405 reputation
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age 27
visits member for 3 years, 11 months
seen Jul 17 at 22:18

Apr
7
awarded  Yearling
Oct
16
comment The transcendence degree of a separably generated field extension K/L
There is an excellent discussion of separating transcendence bases in Matsumura's book "Commutative Algebra". See in particular section 27. (Note that Matsumura's "Commutative Algebra" is distinct from his "Commutative Ring Theory", although they have some significant overlap with one another.)
Jun
28
comment Subjects studied in number theory
Regarding your first sentence: I recently learned of a (to me) surprising connection between number theory and PDE. In the paper arxiv.org/pdf/0705.2891v7.pdf, Prasad and Rapinchuk show that a PDE condition relating two locally symmetric spaces (isospectrality: the respective Laplacians have the same eigenvalues) yields a rather number-theoretic relation between the two spaces (commensurability). Their results imply that one could in principle use PDE to prove a certain subgroup of, say, GL(n,Q) is arithmetic (= commensurable with GL(n,Z)), a decidedly number theoretic property.
Jun
28
comment Important papers in arithmetic geometry and number theory
Prof. Emerton: what a joy that Wyman paper is! Thank you for posting a link to it.
May
31
comment What is a Number Theorist
... "you wcouldn't understand; it's complicated", in response to an inquiry about their faith. Likewise, mathematicians need not patronize those who ask about their research. We may fail to convince a non-mathematician that our efforts are worthwhile. We will probably even fail to get across the basic idea of what we find challenging in our field. Yet we surely each have an answer to the question, "Why is my mathematics valuable or interesting?" We owe it to ourselves to share the answer with anyone who is curious enough to ask.
May
31
comment What is a Number Theorist
... and to make an honest effort to explain to them what exactly it is that you find interesting about the little part of the mathematical universe that you call home. I often fail when I make such an effort. But I strongly believe the effort it worthwhile. Here's an analogy. As a non-religious person, I have a hard time really "getting" where believers are coming from when they discuss their faith. But I hardly think this means I should avoid asking about the religious aspect of their intellectual and spiritual life. In fact, I would feel condescended to if they were just to say [ctd.]
May
31
comment What is a Number Theorist
If your bio friend can't say roughly what property of the microbe he or she is studying makes said microbe more interesting than the zillions of other microbes, do you not have reason to doubt the intellectual value of his or her endeavors? And if you do not bother to give him/her the chance to explain -- because "you don't care" about stupid microbes -- are you not altogether poorer for it, in the event that he/she actually has a very good reason why these microbes are interesting? I think it is very legitimate to take people at face value when they ask about your mathematics [ctd.]
Aug
18
awarded  Yearling
Sep
26
comment Best book ever on Number Theory
broken link, as far as i can tell
Sep
26
awarded  Critic
Sep
23
comment Topological group: Multiplying two loops is homotopic to linking these paths?
en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_argument
Sep
23
answered Why is $\mathbb{Q}(t,\sqrt{t^3-t})$ not a purely transcendental extension of $\mathbb{Q}$?
Sep
21
comment How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized continued fraction?
I think you could shunt this over to mathoverflow
Aug
30
revised Recovering the two SU(2) matrices from SO(4) matrix
deleted 1398 characters in body
Aug
30
comment Recovering the two SU(2) matrices from SO(4) matrix
whoops you're right. I'll edit
Aug
30
comment Recovering the two SU(2) matrices from SO(4) matrix
Oh this is much easier than my description! Go quaternions!
Aug
30
answered Recovering the two SU(2) matrices from SO(4) matrix
Aug
29
comment How are gauge transformations of a $G$-bundle related to the adelic points of $G$?
I would... but in fact this has been answered to my satisfaction. Ramras's Atiyah-Bott reference seems like a good place for me to learn what the deal with gauge groups is (thanks! I've only glanced at it so far...) and Matt E gave a nice explanation of the link between G-bundles and $G(\mathbf{A})$. I see no need to repost on MO.
Aug
28
comment Using Gröbner bases for solving polynomial equations
I don't really understand how these computations work. But what I do understand, I learned from a paper of Mumford and Bayer, called "What can be computed in algebraic geometry?". It's really good!
Aug
28
comment How are gauge transformations of a $G$-bundle related to the adelic points of $G$?
Thanks! Someone told me offline about the double coset space parametrizing bundles. If anyone is curious about the ground field $k$, apparently either $k$ finite or $k$ algebraically closed ensures that this bijection works.