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# Will Sawin

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## Registered User

 Name Will Sawin Member for 1 year Seen 4 hours ago Website Location Princeton Age 19
I am a graduate student at Princeton studying arithmetic algebraic geometry.
 1d revised What fraction of n-point sets in the unit ball have diameter smaller than 1?edited body 1d revised What fraction of n-point sets in the unit ball have diameter smaller than 1?added 14 characters in body; added 3 characters in body 1d revised What fraction of n-point sets in the unit ball have diameter smaller than 1?added 207 characters in body 1d answered What fraction of n-point sets in the unit ball have diameter smaller than 1? 1d revised Сonvergence of the sumadded 7 characters in body 1d answered Convergence to a k-dimensional Gaussian vector 1d comment When is a 0-1 matrix a one-intersection incidence matrix?Yes I thought about that configuration as well. The dual configuration is realizable, so if it's not it's a counterexample to symmetry. I currently see neither a way to realize it nor a method of proof it can't be. 1d comment Permutations of $(Z/pZ)^*$A very simple heuristic: This problem is about the complement of $p^2(p-1)^2/4$ hyperplanes in a $(p-1)^2$-dimensional projective space. So the "expected" number of points, if the hyperplanes are randomly located, is $p^{ (p-1)^2} (1-{1/p} )^{ p^2(p-1)^2/4}$, which goes to $0$ rapidly, so we should not expect there to be any solutions beyond the obvious ones, even without any good reason. 1d comment Braided coverings and braided monodromyThis link, giving explicit models for these classifying spaces, might be helpful: mathoverflow.net/questions/56363/… In particular, it states that a braided cover is just a cover with an embedding up to isotopy in $X \times \mathbb R^2$. 2d accepted Potentially good, semi-stable reduction => good reduction ? Jun16 comment Potentially good, semi-stable reduction => good reduction ?This is a special fact about the etale cohomology of a curve. $H^0$ is trivial, $H^2$ is a Tate twist of trivial by Poincare duality, and $H^1$ is the Tate module of the Jacobian, so since the Jacobian of a torsor for $E$ is just $E$, it's the same. Jun16 comment When is a 0-1 matrix a one-intersection incidence matrix?The smallest two matrices which are not linearly representible for trivial reasons are the Fano plane and affine space over $\mathbb F_3$ minus a point. You have shown how to draw the Fano plane. I think I know how to draw affine space over $\mathbb F_3$ minus a point, I'm going to figure out how to make a pretty picture to show you. This means that any bad configuration must have at least 9 vertices, decreasing the gap to just 4. I actually don't know if the dual result is true, that each bad configuration must have at least 9 edges. Jun16 comment Potentially good, semi-stable reduction => good reduction ?What are P and E? Jun16 comment inner product of two gaussian random vectors?The only situation where the cumulative density functions are close but the probability density functions are far is if at least one of the probability density functions oscillates a lot. But since both probability density functions are clearly monotonic on the positive and negative numbers, they cannot oscillate very much. Jun16 comment Potentially good, semi-stable reduction => good reduction ?If it had good reduction you could lift a point over the residue field using Hensel's lemma, and all genus $0$ curves over finite fields have points. Alternately, by a well-known fact there are two genus $0$ curves over a local field, one with good reduction and rational points and one with bad reduction and without rational points. Jun16 answered Potentially good, semi-stable reduction => good reduction ? Jun14 accepted Positive subspaces of quadratic forms Jun14 comment Positive subspaces of quadratic formsIt's an equation of the form $x_1^2-x_2^2 -x_3^2 - \dots x_{n+1}^2$. I consider this a solid bicone by definition. The point is that all quadratic forms of a given signature are equivalent. Jun14 revised Positive subspaces of quadratic formsdeleted 1 characters in body; edited body Jun14 comment Does this modified Hasse principle hold for curves?This is inspired by work of Manjul Bhargava on hyperelliptic curves. I guess this is the universal descent obstruction for finite abelian groups? Jun14 answered Positive subspaces of quadratic forms