Joseph Ferrar – In Memorium

I learned last month from Facebook that Professor Joseph Ferrar passed away in May of last year. While I do not feel particularly capable to speak of him, I feel compelled to write a few words about my experience with him and the impact he had on my life.

Unfortunately, I was not blessed to have a particularly strong relationship with him, and my memory of certain events are not as clear as I wish as we last spoke more than ten years ago. He was not my thesis advisor or any officially distinguished professor for me beyond teaching two algebra courses. I know very little of his professional career. Some small searching led to a book he edited and his MathSciNet page. He was department chair at Ohio State University for a time, as he mentioned, and perhaps the most popular work mentioning him is the Acknowledgments of William Dunham’s Journey through Genius (emphasis mine):

To the Ohio State University, and particularly to its Department of Mathematics, go my sincere thanks for their warm hospitality while, as a visiting faculty member in their midst, I was writing this book. I shall always appreciate the kindness of Department Chair Joseph Ferrar and of the Leitzels — Joan and Jim — who were unfailingly helpful and supportive during my two-year visit.

My only experience with him was in two algebra courses and a reading I did with him for a quarter on Lie algebras, reading through portions of James Humphreys’s Introduction to Lie Algebras and Representation Theory.

He began the course with the Peano axioms of arithmetic, which despite my third time seeing was fresh because he started at the beginning with the successor function. Prior to this, I only saw formulations starting with ring-theoretic axioms. He introduced the idea of formal differences in a homework assignment, which I would later study at greater length as the Grothendieck group. I remember asking him about polynomial rings and matrices and him answering that it only worked if the matrices commute. An important concept and one that I would consider again in the future. Despite not remembering much, I think back on that course fondly. I had a difficult time with Galois theory, so I have little to say about that second class, with one exception. Late in the second course, I attended a talk on Hopf algebras and had some difficulty understanding it. He gave a lecture soon after on the topic. While I was more inclined towards abstract algebra in the first place, it was his courses that made me want to become an algebraist.

After a random question about the Lie group E8 prompted by a housemate, he offered to do a reading with me on Lie algebras. I agreed, and we met weekly. In retrospect, I was a terrible student. I did the readings last minute, and had offered very little in the way of curiosity or insight. He was a patient teacher and despite the fact that there were no stake in doing these readings, he gave me one-on-one lectures on the subject.

I asked him for a letter of recommendation and advice on my cover letter. I had written that I wanted to become an algebraist, but he told me gently that rather than insisting on a particular field of research from the start, I should keep my mind open to possibilities. There are fields that are adjacent to algebra, and going to a school like the University of Oregon might have topics like functional analysis which had plenty of similarities to algebra. His words turned out to be prophetic, as I ended up in operator algebras, a branch of functional analysis.

There were occasions during graduate school and after that I wanted to contact him again and tell him what an impact he had on me. Part of me was procrastinating, but I think I didn’t want to, because in my mind, I didn’t really accomplish anything of significance. But while I was pitying myself, I lost the chance to express my gratitude to him.

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Reflections on the Last Month

So I managed to blog every day in November with varying degrees of quality. So what did I learn? First, while writing every day might be a good idea, blogging every does not seem to be great. I felt quite burnt out by the end as evidenced by some of my latter posts. Second, while blogging every day does force me to avoid procrastination through excess preparation, it also means that ideas that require some more time to develop don’t get that. I don’t have a solution to resolve this tension, but it’s something I should think about. Third, I think I want to transition to talking about mathematics strictly on this blog. This was difficult to attempt in the previous month, since writing every day was difficult even without restricting the subject matter. It seems that the readers agree, since more views came from mathematical posts.

So what’s in store for this blog? I can’t make any promises, but here’s some plans.

  1. Progress on research. I still haven’t published my approximate diagonalization paper. I have some hopes of doing some further research along these lines, and as I get some insight I hope to write about it.
  2. Math education (broadly defined). I hope to (at least privately) write some lecture notes for possible future use or for hypothetical classes, since I am currently unemployed. I may also discuss either actual math education research or some popular mathematical videos or other media.
  3. Math in Plain English. I might try revisiting my own attempts at writing about mathematics for people without a technical bent. My original motivation for this series is gone, but it might be worthwhile, nonetheless.
  4. Mathematical reading. I’d like to read some mathematical texts, including some classics. I have a large backlog, so hopefully I’ll develop some thoughts while reading and might make some insights.
  5. The use of categories in analysis. I made a small post about this, but I think there’s some untapped potential here that I want to understand. Of course, there’s a lot that already done which I need to get caught up on.

Thank you for reading this past month. If all goes well, I hope to see you again soon.

Starting in the Middle

This was a challenge. Unintentional, of course. Since the creators of television shows made the assumption that you know things you don’t, it’s up to you to try to figure things out. And this is exciting, because every interaction and every plot point becomes a clue for understanding the pieces that you missed. Some shows make it easier than others. But it’s never completely trivial.

It reminds me of life actually. We are thrown in the middle of things without a clear context for understanding what’s going on. We have to make assumptions and try to deduce the nature of things despite the fact that everyone else seems to know something that we do not. Slowly as the story develops, we begin to understand more of what’s going on and what we missed out on. We might have to revise our original thoughts. So this experience of mine was like a metaphor for life.

