Logic and C∗\mathrm{C}^*-algebras: set theoretical dichotomies in the theory of continuous quotients

Given a nonunital $\mathrm{C}^*$-algebra $A$ one constructs its corona algebra $\mathcal M(A)/A$. This is the noncommutative analog of the \v{C}ech-Stone remainder of a topological space. We analyze the two faces of these algebras: the first one is given assuming CH, and the other one arises when Forcing Axioms are assumed. In their first face, corona $\mathrm{C}^*$-algebras have a large group of automorphisms that includes nondefinable ones. The second face is the Forcing Axiom one; here the automorphism group of a corona $\mathrm{C}^*$-algebra is as rigid as possible, including only definable elementsComment: This is the author's Ph.D. thesis, defended in April 2017 at York University, Toront

Nonlinear Analysis and Optimization with Applications

Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world

Effective statistical physics of Anosov systems

We present evidence indicating that Anosov systems can be endowed with a unique physically reasonable effective temperature. Results for the two paradigmatic Anosov systems (i.e., the cat map and the geodesic flow on a surface of constant negative curvature) are used to justify a proposal for extending Ruelle's thermodynamical formalism into a comprehensive theory of statistical physics for nonequilibrium steady states satisfying the Gallavotti-Cohen chaotic hypothesis.Comment: 38 pages, 17 figures. Substantially more details in sections 4 and 6; new and revised figures also added. Typos and minor errors (esp. in section 6) corrected along with minor notational changes. MATLAB code for calculations in section 16 also included as inline comment in TeX source now. The thrust of the paper is unaffecte

The definable content of homological invariants I: Ext\mathrm{Ext} & lim1\mathrm{lim}^1

This is the first installment in a series of papers in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information. To effect this enrichment, we show that many of these invariants can be naturally regarded as functors to the category of groups with a Polish cover. The resulting definable invariants provide far stronger means of classification. In the present work we focus on the first derived functors of $\mathrm{Hom}(-,-)$ and $\mathrm{lim}(-)$. The resulting definable $\mathrm{Ext}(B,F)$ for pairs of countable abelian groups $B,F$ and definable $\mathrm{lim}^{1}(\boldsymbol{A})$ for towers $\boldsymbol{A}$ of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable $\textrm{Ext}(-,\mathbb{Z})$ is a fully faithful contravariant functor from the category of finite rank torsion-free abelian groups $\Lambda$ with no free summands; this contrasts with the fact that there are uncountably many non-isomorphic such groups $\Lambda$ with isomorphic classical invariants $\textrm{Ext}(\Lambda,\mathbb{Z})$. To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover and we prove several rigidity results for non-Archimedean abelian groups with a Polish cover. A special case of our main result answers a question of Kanovei and Reeken regarding quotients of the $p$-adic groups. Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem $\mathcal{R}(\mathrm{Aut}(\Lambda)\curvearrowright\mathrm{Ext}(\Lambda,\mathbb{Z}))$ of classifying all group extensions of $\Lambda$ by $\mathbb{Z}$ up to base-free isomorphism, when $\Lambda =\mathbb{Z}[1/p]^{d}$ for prime numbers $p$ and $d\geq 1$.Comment: Typos fixed, and a table of contents adde

Hyers-Ulam-Rassias stability of nonlinear integral equations through the Bielecki metric

We analyse different kinds of stabilities for classes of nonlinear integral equations of Fredholm and Volterra type. Sufficient conditions are obtained in order to guarantee Hyers-Ulam-Rassias, $\sigma$-semi-Hyers-Ulam and Hyers-Ulam stabilities for those integral equations. Finite and infinite intervals are considered as integration domains. Those sufficient conditions are obtained based on the use of fixed point arguments within the framework of the Bielecki metric and its generalizations. The results are illustrated by concrete examples

Game theory

game theory