I graduated from the undergraduate program in mathematics with a bachelor of science with a specialized honours in mathematics and a minor in computer science. Since then, I've visited the University of Guelph web site, and there is no mention of former faculty members anywhere. Consequently, I thought it would be nice to mention some of them here.
Bruno Ernest (Ernie) Frejer
In my first semester, I needed to pick a fifth course, and my peers I was living with (Jim Morey and Thomas Cook) suggested I take a second-year mathematics course: 63-200 Set Theory. This was an amazing course taught by Ernie, and he taught it very well. This course introduced me to some very interesting mathematical concepts, and is one of the two reasons I switched from chemical physics to mathematics. In addition to the axiomatic method, I was always entranced by the observation that the ordered pair (x, y) can be represented by the set {{x}, {x, y}}, and it wasn't until over a decade later that this was a critical problem with Russell and Whitehead's Principia Mathematica where they did not know that ordered pairs can be represented as sets, and thus introduced what is now called ramified type theory to compensate. Ernie was a founding member of the department when it was started in 1965. In the CMA Notes of September 2002, it states that "He was a meticulous teacher, always well-organized, clear and fair. He had the distinction of teaching the greatest variety of courses in the department, from first year algebra to fourth year tensor analysis." I believe this is a reasonable summary.
Ernie was a quiet but excitable instructor, and I enjoyed both his lectures and talking to him outside his lectures. He had this peculiar fascination with Edgar Cayce, which is I guess no better or worse than having a peculiar fascination with Jesus. I recall that I was retiring from the University of Guelph close to when I was finishing my undergraduate studies, and he gave me a number of his older text books as well as his books on Edgar Cayce.
I really have found very little other information about Ernie: birth year, place of birth, etc. From an older undergraduate calendar from Guelph, I have that even in 1989, his title was Professor Emeritus and he had an undergraduate B.Sc. from McMaster and a M.Sc. and M.Phil from Waterloo. I haven't even found an obituary. There does not appear to be anyone else in Ontario with the last name Frejer. If anyone finds this and has some more information about this remarkable person, please contact me.
I did find my course notes from 63-200 Set Theory. I still have my course outline:
63-200 Set Theory
COURSE OUTLINE
Fall Semester 1990
Instructor: B.E. Frejer, Room 514, MacNaughton
Text: Theory and problems of Set Theory and Related Topics by S. Lipschutz. The Main purpose of this book is to give the student additional examples and problems. A complete set of accurate class notes will be especially useful and necessary for success in this course.
Course Content: This course by its very nature has to be theoretical. We will investigate certain concepts, notations and terminologies which are basic to most all fields of mathematics. Different types of proofs will be used; both deductive and inductive; direct and indirect.
Problem Set I will include: sets of real numbers, Algebra of Sets. Generalized Operations and Partitions, Principle of Mathematical Induction. Ordered Pairs and the Cartesian Product.
Problem Set II will include: Properties and Theorems concerning Relations, functions, Images and Inverse Images under a Mapping.
Problem Set III will include: Equivalent Sets. Cardinal Arithmetic, Equivalence Relations.
Problem Set IV will include: Partially, Totally and Well Ordered Sets. Axiom of Choice and Zorn's Lemma.
Tests: There will be three 50 minute tests in class on the following dates: October 4, November 1 and November 26.
Approximately 70% of the problems for these tests will be identical to a) problems contained in the problem sets or to b) examples or theorems presented in class. Each test is worth 16% and the final exam is worth 52%.
Note: If for any reason a student misses a test the 16 % will be added to the final exam.
Here is his Grade 13 year-book photograph from the St. Catharines Collegiate Institute around 1956 and write up:
Here is one of the few followers of Grade 13 German. Ernie is going to take Math and Physics at McMaster, and in a few years we may see him in a lab next to Mr. Doolittle.
The reference confuses me, as "Mr. Doolittle" is a likely reference to a character in the play Pygmalion, but I would have thought they were actually referring to Dr. Dolittle.
In Chess Life, Volume VIII, No. 13 from Friday, March 5, 1954 (see here), under the article titled "Toronto Hobby Show Features Chess, Match Bondarevsky-Anderson by Cable" with the relevant text
The Hobby Show Tournament was an 11-player round robin won by Joseph Pochmurskyj with 9-1, while tied for second with 8-2 each were E. Freier, G. McLeod, and G. Coyne... In the two Junior events, Peter Bates of North Toronto Collegiate was first in the boys' event with Ernest Freier second and Walter Boychuk third, both from St. Catharines Collegiate.
Here is another photograph I found in At Guelph, Volume 34, Number 2, January 10, 1990.
Silver Anniversary
Twenty employees who have worked on campus for 25 years were honored recently. Seated ... . Standing are ... Ernie Frejer, Mathematics and Statistics.
Ernie is the fourth person standing from the left.
