Real and Complex Analysis

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Methods Of Modern Mathematical Physics

The concept of invariance with respect to transformations of a group is one of the most important and successful ideas of nineteenth century mathematics. After the use of coordinates had dominated many branches of mathematics and physics for centuries, a critical review of these methods was initiated by a new look on its foundations.

The General Theory

Lectures on the Combinatorics of Free Probability: The free commutator

The history of the quadratic stochastic operators can be traced back to the work of Bernshtein (1924). For more than 80 years, this theory has been developed and many papers were published. In recent years it has again become of interest in connection with its numerous applications in many branches of mathematics, biology and physics. But most results of the theory were published in non-English journals, full text of which are not accessible. In this paper we give all necessary definitions and a brief description of the results for three cases: (i) discrete-time dynamical systems generated by quadratic stochastic operators; (ii) continuous-time stochastic processes generated by quadratic operators; (iii) quantum quadratic stochastic operators and processes. Moreover, we discuss several open problems.

Quadratic stochastic operators and processes: results and open problems

This text is the first complete study and monograph dedicated to singular traces. For mathematical readers the text offers, due to Nigel Kalton's contribution, a complete theory of traces on symmetrically normed ideals of compact operators. For mathematical physicists and other users of Connes' noncommutative geometry the text offers a complete reference to Dixmier traces and the deeper mathematical features of singular traces. An application section explores the consequences of these features, which previously were not discussed in general texts on noncommutative geometry.

Singular Traces: Theory and Applications

http://iopscience.iop.org/article/10.1070/RM2004v059n03ABEH000744/pdf

On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex

Banach limits and traces on L1

Let M be a semi-finite von Neumann algebra and let f : R → C be a Lipschitz function. If A,B ∈ M are self-adjoint operators such that (A,B) ∈ L1(M), then

Weak type commutator and Lipschitz estimates: resolution of the Nazarov-Peller conjecture

In this paper, we propose a new approach to Cwikel estimates both for the Euclidean space and for the noncommutative Euclidean space.

Dmitriy Zanin

Papers

Singular Traces: Theory and Applications

On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex

Banach limits and traces on L1

Weak type commutator and Lipschitz estimates: resolution of the Nazarov-Peller conjecture

Cwikel estimates revisited