Undergraduate student of mathematics. Currently interested in algebraic topology. Also studying algebraic geometry. Currently topic of study: the Goerss-Hopkins obstruction theory.
I’ve been considering making posts out of some musings on the notion of “work” (in the more general sense of the word, not in the sense of referring to a job) in society, but instead of creating a high-effort post like that, I’m going to make a low-effort post in an effort to create a habit of posting at all.
So, I’m going to list my current plans of study in various areas of math that I’m studying in some sort of effort to organize and convince myself to not pick up any more threads of study until I finish the ones I’m currently pursuing.
Algebraic Topology:
This is my flagship subject. Currently I’m starting up a pretty ambitious research project with my advisor. We’re trying to utilize the obstruction theory in the monstrous paper “Moduli Problems for Structured Ring Spectra” by Goerss and Hopkins to investigate E_∞ structures on truncated Brown-Peterson spectra. As I understand it, this problem has been open for a long time, and not much progress has been made. BP<-1> and BP<0> are essential trivial, and BP<1> was somehow teased out of connective K-theory. Recently, work of Lawson and Naumann has shown that there is a unique E_∞ structure on BP<2> at the prime 2. While BP<1> was obtained from K-theory, which lives at chromatic height 1, BP<2> comes from tmf, which lives at height 2. This means that similar methods seem pretty hopeless for higher n, as we know so little about higher chromatic heights.
In any case, I’m pretty hyped for this project and intend to spend most of my time on it this semester. It will be really cool if I get good results, as this is a problem that people care about.
Beyond that, I really want to get up to date on the chromatic stuff, and my intention is to start with Ravenel’s green book (“Complex Cobordism and the Stable Homotopy Groups of Spheres”, though he wanted to call it “The Music of the Spheres”) and the “Complex Oriented Cohomology Theories and the Language of Stacks” notes.
Algebraic Number Theory:
I’ve been studying some algebraic number theory lately as a prelude to elliptic curves stuff, and it’s really cool stuff! Currently I’m reading Milne’s notes on algebraic number theory, and I’d love to read his class field theory stuff at some point. Short term, I think I’ll stick to picking up the basics. With all the links between algebraic topology and number theory popping up, I’d like to be poised to get a serious education in this stuff if it turns out to be useful.
Algebraic Geometry:
For now, devoting a serious amount of time to the algebraic geometry class I’m taking will suffice. I should do a nontrivial amount of Hartshorne as well, as it has a lot more material on sheaves and their cohomology than Liu, which we’re using the in the class. I would really like to cover some stuff on stacks on descent at some point.
Toric varities seem like a cute bunch of concrete examples, so I hope to find time to read Fulton’s “Introduction to Toric Varities” at some point.
Arithmetic Geometry:
I plan to start reading Silverman’s book on elliptic curves during my spare time once I finish with Milne’s notes on algebraic number theory. I also intend to read something about their moduli stack at some point in the future, as it plays a fundamental role in the construction of topological modular forms (tmf).
Over the summer, I’d like to work through the later chapters of Liu on arithmetic geometry and surfaces with my algebraic geometry professor.
He’s quite the expert on Shimura varieties and their compactifications, and I’d like to learn as much as I can about them from him while I’m here. They were used in the construction of topological automorphic forms (TAF) by Behrens and Lawson, and I’d really like to understand TAF. Apparently the compactification of the moduli stack of elliptic curves is used in an important way in the theory of topological modular forms, and I’m curious as to whether compactifications of Shimura varieties could be used similarly for TAF!
Homological Algebra:
I should really give Weibel a serious read. I feel that I have a good grasp of the basics, but I still have a lot to learn.
Higher Categories:
I’d really like to read Lurie’s “Higher Topos Theory” sometime. ∞-categories are really cool, and what little I know about quasicategories is already really enlightening!
Complex Analysis:
I’m about halfway through my casual read of Stein and Shakarchi’s book. It’s been fun. After I finish it, I’ll pick up some stuff about modular forms because tmf, also number theory is cool.
Real Analysis:
I’m taking a class in this, but I’m actually not feeling super motivated about it. I’ll work hard enough to get a good understanding of the material and thus an easy A, but I won’t be doing extracurricular stuff on it.
Manifolds:
I intend to follow along in Lee’s “Introduction to Smooth Manifolds” with the graduate manifolds class during the upcoming semester. Should be pleasant, and I’m happy to gain some familiarity with this stuff.
