Sums of Reciprocals of Fractional Parts and Multiplicative Diophantine Approximation
Author: Victor Beresnevich
Publisher: American Mathematical Soc.
Published: 2020-04-03
Total Pages: 77
ISBN-13: 1470440954
DOWNLOAD EBOOK →Lanterna Rally
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Dynamics and Analytic Number Theory
Author: Dzmitry Badziahin
Publisher: Cambridge University Press
Published: 2016-11-10
Total Pages: 341
ISBN-13: 1316817776
DOWNLOAD EBOOK →Written by leading experts, this book explores several directions of current research at the interface between dynamics and analytic number theory. Topics include Diophantine approximation, exponential sums, Ramsey theory, ergodic theory and homogeneous dynamics. The origins of this material lie in the 'Dynamics and Analytic Number Theory' Easter School held at Durham University in 2014. Key concepts, cutting-edge results, and modern techniques that play an essential role in contemporary research are presented in a manner accessible to young researchers, including PhD students. This book will also be useful for established mathematicians. The areas discussed include ubiquitous systems and Cantor-type sets in Diophantine approximation, flows on nilmanifolds and their connections with exponential sums, multiple recurrence and Ramsey theory, counting and equidistribution problems in homogeneous dynamics, and applications of thin groups in number theory. Both dynamical and 'classical' approaches towards number theoretical problems are also provided.
Global Smooth Solutions for the Inviscid SQG Equation
Author: Angel Castro
Publisher: American Mathematical Soc.
Published: 2020-09-28
Total Pages: 89
ISBN-13: 1470442140
DOWNLOAD EBOOK →In this paper, the authors show the existence of the first non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation.
Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case
Author: Jacob Bedrossian
Publisher: American Mathematical Soc.
Published: 2020-09-28
Total Pages: 154
ISBN-13: 1470442175
DOWNLOAD EBOOK →The authors study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. They prove that for sufficiently regular initial data of size $epsilon leq c_0mathbf {Re}^-1$ for some universal $c_0 > 0$, the solution is global, remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the Couette flow as $t rightarrow infty $. For times $t gtrsim mathbf {Re}^1/3$, the streamwise dependence is damped by a mixing-enhanced dissipation effect and the solution is rapidly attracted to the class of ``2.5 dimensional'' streamwise-independent solutions referred to as streaks.
The Riesz Transform of Codimension Smaller Than One and the Wolff Energy
Author: Benjamin Jaye
Publisher: American Mathematical Soc.
Published: 2020-09-28
Total Pages: 97
ISBN-13: 1470442132
DOWNLOAD EBOOK →Fix $dgeq 2$, and $sin (d-1,d)$. The authors characterize the non-negative locally finite non-atomic Borel measures $mu $ in $mathbb R^d$ for which the associated $s$-Riesz transform is bounded in $L^2(mu )$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known. As an application, the authors give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-Delta )^alpha /2$, $alpha in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.
Conformal Graph Directed Markov Systems on Carnot Groups
Author: Vasileios Chousionis
Publisher: American Mathematical Soc.
Published: 2020-09-28
Total Pages: 153
ISBN-13: 1470442159
DOWNLOAD EBOOK →The authors develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, they develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen's parameter. They illustrate their results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the non-real classical rank one hyperbolic spaces.
Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
Author: Lisa Berger
Publisher: American Mathematical Soc.
Published: 2020-09-28
Total Pages: 131
ISBN-13: 1470442191
DOWNLOAD EBOOK →The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $mathbb F_p(t)$, when $p$ is prime and $rge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $mathbb F_q(t^1/d)$.
Filtrations and Buildings
Author: Christophe Cornut
Publisher: American Mathematical Soc.
Published: 2020-09-28
Total Pages: 150
ISBN-13: 1470442213
DOWNLOAD EBOOK →The author constructs and studies a scheme theoretical version of the Tits vectorial building, relates it to filtrations on fiber functors, and uses them to clarify various constructions pertaining to affine Bruhat-Tits buildings, for which he also provides a Tannakian description.
The Mother Body Phase Transition in the Normal Matrix Model
Author: Pavel M. Bleher
Publisher: American Mathematical Soc.
Published: 2020-09-28
Total Pages: 144
ISBN-13: 1470441845
DOWNLOAD EBOOK →In this present paper, the authors consider the normal matrix model with cubic plus linear potential.
Affine Flag Varieties and Quantum Symmetric Pairs
Author: Zhaobing Fan
Publisher: American Mathematical Soc.
Published: 2020-09-28
Total Pages: 123
ISBN-13: 1470441756
DOWNLOAD EBOOK →The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown to be a coideal subalgebra of the quantum group of finite type $A$.
