Mathematics is a strange subject. Even after an undergraduate degree in pure math, you won’t be able to read research papers and dissertations. Ph.D. students can’t understand what their fellow candidates are working on. So, if one is serious about making mathematics their career, they ought to go where other mathematicians go. One of those places is the University of Toronto.
But there’s a problem with its satellite campus, the one located in Mississauga, an hour away. “Downtown is for math; Mississauga is for teaching.” Says Noriko Yui, a professor who taught in the math department at the University of Toronto Mississauga (UTM) from 1982 to 1987. Erindale College, which would later become UTM, was founded in 1967. And who wanted to go there when they could be in Toronto? Yet, somehow, we all come here, professors included.
Unfortunately, in mathematics, we only remember the math itself, bar a few exceptions such as Galois creating a field of math before dying in a duel over a woman at 20, or witty retorts such as Fermat claiming he can prove a result, which was proved about 350 years after his death. Remarkable results are remembered. Teachers and mentors are forgotten. I wanted to see what these mentors from the past are like, all while witnessing the transformation of UTM’s math department.
I started with Peter Fantham. Entering his name in the search bar, only a memorial scholarship bearing his name appears, where it’s written, “Professor Peter Fantham loved U of T Mississauga (formerly Erindale College) and was well liked by the students, staff and faculty.” He passed away on June 11, 1992, and his wife Elaine Fantham established the award in 1993 for UTM students.
Peter Henry Holman Fantham was born on February 17, 1931, in Liverpool, got his Ph.D. from the University of Oxford in 1957, and had 4 Ph.D. students of his own. Fantham married Elaine Crosthwaite after they met at Oxford. They had two kids, Julia and Roy, by the time they moved to Toronto. Elaine Fantham was a classics professor who ‘[r]e-drew the map of Latin studies’ and went to Princeton after Peter’s death, before retiring as a full-time professor and going back to Toronto, until her death on July 11, 2016.
Elaine was “[d]escribed by her colleagues as incredibly loyal on both a personal and professional level, and one of the most learned people known in regards to not only the subject of classical antiquity but many other subjects as well.” She enjoyed reading, classical music, talking with family, friends and, former students. She enjoyed movies like The Man Who Would Be King, The Girl with the Pearl Earring, Quai des Brumes and, Les Choristes. And, most interestingly, she had a ‘particular fondness for frogs, good quality scotch and lively conversation’.
But we should return to the subject at hand, Peter. While Fantham wasn’t too invested in research, he still knew his way around math. He was “an editor of the Canadian Journal of Mathematics from 1971 to 1976” and some of his work was published posthumously. His Ph.D. was titled Certain Problems in Algebraic Topology, a field of math that uses algebra to study topological invariants. Think of a pillowcase with objects inside it. How can you tell what the objects look like without seeing them? Well, you use your other senses, in this case, maybe touch. So, imagine that touch is ‘algebra’, and the objects (and their properties) in the pillowcase we want to study are the ‘topological invariants’.
He taught calculus and linear algebra, differential equations, and philosophy of mathematics. His course evaluation for calculus and linear algebra from 1970 mentions that:
Students found the course content and subject area very interesting, but found the lectures difficult to follow. Questions arising in class were handled satisfactorily, and Dr. Fantham was usually available for consultation. His expectation of the student was realistic according to the majority of them, but others said it was ‘excessive’ and also ‘too low’.
Fantham’s character remains a silhouette in the mist. I find weird things about him, like the fact that he was the faculty advisor in 1991 and that he was also supposed to be the discipline representative and faculty advisor in the 1992 to 1993 academic year. Here’s the closest thing I could find written down to attest to his character, an acknowledgment from one of his four Ph.D. students. In the student’s thesis, the following words are printed:
I wish to record my indebtedness to the following:
Firstly, about the thesis supervisor Professor P. H. H. Fantham, I shall say that he took a piece of dry earth and then like a skilled potter gave shape to it. Anyone around his office knew well how much time and energy he spent on me. My gratitude to him is beyond expression. I shall remember his paternal care all my life.
Md. Mahatabuddin wrote this. Born in India in 1930, he achieved his Ph.D. in mathematics in 1973. I can’t find anything else about him. Unfortunately, his other three Ph.D. students’ theses mention him sparingly in their acknowledgment sections, only shedding a phrase of gratitude. Regarding a personal account, I do have something compelling. I bring to the stand, Joe Repka.
