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moshchevitin [2020/07/04 10:00]
Уланский Евгений Алесандрович [Lectures, new and old]
moshchevitin [2022/02/28 17:00] (текущий)
Мощевитин Н.Г. [Lectures and Events, new and old]
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 ====== Спецкурсы и спецсеминары Николая Германовича Мощевитина ====== ====== Спецкурсы и спецсеминары Николая Германовича Мощевитина ======
-Написать [[Мощевитин Николай Германович|Николаю Германовичу]] можно по почте: **moshchevitin@gmail.com** ​или в скайп: Nikolaus Moshchevitin\\+Написать [[Мощевитин Николай Германович|Николаю Германовичу]] можно по почте: **moshchevitin@gmail.com** ​ 
 \\ \\
 ====== Online seminar "​Diophantine Analysis"​ ====== ====== Online seminar "​Diophantine Analysis"​ ======
Строка 10: Строка 10:
 If you are interested in participating please contact Nikolay Moshchevitin (**moshchevitin@gmail.com**).\\ If you are interested in participating please contact Nikolay Moshchevitin (**moshchevitin@gmail.com**).\\
  
-===== Lectures, new and old =====+===== Lectures ​and Events, new and old ===== 
 + 
 + 
 + 
 + ​**2022** 
 + 
 + 
 +CANCELLED **1**. Thursday, March 10  at 15:00 Moscow time (GMT+3). ​  ​CANCELLED 
 +\\ **Speaker:​** Wolfgang Steiner 
 +\\ **Title:​** ​  On the second Lyapunov exponent of some multidimensional continued fraction algorithms 
 +\\ **Abstract:​**  
 +We study the strong convergence of certain multidimensional continued fraction algorithms. In particular, in the two- and three-dimensional case, we prove that the second Lyapunov exponent of Selmer'​s algorithm is negative and bound it away from zero. Moreover, we give heuristic results on several other continued fraction algorithms. Our results indicate that all classical multidimensional continued fraction algorithms cease to be strongly convergent for high dimensions. The only exception seems to be the Arnoux-Rauzy algorithm which, however, is defined only on a set of measure zero. 
 +This is joint work with Valérie Berthé and Jörg Thuswaldner.  
 + 
 +  
 + 
 + \\ 
 + 
 +------------------------------------------------------------------------------------------------ 
 + 
 +** Online conference "DAYS of TRANSCENDENCE"​** 
 + 
 +in occasion of Professor ​ Yuri Nesterenko'​s 75th birthday 
 + 
 +January 31 - February 02, 2022 
 +  
 +All the information about the meeting is [[nesterenko75|here]]. 
 + 
 +------------------------------------------------------------------------------------------------- 
 + 
 + \\ 
 + 
 + ​**2021** 
 + 
 + 
 +**18**. Tuesday, December 21  at 15:00 Moscow time (GMT+3). 
 +\\ **{{https://​disk.yandex.ru/​i/​Ob6gl5smcRl2rg|Slides of the lecture}}** 
 +\\ **[[https://​disk.yandex.ru/​d/​tepuhs5y6XOLyg|Recording of the lecture]]** 
 +\\ **Speaker:​** Iskander Aliev 
 +\\ **Title:​** ​ Proximity and sparsity of integer points in rational polyhedra 
 +\\ **Abstract:​** 
 + We will discuss proximity and sparsity of integer points in a rational polyhedron P given in the standard form P = {x: Ax = b, x nonnegative},​ where A is an integer mxn matrix and b is an integer vector. The proximity-type results estimate the distance from a vertex of P to the set of its integer points. The sparsity-type results provide bounds for the size of support of integer points in P.  We will show that the proximity and sparsity are closely connected and establish a transference bound that implies, in certain scenarios, a drastic improvement on previously known results. 
 + 
 +  
 + 
 + 
 +**17**. Tuesday, December 14 at 15:00 Moscow time (GMT+3). 
 +\\ **{{https://​disk.yandex.ru/​i/​uU8Gkcfypd2IxQ|Slides of the lecture}}** 
 +\\ **[[https://​disk.yandex.ru/​i/​MC-AXLZ0pcoyzA|Recording of the lecture]]** 
 +\\ **Speaker:​** V. Vinay Kumaraswamy 
 +\\ **Title:​** ​ Some new results in effective diophantine approximation 
 +\\ **Abstract:​** In this talk, I will discuss recent joint work with A. Ghosh on 
 +the problem of finding ‘small’ solutions to inequalities involving temary 
 +diagonal quadratic forms. We study this problem on average over a 
 +one-parameter family of such forms. Building on work by Bourgain, I will 
 +present new results examining this problem over the primes, as well as a 
 +generalisation to inhomogeneous forms. 
