Related papers: Ma\~n\'e's conjectures in codimension one
In a very celebrated paper A. Connes has formulated a conjecture which is now one of the most important open problem in Operator Algebras. This importance comes from the works of many mathematicians who have found some unexpected equivalent…
The counting and (upper) mass dimensions are notions of dimension for subsets of $\mathbb{Z}^d$. We develop their basic properties and give a characterization of the counting dimension via coverings. In addition, we prove Marstrand-type…
The codimension-three conjecture states that any regular holonomic module extends uniquely beyond an analytic subset with codimension equal to or larger than three. We give a proof of this conjecture.
We conjecture that the exceptional set in Manin's Conjecture has an explicit geometric description. Our proposal includes the rational point contributions from any generically finite map with larger geometric invariants. We prove that this…
In this paper we consider the remaining cases of Hebey-Vaugon conjecture.
We develop techniques to deal with monotonicity of sequences z_{n+1}/z_n and \sqrt[n]{z_n}. A series of conjectures of Zhi-Wei Sun and of Amdeberhan et al. are verified in certain unified approaches.
We show how the geometry of a 1-motive $M$ (that is existence of endomorphisms and relations between the points defining it) determines the dimension of its motivic Galois group ${\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}(M)$. Fixing periods…
For $n \geq 2$ we construct a measurable subset of the unit ball in $\mathbb{R}^n$ that does not contain pairs of points at distance 1 and whose volume is greater than $(1/2)^n$ times the volume of the ball. This disproves a conjecture of…
We prove Manin's conjecture for bi-equivariant compactifications of unipotent groups.
In this paper, we give an expository presentation of the paper of Olivier Mathieu. The paper of Mathieu proves that a Lie group-theoretic conjecture implies the Jacobian Conjecture. To give Mathieu's proof, we first review the required…
Let $\mathcal{D}=(d_n)_{n=1}^\infty$ be a bounded sequence of integers with $d_n\ge 2$ and let $(i, j)$ be a pair of strictly positive numbers with $i+j=1$. We prove that the set of $x \in \RR$ for which there exists some constant $c(x) >…
In this article, we verify the additivity for rank of a sum of coprime monomials and bivariate polynomials generalizing the result in (\cite{CCG}). We also show similar results hold for cactus rank.
In this paper, we confirm the Fino-Vezzoni Conjecture for unimodular Lie algebras which contain abelian ideals of codimension two, a natural generalization to the class of almost abelian Lie algebras. This provides new evidence towards the…
We prove Zhang's Dynamical Manin-Mumford Conjecture and Dynamical Bogomolov Conjecture for dominant endomorphisms of (P^1)^n. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system,…
We construct solutions to Monge-Amp\`ere equations whose Monge-Amp\`ere measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present…
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
In this paper, we find a class of Carrollian and Galilean contractions of (extended) BMS algebra in 3+1 and 2+1 dimensions. To this end, we investigate possible embeddings of 3D/4D Poincar\'{e} into the BMS${}_3$ and BMS${}_4$ algebras,…
We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes' Embedding Conjecture as well as the Measurable Determinant Conjecture of L\"uck, Sauer and Wegner hold for treeable equivalence relations.
The author introduces a conjecture about Makar-Limanov invariants of affine unique factorization domains over a field of characteristic zero. Then the author finds that the conjecture does not always hold when $\mathbbm{k}$ is not…
In this paper we give a proof of the Manickam-Mikl\'os-Singhi (MMS) conjecture for some partial geometries. Specifically, we give a condition on partial geometries which implies that the MMS conjecture holds. Further, several specific…