Jonah Sinick
|
Registered User
|
I was a graduate student at University of Illinois at Urbana Champaign under the direction of Nathan Dunfield, finishing a PhD in hyperbolic geometry and number theory in 2011. I can be reached at jsinick@gmail.com.
|
|
May 6 |
comment |
Modern Mathematical Achievements Accessible to Undergraduates Is it fair to characterize the use of the Jones Polynomial to distinguish knot as tremendous progress in mathematics? My impression is that its primary source of interest is in its deeper significance. |
|
Apr 29 |
comment |
Grothendieck 's question - any update? I believe that this question was answered in the affirmative by Jean-Pierre Serre, but no longer remember where I read that, and may be misremembering. As for the corresponding question for cohomology groups, see pg. 6 of arxiv.org/pdf/math/0210327v1.pdf |
|
Apr 4 |
comment |
New Geometric Methods in Number Theory and Automorphic Forms The description doesn't explain what breakthroughs it's referring to. This makes it hard to know what the authors have in mind. |
|
Apr 3 |
revised |
Possible reasons why the image of 0 in the modular parametrization would always be O for a family of quadratic twists of elliptic curves added 425 characters in body; added 20 characters in body |
|
Apr 3 |
answered | Possible reasons why the image of 0 in the modular parametrization would always be O for a family of quadratic twists of elliptic curves |
|
Mar 31 |
comment |
How did Takagi prove Kronecker’s Jugendtraum for Q(i)? ...because the class number is 1, and the unit group is finite. |
|
Mar 31 |
comment |
How did Takagi prove Kronecker’s Jugendtraum for Q(i)? I would imagine that you can mimic the proof of the Kronecker-Weber theorem. |
|
Mar 28 |
comment |
Have we ever proved any non-solvable case of reciprocity without the Langlands program ? If I understand correctly, Eichler proved the theorem for X_0(11) specifically rather than X_0(N). (The former is an elliptic curve, whereas in the higher genus case one has to look at the Jacobian...) |
|
Mar 28 |
revised |
A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface? added 1 characters in body; edited title |
|
Mar 27 |
comment |
The existence of meromorphic functions on Riemann surfaces @Ian: I guess what I want to do is (i) show that the collection of genus g Riemann surfaces with a meromorphic map f to the sphere is dense in the moduli space of genus g Riemann surfaces and then (ii) show that an infinitesimal deformation X' of a Riemann surface X with a meromorphic map to the sphere itself has a meromorphic map to the sphere by infinitesimally deforming f. Presumably this is what the analytic proofs implicitly do: I'll have to think about how they correspond to this mental model. |
|
Mar 26 |
comment |
The existence of meromorphic functions on Riemann surfaces wwcanard — Ok, so I agree that algebraic arguments won't be relevant. Still, one can hope for a more geometric argument. |
|
Mar 26 |
comment |
The existence of meromorphic functions on Riemann surfaces I guess one could argue that even the proof in the genus 1 case requires hard analysis (in that one has to verify the convergence of infinite series), but somehow it's still quite conceptual. |
|
Mar 26 |
comment |
The existence of meromorphic functions on Riemann surfaces Thanks Ian. If I understand correctly, the fact that 1-forms have holomorphic representatives requires the Hodge Theorem, which in turn is usually proved via the theory of elliptic operators, though I found this question mathoverflow.net/questions/28265/… (which will take some time to digest). The only proofs of the uniformization theorem that I've seen use hard analysis. Maybe the use of hard analysis is inevitable. |
|
Mar 26 |
comment |
The existence of meromorphic functions on Riemann surfaces They're not, but you can still consider the moduli space of algebraic curves of given genus and the moduli space of Riemann surfaces of a given genus and to prove that they're the same. The spaces have the same number of dimensions and the former lies in the latter. Maybe you could use some sort of continuity argument to prove that the former can't be a proper subset of the latter. |
|
Mar 26 |
asked | The existence of meromorphic functions on Riemann surfaces |
|
Mar 6 |
asked | Counting smooth structures on manifolds |
|
Mar 4 |
awarded | ● Nice Question |
|
Feb 28 |
comment |
“Must read ”papers on analytic number theory Are you interested in additive number theory? Multiplicative number theory? Nonabelian harmonic analysis? |
|
Feb 28 |
comment |
“Must read ”papers on analytic number theory The proof of the Brauer–Siegel theorem (e.g. as presented in Lang's book), though probably that's something that you'll do regardless / have already done. I know some people who like Soundararajan and Ono's paper titled "Ramanujan's ternary quadratic form." |
|
Feb 25 |
awarded | ● Notable Question |
