his ancestors Euler, who was also pretty amazing. So, being in a direct line from both Gauss and Euler (so to speak, anyway) is very nice.

You are from the U.S. Where do you call home over there, and what are your favourite things about it? I’m also keen to hear a bit about your two miniature dachshunds, and see a photo if you are happy to share one!��

I spent the first part of my childhood in San Francisco, California. Then we moved to Omaha, Nebraska. And later I lived for a year in Providence, Rhode Island. So, I’ve lived on both coasts of the U.S. as well as in the middle of the country. But it’s Omaha where I’ve spent most of my life. Omaha is a generally pleasant city of about 1 million people in its metropolitan area (about 500,000 in the city proper). It has a surprisingly vibrant dining and arts scene, more in line with a much larger city; in fact, Omaha has some absolutely wonderful restaurants. And it has two world-class medical centres; in fact, my younger brother is a professor of paediatrics at one of them.

If I were to pick one random and interesting factoid about Omaha it is that we have very extreme weather here. If you don’t like extreme cold and heat, violent thunderstorms, and blizzards, then Omaha might not be the place for you. In the summer it is not uncommon to reach 38 degrees, even as high as 45, with dew points regularly exceeding 25. And in the winter there are usually a few days when the low temperatures bottom out around -20 to -25. Also, Omaha is rather prone to tornadoes (relatively speaking, of course - they are rare events, after all). Omaha was hit by violent tornadoes in 1913 and 1975; the latter, in fact, destroyed my parents’ residence at the time.

And, more recently, the city and/or its suburbs have been struck by less impactful tornadoes in 1986, 2006, 2008, 2014, and 2016. In fact, one of the tornado strikes in 2008 was within three miles of my parents’ house; they had housing insulation and other debris, which had been transported from destroyed houses a few miles to the south, floating in their backyard pool. So, if you like extreme and somewhat “exciting” weather, then Omaha might be a great place to live. If not, then I’m afraid I’d advise looking elsewhere!

And, yes, my dachshunds are the love of my life. I confess that I love dogs and wolves generally, the latter a terribly unfairly maligned yet deeply honourable species. Until recently I had two miniature dachshunds, Maddie and Lily. Lily, or Lord Lillian as I tend to call her, will turn 15 in January and is a black miniature dachshund; she seems to revel in mischief. But, ultimately, she’s a great companion in spite of her predilection for naughtiness.

Maddie, who would have turned 20 next March, was a red miniature dachshund and an amazing dog in so many ways. Tragically, she died rather suddenly a few months ago. She was the most wonderful animal I’ve ever known, and we were very close. I think about her every day, and I miss her more than words can convey. In honour of the recently departed Stephen Sondheim, to misquote slightly a lyric from his musical Sweeney Todd: she was my reason and my life, and she was beautiful.

Images, from top: Dr Chris Goodrich; Chris with his PhD advisor��Al Peterson; Chris' book Discrete Fractional Calculus; Chris with his mini dachshunds Lily and Maddie.��

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December 2021

Interview conducted by Susannah Waters