John A.R. Holbrook
The coolest and most relaxed but intelligent professor in the department. On the first day of undergraduate studies, he introduced himself as "John" and insisted we address him by his given name. He gave us his home phone number and his home address, and students, in times of desperation, sometimes did make use of this. His gray beard and white hair and relaxed attitude would make him ideal as either Gandalf or Dumbledore. I remember on the first day of undergraduate studies, he covered Reimann sums, discrete dynamical systems, statistical distributions and delta-epsilon. We did so much delta-epsilon in that course that it became second nature. I had one really fascinating assignment related to Taylor series, specifically: Show that the limit of the nth derivative of exp(-1/x^2) as x approaches zero is zero. A proof-by-induction inside a proof-by-induction. Figure 1 shows a photograph from the University of Guelph News Bulletin of October 5, 1972. Also from that bulletin:
"Dr. Holbrook is a native of Ontario with family roots in the Guelph area and in Ottawa. He studied mathematics at Queen's University and obtained his doctorate in that field at the California Institute of Technology in 1965. He comes to Guelph after a number of years on the faculty of Caltech and the University of California." John did his own tutorials, and his tutorials were essentially another class, and that was okay.
Figure 1. John A.R. Holbrook.
Barbara Lehman
To be completed. Figure 2 shows a photograph from the University of Guelph News Bulletin of October 5, 1972. "Dr. Lehman completed her undergraduate studies at Vassar College, majoring in Philosophy. She obtained the M.A. in Mathematics from Western Washington State College, and completed her graduate studies at Iowa State University in 1972." I took 63-436 Point-set Topology with her, and sadly, it seems that was the last time that course was taught. I really found that course interesting, for it was the cumulation of a number of courses I had previously taken, including John Holbrook's 66 111 The Calculus, followed by Real Analysis I with Rex Mercer and Real Analysis II. She was using overheads, and when she retired, I asked her for the overheads of her courses. Over the next few years, I went through many of those notes. I really enjoyed her course. It's so fascinating to be able to define open sets, the images of open sets, the pre-images of open sets, etc. Defining openness in ways other than how I previously did was such a fascinating idea, and to see the consequences: what are continuous functions under different definitions of openness. In the course, the text book was James R. Munkress's "Topology: A First Course." Looking through that book, I remember ending with Chapter 5, the Tychonoff Theorem.
Figure 2. Barbara Lehman.
Murray J. Code
Prof. Code is a really cool instructor when he's teaching a course more related to philosophy. I still remember taking his two courses on 63 332 Modern Geometry and 63 333 Differential Geometry. The first was the most interesting, and I still have and enjoy reading the recommended but not required textbook on projective geometry. He was an associate professor and he got his B.Sc. from Queens University, a second B.Sc. from London University, and an M.Sc. from Queens again.
A. McD. (Rex) Mercer
Prof. Mercer taught me Real Analysis I. That was a cool course, and the course text book, "Principles of Mathematical Analysis" by Rudin was both deep and the most expensive book I had purchased up to that time: I recall calculating it was a quarter a page. The course, however, was fascinating with the concepts of open and separated sets. Rex was a very rigorous instructor, and yet he was always friendly and had this beaming smile.
Once again, I cannot find any photographs of Rex. Do all these people disappear into history without any formal acknowledgement?
Pal Fischer
I took a course on measure theory with Pal, and he is such an amazingly friendly person. His office was at the end of the hallway close to the computer lab on the top floor of McNaughton. Pal and John were two great professors at Guelph. A number of his students, I remember, used to think of him as a large teddy bear. He also first introduced me to the APL programming language. He was always enthusiastic about the most theoretical mathematics he was working on, and I took a course from him on measure theory.
Hosh Pesotan
In my 3rd term (still second year) I took Prof. Pesotan's 3rd-year course on Modern Algebra. It, too, was a memorable course, and he was a fascinating instructor. I took that course with Jim Morey and Thomas Cook. I never took another course with Prof. Pesotan, but that one course was sufficient to make him memorable. He was always happy in class and carefully worked through the ideas, definitions and theorems as well as consequences. I recall his black-board style was very clean and well practiced. Because I was a second-year student taking a third-year class, I didn't have anyone to study with, so most of my time was learning the material on my own.
I recall that Israel Nathan Herstein's text book "Abstract Algebra" had the occasionally witticism, something you don't find in every text book. It's one of the few text books I still read and cherish. I understand now that it is a much simplified version of his more advanced "Topics in Algebra" book, but honestly, for a second-year student, "Abstract Algebra," together with Prof. Pesotan's discussion, was a good introduction.
I found my course notes for this course: 63-313. I like the comment I wrote about office hours: TUE 1410-1510 THU 1400-1600 DON'T BOTHER ON FRI. I'm going through my notes, and after all these decades, I still remember some of them. I really enjoyed ring theory, although I didn't appreciate it until later. The topics in my course notes include:
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group theory,
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cyclic groups,
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subgroups,
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cosets,
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group homomorphisms,
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kernel of a group homeomorphism,
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normal subgroups,
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ring theory,
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subrings,
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ideals,
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ring homomorphisms,
Throughout the term, he had a number of handouts. Interestingly enough, today, these would all appear online.
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Function (fct), map or mappings.
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Concept of algebraic systems
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Integers
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Mathematical Induction
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We begin a study in the abstract of an algebraic system known as a "group".
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Cyclic groups.