Professor Repka teaches math downtown. As an undergraduate at U of T in Toronto in the 70s, he entered the math, physics, and chemistry stream with a specialization in higher years. He then went to the United States of America for graduate school and came back to Toronto shortly after. His research involved group theory with partial overlap in physics. Repka describes his research as studying symmetries.
Repka met Fantham for the first time in 1977 as a student in Fantham’s algebraic topology class. In algebraic topology, there are two main objects of study: homology and cohomology. Repka says, “In slightly different contexts, ‘co’ means something just a little bit different, so you could make an argument that they’re misnamed,” or these titles need to be flipped. Someone wrote a textbook that did just that. Called homology cohomology and vice versa. “Peter loved this book… it was nuts to change it.” Repka always used to confuse the two as a result. But Fantham was a great professor. “He knew his stuff, made it interesting, and was fun to talk to.” His classes were lively.
When Repka started working as a professor, he worked alongside Fantham. Fantham didn’t care too much about seniority, so the transition from student to coworker was smooth. “He never seemed to get too bothered by anything.” At a house party, Fantham noticed a snack bowl and unsure about the snacks, slowly munched on them, emptying the whole bowl. The hostess’ face fell when she saw the bowl, “That’s not a bowl of snacks, that’s a potpourri, [a bowl filled with fragrant items] to make a sweet smell in the room.” Fantham demolished it, but “instead of being disgusted or horrified, he said, ‘well, it had tasted really wonderful.’”
Another time, one of Fantham’s coworkers named Peter, at the time the associate chair, kept an aquarium in his office. As an ongoing joke, Fantham used to comment how the fish was the right size for an afternoon snack. One day, Peter put a cracker on the table, scooped out a dead fish after it had died, and offered it to Fantham when he walked in. Fantham was horrified.
Fantham didn’t talk about UTM with Repka, but UTM tried to stand out, rather than being under the main campus’ shadow. The math department held an event to have a picnic outside on campus grounds. As if out of a movie, it rained that day, and no one postponed it. So, they still held a picnic, probably on the sidewalk, because it was drier than the grass. But Fantham, on recounting the story, described it as saying, “It looks like we’re a bus stop, except there’s no bus.”
When Fantham finished his thesis, he had to present it to his department at Oxford. “It was a tough event.” In Fantham’s thesis, he introduced three mathematical objects labeled F, G and H. In the thesis, Fantham wanted to show that H was part of G and that G was part of F. The ordering went H, G, and F. One of the leading faculty members in the algebraic topology asked Fantham a question. Wouldn’t it be sensible to order them alphabetically? Repka says, “That’s a really mean thing to do to somebody in the middle of a presentation.” Changing the labels in the middle of a presentation would cause some slip-ups.
“So, there was a long silence… [Fantham] was sort of a large man” and he put “his hands over his tummy, thought for a minute and said, ‘Professor so-and-so is absolutely right. It would have been better to label them in alphabetical order. However, for the sake of this lecture, the letter F will be pronounced H and will be written with two vertical bars and one horizontal bar, and the letter H will be pronounced F and will be written with one vertical bar and two horizontal bars.’ And he went on his merry way.”
Fantham kept his private life separate from his professional life. Repka wasn’t too close to Fantham. But Fantham was always happy to talk, but also serious about his work. There is one more story about Fantham. One day, in the lounge, two people were arguing about the asteroid that wiped out the dinosaurs. It got heated. Fantham came in, listened for a few minutes, put on his “pretend innocent face, and said in a big loud voice, ‘What I can never understand when the asteroid killed all the dinosaurs, is what the dinosaurs were doing all standing around in the same place.’”
Two guys arguing didn’t realize he was joking and tried to explain, and Fantham just smiled every time. “Anyone else coming up on these guys… might have said something nasty… But [Fantham] didn’t do that. He just made it a joke… he was making fun of them, but in a gentle sort of way… he gently destroyed the whole thing, with a smile on his face.”
Fantham wrote the 25-year history of the mathematics department at UTM. It’s tough to innovate and create an identity when you need to make sure the department gets off the ground. Many of the courses offered or labs created don’t remain today in the same form. Indeed, there always needs to be some first-year calculus and linear algebra classes, and that takes a lot of effort to run.
