Researcher:
Jamneshan, Asgar

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Asgar

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Jamneshan

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Jamneshan, Asgar

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2022 - 20236

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    Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration
    (Polish Acad Sciences Inst Mathematics-Impan, 2023) Tao, Terence; Department of Mathematics; Jamneshan, Asgar; Faculty Member; Department of Mathematics; College of Sciences; 332404
    We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and probability algebras, and commutative C*-algebras and von Neumann algebras equipped with traces, in the "uncountable" set-ting in which no separability, metrizability, or standard Borel hypotheses are placed on these spaces and algebras. In particular, we review the Gelfand dualities and Riesz rep-resentation theorems available in this setting. We also present a canonical model that represents probability algebras as compact Hausdorff probability spaces in a completely functorial fashion, and apply this model to obtain a canonical disintegration theorem and to readily construct various product measures. These tools are useful in applications to "uncountable" ergodic theory (as demonstrated by the authors and others).
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    PublicationOpen Access
    An uncountable ergodic Roth theorem and applications
    (American Institute of Mathematical Sciences, 2022) Schmid, Polona durcik; Greenfeld, Rachel; Iseli, Annina; Jamneshan; Madrid, Jose; Department of Mathematics; Jamneshan, Asgar; Faculty Member; Department of Mathematics; College of Sciences; 332404
    We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements a result of Zorin-Kranich. We establish the following two additional results: First, a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. Second, a new uniformity aspect in the double recurrence theorem for Gamma-systems for uniformly amenable groups Gamma. As a special case, we obtain this uniformity over all Z-systems, and our result seems to be novel already in this case. Our uncountable Roth theorem is crucial in the proof of both of these results.
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    Decoupling for fractal subsets of the parabola
    (Springer Nature, 2022) Chang, Alan; Pont, Jaume de Dios; Greenfeld, Rachel; Li, Zane Kun; Madrid, Jose; Department of Mathematics; Jamneshan, Asgar; Faculty Member; Department of Mathematics; College of Sciences
    We consider decoupling for a fractal subset of the parabola. We reduce studying l(2)L(p) dccoupling for a fractal subset on the parabola {(t , t(2)) : 0 <= t <= 1} to studying l(2)L(p/3) decoupling for the projection of this subset to the interval [0, 1]. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain-Demeter's decoupling theorem for the parabola. In the case when p/3 is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to [0, 1]. Our ideas are inspired by the recent work on ellipsephic sets by Biggs (arXiv:1912.04351, 2019 and Acta Arith. 200(4):331-348, 2021) using nested efficient congruencing.
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    An uncountable Mackey-Zimmer theorem
    (Institute of Mathematics of the Polish Academy of Sciences, 2022) Tao, Terence; Department of Mathematics; Jamneshan, Asgar; Faculty Member; Department of Mathematics; College of Sciences; 332404
    The Mackey–Zimmer theorem classifies ergodic group extensions X of a measure-preserving system Y by a compact group K, by showing that such extensions are isomorphic to a group skew-product X?Y??H for some closed subgroup H of K. An analogous theorem is also available for ergodic homogeneous extensions X of Y, namely that they are isomorphic to a homogeneous skew-product Y??H/M. These theorems have many uses in ergodic theory, for instance playing a key role in the Host–Kra structural theory of characteristic factors of measure-preserving systems.The existing proofs of the Mackey–Zimmer theorem require various “countability”, “separability”, or “metrizability” hypotheses on the group ? that acts on the system, the base space Y, and the group K used to perform the extension. In this paper we generalize the Mackey–Zimmer theorem to “uncountable” settings in which these hypotheses are omitted, at the cost of making the notion of a measure-preserving system and a group extension more abstract. However, this abstraction is partially counteracted by the use of a “canonical model” for abstract measure-preserving systems developed in a companion paper. In subsequent work we will apply this theorem to also obtain uncountable versions of the Host–Kra structural theory.
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    PublicationOpen Access
    An uncountable Furstenberg-Zimmer structure theory
    (Cambridge University Press (CUP), 2022) Department of Mathematics; Jamneshan, Asgar; Faculty Member; Department of Mathematics; College of Sciences; 332404
    Furstenberg-Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure-preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogs of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg-Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi.
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    An uncountable Moore-Schmidt theorem
    (Cambridge University Press (CUP), 2022) Tao, Terence; Department of Mathematics; Jamneshan, Asgar; Faculty Member; Department of Mathematics; College of Sciences; 332404
    We prove an extension of the Moore-Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact Hausdorff abelian group. The proof relies on a 'conditional' Pontryagin duality for spaces of abstract measurable maps.