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Спецсеминар "Графы на поверхностях и кривые над числовыми полями"

2019 – 2020 гг


Fedor Pakovich (Ben Gurion University of Negev), Recomposing rational functions



David Roberts (University of Minnesota Morris), Hypergeometric Belyi Maps

Abstract: A classical hypergeometric function {}_n F_{n-1)(alpha,beta,t) and an auxiliary prime number determine a Belyi map. We will give an overview of this class of Belyi maps, and why it is important to explicitly compute them. To illustrate this theory, we present a new computation of a degree 120 hypergeometric Belyi map with monodromy group Sp_8(F_2).


David Torres-Teigell (Universidad Autónoma de Madrid), Teichmüller curves: where to find them and how to cut them out

Abstract: Teichmüller curves are totally geodesic curves inside the moduli space of Riemann surfaces. They can always be seen, via certain Prym-Torelli map, as Kobayashi geodesics inside a Hilbert modular variety parametrizing abelian varieties with real multiplication. Our main objective is to cut Teichmüller curves out as the vanishing locus of a Hilbert modular form. The building blocks of these modular forms turn out to be certain theta functions and their derivatives, which can be made very precise. This calculation allows us to compute the Euler characteristics of the Teichmüller curves and the Masur-Veech volume of the affine invariant manifold where they live. In this talk we will give an overview of the subject and focus on the case of gothic Teichmüller curves.


Leonardo Zapponi (Sorbonne Université), Belyi theorem in positive characteristic and its applications.

Abstract: The first part of the talk is a review of the different aspects of Belyi theorem, from the original characteristic 0 case, through its odd characteristic version, concluding with its recent proof in characteristic 2. The second part is an illustration of these results and their applications for elliptic curves, involving cellular decomposition of moduli spaces and realizations of modular curves as Hurwitz spaces.


Alexander Mednykh (Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia), Volumes of two-bridge knots in spaces of constant curvature

Abstract: We investigate the existence of hyperbolic, spherical or Euclidean structure on cone manifolds whose underlying space is the three-dimensional sphere and singular set is a given two-bridge knot. For two-bridge knots with not more than 7 crossings we present trigonometrical identities involving the lengths of singular geodesics and cone angles of such cone manifolds. Then these identities are used to produce exact integral formulae for volume of the corresponding cone manifold modeled in the hyperbolic, spherical and Euclidean geometries.


George Shabat, On the Belyi height

Abstract: Belyi height of a complex curve is defined as the smallest possible degree of a Belyi function on it. For a fixed genus it is considered as a function on the moduli space; according to Belyi theorem, the Belyi height of a curve is finite if and only if the curve is defined over the field of algebraic numbers.

Belyi height will be compared with the other heights and with the Kolmogorov complexity. Some examples due to the speaker and to Leonardo Zapponi will be presented. The recent result by Ariyan Javanpeykar and John Voight on the algorithmic computability of the Belyi height will be formulated and the algorithmic aspects of the passport realizability discussed.


Konstantin Golubev (ETH Zürich), High-Dimensional Expanders and Property Testing

Abstract: Expander graphs can be defined in a number of equivalent ways, each of which however gives rise to a different notion when generalized to higher dimensions. In my talk, I will describe one of them, F2-coboundary expansion, and describe its connection to the theory of Property Testing.


Vasilisa Shramchenko (Université de Sherbrooke), Poncelet theorem and Painlevé VI equation

In 1995 Hitchin constructed explicit algebraic solutions to the Painlevé VI (1/8,-1/8,1/8,3/8) equation starting with any Poncelet trajectory, that is a closed billiard trajectory inscribed in a conic and circumscribed about another conic. In this talk I will show that Hitchin's construction is actually the Okamoto transformation between Picard's solution and the general solution of the Painlevé VI (1/8,-1/8,1/8,3/8) equation. Moreover, this Okamoto transformation can be written in terms of an Abelian differential of the third kind on the associated elliptic curve, which allows to write down solutions to the corresponding Schlesinger system in terms of this differential as well. This solution of the Schlesinger system admits a natural generalization to hyperelliptic curves.