 + 
 + 
 +**16**. Tuesday, November 30 at 15:00 Moscow time (GMT+3). 
 +\\ **{{https://​disk.yandex.ru/​i/​1hdzcZ6BxFn05w|Slides of the lecture}}** 
 +\\ **[[https://​disk.yandex.ru/​i/​Sbu9J3wUsMTGEw|Recording of the lecture]]** 
 +\\ **Speaker:​** Anurag Rao 
 +\\ **Title:​** ​ On a theorem of Davenport-Schmidt on Dirichlet improvable pairs 
 +\\ **Abstract:​** 
 + In 1970, Davenport and Schmidt studied a Diophantine property of pairs of real numbers; it concerned those pairs for which the classical Dirichlet theorem can be improved. They showed that the set of Dirichlet-improvable pairs, while small in the sense of having zero Lebesgue measure, has full Hausdorff dimension. We study a similar Dirichlet-improvable property, where the approximations are made using an arbitrary norm rather than the supremum norm, and show the same result. To this end this, we recast the Dirichlet-improvable property into a dynamical property of certain orbits in the space of unimodular lattices, and prove a Hajos-Minkowski type result in the geometry of numbers. This is joint work with Dmitry Kleinbock. 
 + 
 +**15**. Thursday, November 18, at 15:00 Moscow time (GMT+3). 
 +\\ **{{https://​disk.yandex.ru/​d/​ZyfCYlBNQlW72Q|Slides of the lecture}}** 
 +\\ **[[https://​disk.yandex.ru/​i/​ioztM8mQJEQRew|Recording of the lecture]]** 
 +\\ **Speaker:​** Stephane Fischler 
 +\\ **Title:** Linear independence of odd zeta values using Siegel'​s lemma 
 +\\ **Abstract:​**  
 +Conjecturally,​ 1 and all values of the Riemann zeta function at odd  
 +integers $s\geq 3$ are linearly independent over the rationals (and these  
 +zeta values are, therefore, irrational). However, very few is known in  
 +this direction. Ap\'​ery proved in 1978 that $\zeta(3)$ is irrational;  
 +Ball-Rivoal proved in 2001 that for any $\epsilon >0$, at least  
 +$(1-\epsilon) (\log s) / (1+\log 2)$ numbers among 1, $\zeta(3)$,  
 +$\zeta(5)$, ..., $\zeta(s)$ are linearly independent over the rationals,  
 +when $s$ is odd and large enough in terms of $\epsilon$. In this lecture  
 +we shall explain how this lower bound can be improved to $0.21  
 +\sqrt{s/​\log s}$. The strategy is to replace explicit constructions with  
 +the use of Siegel'​s lemma. 
 + 
 +**14**. Tuesday, October 26, at 13:00 Moscow time (GMT+3). 
 +\\ **[[https://​disk.yandex.ru/​i/​O7V7y89LS3TMdQ|Recording of the lecture]]** 
 +\\ **Speaker:​** Mumtaz Hussain 
 +\\ **Title:** Improvements to Dirichlet’s theorem and limit theorems for sums of partial quotients 
 +\\ **Abstract:​** By using the continued fractions, Khintchine and Jarnik’s theorems are concerned with the growth of large partial quotients whereas, recently, it has been shown that improvements to Dirichlet’s theorem is concerned with the growth of the product of consecutive partial quotients. I will present a near complete picture for the size of the set of Dirichlet non-improvable numbers, ​ weak and strong laws for the partial sums of the product of consecutive partial quotients, ​ and its increasing rate from a multifractal point of view. 
 + 
 +**13**. Tuesday, October 19, at 15:00 Moscow time (GMT+3). 
 +\\ **{{ https://​disk.yandex.ru/​d/​vYSUh9RzTgy3VQ|Slides of the lecture}}** 
 +\\ **[[https://​disk.yandex.ru/​i/​u-4a705uUghdNA|Recording of the lecture]]** 
 +\\ **Speaker:​** Leonhard Summerer 
 +\\ **Title:** On simultaneous approximation to m numbers 
 +\\ **Abstract:​** ​ {{https://​disk.yandex.ru/​i/​ILb1cZ1W3_CHYg|IS HERE}} 
 +  
 + 
 +**12**. Tuesday, October 05, at 15:00 Moscow time (GMT+3). 
