Related papers: Threefold Thresholds

We obtain a new bound on the second moment of modified shifted convolutions of the generalized 3-fold divisor function, and show that, for applications, the modified version is sufficient.

The study of the additive volume of sets can be reduced to the case of one-dimensional sets. The exact values of the volume of extremal sets are given as a conjecture.

We present a counter-example to the recent claim that supermultiplets of N-extended supersymmetry with no central charge and in 1-dimension are specified unambiguously by providing the numbers of component fields in all available…

We show that for every integer $k\geq3$, the set of Tur\'an densities of $k$-uniform hypergraphs has an accumulation point in $[0,1)$. In particular, $1/2$ is an accumulation point for the set of Tur\'an densities of $3$-uniform…

Assuming a lower bound on the dimension, we prove a long standing conjecture concerning the classification of global solutions of the obstacle problem with unbounded coincidence sets.

We improve the well-known Szemer\'edi-Trotter incidence bound for proper 3--dimensional point sets (defined appropriately)

We first introduce a notion of convex structure in generalized metric spaces, then we introduce tripartite contractions, tripartite semi-contractions, tripartite coincidence points, as well as tripartite best proximity points for a given…

We prove Sklar's theorem in infinite dimensions via a topological argument and the notion of inverse systems.

We outline the proof that non-triangulable manifolds exist in any dimension greater than four. The arguments involve homology cobordism invariants coming from the Pin(2) symmetry of the Seiberg-Witten equations. We also explore a related…

For any j_1,...,j_n>0 with j_1+...+j_n=1 and any x \in R^n, we consider the set of points y \in R^n for which max_{1\leq i\leq n}(||qx_i-y_i||^{1/j_i})>c/q for some positive constant c=c(y) and all q\in N. These sets are the `twisted'…

We give a new elementary proof of the parallelizability of closed orientable 3-manifolds. We use as the main tool the fact that any such manifold admits a Heegaard splitting.

We prove extension-dimensional versions of finite dimensional selection and approximation theorems. As applications, we obtain several results on extension dimension.

We prove the topological analogue of the period-index conjecture in each dimension away from a small set of primes.

Using standard methods (due to Janson, Stein-Chen, and Talagrand) from probabilistic combinatorics, we explore the following general theme: As one progresses from each member of a family of objects ${\cal A}$ being "covered" by at most one…

We obtain upper bounds on the number of singular points of factorial terminal Fano threefolds.

The weight-monodromy conjecture claims the coincidence of the weight filtration and the monodromy filtration, up to shift, on the $l$-adic \'etale cohomology of a proper smooth variety over a complete discrete valuation field. Although it…

We prove that the degree of a nonconstant morphism from a smooth projective 3-fold $X$ with N\'{e}ron-Severi group ${\bf Z}$ to a smooth 3-dimensional quadric is bounded in terms of numerical invariants of $X$. In the special case where $X$…

In this paper, we show that the maximum number of points in $d\geq3$ dimensions determining exactly 2 distinct triangles is $2d$. We further show that this maximum is uniquely achieved by the vertices of the $d$-orthoplex. We build upon the…

We prove the Nonvanishing Theorem for threefolds over an algebraically closed field $k$ of characteristic $p >5$.

This paper considers some different measures for how additively structured a convex set can be. The main result gives a construction of a convex set $A$ containing $\Omega(|A|^{3/2})$ three-term arithmetic progressions.