Related papers: Sarnak's Conjecture -- what's new
We present a history of the Baum-Connes conjecture, the methods involved, the current status, and the mathematics it generated.
We show that Sarnak's conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.
In this note, while giving an overview of the state of art of the well known Hadamard conjecture, which is more than a century old and now it has been established by using the methods given in the two papers by Mohan et al [6,7].
We develop the intersection theory associated to immersed, oriented and mea- sured solenoids, which were introduced in arXiv:0910.2836.
We give a survey on recent development of the Novikov conjecture and its applications to topological rigidity and non-rigidity. .
We give a number theoretic proof of the integrality of certain BPS invariants of knots. The formulas for these numbers are sums involving binomial coefficients and the M\"obius function. We also prove a conjecture about further divisibility…
This is part one of a series of papers. In this series of papers, we consider problems analogous to the Oppenheim conjecture from the viewpoint of prehomogeneous vector spaces.
Two conjectures recently proposed by one of the authors are disproved
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…
We announce numerous new results in the theory of orthogonal polynomials on the unit circle.
Recent years have seen dramatic progress in the study of joint ergodicity, i.e. a scenario in which a multiple ergodic average converges in norm to the product of integrals of individual functions. This survey, accompanying the talk given…
We survey some old and new results on strong variants of Chang's Conjecture and related topics.
This is an informal paper presenting historical results around the recent paper of the author about Lang's Conjecture and torsion of elliptic curves. This paper also discusses a few aspects of the proof.
We formulate the generalized Sarnak's M\"obius disjointness conjecture for an arbitrary number field $K$, and prove a quantitative disjointness result between polynomial nilsequences $(\Phi(g(n)\Gamma))_{n\in\mathbb{Z}^{D}}$ and aperiodic…
We show that some mathematical results and their negations are both deducible. The derived contradictions indicate the inconsistency of current mathematics. This paper is an updated version of arXiv:math/0606635v3 with additional results…
We survey some recent developments and give a list of open problems regarding multiple recurrence and convergence phenomena of $\mathbb{Z}^d$ actions in ergodic theory and related applications in combinatorics and number theory.
Given a topological dynamical system $(X,T)$ and an arithmetic function $\boldsymbol{u}\colon\mathbb{N}\to\mathbb{C}$, we study the strong MOMO property (relatively to $\boldsymbol{u}$) which is a strong version of…
We study the spectral disjointness of the powers of a rank-one transformation. For a large class of rank-one constructions, including those for which the cutting and stacking parameters are bounded, and other examples such as rigid…
We obtain similar types of conclusions as that of Br\"{u}ck [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the…
We discuss recent progress many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.