 +\\ **{{https://​disk.yandex.ru/​d/​zQ_Nhv043rAhWQ|Slides of the lecture}}** 
 +\\ **[[https://​disk.yandex.ru/​d/​EB-lpagiX6P0eg |Recording of the lecture PART 1]]** 
 +\\ **[[https://​disk.yandex.ru/​d/​CyI0zZyXeyMz4g |Recording of the lecture PART 2]]** 
 +\\ **Speaker:​** Byungchul Cha 
 +\\ **Title:** Intrinsic Diophantine Approximation of spheres 
 +\\ **Abstract:​** Let $S^1$ be the unit circle in $\mathbb{R}^2$ centered at the origin and let $Z$ be the set of rational points on $S^1$. We give a complete description of an initial discrete part of the Lagrange spectrum of $S^1$ in the sense of intrinsic Diophantine approximation. This is an analogue of the classical result of Markoff in 1879, where he characterized the most badly approximable real numbers via the periods of their continued fraction expansions. Our proof can be adapted to give similar results for a different intrinsic approximation of $S^1$. Namely, we prove a similar result when $S^1$ is the unit circle $|z| = 1$ in the complex plane and $Z$ is the $\mathbb{Q}(\sqrt{-3})$-rational point of $S^1$. We also study intrinsic Diophantine approximation of a two-sphere $S^2$. We consider three different pairs $(X_i, Z_i)$ for $i=1, 2, 3$, where $X_i$ is a two-sphere $S^2$ (embedded differently in a Euclidean space) and $Z_i$ is a countable dense subset of $X_i$. In each case, we determine an initial discrete part of its Lagrange spectrum.  
 +This is joint work with Dong Han Kim. 
 + 
 +**11**. Wednesday, September 29, at 16:00 Moscow time (GMT+3). 
 +\\ **{{https://​disk.yandex.ru/​i/​FMBE7qkAtyuBcA|Slides of the lecture}}** 
 +\\ **[[ https://​disk.yandex.ru/​i/​Uq5wlZ4SqiQVOg|Recording of the lecture]]** 
 +\\ **Speaker:​** Felipe Alberto Ramírez 
 +\\ **Title:** On an inhomogeneous,​ nonmonotonic version of the Khintchine-Groshev theorem 
 +\\ **Abstract:​** The classical Khintchine-Groshev theorem (for approximation of systems of $m$ linear forms in $n$ variables) makes a monotonicity assumption on the approximating function, but it is known that that monotonicity assumption can actually be omitted from the statement whenever $nm>1$. The motivating problem for this talk is to remove the corresponding monotonicity assumption from the inhomogeneous version of the Khintchine-Groshev theorem. In joint work with Demi Allen, we take a point of view that allows us to remove the monotonicity assumption whenever $nm>2$. Informally speaking, the idea is that the approximation sets arising in this problem should become "more probabilistically independent"​ as the number of variables $n$ increases. After quantifying this point of view, we find that it is enough to understand what happens in the cases where $n=1$, that is, in the setting of simultaneous approximation where we have Khintchine'​s theorem and its inhomogeneous analogue. In a sense, we show that "​Khintchine"​ implies "​Khintchine-Groshev."​ 
 + 
 +\\ 
 + 
 +------------------------------------------------------------------------------------------------ 
 + 
 +** Online conference "​DIOPHANTINE ANALYSIS AND RELATED TOPICS"​**. ​ May 31 - June 04  
 +\\ **{{http://​mjcnt.phystech.edu/​conference/​diophantine_analysis_2021/​|Site of the conference}}** 
 +\\ You can find the slides and the recordings of most of the lectures ​ [[DART2012 slides and recordings|here]]. 
 + 
 +------------------------------------------------------------------------------------------------- 
 + 
 + 
 +\\ 
 + 
 + 
 +**10**. Tuesday, May 11, at 14:00 Moscow time (GMT+3).  
 +\\ **{{https://​disk.yandex.ru/​i/​CFXQHS8npYt7jw|Slides of the lecture}}** 
 +\\ **[[ https://​disk.yandex.ru/​i/​1kkFSo4rEEkiUg|Recording of the lecture]]** 
 +\\ **Speaker:​** Agamemnon Zafeiropoulos 
 +\\ **Title:** Multiplicative Inhomogeneous Diophantine Approximation 
 +\\ **Abstract:​** Let $γ,δ \in \mathbb{R}$. We establish a metric quantitative result on $n ||nα-γ|| || nβ-δ|| $, where: 
 + 
 + - α lies in a subset of [0,1] of Lebesgue measure 1 
 + 
 + - β lies in a set of badly approximable numbers of Hausdorff dimension 1. 
 + 
 +We also generalise a relevant theorem of Haynes, Jensen and Kristensen. (Based on joint work with Sam Chow). 
 + 
 +**9**. Thursday, May 06, at 15:00 Moscow time (GMT+3).  
