An abstract algebraic approach to homogeneous linear differential equations with constant coefficients, Part I

Section 0: Introduction

Upon learning about the Weyl algebra and the notion of D-modules, the algebraically minded person may have the thought, “how much calculus and differential equation theory can I do with this?” This mild curiosity grew to a weird obsession over the years (despite not actually doing much mathematically with it) due to a compelling case of what it says about homogeneous linear ordinary differential equations with constant coefficients.

Much of my mathematical interest I would say lies at the intersection of algebra and analysis. Unfortunately, as in the case here, I find myself separating things out between the two and, in this case, showing that certain seemingly analytic results are wholly algebraic. But I think that there’s a significance to that idea to know when in analysis, we are engaging in analytic methods and when we are engaging in algebraic methods (or when we are using both in a fascinating way). Topology is left out here, not intentionally. Sorry topologists, maybe one day I’ll talk about your stuff.

Recall a standard argument used with homogeneous second order linear ordinary differential equations with constant coefficients. We assume that solutions of the form y = e^{ax} exist and proceeding with chain rule, obtain a polynomial called the characteristic polynomial (for linear algebraic reasons). Roots of the polynomial, provided that they are distinct, provide two linear independent solutions, whose linear span forms a two-dimensional subspace of the solution space, which in turn means by the uniqueness theorem that we have the whole solution space.

This is logically precise proof, and I even taught it with this method, but I have issues with it. The big one is the assumption of solutions (as smarter people let me know is called an ansatz) of a particular form bugs me. A proof that assumes less is in my opinion cleaner than a proof makes seemingly magical assumptions (see that one xkcd comic). Now people will say that this assumption is magical, but completely reasonable. Which maybe? But it does demand a non-rigorous “intuition” that personally I think should be avoided when possible. My main issue is this, the assumption of the solution is only an excuse to obtain the polynomial in a precise manner. If we think about things algebraically (the more used path here is the linear algebraic path for good reason), then we see that the polynomial is either the characteristic polynomial of a matrix or a polynomial of an operator (the approach advocated here).

Now let’s talk about an alternative. Somehow, I have the psychological issue of calling this a proof, even though my claim is that in the end it is a proof. By treating the differential equation as a polynomial in the differential operator, we can factor it. The factors have correspond to their own first-order differential equations, which are easily solvable. Then we can combine the solutions by taking linear combinations of the solutions. Use uniqueness as above and we’re done.

There are a few issues here too. One, are we allowed to treat differential equations as polynomials of an operator? (Yes, we can.) Two, are we allowed to factor? What’s multiplication here? (Also, yes. Multiplication is composition of operators which is why the second derivative is the “square” of the operator.) This seems like abstract nonsense? (Please, I promised myself not to get angry today.) This doesn’t seem super precise? (That’s not a question, but you’re right.) Why do we take sums of solutions when we’re looking at multiplicative factors? (Um… Do you know ring theory? No? … Magic?)

As I said before, I want to put that latter argument on firmer mathematical footing. But for that I will need a little bit of (supposedly elementary) ring theory. I should emphasize here that I’m not trying to say you need ring theory to do this. If you know ring theory fairly well, this theory becomes a nice application of algebraic ideas, showing that this rather easy result of differential equations is in the domain of algebra with the transcendental functions only showing up at the end.

Section 1: Definitions and Motivation

The first Weyl algebra is defined as the quotient of the free algebra (over the complex numbers) generated by two elements M and D by the ideal generated by DM - MD - 1. As a first encounter, this probably seems a bit strange. What does this have to do calculus? In this case, D represents the differentiation operator (we are currently only working in single-variable calculus) and M represents the multiplication operator by the function f(x) = x. The relation arises from the product rule and noticing that if acting on some suitable function space, then

DM(f(x)) = D(xf(x)) = xf'(x) + f(x) = (MD + 1)(f(x)).

(Some people will appropriately be angry at me for my notation conflating a function and its evaluation. Please forgive me, I did it for clarity and brevity.) The (first) Weyl algebra is an interesting algebra, introduced to me because it is an example of a simple ring which isn’t a matrix ring. But we will be unconcerned with these aspects (at least for now).