|
Feb 20 |
revised |
Functional equations of zeta functions over global fields added 172 characters in body |
|
Feb 20 |
asked | Functional equations of zeta functions over global fields |
|
Jan 28 |
revised |
What do theta functions have to do with quadratic reciprocity? added 1 characters in body |
|
Jan 28 |
answered | What do theta functions have to do with quadratic reciprocity? |
|
Jan 23 |
comment |
The Riemann Hypothesis and the Langlands program The first paragraph of Langlands' 1978 ICM lecture "L-functions and Automorphic Representations" publications.ias.edu/rpl/paper/65 reads "Introduction. There are at least three different problems with which one is confronted in the study of L-functions: the analytic continuation and functional equation; the location of the zeroes; and in some cases, the determination of the values at special points. The first may bethe easiest. It is certainly the only one with which I have been closely involved." |
|
Jan 19 |
comment |
The Riemann Hypothesis and the Langlands program @ GH – Yes, I'm aware of this, I glossed over this for brevity. Feel free to edit my answer if you'd like. |
|
Jan 19 |
comment |
The Riemann Hypothesis and the Langlands program I don't think that our converse theorems are strong enough for what I wrote to be provably true (in general), but I think it's true. What do you mean when you write "many sorts of motivic L-functions could non-disprovably have analytic continuations without automorphicity"? |
|
Jan 19 |
answered | The Riemann Hypothesis and the Langlands program |
|
Jan 19 |
comment |
The Riemann Hypothesis and the Langlands program @ Agol – I don't really know anything about this, but see the second paragraph of this page books.google.com/… of Iwaniec's "Spectral Methods of Automorphic Forms" for Iwaniec's take on the situation. |
|
Jan 14 |
awarded | ● Popular Question |
|
Jan 4 |
awarded | ● Popular Question |
|
Jan 2 |
comment |
Algorithm for determining whether two polynomials have the same splitting field This is nice. I don't immediately see why the effective Chebotarev density theorem tells you when you're done. But it's nice. wccanard's answer seems potentially more conceptually primitive, but I guess one would need to take a close look at the algorithm to be sure. |
|
Dec 31 |
asked | Algorithm for determining whether two polynomials have the same splitting field |
|
Dec 28 |
comment |
Applications of Iwasawa Theory Thanks David! As before, I need to learn French... What other reading would you recommend on this topic? The original Coates-Wiles paper? |
|
Dec 28 |
asked | Applications of Iwasawa Theory |
|
Dec 27 |
comment |
Characterizing primes that split completely vs. primes with a given splitting behavior @ Chandan - I added "abelian" in response to your comment. I think you're misreading my question. I didn't ask what the characterization of the splitting behavior of primes is - I asked whether the characterization follows from the characterization of primes that split completely @ Emerton - Thanks for the response. |
|
Dec 27 |
comment |
Characterizing primes that split completely vs. primes with a given splitting behavior Can you elaborate? All of the extensions that I have in mind are abelian. It's still interesting to ask about the discrepancy in the nonabelian case, but I'll change my question to make it more focused. |
|
Dec 27 |
comment |
Characterizing primes that split completely vs. primes with a given splitting behavior @ Chandan - why? |
|
Dec 27 |
comment |
Characterizing primes that split completely vs. primes with a given splitting behavior @Ari - Fixed. I had Galois extensions in mind. |
|
Dec 27 |
revised |
Characterizing primes that split completely vs. primes with a given splitting behavior added 7 characters in body |
|
Dec 27 |
asked | Characterizing primes that split completely vs. primes with a given splitting behavior |
|
Dec 22 |
awarded | ● Popular Question |
|
Dec 20 |
awarded | ● Nice Question |
|
Dec 20 |
comment |
Liouville’s theorem with your bare hands @ Ralph - Thanks for the new proof! |
|
Dec 20 |
comment |
Liouville’s theorem with your bare hands @ Andy - for my purposes I only need the result for power series, so I'm in good shape :) |
|
Dec 20 |
comment |
Liouville’s theorem with your bare hands @ Ralph - Good question |
|
Dec 20 |
comment |
Liouville’s theorem with your bare hands Thanks Ian. I hadn't seen this before. My first take on the proof is that it's somewhat unnatural even if elementary, but I'll think about it some more. |
|
Dec 20 |
comment |
Liouville’s theorem with your bare hands Cool, very nice. I think that this is what I was looking for. |
|
Dec 20 |
comment |
Liouville’s theorem with your bare hands This is new to me / interesting, but I'm not sure if it's any more direct than the proof by Cauchy's theorem :) |
|
Dec 20 |
asked | Liouville’s theorem with your bare hands |