 +\\ **{{https://​disk.yandex.ru/​i/​30FgwyM2aLjDTw|Slides of the lecture}}** 
 +\\ **[[https://​disk.yandex.ru/​i/​EEj5W9vsUi4vJg|Recording of the lecture]]** 
 +\\ **Speaker:​** Andreas Strömbergsson 
 +\\ **Title:** Towards a zero-one law for improvements to Dirichlet'​s Theorem in general dimension 
 +\\ **Abstract:​** Let psi be a decreasing function defined on all large positive real numbers. 
 +We say that a real m times n matrix Y is "​psi-Dirichlet"​ if for every sufficiently 
 +large real number t one can find integer vectors p (m-dim) and q (n-dim),  
 +with q non-zero, satisfying |Yq-p|^m<​psi(t) and |q|^n<​t 
 +(where the bars denote supremum norm on vectors). 
 +This property was introduced by 
 +Kleinbock and Wadleigh in 2018, and it generalizes the property of Y being 
 +"​epsilon-Dirichlet improvable"​ which has been studied by several people, 
 +starting with Davenport and Schmidt in 1969. We will present results 
 +giving sufficient conditions on psi to ensure that the set of psi-Dirichlet  
 +matrices Y has zero, resp., full measure. 
 +This is joint work with Dmitry Kleinbock and Shucheng Yu. 
 +  
 +**8**. Thursday, April 08, at 17:00 Moscow time (GMT+3). 
 +\\ **{{https://​disk.yandex.ru/​i/​q8zkxs8oWOCDWw|Slides of the lecture}}** 
 +\\ **[[https://​disk.yandex.ru/​d/​AckBGDeLpVPmnw|Recording of the lecture]]** 
 +\\ **Speaker:​** Lenny Fukshansky 
 +\\ **Title:** Representing integers by multilinear polynomials 
 +\\ **Abstract:​** 
 +Given a homogeneous multilinear polynomial F(x) in n variables with integer coefficients,​ we obtain some sufficient conditions for it to represent all the integers. Further, we derive effective results, establishing bounds on the size of a solution x to the equation F(x) = b, where b is any integer. For a special class of polynomials coming from determinants of rectangular matrices we are able to obtain necessary and sufficient conditions for such an effective representation problem. This result naturally connects to the problem of extending a collection of primitive vectors to a basis in a lattice, where we present counting estimates on the number of such extensions. The talk is based on joint works with A. Boettcher and with M. Forst. 
 + 
 +**7**. Thursday, March 25, at 18:00 Moscow time (GMT+3). 
 +\\ **{{https://​disk.yandex.ru/​i/​_ZmjM4uLso46nw|Slides of the lecture}}** 
 +\\ **[[https://​disk.yandex.ru/​i/​tr0WeW02Dkdo4g|Recording of the lecture]]** 
 +\\ **Speaker:​** Dmitriy Bilyk 
 +\\ **Title:** Discrete minimizing measures 
 +\\ **Abstract:​** In many natural examples, measures or point configurations minimizing various energy functionals tend to be  uniformly distributed. However, a peculiar effect is observed for certain energies, especially with attractive-repulsive potentials: minimizing measures are not spread out over the domain, but rather turn out to be discrete (or, at least, supported on small sets). The collections of  points ​ that arise as discrete minimizers are often closely related to various objects and problems of discrete geometry: spherical designs, optimal codes, tight frames, equiangular lines, mutually unbiased bases, etc. We shall discuss this phenomenon and its applications,​ as well as the interplay between energy minimization and discrepancy theory.  
 + 
 + 
 +**6**. Tuesday, March 23, at 15:00 Moscow time (GMT+3). 
 +\\ **{{https://​disk.yandex.ru/​i/​uxA9GA5cbDD2vA|Slides of the lecture}}** 
 +\\ **[[https://​disk.yandex.ru/​i/​otS59tp4SsdJ6w|Recording of the lecture]]** 
 +\\ **Speaker:​** Verónica Becher 
 +\\ **Title:** Nested perfect necklaces and normal numbers 
 +\\ **Abstract:​** M. B. Levin used Sobol-Fauré low discrepancy sequences with Pascal triangle matrices modulo 2 to construct a real number x such that the first N terms of the sequence (2^n x mod 1)_{n≥1} have discrepancy O((log N)^2/N). This is the lowest discrepancy known for this kind of sequences. In this talk I will  show that  Levin’s construction can be characterized ​ in terms of nested perfect necklaces, which are a variant of the classical de Bruijn sequences. Moreover, ​  every real number x whose binary expansion is the concatenation of nested perfect necklaces of exponentially increasing order satisfies that the first N terms of (2^n x mod 1)_{n≥1} have discrepancy O((log N)^2/​N). ​ For the order being a power of 2,  we know  the exact number of nested perfect necklaces and an explicit method based on matrices to construct each of them.  
