bio | website | math.uga.edu/~pete |
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location | Athens, GA | |
age | 38 | |
visits | member for | 4 years, 10 months |
seen | yesterday | |
stats | profile views | 25,889 |
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.
Aug 24 |
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Proofs of the Chevalley-Warning Theorem
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Aug 23 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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What are the most misleading alternate definitions in taught mathematics?
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Jul 2 |
awarded | Inquisitive |
Jul 2 |
awarded | Curious |
Jun 24 |
answered | Class number for binary quadratic forms discriminant $\Delta$ to class number $\mathbb Q(\sqrt \Delta)$ |
Jun 23 |
comment |
Are all Hawaiian Earrings homeomorphic?
@MarkBell: This is one of those responses that is ridiculously not worth the wait, but: for me, simple connectedness and the fundamental group is algebraic topology, not general topology. I wouldn't want to argue for this: certainly many general topology texts, e.g. Munkres's end with a chapter on this material (I would say that they end with an introduction to algebraic topology...) This is just by way of explaining my ancient comment. |
Jun 22 |
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Why is a topology made up of 'open' sets?
@David: I find your remark curious. If you only care about Tychonoff spaces, it follows that you don't care about the Sierpinski space. So why comment on it? |
Jun 22 |
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Is there an algebraic proof of the infinitude of primes?
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