Related papers: The Baum-Connes conjecture: an extended survey
In this paper, we introduce a notion of stable coarse algebras for metric spaces with bounded geometry, and formulate the twisted coarse Baum--Connes conjecture with respect to stable coarse algebras. We prove permanence properties of this…
This chapter surveys advances in the field of Bayesian computation over the past twenty years, with missing data. It also contains some novel computational entries on the double-exponential model that may be of interest per se.
Let $G$ and $H$ be finitely generated groups. In this paper, we prove the quantitative coarse Baum--Connes conjecture for the free product $G* H$ under the assumption that the conjecture holds for both $G$ and $H$.
This is a survey of recent developments in combinatorics. The goal is to give a big picture of its many interactions with other areas of mathematics, such as: group theory, representation theory, commutative algebra, geometry (including…
Causal inference is a critical research topic across many domains, such as statistics, computer science, education, public policy and economics, for decades. Nowadays, estimating causal effect from observational data has become an appealing…
This is a brief review of the present status, of some recent developments and of the open challenges in string/M theory.
Fontaine-Mazur Conjecture is one of the core statements in modern arithmetic geometry. Several formulations were given since its original statement in 1993, and various angles have been adopted by numerous authors to try to tackle it.…
We give new proofs on Arnold Chord Conjecture and Weinstein Conjecture in M\times C which generalizes the previous works.
We show that the classical Baum-Connes assembly map is quantitatively an isomorphism for a class of lacunary hyperbolic groups, and we explain how to see that this class contains many examples of groups that contain graph sequences of large…
In this paper we adopt a geometric point of view regarding a famous conjecture due to Littlewood in diophantine approximation of real numbers. Following the spirit of the geometric theory of continued fractions, we give a sufficient…
Let F be a global field, A its ring of adeles, G a reductive group over F. We prove the Baum-Connes conjecture for the adelic group G(A).
After a short review of the Method of Recursive Counting we introduce a general algebraic description of recursive lattice building. This provides a rigorous framework for discussion of method's limitations.
We give the counter-examples related to a Gaussian Brunn-Minkowski inequality and the (B) conjecture.
This memoir is a survey of theorems and inequalities which have grown out of, and extended, the seminal estimate of Montgomery \cite{HM70} \begin{multline*} V(x,Q)=\sum_{q\le Q}\sum_{\substack{a=1\\ (a,q)=1}}^q \left| \psi(x;q,a) -…
We give combinatorial generalizations of the Cayley-Bacharach theorem and induced map.
We provide empirical evidence for the Erd\H{o}s-Straus conjecture by improving computational bounds to $10^{18}$ and by evaluating the solution-counting function $f(p)$ for this conjecture.
Inference and prediction are fundamental to the study of complex systems, where network data are often incomplete, inaccurate or obtained indirectly. In this paper, we review recent advances in network sampling and comparison, as well as in…
We review what is known about the Hodge conjecture for abelian varieties, with some emphasis on how Mumford-Tate groups have been applied to this problem.
We present an overview of how certain computational tools currently interact with mathematical practice, and reflect on the implications for research mathematics in the short to medium term, as the field navigates the emerging age of AI and…
In the paper different kinds of proof of a given statement are discussed. Detailed descriptions of direct and indirect methods of proof are given. Logical models illustrate the essence of specific types of indirect proofs. Direct proofs of…