Jun14 asked Does this modified Hasse principle hold for curves? Jun14 revised Finite field hypergeometric functions.added 131 characters in body Jun14 accepted Koszul complex of a variety inside a product Jun14 answered Finite field hypergeometric functions. Jun13 answered Koszul complex of a variety inside a product Jun13 comment explicity equations for curves in the projective space1. The degree $6$ rational curve comes from a the Veronese embedding $\mathbb P^1 \to \mathbb P^6$ and a nontrivial projection $\mathbb P^6 \\ \mathbb P^2 \to \mathbb P^3$. Given an explicit projection (essentially 3 degree 6 polynomials in 2 variables) you could try to search for explicit equations by searching for relations among the variables. 2. If you add some singularities, you have to decide if you care about the arithmetic or geometric genus. The complete intersection family has arithmetic genus $4$, but I believe it has members of geometric genus $0$ with simple singularities. Jun13 comment Counter example of upper semicontinuity of fiber dimension in classical algebraic geometryYes, good point. To solve this, connect the point to $\mathbb A^2$ using a line. Jun12 answered Counter example of upper semicontinuity of fiber dimension in classical algebraic geometry Jun12 comment Who is the commutator sheaf?If $G$ is over a finite field, then the trace of Frobenius of this complex is just a class function. Applied to an element of the group, this class function just counts the number of pairs of elements whose commutator is that element. This class function exists in every finite group , and can always be written in terms of characters. How to write it in terms of characters is a purely group-theoretic problem. Do you know how to solve that problem? Jun11 comment Variation the definition of toric varietiesThis action factors through the action of another torus. So I think you just end up with the same definition. Jun10 accepted Zeros of compositions of polynomials and derivatives Jun10 accepted notion of torsor defined by exact sequence Jun10 answered notion of torsor defined by exact sequence Jun9 answered Multivariate analogue of Vandermonde determinant Jun9 comment What does this notation mean: matrix norm with a two-number subscriptPerhaps it means the maximum value of the $L_1$-norm of $Wx$ divided by the $L_2$-norm of $x$ over all nonzero $x$. Jun7 accepted One question on first Stiefel-Whitney class Jun7 answered One question on first Stiefel-Whitney class Jun7 revised One question on first Stiefel-Whitney class added 8 characters in body Jun6 accepted Helly’s Theorem for Biconvex Sets Jun5 answered for which truth-operations f can f-membership in a prime ideal be represented by a polynomial? Jun4 answered Minimal representation of a polynomial as a linear combination of squares Jun4 answered probability of having linearly independent sparse vectors over finite fields Jun3 comment How to determine “genericness” of an element of a family of algebraic varieties?I'm not sure which conditions are needed to make this true. Jun3 comment How to determine “genericness” of an element of a family of algebraic varieties?I'm guessing the best cohomological way to detect genericity is going to be the vanishing cycle sheaf. You want a statement like: If there are no vanishing cycles, the fiber is homeomorphic to generic $X_t$. Jun3 answered What is the largest possible operator norm of a sparse (0,1)-matrix? Jun3 comment Construction of the spectral sequence of Katz/OdaI think all that's going on is complexes consisting of injective sheaves get sent to complexes consisting of acyclic sheaves. Jun2 comment Construction of the spectral sequence of Katz/OdaThis is how you get an equality of functors in the category of complexes. If you want a spectral sequence, from this perspective, it's the spectral sequence for the hypercohomology of $\Gamma_s$. Jun2 answered Construction of the spectral sequence of Katz/Oda Jun2 comment When is a 0-1 matrix a one-intersection incidence matrix?No. I am claiming that if $M$ is not realizable by straight lines, then removing such a row or column will not make it realizable. The reason for this should hopefully be clear, but since some things are realizable by curves and not straight lines, this can only provide a lower bound. By the way, I think the next couple arrangements to check are affine space over $\mathbb F_3$ and affine space over $\mathbb F_3$ minus a point. Can these be realizes by curves?