 +  
 + 
 +**5**. Tuesday, March 16, at 15:00 Moscow time (GMT+3). 
 +\\ **{{https://​disk.yandex.ru/​i/​1P7RfVY0Hg7Lxg|Slides of the lecture}}** 
 +\\ **[[  https://​disk.yandex.ru/​d/​V4efeZHWCNvOkg|Recording of the lecture]]** 
 +\\ **Speaker:​** Alexander Gorodnik 
 +\\ **Title:** Can one count the "​sound"​ of the space? 
 +\\ **Abstract:​** We will explore some connections between Arithmetic (counting Diophantine solutions) and Analysis (spectral decomposition). 
 + 
 +  
 +  
 +**4**. Tuesday, March 02, at 17:00 Moscow time (GMT+3). 
 +\\ **[[ https://​disk.yandex.ru/​d/​FBihhsHtQVB_jw|Recording of the lecture]]** 
 +\\ **Speaker:​** Alex Kontorovich 
 +\\ **Title:** Diophantine Analysis on Groups 
 +\\ **Abstract:​** ​ We will describe some problems in geometry/​arithmetic/​group theory that can be attacked by methods from Diophantine analysis. 
 + 
 + 
 +**3**. Tuesday, February 16, at 14:00 Moscow time (GMT+3). 
 +\\ **{{ https://​disk.yandex.ru/​i/​zDuEWwZo6Hm-5Q|Slides of the lecture}}** 
 +\\ **[[ https://​disk.yandex.ru/​i/​F9pM-RWuJHHg7A|Recording of the lecture]]** 
 +\\ **Speaker:​** Anish Ghosh 
 +\\ **Title:** Diophantine approximation on varieties 
 +\\ **Abstract:​** I will discuss some recent results in the theory of intrinsic Diophantine approximation on varieties with an emphasis on quantitative results. 
 +  
 + 
 +**2**. Thursday, January 21, at 16:00 Moscow time (GMT+3). 
 +\\ **{{ https://​yadi.sk/​i/​UVXE5vwD5xRtSQ|Slides of the lecture}}** 
 +\\ **[[ https://​yadi.sk/​i/​h3LE0bDtPual0w|Recording of the lecture]]** 
 +\\ **Speaker:​** James Maynard 
 +\\ **Title:** Simultaneous small fractional parts of polynomials 
 +\\ **Abstract:​** Given k real numbers \alpha_1,​..,​\alpha_k,​ how well can you simultaneously approximate these real numbers with squares? More specifically,​ how small can you make all the fractional parts \{n^2\alpha_1\},​ ... ,​\{n^2\alpha_k\}$ using integers n<x? Classical work of Schmidt shows that there is an integer n<x such that all of these fractional parts are at most x^{-c/k^2} (for some constant c>0), uniformly for every choice of \alpha_i. I'll present some recent work which improves this bound to x^{-c/k}, which is optimal up to the value of the constant c>0 and similar results for more general polynomials instead of the squares. The improvement comes from a using fun blend of additive combinatorics and geometry of numbers. 
 + 
 +  
 +**1**. Thursday, January 14, at 17:00 Moscow time (GMT+3). 
 +\\ **{{https://​yadi.sk/​i/​Goeh8ksZtmzCsQ|Slides of the lecture}}** 
 +\\ **Speaker:​** Polina Vytnova 
 +\\ **Title:** Computing Hausdorff dimension of sets of continued fractions 
 +\\ **Abstract:​** We will present a simple and practical approach to get rigorous bounds 
 +on the Hausdorff dimension of limits sets of some one dimensional Markov iterated 
 +function schemes which naturally arise in number theory, in particular in the study 
 +of Markov and Lagrange spectra and in questions related to Zaremba conjecture 
 + 
 +\\ 
 + 
 + 
 +------------------------------------------------------------------------------------------------ 
 + 
 +\\  
 + 
 +**2020** 
 + 
 +  
 + 
 + 
 + 
 + ​**21**. Tuesday, December 15, at 14:00 Moscow time (GMT+3). 
 +\\ **[[https://​yadi.sk/​i/​txFPfsIgfVe8Eg|Recording of the lecture]]** 
 +\\ **Speaker:​** Carlos Matheus Santos 
 +\\ **Title:** On the intermediate portions of the Lagrange and Markov spectra 
 +\\ **Abstract:​** ​ The classical Lagrange and Markov spectra are closed subsets of the real line arising naturally in the study of Diophantine approximations of real numbers and certain indefinite binary quadratic forms. After the seminal works of A. Markov from 1879, the structure of these spectra were heavily investigated by many authors and, in particular, we know that these spectra begin with an explicit increasing sequence \sqrt{5} < \sqrt{8} < ... converging to 3, and end with an explicit half-line [4.5278..., \infty).  