Due to the nature of television and mistaking unlabeled disks, I’ve had the experience of starting a television show from the middle at many points in time. Despite this always being a rather strange mistake, I’ve grown to appreciate these experience. (Though I don’t do so intentionally.) There’s something exciting about not knowing the context about what’s going on and having to puzzle it out.

The Difficulty in Talking about AI

Automation and AI are important topics to discuss. One major aspect is what we will do about the increasing unemployment once jobs currently done by humans are increasingly done by automation. Of course, this is a discussion that seems largely unnoticed by the people who should be most concerned.

But among techno-optimists and luddites alike, the main challenge about these conversations is that it’s not clear what the technology is capable of, what it will be capable in any given timeframe, and if there are any limits to it at all. Both parties seem to make the assumption that anything is possible. And that given time, any task can be automated.

Now, I don’t know much about the subject, but I have read a little about the progress of this technology. One of the things I noticed is the fact that the computers have a very narrow understanding about many things. The detection of disease in plants is determined by the color and texture of the leaves. A self-driving car determines the lanes of a road by reducing a photo to a monochromatic image that has a difficult time with shadows. I became aware of the fact that computers have their own set of limitations.

More generally, it seems that since algorithms require some end, open-ended projects would be limited. For example, if you want an algorithm to record what’s interesting during a sports game, then you would need an a priori way of determining what’s “interesting” (which could be conducted with a separate algorithm), but it wouldn’t understand something unexpected if it’s not within the parameters set forth.

Or would it? I don’t know. The trouble with talking about automation is that the details matter, and it’s not clear how much I or the next person knows about the matter.

Unappreciated Exposition

During my time in China, I often had a conversation with various people about the completion of the Elliott classification program for nuclear C*-algebras. Often, the main question was: Who’s going to write the book on the topic? The general answer seemed to be no one. Despite the usefulness of a good exposition on the subject, it seems that there is no incentive for writing such a book. And there is no incentive because no math department will consider such a book to be a worthwhile research accomplishment. And my chain of whys comes to a sudden halt here. I don’t know why expositionary work wouldn’t be rewarded (or even if this is always true). The effort to read the proofs and to write a text for a slightly more general audience (by which I mean, more than the handful of people responsible for the proof) is not worth the opportunity cost of conducting original research.

It may be the case that this is a particular problem in my field. I recall in my undergraduate days there was an attempt to write an exposition of the classification of finite simple groups (a much more difficult project, I presume). This may have more to do with where C*-algebras are right now, rather than being a permanent feature. Honestly, I don’t know, but it is disappointing to me that exposition of “known” results is less significant than even marginal original research.

Discovery vs. Invention: A Confusion

I have a strange relationship with philosophy, which I think is the case for most people who have an amateur interest in the subject. Many of the topics and ideas seem fascinating, but many other ideas seem like pointless semantics. In my case, most of the big topics in the philosophy of math are uninteresting. In particular, is math invented or discovered? Or in other words, are mathematical concepts a type of platonic ideal, or are they a construct of human imagination?

I think many mathematicians (particularly pure mathematicians) have to some degree a belief in mathematical concepts as platonic ideals. People who do mathematics for its own sake, myself included, don’t often feel that they are playing an elaborate man-made game, but that there is something organic (for lack of a better word) about the study of mathematics. The emergence of deeper patterns is the main draw for studying mathematics for me.

I suppose it’s the mathematician nature in me that always goes back to the question, “What is existence?” whenever I hear this question. And somehow, trying to determine what existence is prior to what seems like a hopelessly futile task. Regardless of whether material existence is prior to abstract concepts or vice versa, it doesn’t affect how we would go about studying math. It seems to me that these philosophical ideas are two distinct systems of axioms which have different theorems but are the same for all tangible purposes.

But perhaps the main point is that by expressing different ideas for the ultimate questions about reality, we put into context our own beliefs about the matter, and we are made aware of our unconscious assumptions about reality. And we are confronted with the idea that our beliefs aren’t at all universally held.

Thankful

It’s been a difficult year for me, and it’s not looking good for the next year. Nonetheless, in the spirit of the holiday, I am listing ten things that I’m grateful for.

  1. I am fortunate to be able to stay with my family this holiday season. It’s the first time I’ve been home for Thanksgiving for nearly a decade.
  2. I had the great fortune to be raised in a loving home. My parents encouraged me to study hard, yet gave me the freedom to be who I wanted to be.
  3. I had a great time in the summer in Shanghai, despite the heat. I met with so many great people and had lots of great conversations over dinners.
  4. I met many of my relatives this year, many of whom it’s difficult to meet, since they live so far away.
  5. I came back from my postdoc in Shanghai safely.
  6. I made some friends during my time in Shanghai.
  7. I live in relative peace and safety without risk of violence or crime.
  8. Access to the Internet makes me feel connected to the rest of the world.
  9. There are many things worth studying and exploring.
  10. I live in an active democracy, where there are checks on the powers of every level of government with a citizenry ready to defend its country in a peaceful manner.

I hope you all have had a great day.