Alexander Zvonkin (LaBRI, University of Bordeaux), Construction of regular maps from their small quotients

Every bicolored map may be represented by a triple of permutations (x,y,z) acting on the set E of edges and such that xyz=1. Here the cycles of x are black vertices, the cycles of y are white vertices, and the cycles of z are faces. To every map one can associate two groups: the monodromy group G=<x,y,z>, and the automorphism group H. A map is called regular if these two groups are isomorphic. In this case the set E of edges can be identified with the group, and this group acts on itself by multiplications. Thus, a construction of a regular map, even a large one, may be reduces to a construction of a group with desired properties, and this group may be constructed as a monodromy group of another map, often much smaller.

As an example of special interest we will consider Hurwitz maps. In 1893, Hurwitz proved that for a map of genus g>1 the order of its automorphism group is bounded by 84(g-1). Hurwitz maps are interesting not only because they are very symmetric but also because they are very rare. Marston Conder (Aucland) classified all regular maps of genus from 2 to 101. Their number is more 19 thousand, and only seven of them are Hurwitz.

This is a joint work with Gareth Jones (Southampton).


George Shabat, Is it possible to understand Mochizuki?

Shinichi Mochizuki is the author of a great amount of long and very interesting texts; every month this corpus is growing. The author claims to have developed the «Inter-universal Teichmüller theory», from which, e.g., the ABC-conjecture follows.

It will be shown that the ordinary procedure of comprehension of Mochizuki's texts is close to impossible. However, some of his results are clearly formulated and related to the traditional subjects of our seminar, These ones will be formulated, and then the attempt to enter the fascinating Mochizuki's mathematical world will be made.


Hartmut Monien (University Bonn), Dessins d'enfants and modular curves  associated to the sporadic group Co3 and Janko 2.  

Dessins d'enfants and their realization as Belyi maps of compact Riemann surfaces were originally discovered by Felix Klein. Their importance and relevance was finally understood by Alexander Grothendieck who rediscovered and named them in his «Esquisse d'un programme» in 1984. The most important aspect of dessins is the operation of the absolute Galois group on them. Accordingly, dessins d'enfants provide fascinating insights and fundamental links between different fields of mathematics like inverse Galois theory, Teichmüller spaces, hypermaps, algebraic number theory and mathematical physics. The related problem of the construction of Riemann surfaces with given automorphism group turns out to be rather challenging. 


Г.Б. Шабат, Верификация длинных доказательств: мечты, планы и реальность

Речь в основном пойдет о незавершенном проекте Владимира Воеводского (предварительный итог которого подведен в коллективной монографии [1]), в котором предполагалось существенно расширить взаимодействие математиков c компьютерами при построении и проверке доказательств.

После краткого обзора унивалентных оснований математики внимание будет сосредоточено на проблемах, возникающих в связи с доказательствами, традиционное понимание которых затруднено или невозможно по причине их длины и сложности. Будут приведены примеры; позиции докладчика будут критически сопоставлены с положениями известного провокационного текста Николая Вавилова [2].

В заключение будут высказаны соображения о формализации преподаваемой математики.

[1] Homotopy Type Theory: Univalent Foundations for Mathematics. Univalent Foundations Project, Institute for Advanced Study, 2013. (465 pages) arXiv: 1308.0729

[2] Nikolai Vavilov. Reshaping the metaphor of proof. Philosophical Transactions of the Royal Society A. Mathematical, Physical, and Engineering Sciences, 2019. DOI: 10.1098/rsta.2018.0279

Заседание будет совмещенным с Научно-исследовательским семинаром по математической логике кафедры Математической логики и теории алгоритмов, ссылка на запись семинара.


Г.Б. Шабат, Деформации пар Белого

01.04.2020 Начало: 16.00!

1. Дима Звонкин, Несколько классов когомологий на Mbar_{g,n}, вычисленных с помощью классификации полупростых когомологических теорий поля.

Я приведу несколько примеров естественных классов когомологий на пространстве модулей стабильных кривых и формул для них. Все эти формулы по виду похожи друг на друга, и это не случайно: все они получены применением одной и той же классификации полупростых когомологических теорий поля, разработанной Гивенталем и Телеманом.

2. Разное


Г.Б. Шабат, Арифметико-геометрическое среднее и семейства Фрида.