 +\\ 
 +On the other hand, several interesting questions about the intermediate portions of the Lagrange and Markov spectra are still open despite the progress obtained by Perron, Hall, Freiman, Flahive, ... . In this talk, we will discuss some dynamical tools which might be relevant in piercing some of the mysteries about the structure of the intermediate regions in the classical spectra. 
 +  
 + 
 +**20**. Thursday, December 10, at 18:00 Moscow time (GMT+3). 
 +\\ **{{http://​rivoal.perso.math.cnrs.fr/​articles/​beamermoscou.pdf|Slides of the lecture}}** 
 +\\ **Speaker:​** Tanguy Rivoal 
 +\\ **Title:​** ​ Algebraic values of E-functions 
 +\\ **Abstract:​** I will present an algorithm that performs the following tasks: given an  
 +E-function F (in the restricted sense) as input, it outputs the finite  
 +list of algebraic points A such that F(A) is algebraic, together with  
 +the list of the corresponding values F(A). 
 +\\ 
 +This is a joint work with Boris Adamczewski. 
 + 
 + 
 + 
 + ​**19**. Tuesday, December 08, at 14:00 Moscow time (GMT+3). 
 +\\ **Speaker:​** Dong Han Kim 
 +\\ **Title:** Intrinsic Diophantine Approximation of circles and spheres 
 +\\ **Abstract:​** ​ Let $S^1$ be the unit circle in $\mathbb{R}^2$ centered at the origin. 
 +We study the intrinsic Diophantine approximation of $S_1$ and give a 
 +complete description of the discrete part of the Lagrange spectrum. 
 +We also consider the intrinsic Diophantine approximation of other 
 +circles and spheres. 
 +This talk is based on joint work with Byungchul Cha. 
 +  
 + ​**18**. Tuesday, December 01, at 14:00 Moscow time (GMT+3). 
 +\\ **[[https://​yadi.sk/​i/​W1xTnSGSDGWdqA|Recording of the lecture]]** 
 +\\ **Speaker:​** Faustin Adiceam 
 +\\ **Title:** What does a sunflower look like in dimension 4? 
 +\\ **Abstract:​** A more formal title for the talk could be: “the concept of bad approximability in geometric discrepancy”. Inspired by a most classical problem in phyllotaxis (the study of the arrangement of leaves on a plant stem), the goal will be to answer a question which has appeared in several places in the literature, namely: does there exist a spiral Delone set in any dimension? A typical example of such a point set visible in nature is the sunflower. 
 +Here, a point set is said to be a spiral if it is obtained by suitable radial stretches of a spherical sequence. It is furthermore Delone if, roughly speaking, it is both well-spaced and dense in the Euclidean space. 
 +We will see that the answer to the above question is related to several problems in the theory of the irregularity of distributions,​ in geometry (packing and covering of given objects, properties of Platonic solids…) and in Diophantine approximation. It is also closely related to the problem of equidistributing points on a sphere, a well-studied question to which many authors (including Arnol’d, Sarnak, Lubotsky, Phillips, Oh and Gorodnik) have contributed. 
 +Time permitting, several open problems will conclude the talk. Joint work with Ioannis Tsokanos (University of Manchester). 
 + 
 +  
 +**17**. Thursday, November 12, at 15:00 Moscow time (GMT+3). 
 +\\ **{{:​moscou-fischler.pdf|Slides of the lecture}}** 
 +\\ **Speaker:​** Stéphane Fischler 
 +\\ **Title:​** ​   Linear independence of values of G-functions 
 +\\ **Abstract:​** G-functions are a class of functions introduced by Siegel in 1929; they  
 +include polylogarithms and $_{p+1} F_p$ hypergeometric series with  
 +rational parameters. Given a transcendental $G$-function $F(z)$ and an  
 +algebraic number $z_0$, it is in general a very difficult problem to  
 +determine whether $F(z_0)$ is algebraic or transcendental,​ and even  
 +whether it is rational or not. Seminal results due to Siegel, Galochkin,  
 +Bombieri, Chudnovsky, André and others apply to $F(z_0)$ if $z_0$ is  
 +sufficiently close to $0$, for instance proving its irrationality. On the  
 +opposite, the point of view in this lecture (based upon a joint work with  
 +Tanguy Rivoal) is to fix a non-zero algebraic point $z_0$, and to prove  
 +irrationality results about $F(z_0)$ for some $G$-functions $F(z)$ in a  
 +given family. This follows the approach of Ball-Rivoal,​ that yields the  
 +irrationality of $\zeta(s)$ (i.e., the value at $z_0=1$ of the $s$-th  
 +polylogarithm) for infinitely many odd integers $s$. 
