40,726 reputation
5127276
bio website math.uga.edu/~pete
location Athens, GA
age 38
visits member for 4 years, 10 months
seen yesterday
assistant associate prof of math @ University of Georgia.

Primary research interests: number theory, arithmetic geometry, Galois cohomology

Secondary research interests: field theory, commutative algebra, general topology, model theory, and various combinations thereof

I also have an enduring interest in mathematical exposition.

Sep
12
accepted What is the status of the Gauss Circle Problem?
Sep
10
awarded  Necromancer
Sep
10
comment What is the status of the Gauss Circle Problem?
@Lucia: Yes, I had seen it. Thanks for reminding me.
Sep
10
answered What is the status of the Gauss Circle Problem?
Sep
10
revised What is the status of the Gauss Circle Problem?
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Aug
24
revised Proofs of the Chevalley-Warning Theorem
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Aug
23
comment Proofs of the Chevalley-Warning Theorem
In the realm of refinements on the $p$-adic congruences, here is a recent good-looking paper: arxiv.org/abs/1408.3224. Very recent: it appeared after your answer!
Aug
23
revised Proofs of the Chevalley-Warning Theorem
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Aug
23
awarded  Nice Answer
Aug
23
revised Proofs of the Chevalley-Warning Theorem
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Aug
22
revised Proofs of the Chevalley-Warning Theorem
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Aug
22
answered Proofs of the Chevalley-Warning Theorem
Aug
2
comment Fields with trivial automorphism group
@dke: Thanks, these are very interesting results.
Jul
30
awarded  Nice Question
Jul
26
comment Examples of famous 'workhorse' theorems
Of course experiences differ: when I was a grad student, I ran into someone I had taught at an academic summer program for teenagers: he was then a second year undergraduate at my school. I asked him what he was doing, and he told me he was taking a reading course on derived categories. Somewhat floored, I replied that it was remarkable that he had received exposure to all the prior material that would motivate someone to study this. In what context had he learned about spectral sequences, for instance? Answer: "Yeah, I'm starting to think I should have studied those first."
Jul
26
comment Examples of famous 'workhorse' theorems
In my experience, spectral sequences are so technical that most people who learn them get introduced to them in roughly the following way: "Now I'm going to tell you about spectral sequences. They're going to look horrible at first, but the point is that they can be used to immediately derive [this very useful thing]. In fact, maybe the first time through you should just assume that the spectral sequences degenerate immediately and see what that gives you..."
Jul
26
comment Examples of famous 'workhorse' theorems
I would not describe the Mayer-Vietoris sequence as " either uninteresting at first glance or too imposing to be understood by non-experts". On the contrary, I think most students of topology get exposed to the complicated definitions for co/homology and get increasingly nervous, wondering "How am I ever going to compute the co/homology of anything??" Then they see Mayer-Vietoris and think "Oh! Thank goodness." Spectral sequences are more technical, I agree, but their usefulness is still pretty apparent even to beginners in the area.
Jul
11
comment What are the most misleading alternate definitions in taught mathematics?
@Cam: A very belated reply: what you suggest is very good...You are bailing out of directly defining the multiplication operation and instead defining the $\mathbb{Z}$-module structure (i.e., you are using the one ring). Once they understand that, of course the next thing you say is that -- good news! -- the quantity $(a \in \mathbb{Z}) \cdot (b \in \mathbb{Z}/n\mathbb{Z})$ depends only on the residue class of $a$ modulo $n$, so it gives a multiplication operation. This is fully consistent with what I was advocating and really not "clock multiplication" (which means nothing, I fear...).
Jul
11
revised What are the most misleading alternate definitions in taught mathematics?
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Jul
2
awarded  Inquisitive