35,054 reputation
256129
bio website math.princeton.edu/~wsawin
location Princeton, NJ
age 21
visits member for 3 years, 9 months
seen 56 mins ago

I am a graduate student at Princeton studying arithmetic algebraic geometry.


20h
revised Idea of using etale site
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20h
comment Idea of using etale site
@DonuArapura Right, I didn't meant to say that it does not involve any analogy to singular cohomology, only that it does not follow directly from the analogy, like the rationality follows directly from the analogue of the Lefschetz fixed point formula (+ finiteness).
20h
comment Idea of using etale site
@paulgarrett Yes, that's correct (or use the pro-etale site, which was invented much later). I'm used to only thinking about the $\ell$-adic cohomology.
21h
comment Minimum value of $|p(1)|^2+|p(2)|^2 +…+ |p(n+3)|^2$ over all monic polynomials $p$
why $n+3$ values?
21h
comment Symplectic form on the third symmetric power of a plane
Good point. I think the actual characterization is that for this to be nondegenerate, $2m+2$ must have a single nonvanishing digit in base $p$ notation - i.e. it is a power of $p$ times an integer less than $p$.
22h
comment Idea of using etale site
@Wkf After that I would try showing that, more generally, $H^1$ of a variety is twice the dimension of the Albanese, and then to show that $H^2$ of a curve is one-dimensional. I would consider that enough basic tests and then try to prove some theorems about a cohomology theory that passed the tests.
22h
comment Idea of using etale site
@wkf I don't think there's a formal statement. The first basic test I would try is whether $H^1$ of a smooth projective curve of genus $g$ is $2g$-dimensional. Grothendieck said that if you have a good theory of $H^1$ you should be able to find a good theory of all the other cohomology groups, so that might be enough.
22h
comment Idea of using etale site
@SamHopkins No, I mean to say that the coefficients of the cohomology theory are in a field of characteristic zero. The varieties, I should have said, are characteristic $p$.
23h
answered Idea of using etale site
1d
comment Model over DVR for smooth projective curves
note that the uniqueness requires stability, which includes a condition on not having certain irreducible components. However I guess having one irreducible component always implies this
1d
comment Goldbach's problem in algebraic number fields
The fact that a prime times a unit is a prime is significant and might make some formulations of the problem easier.
1d
comment Probability a random matrix contains a short integer vector in its kernel
You can compute the probability for any fixed short vector using the circle method. Then summing over all short vectors should give the answer. I don't know how short they have to be for this method to work.
2d
comment Independent Generic Curves in the Projective Plane
One definition is that, not only are they generic, i.e. the coefficients of each curve satisfy no polynomial relations, but also the coefficients of the curves together satisfy no polynomial relations with each other. So the curves are defined over disjoint transcendental extensions of the base field. Does this fit with the paper?
2d
answered Algorithms for calculating R(5,5) and R(6,6)
2d
comment Algorithms for calculating R(5,5) and R(6,6)
Is a quantum computing algorithm to calculate random numbers actually known? $R(m,2)$ is trivial and $R(3,3)$ is pretty easy as well. My understanding is that these quantum optimization algorithms scale very poorly.
Jul
1
comment When are the powers of 2 sum-free mod n?
For most $n$ it's probably not, or at least for most prime $n$, because on average powers of $2$ are at least conjecturally a positive proportion of numbers mod $p$.
Jul
1
revised Can phase significantly concentrate a function's spectrum?
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Jul
1
comment Can phase significantly concentrate a function's spectrum?
@DustinG.Mixon For any function $f$, if we let $f_n$ be $f$ composed with multiplication by $n$ then there is a simple formula for $\hat{f}_n$.
Jun
30
revised Can phase significantly concentrate a function's spectrum?
added 603 characters in body
Jun
30
revised Can phase significantly concentrate a function's spectrum?
added 465 characters in body