As noted in our motivation above, we are concerned about the actions of the Weyl algebra or in other words (left) modules over the Weyl algebra. Considering these ideals allows us to follow in the footsteps of D-module theory, but with far less substantial concerns. There are many natural candidates for a preferred module: the polynomials (maybe too small), the formal power series (maybe too big), and the set of infinitely differentiable functions with domain of real numbers (maybe too much analysis is involved). Our approach is to try to be as vague as possible about what module we are considering and make assumptions about what we need on the fly. Note that despite the fact that all our examples are rings, not every interesting example of a function space is a ring, and we will not make the assumption that our module is a ring. It is, I think, safe to say that most useful function spaces are vector spaces since scalar multiples of “nice” functions are also similarly “nice.” We may make the tacit assumption that the module under consideration is a vector space.

Now let a homogeneous linear ordinary differential equation with constant coefficients be given by:

y^{n} + a_{n-1} y^{n-1} + ... + a_1 y' + y = 0.

Let P(X) = X^n + a_{n-1}X^{n-1} + ... a_1 X + a_0. Then the differential equation can be written in the form: p(D)y = 0, or to use more module language, solutions of the differential equations are the p(D)torsion elements of the module. Because I can’t seem to find notation for what I want (please comment with a reference if you have one!), given an element r of a ring R and a left R-module N, I will refer to the subgroup (if R is commutative, then it is a submodule) of r-torsion elements by \mathrm{Tor}(r), or more set-theoretically:

\mathrm{Tor}(r) = \{ n \in N \colon r.n = 0 \}

I claim that understanding the structure of the subgroup \mathrm{Tor}(p(D)) will give us the result about solutions to the associated differential equation. But first we need some basic results about these torsion subgroups.

Section 2: r-Torsion Subgroups

Lemma 2.1. Let R be a ring, N a left R-module. Let r and s be elements of R such that r and s commute, i.e. rs = sr. Then

\mathrm{Tor}(r) + \mathrm{Tor}(s) \subseteq \mathrm{Tor}(rs).

The proof is left as an exercise to the reader. (It’s easy; use commutativity.)

Lemma 2.2. Let R be a ring, N a left R-module. Let r and s be elements of R such that the left ideal generated by r and s is R,  i.e. Rr + Rs = R. Then

\mathrm{Tor}(r) \cap \mathrm{Tor}(s) = \{0\}.

Consequently, the sum \mathrm{Tor}(r) + \mathrm{Tor}(s) is an internal direct sum.

Proof: Let a and b be elements of R such that ar + bs = 1. Let n \in \mathrm{Tor}(r) \cap \mathrm{Tor}(s). So r.n = s.n = 0. So n = (ar+bs).n = a(r.n) + b(s.n) = 0. Or n = 0. Therefore, \mathrm{Tor}(r) \cap \mathrm{Tor}(s) = \{0\}.

The condition on the following lemma is a bit awkward.

Lemma 2.3. Let R be a ring. Let r and s be elements of R such r and s commute. If there exist elements a and b of R that commute with both r and b such that ar + bs = 1, then

\mathrm{Tor}(rs) = \mathrm{Tor}(r) \oplus \mathrm{Tor}(s).

Proof: From the previous lemmata,

\mathrm{Tor}(r) + \mathrm{Tor}(s) \subseteq \mathrm{Tor}(rs),

and the sum in the right-hand side is an internal direct sum. It suffices to show

\mathrm{Tor}(rs) \subseteq \mathrm{Tor}(r) \oplus \mathrm{Tor}(s).

Let a and b be elements of R such that ar + bs = 1. Let n \in \mathrm{Tor}(rs). Let n_1 = (bs).n and n_2 = (ar).n. Notice that

(bs).n + (ar).n = (ar + bs).n = 1.n = n.

