A theorem of Göttsche establishes a connection between cohomological invariants of a complex projective surface S and corresponding invariants of the Hilbert scheme of n points on S. This relationship is encoded in certain infinite product q-series which are essentially modular forms. Here we make use of the circle method to arrive at exact formulas for certain specializations of these q-series, yielding convergent series for the signature and Euler characteristic of these Hilbert schemes. We also analyze the asymptotic and distributional properties of the q-series' coefficients.
The recent success of distributed word representations has led to an increased interest in analyzing the properties of their spatial distribution. Several studies have suggested that contextualized word embedding models do not isotropically project tokens into vector space. However, current methods designed to measure isotropy, such as average random cosine similarity and the partition score, have not been thoroughly analyzed and are not appropriate for measuring isotropy. We propose IsoScore: a novel tool that quantifies the degree to which a point cloud uniformly utilizes the ambient vector space. Using rigorously designed tests, we demonstrate that IsoScore is the only tool available in the literature that accurately measures how uniformly distributed variance is across dimensions in vector space. Additionally, we use IsoScore to challenge a number of recent conclusions in the NLP literature that have been derived using brittle metrics of isotropy. We caution future studies from using existing tools to measure isotropy in contextualized embedding space as resulting conclusions will be misleading or altogether inaccurate.
Let 1 , 2 , . . . be a countable collection of lines in R d . For any t ∈ [0, 1] we construct a compact set Γ ⊆ R d with Hausdorff dimension d − 1 + t which projects injectively into each i , such that the image of each projection has dimension t. This immediately implies the existence of homeomorphisms between certain Cantor-type sets whose graphs have large dimensions. As an application, we construct a collection E of disjoint, non-parallel k-planes in R d , for d ≥ k + 2, whose union is a small subset of R d , either in Hausdorff dimension or Lebesgue measure, while E itself has large dimension. As a second application, for any countable collection of vertical lines w i in the plane we construct a collection of nonvertical lines H, so that F , the union of lines in H, has positive Lebesgue measure, but each point of each line w i is contained in at most one h ∈ H and, for each w i , the Hausdorff dimension of F ∩ w i is zero.
Fix a non-CM elliptic curve E/Q, and let aE(p) = p + 1 − #E(Fp) denote the trace of Frobenius at p. The Sato-Tate conjecture gives the limiting distribution µST of aE(p)/(2 √ p)within [−1, 1]. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval I ⊆ [−1, 1], let pI,n denote the nth prime such that aE(p)/(2 √ p) ∈ I.We show lim infn→∞(pI,n+m −pI,n) < ∞ for all m ≥ 1 for "most" intervals, and in particular, for all I with µST (I) ≥ 0.36. Furthermore, we prove a common generalization of our bounded gap result with the Green-Tao theorem. To obtain these results, we demonstrate a Bombieri-Vinogradov type theorem for Sato-Tate primes.
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