Math year one. Classes

I’ve started a Master’s degree in Mathematics. This post is a summary of the classes I’ve taken in the prequel year. The 2020 / 2021 session will be “the master proper”. There’s a higher level post on my personal blog.

Galois Theory

This was my greatest fear and my biggest challenge. I had never seen abstract algebra at the university level, and though linear algebra is a sibling, there’s just so much modern algebra to absorb. This subject was offered to junior and senior undergraduates that had taken a year of Algebraic Structures, where they’d have learned about rings, groups, fields and the like.

The first month was a quick review of rings and ideals which had me playing catch-up. I found ideals a very interesting mathematical object. A formalization of the properties of multiples (as in, the addition of multiples of 3 is a multiple of 3, and the multiplication of a multiple of 3 times any number is also a multiple of 3.) And an analog, for rings, of the concept of a Normal Subgroup.

Here’s my informal summary of Galois Theory: we want to see whether a given polynomial equation with integer coefficients can be solved algebraically (with sums, products and roots.)
The simplest equations can be solved with rational numbers:
for example \( 4x + 5 = 0\) solves to \( x=-5/4\).
It’s natural to start with the rational numbers, then. Next, we add some extra numbers to enable us to solve more equations. For example, we could add a certain square root. This would allow us to solve some quadratic equations.
One key is what we mean by “add some extra numbers”. This is called a field extension. If we add, say, \( \sqrt{2}\), to the rationals, we expect the rules of arithmetic to remain the same. Indeed, the definition of field extension ensures the rules hold.
Here’s the key insight: field extensions generated by adding roots have certain symmetry properties.

Symmetry is studied with groups, in particular the Galois group: transformations that map a field extension onto itself, while preserving arithmetic operations.
An example: in order to solve a certain quadratic equation, we have added \( \sqrt{2}\) to the rationals. It turns out that if \( a + b \sqrt{2}\) is a solution of our equation, then so is \( a - b \sqrt{2}\) (for equations with real coefficients). The mapping \( a+b\sqrt{2} \to a-b\sqrt{2}\) is a transformation that maps the field extension to itself, and preserves algebraic operations. This mapping, then, belongs to the Galois Group of our equation.
Once he created groups, Galois saw that the group of symmetries of fields extended by adding roots always have sub-symmetries (normal subgroups.) And that some equations of degree 5 or higher cannot have any such sub-symmetries, hence they cannot be solved by radicals.

What a work of genius. We’ve spent 3 months studying work that Galois completed by age 20, before his death in a duel.

Materials

• John Stillwell’s Elements of Algebra was invaluable. This thin book has a narrative thread that motivates material. It is not formal, yet it is rigorous at the same time.

• Serge Lang’s Undergraduate Algebra was useful, as it was broader than Stillwell, and had a treatment closer to that of my professor. But I still went to Stillwell to really understand the concepts.

• Michael Artin’s Algebra I used for the background on groups, rings and ideals. It is great, but heavy-going if you just want to dip in for a casual reference.

• Benedict Gross’s Video Lectures were where I learned about group theory on my own a couple of years back. The lectures on rings and ideals were extremely useful this time around. As an added benefit, Michael Artin’s book is followed.

Topology

This is a world-building subject. Some math topics are practical and allow you to compute stuff. Topology sets the stage in which you can discuss matters that depend on proximity.

The topology we studied was mostly “point set” topology, then a little Algebraic Topology thrown in at the end.

Topology is a perfect subject to show the power and pervasiveness of set theory in modern math. Starting with the proximity relations used in calculus, you abstract them into collections of sets, and you build from there.

In calculus you used \( \epsilon - \delta\) (“epsilon-delta”) arguments, all about a neighborhood of points around a point \( x\) that are less than \( \delta\) away from it. In topology you cut straight to the set of points “near” \( x\) and call them neighborhoods, then axiomatize back to the building blocks needed to define neighborhoods.

Most of the work in point set topology is set reasoning, which is quite verbal. There is not that much computation involved in this course.

You learn about open sets, boundaries, compact sets, the Hausdorff condition. All these are fundamental objects in analysis later on, but they are interesting themselves.

In particular, the definition of compact sets seems to have been created by a sadist, but once you start proving theorems, you understand the brilliance of the definition.

Materials

• Munkres’s Topology is the classic here, for good reason. It goes point by point over the definitions, and shows you why they work, and why something simpler won’t cut it. Our professor followed Munkres closely, which made this an easy subject to study for.
• Allen Hatcher’s Topology Notes (available from his website), are much more concise and livelier, recommended as additional reading. However, I think Munkres is not replaceable.

Measure Theory and Integration

In a similar way to Topology, Measure Theory applies the force of set theory to a problem in analysis. If topology takes on the question of closeness and continuity, Measure Theory takes on measure (duh).

Another world-building subject.

As it turns out, not all sets have a meaningful measure that can be assigned to them. This was proven by Vitali in 1905, and it’s a good surprise.

The first order of business then is to define which sets are measurable. Then which functions are measurable, and from then, you build the Lebesgue integral.