 + 
 + 
 +  
 + 
 +**16**. Tuesday, November 10, at 14:00 Moscow time (GMT+3). 
 +\\ **[[https://​yadi.sk/​i/​1DIrUwIgQyERAA|Recording of the lecture]]** 
 +\\ **Speaker:​** Sam Chow 
 +\\ **Title:​** ​   Dyadic approximation in the Cantor set 
 +\\ **Abstract:​** ​ We investigate the approximation rate of a typical element of the Cantor set by dyadic rationals. This is a manifestation of Furstenberg'​s "times two times three" phenomenon, and is joint work with Demi Allen and Han Yu. 
 + 
 + 
 + 
 +**15**. Tuesday, October 06, at 16:00 Moscow time (GMT+3). 
 +\\ **[[https://​yadi.sk/​i/​_YXLiC7qqeaEIg|Recording of the lecture]]** 
 +\\ **Speaker:​** Alan Haynes 
 +\\ **Title:​** ​   Gap theorems for linear forms and for rotations on higher dimensional tori 
 +\\ **Abstract:​** ​ This talk is based on joint work with Jens Marklof, and with Roland Roeder. The three distance theorem states that, if x is any real number and N is any positive integer, the points x, 2x, … , Nx modulo 1 partition the unit interval into component intervals having at most 3 distinct lengths. We will present two higher dimensional analogues of this problem. In the first we consider points of the form mx+ny modulo 1, where x and y are real numbers and m and n are integers taken from an expanding set in the plane. This version of the problem was previously studied by Geelen and Simpson, Chevallier, Erdős, and many other people, and it is closely related to the Littlewood conjecture in Diophantine approximation. The second version of the problem is a straightforward generalization to rotations on higher dimensional tori which, surprisingly,​ has been largely overlooked in the literature. For the two dimensional torus, we are able to prove a five distance theorem, which is best possible. In higher dimensions we also have bounds, but establishing optimal bounds is an open problem. 
 + 
 +**14**. Tuesday, September 29, at 14:00 Moscow time (GMT+3). 
 +\\ **[[https://​yadi.sk/​i/​huaO6GzPY0XExA|Recording of the lecture]]** 
 +\\ **Speaker:​** Florin Boca 
 +\\ **Title:​** ​   Distribution of reduced quadratic irrationals (QIs) of even and of backward type 
 +\\ **Abstract:​** ​  ​Reduced QIs arising from the regular CF are closely related with the Pell  
 +equation and with closed geodesics on the modular surface. By a classical result of  
 +Pollicott, they are known to be equidistributed with respect to the Gauss probability  
 +measure, when ordered by their corresponding closed primitive geodesics length.  
 +This talk will consider the reduced quadratic irrationals arising from the even CF and  
 +the backward CF, where similar (and effective) equidistribution results with respect to  
 +the invariant (infinite!) invariant measures have been established in recent joint work  
 +with M. Siskaki. 
 + 
 +**13**. Tuesday, September 22, 2020, at 14:00 Moscow time (GMT+3). 
 +\\ **{{:​bcz2talk2020.pdf|Slides of the lecture}}** 
 +\\ **Speaker:​** ​ Yitwah Cheung 
 +\\ **Title:​** ​   Mixing properties of the BCZ map 
 +\\ **Abstract:​** ​  The BCZ map was introduced by F. Boca, C. Cobeli and A. Zaharescu in their investigations of  the statistical properties of the Farey sequences. ​ This is a piecewise linear map of a certain ​ right triangle whose connection to the horocycle flow on the modular surface was discovered by Athreya and myself, using which a newfound understanding of known results about Farey sequences was obtained, e.g. the distribution of gaps found by Hall in 1970 can be derived as the push-forward of Haar measure under the roof function of the BCZ map.  In this talk I will report on joint work with Anthony Quas and Yiwei Zhang in our understanding of the mixing properties of the BCZ map.  Specifically,​ I will explain how a miraculous property of the BCZ map, which we call infinitesimal self-similarity,​ allows us to prove that the BCZ map is weakly mixing. ​  
 +The question of strong mixing remains open, correcting a claim I made earlier this year.   
 +I will also describe a reformulation of the Riemann Hypothesis in terms of a BCZ cocycle.
  
 **12**. Thursday, July 09, 2020,  at 15:00 Moscow time (GMT+3). **12**. Thursday, July 09, 2020,  at 15:00 Moscow time (GMT+3).