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Group homomorphism.
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Defn of a ring.
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Multiplication rules in Rings.
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Ring homomorphisms.
It seems that after one of the term tests, someone (probably Jim Morey) "wat did you get on the test?" "19 1/2" "that's not an A! I'm disappointed. (I beat you)" After this, I wrote "COMPLEX?", likely referring to the 3rd-year complex analysis course we were taking simultaneously, and he wrote "M.Y.O.B.". I don't have my commentary on the day-to-day notes, because at the end of each week, I took all my course notes and rewrote them in a formal way, and then discarded the notes I took in class. I found this useful, as I found mistakes and it gave me a chance to focus again on the material. To identify my formal notes, I always drew some kind of flower in the top right-hand corner of the page, and carefully numbered the pages.
Eleanor Chin-Hwa Chu
She passed away? She was so young. I remember her teaching me my first course on numerical methods. She was so enthusiastic about the material.
She was awarded full professorship in July, 2001.
George Leibbrandt
Well, as George was a founding member of the Perimeter Institute, I suspect that most people know George or of George. He was a careful and meticulous instructor. I had two courses with him: The first was 63-317 Special Functions, which was a good course. I remember and still have a project I did on the Mathieu functions. The second was a course on tensor analysis. That course scared me. On the first day of class, there were about fourteen students. By the end of the second week, that had reduced to three: me, Karn Kallio and a graduate student. One day, George asked if we understood Maxwell's equations. The grad student and Karn nodded, so I was told to go study the equations that evening. On another day, he complained that the Registrar's Office doesn't like it if the class average is too high, so I thought: "Okay, I'm doomed. Karn will get his usual 100, the graduate student will get 80, and to get a 75 average, I will get 55." Fortunately, that didn't happen.
As an aside, I wonder what happened to Karn. He was quite the inspiration both in school and life, but also the discipline of never using an elevator. He lived on the twelfth floor, and he did not once use the elevator. Since that year, I never used the elevator again unless it was incredibly high up or if I was with someone who needed to use one. He went on to do a masters at Simon Fraser titled "Static Potential Between Adjoint "Quarks" on Coarse Lattices in SU(2) Gauge Theory."
Dave Fieldhouse
He was friendly and I did take two courses from him: the theory of groups and the theory of rings. These were, however, at a much more intermediate level than I would have liked as Guelph really didn't have that many students interested in pure mathematics, so the classes had to appeal to a larger group of students, and this would be difficult if it went too far into depth. He was always happy to discuss topics outside of class. He did teach me linear algebra, and while I really enjoyed the course, it did teach me to never use multiple choice on any final examination. It was a funny question: What is the determinant of this 5 x 5 matrix: a) 12 b) -12 c) 18 d) -18 e) none of the above. I guess I got that question correct, as I did get 100 in that course. Still, if you did everything else correct and made one small mistake, you'd get zero.
Who else?
I wish I remember who taught me microbiology. I'm embarrassed I don't remember his name. The course was as much an introduciton to immunology as it was to microbiology, although the two are linked, and I happened to have purchased Kuby's book on Immunology to get a better understanding of the relevant material. That is where I learned how the immune system works, and what I've learned today is every single anti-vaxxer can more or less not be convinced of the nonsense of their bullshit, so I don't try to talk to them anymore; I only describe what I know at a very high level to anyone who is wondering whether or not the bullshit the anti-vaxxer is spewing makes any sense (it doesn't).
I also wish I remembered who taught me a course on non-parametric statistics. That was an incredibly important course, and very worth while. I think it was Prof. Malek, but after almost 30 years, I don't remember.
It seems that so many courses I took at Guelph were removed from the calendar the year I left. I guess I got lucky.
Who outside of mathematics?
Most of the individuals above are associated with the Department of Mathematics and Statistics, but specifically mathematics. I'll add other very cool faculty members, including J.J. Hubert and professors in CS.
Funny stories
Accents are fun. In one case, an instructor said "ress-si-prok-al" when pronouncing the word reciprocal, and so I just assumed that what the instructor said was the name of the operation in question. It was not until the start of the second month that the instructor said it while writing some letter raised to the power -1 that it dawned on me; I felt really stupid. I also understood now that sometimes I need to take care when speaking foreign languages. I still accidentally pronounce "Hamburg" as "Hamberg," and for someone who doesn't speak German, they sound essentially the same, but the real name means the fortress of Hamma, while the other means the mountain of Hamma. It would be no different if someone referred to "Oxfort," it may take a second or two to realize the person meant "Oxford."
Another instructor had a habit of pronouncing "fuck-tor-ee-al" when pronouncing factorial. The course I took from this instructor did not use factorials, but this is a story passed on to me by a number of students taking this instructor's first-year course. Oh yes, one would also "fuck-tor-ize" a polynomial.
Mind you, my own story about my accent is humorous, as I have a standard Canadian accent, although I use a number of English expressions from Monty Python and Blackadder. A graduate came back to campus after almost a decade and we went for lunch at the Grad House. He expressed surprise that I had lost my accent, and when I inquired, he distinctly remembered me having a strong British accent, in addition to using British idioms and expressions.