This subject feels a bit finicky in the beginning. The Carathéodory construction of the outer measure seems like it’s hanging by a thread. You need quite a bit of machinery just to get going.

However, once it gets going, it is a very pleasant subject, and it builds a nice theory where the Riemann integral always seemed rough and ad-hoc.

Materials

• My professor, who was brilliant, recommended Folland. I decided against that book, however, as I did not want to buy yet another analysis book.
• My main book ended up being Bartle’s The Elements of Integration and Lebesgue Measure. This is a well-written book, with good exercise collections. A rarity. Highly recommended.
• I also checked Terrence Tao’s book on measure, which I found somewhat dry.
• Stein & Shakarchi’s Real Analysis. I loved the parts of Stein & Shakarchi I read, and will probably buy the book to have at home. Stein was more advanced than I needed, though, and Bartle was better suited for the class at hand.

Probability with Measure Theory

This is the one subject I have not gotten much out of, in this year. I didn’t enjoy the lectures, and this was aggravated when COVID-19 struck, and we moved the lectures online.

I found this a hodge-podge of some analysis and measure theory sprinkled with some probability theory general know-how.

One day I’ll look into it again.

Commutative Algebra

I signed up for this subject as preparation for Algebraic Curves next year. My professor, a non-specialist with a background in logic, was precise, exacting, somewhat tough. I was not inspired. However, the subject itself needs better motivation, or perhaps should be introduced later in the curriculum.

I suspect that once I take Algebraic Curves, I’ll be grateful for having taken Commutative Algebra. However, at the moment, I think the subject suffers from the cart-before-horse problem that can at times plague algebra.

The COVID-19 forced move to online lectures also did not help.

The course is the study of a collection of objects that are useful in algebraic geometry and number theory: Ideals, prime ideals, Noetherian rings, modules, varieties and Zariski topology, local rings etc.

As is the custom in modern math, and especially algebra, a formalistic approach is taken, and proofs are squeaky clean, yet seem to spring from thin air. “My son, one day this will all make sense.”

Materials

On my professor’s advice, I bought Reid’s Commutative Algebra. Interestingly, Reid complains about the un-motivated approach to algebra often taken… and then falls short himself. I think this book is not a good first book, or perhaps it just assumes previous culture I didn’t have.
This summer I’m reading an elementary book on algebraic curves. I’m hoping to see where some of the definitions in CA are coming from.

My professor took much care and prepared a set of notes. They were a good complement to Reid. Less imaginative but also less jumpy.

Differential Geometry

The third world-building subject this year. It was at the same time better motivated than topology and measure theory, as a growth of previous experience with curves and surfaces, and yet required a lot more machinery to get going.

Due to COVID-19, the lectures moved online half way through, the pace slowed, and we did not get far into the Geometry part of this course, emphasizing the Differential part.
I.e. this has really been a course on differentiable manifolds, which are the generalizations to higher dimensions of curves and surfaces.

There is beautiful intuition dripping from the subject matter. But it does require a substantial amount of machinery, in the form of definitions, before it gets going meaningfully.

First we need to define the objects under study: i.e. differential manifolds. Here we use a mix of analysis (differentiability), geometry (euclidean spaces) and topology (homeomorphisms, numerability, the Hausdorff condition).
Then, we need to abandon old intuitions. A differential manifold may not resemble anything we think of as a geometrical object. E.g. the projective plane, and certain sets of square matrices, are differential manifolds.

The biggest leap in abstraction is the tangent space. The tangent space had been studied in previous courses as a space of tangent vectors to a curve/surface. In this course, the tangent space is re-imagined as the space of operators taking functions to numbers, and obeying a certain computational law (the Leibniz condition).
On the heels of this new definition, the change of coordinates formulas of calculus become absolutely trivial.

As my professor remarked, what we do in DG is learn to operate with curved coordinates.

The subject also bears the great influence of the set concept. If in topology, continuous applications inverse-map open sets to open sets, and in integration theory, measurable functions inverse-map measurable sets to measurable sets, in DG, smooth maps convert smooth functions back to smooth functions via pullback.

Materials

The professor prepared an extensive set of notes, worked out exercises, and very stimulating classes. Probably the best professor I’ve had this year. The classes were laden with mentions to other areas of math, to try to show where differential geometry fits in, and how it clarifies and formalizes loose intuitions. His love for the subject was infectious.

I loved this professor, and I’m aware of some classmates who did too. But this type of professor tends to be much less popular, on average, than those who stick to the subject matter and are greyer. To each his own.

Additionally, I used two books:

• Tu’s An Introduction to Manifolds, which was most useful, and is, true to advertisement, self contained, and good for undergraduates. At times it feels almost too clear, like cheating. High praise.
• Spivak’s A Comprehensive Introduction to Differential Geometry. Spivak is one of the very best writers in math. While his book is at points obscure or messy, he keeps a great narrative thread. I only read a portion of Spivak, but it made a difference to my understanding. I will surely use it again next year.