Every year, a small change, a new hire, or a new course causes slight changes. Recording every change and its impact is something I can’t do. The changes from 1967 to 1992 are visible, but on a year-to-year basis, I can’t tell you what happened. But they grew from two faculty members to twelve. From 1992 to 2007, the number of courses grew from 21 to about 30, yet the number of faculty listed in the academic calendar stayed the same. But Postdocs and visiting positions don’t get mentioned on the calendar. Indeed, in the 40th anniversary in 2007, it says that the “faculty complement has doubled since 2000.” It seems that people took more math. After 40 years of hard work, the campus changed from fledgling to adult. And every adult gains autonomy.
UTM’s math department was under Toronto’s math department. In 2007, G. Scott Graham wrote that “Computer Science is now back in bed with Mathematics and for good measure, has been joined by Statistics to form a tripartite structure known as the Department of Mathematical and Computational Sciences. (Tripartite seems to be the appropriate graph-theoretic term to use here, in place of what some would call a three-way.)” As a result, they needed an actual chair for the department. But how do they lure someone?
Vijaya Kumar Murty always liked math as a kid. His brother Ram, also a celebrated mathematician, encouraged Kumar and vice versa. They helped and worked with each other to pursue their shared love of math. Even now, they co-publish papers together. “We sort of inspired each other,” Kumar Murty says.
Murty went to Carleton for his undergraduate, Harvard for his Ph.D., was a postdoc at Princeton for a while, went to the Tata Institute in Mumbai, Concordia and finally Toronto at U of T. In the late 90’s he started getting interested in information security. Wanting to start his research lab, he created the GANITA Lab.
“Downtown, the hardest thing to get is space… so the chair at the time said ‘it’s a good idea you’re going to set up a lab, it’s great. I can give you this little closet.’” So, he crammed 4 students in this tiny closet, before deciding, “I can’t continue like this.” That was exactly what UTM needed, an undeniable advantage, space. The dean at the time gave Murty space for his lab, even willing to turn a wet lab for the sciences into a dry lab for mathematics.
“UTM is friendly… you just feel it… it’s not that people here [Downtown] are rude or anything like that, but there [at UTM], there’s a kind of closeness that you feel.” Murty is now the director of the Fields Institute located in Toronto, one of the greatest math research spaces in the world.
Having lived in Toronto, Murty never went to UTM nor visited Mississauga. Suddenly, he was chair there. “The first day or two, I thought there’s no work to be done… I can just do my math… then it suddenly dawned on me. No, actually there’s lots going on.” Duties must get done, but opportunities are a gift, and there were a lot of opportunities at UTM. The administration and faculty fully supported him.
In Toronto, everyone stayed in their departments, with less collaboration between departments. But at UTM, the “disciplinary silos are less pronounced.” This was one of the greatest advantages UTM had over Downtown. Growth was possible at UTM, and interdisciplinary studies could shine. For example, Murty created a calculus class for life science students, which still exists today.
But how does one build interdisciplinarity? “Students think they’re the lowest on the totem pole. I think they’re quite high up.” As students, you can learn from the best in so many fields. You “can knock on people’s doors and talk to them, and they will spend time with you. That’s what they do.” Your mind is a garden, and you need to cultivate it. Start a club, start or organize a seminar. Read journals. Take those opportunities presented to you. And when you ask, or have an idea, all you need is someone to listen. To decide, yeah, a calculus course for life science students is a good idea.
U of T couldn’t sacrifice their quality because of campus and UTM had to grow. They need strong faculty and TAs to maintain a high standard. Regarding the math department, they ended up hiring lots of faculty. In one year, they got three more faculty members. But how did he do it?
“I said graduate students can get their TA assignments at Mississauga… and I’ll pay for your bus tickets…I give you as many cookies as you like… all my seminars, I used to make sure there’s a coffee break… it doesn’t cost anything at all… and it really builds a group.” As a result, graduate students realize how nice UTM is. It’s quiet. Work gets done. And no doubt that professors felt likewise. When one can commute between campuses to get the best of both worlds even if they’re stuck on one side, what’s the big deal with being on a different campus?
The next year, Murty was offered department chair in Toronto. Murty could have taken either option, but he chose Downtown because it fit his lifestyle better. Despite this choice, he was satisfied with his experience at UTM. “Even now when I go back, I get this warm greeting. I’m happy to see them. They’re happy to see me… I’m very lucky I had that experience. I sometimes tell my colleagues you really need to go out, spend some at UTM, maybe at Scarborough.”
