Related papers: On the Vinogradov mean value

For an elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication, Lang and Trotter conjectured that the number of primes $p <X$ at which $E$ has a supersingular reduction is asymptotically equal to $c\sqrt{X}/\log X$, where $c>0$…

In 2014, Keevash famously proved the existence of $(n,q,r)$-Steiner systems as part of settling the Existence Conjecture of Combinatorial Designs (dating from the mid-1800s). In 2020, Glock, K\"uhn, and Osthus conjectured a minimum degree…

We prove a general decomposition theorem for the modal $\mu$-calculus $L_\mu$ in the spirit of Feferman and Vaught's theorem for disjoint unions. In particular, we show that if a structure (i.e., transition system) is composed of two…

Two conjectures recently proposed by one of the authors are disproved

In the famous paper of Deligne and Mumford, they proved that a proper hyperbolic curve over a discrete valuation field has stable reduction if and only if the Jacobian variety of the curve has stable reduction in the case where the residue…

We present several results related to statistics for elliptic curves over a finite field $\mathbb{F}_p$ as corollaries of a general theorem about averages of Euler products that we demonstrate. In this general framework, we can reprove…

A previous result about the decategorified bordered (sutured) Heegaard Floer invariants of surfaces glued together along intervals, generalizing the decategorified content of Rouquier and the author's higher-tensor-product-based gluing…

This is the author's 2008 thesis from the University of Chicago. We generalize the notion of the Clifford index to an arbitrary very ample line bundle and show how it determines when a curve and its various secant varieties have…

Using the recent result of Bourgain, Demeter and Guth on Vinogradov's mean value, a number of new results about small fractional parts of polynomials and fractional parts of additive forms are obtained. These improve work of Baker, Cook,…

More than a century ago, G. Kowalewski stated that for each n continuous functions on a compact interval [a,b], there exists an n-point quadrature rule (with respect to Lebesgue measure on [a,b]), which is exact for given functions. Here we…

We first prove a general gluing theorem which creates new nondegenerate constant mean curvature surfaces by attaching half Delaunay surfaces with small necksize to arbitrary points of any nondegenerate CMC surface. The proof uses the method…

We study normal analytic compactifications of C^2 and describe their singularities and configuration of curves at infinity, in particular improving and generalizing results of (Brenton, Math. Ann. 206:303--310, 1973). As a by product we…

We formulate a generalization of Vojta's conjecture in terms of log pairs and variants of multiplier ideals. In this generalization, a variety is allowed to have singularities. It turns out that the generalized conjecture for a log pair is…

In this paper, we survey recent developments concerning the stability of naturally defined bundles on curves that play a central role in the deformation theory of the curve.

This is a new and short proof of the main theorem of classical structure tree theory. Namely, we show the existence of certain automorphism-invariant tree-decompositions of graphs based on the principle of removing finitely many edges. This…

This paper is concerned with rational curves on real classical groups. Our contributions are three-fold: (i) We determine the structure of quadratic rational curves on real classical groups. As a consequence, we completely classify…

The famous Sidorenko's conjecture asserts that for every bipartite graph $H$, the number of homomorphisms from $H$ to a graph $G$ with given edge density is minimized when $G$ is pseudorandom. We prove that for any graph $H$, a graph…

Motivated by his studies in knot theory V. Vassiliev introduced $X$-graphs as regular 4-valent graph with a structure of pairs of opposite edges at each vertex. He conjectured the conditions under which $X$-graph can be embedded into a…

In 1983, Nochka proved a conjecture of Cartan on defects of holomorphic curves in CP^n relative to a possibly degenerate set of hyperplanes. In this paper, we generalize the Nochka's theorem to the case of curves in a complex projective…

We consider general integrable curve nets in Euclidean space as a particular integrable geometry invariant with respect to rigid motions and net-preserving reparameterisations. For the purpose of their description, we first give an overview…