The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V. I. Arnold and V. A. Rokhlin. We make an attempt to systematize the principal results in the subject. After an exposition of general tools and results, special attention is paid to surfaces and curves on surfaces.

In this paper, we show that every separable simple tracially approximately divisible C ? $C^*$ -algebra has strict comparison, and it is either purely infinite or stable rank one. As a consequence, (non-unital) finite Z ${\cal Z}$ -stable

We extend upper bounds on the quantum independence number and Shannon capacity of graphs to their counterparts in commuting operator model. introduce a von Neumann algebraic generalization fractional Haemers bound (over $\mathbb {C}$ ...

The truncated moment problem asks to characterize finite sequences of real numbers that are the moments a positive Borel measure on Rn. Its tracial analog is obtained by integrating traces symmetric matrices and main topic this article. solution bivariate quartic with nonsingular 7×7 matrix M2 whose columns indexed words degree 2 was established Burgdorf Klep, while in our previous work we comp...

The Connes Embedding Problem (CEP) asks whether every separable II1 factor embeds into an ultrapower of the hyperfinite II1 factor. We show that the CEP is equivalent to the statement that every type II1 tracial von Neumann algebra has a computable universal theory.

In this note, we show that the theory of tracial von Neumann algebras does not have a model companion. This will follow from the fact that the theory of any locally universal, McDuff II1 factor does not have quantifier elimination. We also show how a positive solution to the Connes Embedding Problem implies that there can be no modelcomplete theory of II1 factors.

A classification in terms of K-theory and tracial states is obtained for those real structures which are compatible with the inductive limit structure of a simple C∗-inductive limit of direct sums of algebras of continous matrix valued functions on an interval.

We show that if \(S\), \(T\) are two commuting automorphisms of a standard Borel space such they generate free \(\mathbb{Z}^2\)-action then \(S\) and do not have same sets real valued bounded coboundaries. also prove weaker form Rokhlin Lemma for \(\mathbb{Z}^d\)-actions.