Since then, UTM’s mathematics department has grown. Renowned faculty in number theory, probability theory, symplectic geometry, and many more that escape my mind. And what’s the result of all of this? I want to tell you about Peter Phan. Originally a life science student wanting to do medical research, he pivoted to math after realizing that life science is memorizing “hundred slides of facts. And then you spit it out on the exam.” Mathematics, however, is more interesting and flexible. In hindsight, he made the right decision.
Another UTM student, Victoria Valeeva, took MAT257, a notoriously hard second-year calculus course that differentiates simple math designed for engineers and computer scientists, and pure math for mathematicians. She then switched to statistics, and published a paper in Nature Communications, before heading to the University of British Columbia to do her Ph.D. It’s in the math department, under a professor who specializes in theoretical and computational chemistry. Math allows you to switch to another field and do well, in a weird way that other fields don’t allow. Countless examples exist. Ilya Sutskever, a co-founder of OpenAI. Johnny Kim, a doctor, Navy Seal, and astronaut, started in math. Catherine Beni, an MD/PhD. All this is to show, that this pathway leads somewhere.
Peter also took MAT257 with Professor Tyler Holden. MAT257 was important to Peter’s development as a person and in math. “I love Tyler. I love that he pushed us so hard. Through 257, I got my first TA ship” which helped in his CV and financial troubles. It was also an introduction to the field he explored and loved in undergraduate, differential geometry.
And it’s not just Tyler Holden. Almost every professor at UTM is similar. It’s rare to find a student who isn’t appreciative of either a math professor or a math TA, even if they’ve only taken a handful of courses or don’t plan to do math in the future. There is one person however, that I unfortunately didn’t meet, who was loved. Alfonso Gracia-Saz.
“Alfonso was a character. He was brilliant. He was a great mathematician… he could have definitely made a research career happen. He was also possibly the most brilliant… teacher I've ever met. His ability to think about how to frame a question and break it into digestible pieces is absolutely unparalleled.” This is the first thing Tyler Holden told me when I asked him about Alfonso.
Alfonso Gracia-Saz was born in Zaragoza, Spain in 1976. He obtained his Bachelor of Science in physics and mathematics from Universidad de Zaragoza and a PH.D in mathematics from the University of California, Berkeley in 2006. He joined the U of T faculty in Toronto in 2013 and passed away on May 6, 2021. The Canadian Mathematical Society awarded him the 2021 Excellence in Teaching Award a few weeks before he passed away.
The department chair at the time, Jeremy Quastel said that Alfonso’s “lectures inspired not only a love of mathematics, but also a rethinking of what was possible in the act of teaching itself.” Another associate professor at the time said that Alfonso “belongs to this rare breed of born teachers that possess not only the knowledge and creativity, but also the warm and dynamic personality that allows him to teach students in such a natural way that the barrier between teacher and student ceases to exist.” So many of his students thank him for kindling the love for math within them.
“The man was very opinionated,” said Tyler, before continuing, “he was very harsh and very mean, sometimes including to me.” Tyler then relates a story of how after a test, the test count and head count was off. “And Alfonso lost it at me.” Tyler went through every test and the student list to find the missing test for the next two or three hours. “This was Alfonso’s way of teaching me, you do not mess up the count.”
Or another story where Tyler asked Alfonso out for drinks, and Alfonso lost it. “He was so upset, which I thought was unfair,” because Tyler didn’t know Alfonso had a bad history with alcohol.
A student doing their Ph.D. at Toronto Metropolitan University, Ryerson at the time, ran exam preparation sessions for all the calculus courses at U of T. He would charge $50 or $100 for every student who showed up. He would provide success packages, “he did the whole thing, and he was cleaning up…He was making a lot of money, and Alfonso made it his personal life mission to destroy this guy… Alfonso reported him to the CRA.” It's the exploitation of students for money that made Alfonso so angry. But this was partially due to the courses and how they were structured, so Alfonso helped alleviate this problem when he started teaching the intermediate level of calculus, MAT137.
But Alfonso was supportive. If he saw you had potential, he would take you under his wing. People remember him fondly because he was a great person. He was a great teacher.
