Related papers: Sarnak's Conjecture -- what's new
Let $\mathbb{T}$ be the unit circle and $\Gamma\setminus G$ the 3-dimensional Heisenberg nilmanifold. We prove that the M\"obius function is linearly disjoint from a class of distal skew products on $\mathbb{T}\times\Gamma\setminus G$.…
These are some notes on the two Milnor conjectures and their proofs (due to Voevodsky, Orlov-Vishik-Voevodsky, and Morel).
Recent developments in supersymmetric unified theories are reviewed, with particular emphasis on supersymmetric grand unification and a brief discussion of recent ideas about extra dimensions.
In this paper we attack the Erdos-Straus conjecture by means of the structure of its solutions, extending and improving the results of a previous paper. Using previous results and supported by the works of Elsholtz and Tao and Monks and…
Presented here are over one hundred conjectures ranging from easy to difficult, from many mathematical fields. I also summarize briefly methods and tools that have led to this collection.
The purpose of this note is to give an affirmative answer to a conjecture appearing in [Integral Transforms Spec. Funct. 26 (2015) 90-95].
This is a survey of recent advances in commutative algebra, especially in mixed characteristic, obtained by using the theory of perfectoid spaces. An explanation of these techniques and a short account of the author's proof of the direct…
The article provides a counterexample to a conjecture by Blocki-Zwonek.
We show that the M\"obius function is disjoint to every analytic skew product dynamical system on the two-torus over a rotation of the circle.
In a recent paper, Merca posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalized $m$-gonal numbers. Extending Merca's work, we complete the proof of these…
The survey presents the main developments obtained over the last decade regarding pointwise ergodic theorems for measure preserving actions of locally compact groups. The survey includes an exposition of the solutions to a number of long…
We retrace the recent history of the Umbral Calculus. After studying the classic results concerning polynomial sequences of binomial type, we generalize to a certain type of logarithmic series. Finally, we demonstrate numerous typical…
This paper proposes a generalized ABC conjecture and assuming its validity settles a generalized version of Fermats last theorem.
The Union Closed Sets Conjecture states that in every finite, nontrivial set family closed under taking unions there is an element contained in at least half of all the sets of the family. We investigate two new directions with respect to…
We present a new conjecture for the $SU_q(N)$ Perk-Schultz models. This conjecture extends a conjecture presented in our article (Alcaraz FC and Stroganov YuG (2002) J. Phys. A vol. 35 pg. 6767-6787, and also in cond-mat/0204074).
In this paper we present the statement of the Firoozbakht's conjecture, some of its consequences if it is proved and we show a consequence of Zhang's theorem concerning the Firoozbakht's conjecture.
In this paper, we show that for any sequence ${\bf a}=(a_n)_{n\in \Z}\in \{1,\ldots,k\}^\mathbb{Z}$ and any $\epsilon>0$, there exists a Toeplitz sequence ${\bf b}=(b_n)_{n\in \Z}\in \{1,\ldots,k\}^\mathbb{Z}$ such that the entropy $h({\bf…
This is an expanded lecture note for "Masterclass on sofic groups and applications to operator algebras" (University of Copenhagen, 5-9 November 2012). It is about algebraic aspects of the Connes Embedding Conjecture. It contains new proofs…
This is an appendix to our paper "An update of the Hirsch Conjecture" (arXiv:0907.1186), containing proofs of some of the results and comments that were omitted in it.
This is a survey about the Skorokhod embedding problem. It presents all known solutions together with their properties and some applications. Some of the solutions are just described, while others are studied in detail and their proofs are…