Строка 33: Строка 368:
  
 **8**. Tuesday, June  02, 2020,  at 15:00 Moscow time (GMT+3). **8**. Tuesday, June  02, 2020,  at 15:00 Moscow time (GMT+3).
 +\\ **[[https://​yadi.sk/​i/​ezES9RyctUfT2w|Recording of the lecture]]**
 \\ **Speaker:​** Erez Nesharim. \\ **Speaker:​** Erez Nesharim.
 \\ **Title:** The set of weighted badly approximable vectors is hyperplane absolute winning. \\ **Title:** The set of weighted badly approximable vectors is hyperplane absolute winning.
Строка 39: Строка 375:
  
 **7**. Thursday, May  28, 2020,   at 15:00 Moscow time (GMT+3). **7**. Thursday, May  28, 2020,   at 15:00 Moscow time (GMT+3).
 +\\ **[[https://​yadi.sk/​i/​RUoVDKVv611nUQ|Recording of the lecture]]**
 \\ **Speaker:​** Johannes Schleischitz \\ **Speaker:​** Johannes Schleischitz
 \\ **Title:** Cartesian products, sumsets and Hausdorff dimension \\ **Title:** Cartesian products, sumsets and Hausdorff dimension
Строка 51: Строка 388:
  
 **6**. Thursday, May 21, 2020,  at 15:00 Moscow time (GMT+3). **6**. Thursday, May 21, 2020,  at 15:00 Moscow time (GMT+3).
 +\\ **[[https://​yadi.sk/​i/​SaLn9LUdhcpFBA|Recording of the lecture]]**
 \\ **Speaker:​** Dmitri Kleinbock \\ **Speaker:​** Dmitri Kleinbock
 \\ **Title:** Geometry and dynamics of improvements to Dirichlet'​s Theorem in Diophantine approximation \\ **Title:** Geometry and dynamics of improvements to Dirichlet'​s Theorem in Diophantine approximation
Строка 57: Строка 395:
  
 **5**. Thursday, May 07, 2020, at 14:30 Moscow time (GMT+3). **5**. Thursday, May 07, 2020, at 14:30 Moscow time (GMT+3).
 +\\ **[[https://​yadi.sk/​i/​5n9EzqbPS37_yg|Recording of the lecture]]**
 \\ **Speaker:​** Victor Bereslevich \\ **Speaker:​** Victor Bereslevich
 \\ **Title:** Badly approximable points on curves are winning \\ **Title:** Badly approximable points on curves are winning
Строка 67: Строка 406:
  
 **4**. Tuesday, May 05, 2020,  at 14:00 Moscow time (GMT+3) **4**. Tuesday, May 05, 2020,  at 14:00 Moscow time (GMT+3)
 +\\ **[[https://​yadi.sk/​i/​FAqwNIW35pCHfA|Recording of the lecture]]**
 \\ **Speaker:​** Dmitri Badziahin \\ **Speaker:​** Dmitri Badziahin
 \\ **Title:** An improved bound in the problem of Wirsing \\ **Title:** An improved bound in the problem of Wirsing
Строка 77: Строка 417:
  
 **3**. Tuesday, April 21, 2020,  at 15:00 Moscow time (GMT+3). **3**. Tuesday, April 21, 2020,  at 15:00 Moscow time (GMT+3).
 +\\ **[[https://​yadi.sk/​i/​_icxOaLKNEQagw|Recording of the lecture]]**
 \\ **Speaker:​** Simon Kristensen ​   \\ **Speaker:​** Simon Kristensen ​  
 \\ **Title:** Arithmetic properties of series of reciprocals of algebraic integers. \\ **Title:** Arithmetic properties of series of reciprocals of algebraic integers.
Строка 97: Строка 438:
 \\ **Title:** New bounds on the covering radius of a lattice \\ **Title:** New bounds on the covering radius of a lattice
 \\ **Abstract:​** We obtain new upper bounds on the minimal density of lattice coverings of R^n by dilates of a convex body K. We also obtain bounds on the probability (with respect to the natural Haar-Siegel measure on the space of lattices) that a randomly chosen lattice L satisfies L + K = R^n. As a step in the proof, we utilize and strengthen results on the discrete Kakeya problem. I will not assume any prior knowledge of lattice coverings. Joint with Or Ordentlich and Oded Regev. ​ \\ **Abstract:​** We obtain new upper bounds on the minimal density of lattice coverings of R^n by dilates of a convex body K. We also obtain bounds on the probability (with respect to the natural Haar-Siegel measure on the space of lattices) that a randomly chosen lattice L satisfies L + K = R^n. As a step in the proof, we utilize and strengthen results on the discrete Kakeya problem. I will not assume any prior knowledge of lattice coverings. Joint with Or Ordentlich and Oded Regev. ​
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