One of the things Tyler did better than Alfonso was in-class presence. “In the classroom, I was better… and I don’t say this as a disparagement to Alfonso…that was the reason I taught 137.” Someone higher up, either the department chair or Alfonso himself told Tyler to teach 137 “directly opposite Alfonso, like literally, in his time slot… and I said, well, this is dumb. Why would you do this to me.” But Tyler was told he was the only one capable of doing such a thing.
“I always say I’m a teacher because I’m a mediocre mathematician, but you know, I think I’m a pretty darn good teacher.” Tyler doesn’t see teaching as a compromise. He loves teaching more than research. His wife, who had known Tyler since high school knew that he would become a teacher. Tyler did all the extra peer tutoring stuff in high school and during his undergraduate.
As a kid, he would always do math workbooks that his mom got him, and in high school, he loved a lot of subjects. Biology, engineering, computer science, so maybe something in bioinformatics for university would suit him. But it all came back to math and physics, before doing his Ph. D. in pure math.
Tyler’s research ended up coinciding with some of the UTM faculty, and after applying as a postdoctoral, he got accepted. “UTM sort of had at that point… had a queue” of people going from postdoctoral to continuing status to tenure. A position opened for continuing status and Tyler was told to apply only after a year. “You’re not gonna get this, you should apply anyway” for the experience, he was told. He got accepted. “I was very lucky” because most people need to spend four to six years to get continuing status, and since then, he’s been at UTM and has become a household favorite of many students, partially because he taught 257.
257 was usually offered Downtown and not at UTM. When Tyler became a professor, all the upper-year courses, the treat courses for both students and professors alike, were taken. However, UTM decided to offer 257 and MAT157, the first-year version of 257, and Tyler taught it. These courses attract the strongest students and have interesting content. 257 is where “boys become men” and “girls become women.” The students say, “I’ve heard terrible things about this course” but they’ll still take it. But as the professor, you “destroy them. You do your absolute best to tear them into shreds and then rebuild them in the image that you want.”
When Peter took 257, it was online, a very different experience than being in person. Peter says, “I didn’t really enjoy myself during [257], but when you look at it in hindsight, it was like crucial.” Peter didn’t switch to math because he originally found it interesting. As he took more math, he enjoyed it more and more, but it wasn’t passion that caused his pivot. He wanted to do some sort of Ph.D. and thought math was a great move. It allowed him to think differently and tackle any problem. But he needed to do something applied.
“So, I had a lot of phases.” An economics phase, a neuroscience phase, a physics phase, a pedagogy phase, a law phase (where after only a month of subpar studying, he hit the 80th percentile on the test), and a pure math phase. But math wasn’t the play. It’s super competitive. “They’re not only insanely smart, but they’re crazy hard-working, right? They do math for hours a day, and they’re way smarter than me.” It’s discouraging to go up against the greatest competition, especially when your goal is to do something with your research and contribute to society.
So, Peter switched to machine learning, particularly in physics, where he could leverage his strength from math and his physics phase. His mathematical background gave him the ability to do it, despite having limited experience. People find machine learning hard, but he found it easy. It ticked all the boxes. Research that’s intellectually stimulating at the highest level but provides impact and makes money. “You have to think at a high level of where you want to be.” Peter was accepted into the Ph. D. Computer Science program at the University of Toronto, despite only taking two computer science courses during his undergraduate years.
I think a short biography is fitting. As a child, I’ve always been good at math. I never explicitly practiced math or studied. But my interest died down in high school. My lowest mark in high school was in grade eleven math. Even despite this dying interest, I knew I wanted a math major.
I came to UTM, trying to get a computer science degree, and I had to take a course, an introduction to mathematical proofs, for computer science. I fell in love all over again. Every semester onward, I took courses that challenged me. I took MAT257. I took courses in Toronto in my third year, rudely awakened to the reality that life isn’t easy after getting a miserable mark in complex analysis (the name should say enough).
Yet, I am grateful to my professors, for their guidance and their help. This is not an uncommon story. All of us students grew as mathematicians even though many of us won’t pursue math and, more importantly, as people. All I can hope is that those before me felt similarly under the guidance of professors like Peter Fantham, Kumar Murty, and Alfonso Gracia-Saz. Maybe it’s too late to mention it now, but this is my love letter to mathematics. To those who guided me along this beautiful path on which I walk.