And since r.n_1 = r(bs).n = b(rs).n = b.0 = 0 and s.n_2 = s(ar.n) = a(rs).n = a.0 by commutativity, n_1 \in \mathrm{Tor}(r) and n_2 \in \mathrm{Tor}(s).

The previous lemma isn’t quite pretty, but there are some things to point out in its defense. One, the condition is not as unreasonable in light of the previous lemma using a similar formula ar + bs = 1. The commutativity conditions will not play a role in our particular case, since polynomials in D form a commutative subring in which we are operating. The proof includes a moment where a formula is seemingly pulled out of a hat, but in this case, the main thrust of the decomposition comes from writing 1 as a combination of sorts. From there the decomposition is clearer as the part the will be annihilated by r is the factor missing r with a similar situation for s.

Barring this one hiccup, the lemmata we proved hopefully seems like quite reasonable general algebraic results that might be seen in an undergraduate algebra course. With these tools, we return our attention to the specific torsion subgroup related to our differential equation.

Section 3: The Structure of the Subgroup Associated to a Differential Equation

We saw in Section 1 that the solution space of a homogeneous linear ordinary differential equation with constant coefficients is a subgroup Tor(p(D)), where p(X) is the polynomial given by the differential equation.

From Lemma 2.3, we have the following result:

Lemma 3.1. Let q(X) and r(X) be two coprime polynomials. Then

\mathrm{Tor}(q(D)r(D)) = \mathrm{Tor}(q(D)) \oplus \mathrm{Tor}(r(D)).

Consequently, if p(X) = (X - \lambda_1)^{m_1}(X - \lambda_2)^{m_2} ... (X - \lambda_k)^{m_k}, then

\mathrm{Tor}(p(D)) = \mathrm{Tor}((D-\lambda_1)^{m_1}) \oplus \mathrm{Tor}((D-\lambda_2)^{m_2}) \oplus ... \oplus \mathrm{Tor}((D - \lambda_k)^{m_k}).

To simplify matters we can work in the subalgebra generated by D, consisting of polynomials in D. This makes the commutativity conditions vacuous, since we are in a commutative algebra. Since polynomials with complex coefficients form a Euclidean domain, when q(X) and r(X) are coprime, there exist polynomials a(X) and b(X) such that a(X)q(X) + b(X)r(X) = 1. So the ideal generated by q(D) and r(D) is the whole subalgebra.

When P(X) has no multiple roots, we see that

\mathrm{Tor}(p(D)) = \mathrm{Tor}(D-\lambda_1) \oplus \mathrm{Tor}(D-\lambda_2) \oplus ... \oplus \mathrm{Tor}(D - \lambda_k).

So the solution space of the differential equation is given by the direct sum of solutions to first-order differential equations. Taking a look at the summands and returning to analysis, we see that \mathrm{Tor}(D-\lambda_i) is the solution space of the differential equation y' = \lambda_i y, which consist of scalar multiples of the exponential e^{\lambda_i x}. So we conclude with a proof of the fact stated at the top: the general solution of a homogeneous linear ordinary differential equation with constant coefficients is the linear combinations of exponentials whose parameters are given by roots of a polynomial.

In my opinion, this proof and its main ideas make the result clear. One side of the differential equation literally is a polynomial of the differential operator, and the solutions to its factors are summands of the general solution space.

The algebraist may find one (more) complaint, which is this last bit of argument relied on us to know about the exponential function. The algebraist may hope that we can axiomatize somehow the exponential function as a generator of \mathrm{Tor}(D-1). The main issue seems to be how we can make connections between the single exponential (i.e. e^x) and all the other closely related exponential functions e^{\lambda x}. Without a notion of (pre-)composition, I don’t know how this can be done. The alternative, defining infinitely many, a priori unrelated exponentials seems bad. Any thoughts on this matter would be appreciated.

Finally, both in our discussion of the differential equation theory method and the module perspective, we did not address the case with higher multiplicity. I would like to return to that subject, but it requires some more tools and I haven’t quite fleshed out the motivation. I will leave this to a Part II, if I can muster up the strength to write it.

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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.

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.

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.