Related papers: Chang's Conjectures and Easton collapses

The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…

The aim of this paper is to collect some facts about the blowup of Jang's equation. First, we discuss how to construct solutions that blow up at an outermost MOTS. Second, we exclude the possibility that there are extra blowup surfaces in…

We answer a question of Slaman and Steel by showing that a version of Martin's conjecture holds for all regressive functions on the hyperarithmetic degrees. A key step in our proof, which may have applications to other cases of Martin's…

This paper presents some considerations about the Goldbach's conjecture (GC). The work is based on elementary results of the number theory and it provides a constructive method that permits, given an even integer, to find at least a pair of…

We use Koll\'ar's gluing theory to prove the contraction theorem for generalized pairs. In particular, we show that we can run the MMP for any generalized log canonical pairs.

We show that the Andrews-Curtis conjecture holds for all balanced presentations of the trivial group corresponding to Heegaard diagrams of $S^3$.

In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture.

In this short note, we give a simple proof of a Lee-Yang type theorem which appeared in "Lee-Yang theorems and the complexity of computing averages" by Alistair Sinclair and Piyush Srivastava.

We investigate the following surprisingly widespread phenomenon which we call The Rule of Three: in order for a particular kind of commutation relation to hold for subsequences of elements of a ring labeled by any subset of indices, it is…

For a twist knot $\mathcal{K}_{p'}$, let $M$ be the closed $3$-manifold obtained by doing $(p, q)$ Dehn-filling along $\mathcal{K}_{p'}$. In this article, we prove that Chen-Yang's volume conjecture holds for sufficiently large $|p| + |q|$…

We survey the state of the union-closed sets conjecture.

We give a very simple proof of a strengthened version of Chernoff's Inequality. We derive the same conclusion from much weaker assumptions.

We obtain new average results on the conjectures of Lang-Trotter and Sato-Tate about elliptic curves.

In this paper we consider the remaining cases of Hebey-Vaugon conjecture.

In this paper we describe a general strategy for approaching the Weinstein conjecture in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of contact manifolds (planar contact manifolds). We also…

We give three new proofs of the triangle inequality in Euclidean Geometry. There seems to be only one known proof at the moment. It is due to properties of triangles, but our proofs are due to circles or ellipses. We aim to prove the…

We compute the averages over elliptic curves of the constants occurring in the Lang-Trotter conjecture, the Koblitz conjecture, and the cyclicity conjecture. The results obtained confirm the consistency of these conjectures with the…

We give estimates on the number of combinatorial designs, which prove (and generalise) a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. This paper also serves as an expository treatment of our recently developed…

Artin's conjecture is established for all forms that can be realised as a diagonal form on an hyperplane.

In this note, we present a conjecture on intersections of set families, and a rephrasing of the conjecture in terms of principal downsets of Boolean lattices. The conjecture informally states that, whenever we can express the measure of a…