1. Н.Я. Амбург, Г.Б. Шабат, О препринте Манина-Марколли;

2. А.М. Ватузов, TBA


Г.Б. Шабат, Семейство Фрида, связанное с преобразованием Гаусса-Ландена.


1. Г.Б. Шабат, Рациональные функции на кубических кривых (продолжение);

2. Н.М. Адрианов, О симметрических квадратах функций Белого (дополнение).


1. А. Ватузов, Вычисление функций Белого с использованием техники модулярных групп

2. Н.М. Адрианов, О симметрических квадратах функций Белого


1. Г.Б. Шабат, Пары Белого на поверхностях (по Вик. С Куликову)

2. Н.М. Адрианов, О симметрических квадратах функций Белого (продолжение)


1. Ю.Ю. Кочетков, Двойственные четырехугольники на плоскости

2. Н.М. Адрианов, О симметрических квадратах функций Белого (продолжение)


1. Г.Б. Шабат, О хорошей и плохой редукции

2. Н.М. Адрианов, О симметрических квадратах функций Белого (продолжение)


1. Г.Б. Шабат, О семействах детских рисунков и пар Белого

2. Н.М. Адрианов, О симметрических квадратах функций Белого


П.И. Дунин-Барковский (НИУ ВШЭ), Топологическая рекурсия для r-spin чисел Гурвица


1. Г.Б. Шабат, Паспортные многообразия и их размерности (продолжение)

2. Разное.


Г.Б. Шабат, Паспортные многообразия и их размерности (продолжение)


1. Г.Б. Шабат, О реализуемости различных паспортов

2. Н.Я. Амбург (ИТЭФ), Цветные триангуляции и тензорная модель (продолжение)


1. Ю.Ю. Кочетков (НИУ ВШЭ), О вещественных многочленах степени 5 и 6

2. Разное.


1. Г.Б. Шабат, Критическая фильтрация и отображение Ляшко-Лойенги (продолжение)

2. Н.Я. Амбург (ИТЭФ), Цветные триангуляции и тензорная модель


Г.Б. Шабат, Критическая фильтрация и отображение Ляшко-Лойенги

04.09.2019 ВНИМАНИЕ: начало в 18:00

1. Pálfia Miklós, On the recent advances in the multivariable theory of operator monotone functions and means Functional Analysis Research Group, Institute of Mathematics, University of Szeged, Hungary, Sungkyunkwan University, Korea

Abstract: The origins of this talk go back to the fundamental theorem of Loewner in 1934 on operator monotone real functions and also to the hyperbolic geometry of positive matrices. Loewner's theorem characterizing one variable operator monotone functions has been very influential in matrix analysis and operator theory. Among others it lead to the Kubo-Ando theory of two-variable operator means of positive operators in 1980. One of the nontrivial means of the Kubo-Ando theory is the non-commutative generalization of the geometric mean which is intimately related to the hyperbolic, non-positively curved Riemannian structure of positive matrices. This geometry provides a key tool to define multivariable generalizations of two-variable operator means. Arguably the most important example of them all is the Karcher mean which is the center of mass on this manifold. This formulation enables us to define this mean for probability measures on the cone of positive definite matrices extending further the multivariable case. Even the infinite dimensional case of positive operators is tractable by abandoning the Riemannian structure in favor of a Banach-Finsler structure provided by Thompson's part metric on the cone of positive definite operators. This metric enables us to develop a general theory of means of probability measures defined as unique solutions of nonlinear operator equations on the cone, with the help of contractive semigroups of nonlinear operators. We also introduce the recently established structure theory of multivariable operator monotone functions extending the classical result of Loewner into the non-commutative multivariable realm of free functions, providing theoretically explicit closed formulas for our multivariable operator means.

2. F. Pakovich, COMMUTING RATIONAL FUNCTIONS REVISITED Ben Gurion University, Israel

Abstract Let A and B be rational functions on the Riemann sphere. The classical Ritt theorem states that if A and B commute and do not have an iterate in common, then up to a conjugacy they are either powers, or Chebyshev polynomials, or Latt`es maps. This result however provides no information about commuting rational functions which do have a common iterate. On the other hand, non-trivial examples of such functions exist and were constructed already by Ritt. In the talk we present new results concerning this class of commuting rational functions. In particular, we describe a method which permits to describe all rational functions commuting with a given rational function.