27,669 reputation
242102
bio website math.princeton.edu/~wsawin
location Princeton, NJ
age 21
visits member for 3 years, 1 month
seen 4 hours ago

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


Oct
20
comment What is the decomposition group at $p$ in the Galois group unramified outside $\ell?$
One approach would be to find many $n$-dimensional $\ell$-adic Galois representations which are unramified outside $\ell$, where the conjugacy class of $Frob_p$ varies. One can construct these Galois representations from automorphic forms of level a power of $\ell$, then one wants to show that the Forier coefficients vary. I'm not sure whether one can do this.
Oct
7
comment Why do roots of polynomials tend to have absolute value close to 1?
You should expect the absolute value of the $k$th coefficient to be close to the square root of the expectation of the squared absolute value of the $k$th coefficient, which is $\sqrt{ \left(\begin{array}{c} n \\ k \end{array}\right)}$. This is probably the easiest way to figure out the shape of the curve.
Oct
6
comment Does every hypersurface in the projective plane contain a projective line?
@IianSmythe Yes.
Oct
5
answered Does every hypersurface in the projective plane contain a projective line?
Oct
5
comment Why do roots of polynomials tend to have absolute value close to 1?
@მამუკაჯიბლაძე I mean if hypothetically there were a bias in the random number generator such that some angles were less likely, then after taking the product of a large number of linear factors, the value at points on the unit circle near those angles would be much larger than the value elsewhere, causing concentration of mass near those points.
Oct
5
asked How tight is the Weil bound for this exponential sum?
Oct
4
comment Analogy between connections and $\ell$-adic sheaves: what happens with the residue?
Abyankhar's lemma says that the action of the fundamental group on a tamely ramified torsion sheaf factors throughout the quotient coming from the cover where you adjoin the $n$th root of the equation defining the divisor. This group is cyclic. Then for an $\ell$-adic cover you get an action of the inverse limit group, which is infinite cyclic.
Oct
4
answered Analogy between connections and $\ell$-adic sheaves: what happens with the residue?
Oct
4
comment Sieving question
Shouldn't this be at least a positive proportion? A positive proportion are squarefree, then for each $p$ dividing $n$, half the numbers are squares mod $p$, so we should expect half the primes to satisfy this condition, so $\prod_{p \in S} p \geq n^{1/2}$ about half the time. Subtract the $\epsilon$ can only increase this.
Oct
4
comment Why do roots of polynomials tend to have absolute value close to 1?
@მამუკაჯიბლაძე I played with your code with degree 100 polynomials and found many polynomials that look like if you applied the symmetry I describe they would become smooth. So maybe the random functions on the unit circle you get this way tend to have all their mass concentrated somewhere - this seems plausible to me, and would explain my observations for degree $100$. But already at degree $200$ I see this almost never. Could some slight bias in random number generation be compounding? I'm not sure.
Oct
4
answered How to understand a rooting of a dessin d'enfant?
Oct
4
comment Why do roots of polynomials tend to have absolute value close to 1?
A function $\mathbb Z \to \mathbb C$ appears smooth if applying the difference operator to it destroys most of its mass (say, $L^2$ norm), and applying the difference operator more times destroys even more. Taking the Fourier transform (on the unit circle), we get that the function should have most of its $L^2$ norm concentrated near $1$. For the coefficients of a polynomial, the Fourier transform is just evaluating the polynomial on the unit circle. So if you choose random polynomials with random roots that are often near the unit circle but always far from $1$, you should get smoothness.
Oct
4
comment Why do roots of polynomials tend to have absolute value close to 1?
I'm confused by this. Two of your distributions for roots are centrally symmetric around the origin. This gives a symmetry of the distribution for coefficients - for $\alpha$ a complex number of unit norm, you can multiply the $n$th coefficient by $\alpha^n$. This is problematic - if you apply this symmetry to a smooth function, you get a function that is not smooth at all. So why are your functions smooth?
Oct
2
answered Is the cokernel of a map of sheaves a seperated presheaf?
Sep
30
awarded  Explainer
Sep
25
awarded  Yearling
Sep
14
comment Blow-ups and cohomology
I'm guessing $H \cdot E$ corresponds to the line bundle $\mathcal O(1)$ on $W$.
Sep
8
answered Existence of $n$-connected topological groups with $m$-dimensional action extending that of $GL(m)$ on $\mathbb{R}^m$
Sep
8
comment The locus of rational/elliptic curves on a special surface in $\mathbb{P}^3$
Using Riemann-Hurwitz you can show a certain proportion of the zeros of $P(f)$ and $Q(g)$ must be single roots, which means a certain proportion of the zeros of $P(f)$ must be zeros of $Q(g)$ and vice versa. Here I'm thinking of $f,g$ as just rational functions on a curve. One could try to derive a contradiction from that, but I don't immediately see how.
Sep
8
comment An inequality on representations and subgroups of general linear groups over finite field
By the way, the largest $q \equiv 1$ modulo $8$ with $q \leq 43$ is $41$. Is that what you tested up to? Did you mean to type something else?