Some stories about becoming a mathematician

I have a peculiar little day with neither an A-to-Z, a recap, or a Reading the Comics post to publish. It feels illicit somehow. Well, let me share something which I ran across recently. It’s an e-book published by the American Mathematical Society, Living Proof: Stories of Resilience Along the Mathematical Journey. Editors Allison K Henrich, Emille D Lawrence, Matthew A Pons, and David G Taylor.

It’s a collection of short autobiographical essays from mathematicians. The focus is on the hard part. Getting to an advanced degree implies some a lot of work, much of it intellectually challenging. I felt a great relief every essay I found where someone else struggled with real analysis, or the great obstacle of qualifying exams. And I can take some comfort, thinking back at all the ways I was a bad student, to know how many other people were also bad students in the same ways. (Is there anyone who goes on to a doctoral program in mathematics who learns how to study before about two years into the program?)

There are other challenges. It’s not a surprise that American life, even academic life, is harder if you can’t pass as a heterosexual white male. A number of the writers are women, or black people, or non-heterosexual people. And they’re generous enough to share infuriating experiences. One can admire, for example, Robin Blankenship’s social engineering that convinced her algebraic topology instructor that she did, indeed, understand the material as well as her male peers did. One can feel the horror of the professor saying he had been giving her lousy scores because his impression was that she didn’t know what she was doing even as she gave correct answers. And she wasn’t the only person who struggled against a professor grading from the impression of his students rather than their actual work. There is always subjectivity and judgement in grading, even in mathematics, even in something as logically “pure” as a proof. But that …

I do not remember having a professor grading me in a way that seemed out of line with my work. The grade to expect if the grading were done single-blind, with no information about the identity of the exam-taker. But, then, I’m white and male and anyone looking at me would see someone who looks like he should be a mathematician. I think my presentation is, to be precise, “high school physics teacher who thinks this class just might be ready to see an amazing demonstration about something we call the Conservation of Angular Momentum”. That’s near enough to “mathematician” for most needs. This makes many of these essays, to me, embarrassing eye-openings.

So I think the book’s worth reading. And it is a great number of essays, most of them two or three pages. So it’s one you can pick up and read when you just have a few free minutes.

The Equidistribution of Lattice Shapes of Rings: A Friendly Thesis

I can't tell you how excited I am to see a Princeton math PhD thesis that starts like this.https://t.co/O5Ep59HblE pic.twitter.com/3GAzfThdPI

— Federico Ardila (@FedericoArdila) December 8, 2015

So, you’re never going to read my doctoral thesis. That’s fair enough. Few people are ever going to, and even the parts that got turned into papers or textbooks are going to attract few readers. They’re meant for people interested in a particular problem that I never expected most people to find interesting. I’m at ease with that. I wrote for an audience I expected knew nearly all the relevant background, and that was used to reading professional, research-level mathematics. But I knew that the non-mathematics-PhDs in my life would look at it and say they only understood the word ‘the’ on the title page. Even if they could understand more, they wouldn’t try.

Dr Piper Alexis Harron tried meeting normal folks halfway. She took her research, and the incredible frustrations of doing that research — doctoral research is hard, and is almost as much an endurance test as anything else — and wrote what she first meant to be “a grenade launched at my ex-prison”. This turned into something more exotic. It’s a thesis “written for those who do not feel that they are encouraged to be themselves”, people who “don’t do math the `right way’ but could still greatly contribute to the community”.

The result is certainly the most interesting mathematical academic writing I’ve run across. It’s written on several levels, including one meant for people who have heard of mathematics certainly but don’t partake in the stuff themselves. It’s exciting reading and I hope you give it a try. It may not be to your tastes — I’m wary of words like ‘laysplanation’ — but it isn’t going to be boring. And Dr Harron’s secondary thesis, that mathematicians need to more aggressively welcome people into the fold, is certainly right.

You’re not missing anything by not reading my thesis. Although my thesis defense offered the amusing sidelight that the campus’s albino squirrel got into the building. My father and some of the grad student friends of mine were trying without anything approaching success to catch it. I’m still not sure why they thought the squirrel needed handling by them.

A Summer 2015 Mathematics A To Z: ansatz

Sue Archer at the Doorway Between Worlds blog recently completed an A to Z challenge. I decided to follow her model and challenge and intend to do a little tour of some mathematical terms through the alphabet. My intent is to focus on some that are interesting terms of art that I feel non-mathematicians never hear. Or that they never hear clearly. Indeed, my first example is one I’m not sure I ever heard clearly described.

Ansatz.

I first encountered this term in grad school. I can’t tell you when. I just realized that every couple sessions in differential equations the professor mentioned the ansatz for this problem. By then it felt too late to ask what it was I’d missed. In hindsight I’m not sure the professor ever made it clear. My research suggests the word is still a dialect rather than part of the universal language of mathematicians, and that it isn’t quite precisely defined.

What a mathematician means by the “ansatz” is the collection of ideas that go into solving a problem. This may be an assumption of what the solution should look like. This might be the assumptions of physical or mathematical properties a solution has to have. This might be a listing of properties that a valid solution would have to have. It could be the set of things you judge should be included, or ignored, in constructing a mathematical model of something. In short the ansatz is the set of possibly ad hoc assumptions you have to bring to a topic to make it something answerable. It’s different from the axioms of the field or the postulates for a problem. An axiom or postulate is assumed to be true by definition. The ansatz is a bunch of ideas we suppose are true because they seem likely to bring us to a solution.

An ansatz is good for getting an answer. It doesn’t do anything to verify that the answer means anything, though. The ansatz contains assumptions you the mathematician brought to the problem. You need to argue that the assumptions are reasonable, and reflect the actual problem you’re studying. You also should prove that the answer ultimately derived matches the actual behavior of whatever you were studying. Validating a solution can be the hardest part of mathematics, other than all the other parts of mathematics.