Other:
Back when he thought I was a grad student, my algebraic geometry professor told me that I should learn some p-adic Hodge theory some time before I graduate, as there have apparently been big strides in the field recently and he expects it to rear its head in algebraic topology in the nearish future. I have no clue what he means by this, but partially because this recommendation intrigues me and partially because I want to be prepared as people expect a graduating PhD student to be by the time I graduate, I intend to pick some of it up before I graduate. (That sentence got away from me… my defense is that it’s three something in the morning.)
Geez… writing it out like this makes it look like I really am studying way too many things at once.
Yes, but with the caveat that it is still possible that you lose every time. This sounds paradoxical, but it's really just a side effect of how we define what it means for something to have an x% chance of occurring. It actually gets into some reasonably deep mathematics, namely measure theory.
First, let me explain why the chances of you winning at least once are 100%. Suppose that the chances of you winning are P(A) > 0, and that the chances of you losing are P(B) = 1 - P(A) < 1. Then the probability of you not having won after n tries is P(B)^n. The probability of you not having won after an infinite number of attempts will be the limit of this as n goes to infinity. Since the limit as n goes to infinity of a positive real number less than 1 to the nth power is 0, the chances of you not having won after an infinite number of attempts is zero. Thus the chances of you having won at least once are 100%.
Now let me justify my assertion that this does not mean you are guaranteed to win at least once. Consider the following situation: a random real number between 0 and 1 (inclusive) is chosen. What is the probability that it's 0? It must be zero. Why? Suppose that it were 0 < P < 1. Then we would be able to consider the interval I of all real numbers x such that 0 <= x <= P/2. It is clear that a randomly chosen real number between 0 and 1 will be in this interval with probability P/2. But then it is less likely for us to end up in I than for us to end up at a real number in I! This cannot be correct. Thus this probability is 0. But it is still possible for a random number in I to be 0! It is just so incredibly unlikely that it doesn't even deserve to be given a positive probability. For this reason, mathematicians will say that something with a 100% probability of happening will almost surely happen.
If we think about this another way, it becomes somewhat more intuitive. It's clear that the probability a randomly chosen number in I will be in some subset of I will be the "length" of the subset, for some appropriately general notion of length. So really all we're saying here is that a point has zero length, which is intuitive enough.
This leads into measure theory, where we deal with the notions of length, area, volume, and how to appropriately generalize them to cases where it's not entirely clear what the "area" of some set of points should be. (And even then, there are sets that resist our efforts to measure them, see here.*) We may then study probability by defining a "space" where the points are the possible outcomes, and the measure (that is, our generalization of the idea of area) corresponds to the probability that something will happen.
Anyways, that was almost surely (heh) more than you wanted to know. Math major, OUT.
*It is, however, notable that we need to assume the axiom of choice to prove that they exist. Ignore this if you don't know what it means.
Why is this paper using both a w and an \omega in the paragraph? And I'm pretty sure that it used a \omega where it wanted a w in at least one place, unless it really does want to take the sign of a weight function and not a permutation in this determinant.
How did you not realize that this was terrible notation, authors?
I noticed that I hadn't updated Arch over the entire spring semester, so I decided to run a much needed pacman -Syu. The resulting 410 package update was terrifying, and I can only hope that this thing will boot after I next turn it off.
Thinking about transferring to the University of Minnesota. Been reading about the economics programs there and am doing a transfer app. Anyone have any knowledge/personal experience about this school?
I'm a math major at the U of MN, and from what I've heard from friends who are in the econ department, that department is really full of themselves and difficult to work with as an undergraduate. Very few of the undergrad classes are taught by actual professors, and at least the freshman and sophomore classes are ludicrously easy. I've also heard that it's difficult to get any kind of personal treatment, e.g. if you want to take a graduate class you can't just talk to an adviser and get in, instead you have to turn in the relevant paperwork and hope that they let you in. They are also on somewhat bad terms with the College of Liberal Arts, the college in which they are based. Something about wanting more funding, but being unwilling to teach as many classes as CLA wants.
That said, the graduate program is definitely very good, and if you can navigate the bureaucracy there are a lot of elite professors there to work with.
Accepted to the U of MN - Twin Cities combinatorics REU! Considering my age-challenged status and some of the ridiculously smart people who have attended in the past couple of years, I am highly pleased by this.
I have been working on this problem for the past 3 to 4 hours straight, not to mention having been thinking about it for much longer than that. The end is in sight. Thank god.
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