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Спецсеминар "Графы на поверхностях и кривые над числовыми полями"

Семинар проходит по средам в ауд. 14-15 Главного здания, начало в 18:30.


1. А Ватузов, Функции Белого некоторых взвешенных деревьев (продолжение)

2. Г.Б. Шабат, О dessins d’un vieillard по Манину-Марколли


1. А. Моляков, Одно семейство Фрида и замечательные слои в нём

2. О. Шувалова, Модель Эренфеста и плоские поверхности

3. А. Ватузов, Функции Белого некоторых взвешенных деревьев


John Voight (Department of Mathematics, Dartmouth College, USA), Computing Belyi maps: a survey

Abstract: We will survey methods to compute Belyi maps, including some recent developments.


Г.Б. Шабат, Об одном пучке симметричных секстик


A. K. Zvonkin (Bordeaux), In praise of the Bateman-Horn conjecture

Joint work with Gareth Jones, with computational assistance from Jean Bétréma

Let there be a set of polynomials $f_1,\ldots,\f_k\in\mathbb{Z}[t]$. We are interested in the situation when all the values $f_1(t),\ldots,f_k(t)$ are simultaneously prime. The question to which the Bateman-Horn conjecture gives an answer is: for a given $x$, how many $t\le x$ are there for which the above situation occurs.

For example:

(1) when there is only one polynomial $f_1(t)=t$, the answer is given by the Prime Number Theorem (Hadamard and de la Vallée Poussin, 1896);

(2) when there is still one polynomial $f_1(t)=at+b$, Dirichlet (1837) proved that there are infinitely many $t\in\mathbb{N}$ such that $f_1(t)$ is prime; the number of $t\le x$ was established later;

(3) when there are two polynomials $f_1(t)=t$ and $f_2(t)=t+2$, we have the Twin Primes conjecture;

(4) a question which is also in this framework: are there infinitely many projective groups of prime degree?

The Bateman-Horn conjecture predicts, with an astonishing accuracy, the number of the «good» values of $t$. The talk will be given in a mixed Russian-Western style. Namely:

(1) the slides will be in English;

(2) the talk itself will be in English;

(3) but the duration of the talk may turn out to be significantly longer than a polite one hour.


G. B. Shabat, Three facets of dessins d'enfants theory (the 2nd day of the Conference devoted to the memory of Gena Belyi on the occasion of his 70th birthday)

As everybody knows, the 3-points theorem proved by Belyi in 1978 enriched Grothendieck's (with his students') considerations of embeddings of graphs into surfaces; that was the Big Bang of dessins d'enfants theory. I am going to give an overview of the development of this theory during the recent decades, concentrating on the interactions between the algebro-geometric, combinatorial and group-theoretical viewpoints.


1. Ю. Ю. Кочетков (НИУ ВШЭ), Парадокс Банаха-Тарского

2. Ю. Ю. Кочетков (НИУ ВШЭ), Об одной динамической системе на пространстве выпуклых четырехугольников


Г.Р. Челноков (НИУ ВШЭ), Перечисление классов эквивалентности конечнолистных накрытий многообразия Ханце-Вендта


M. Skopenkov (NRU HSE, IITP RAS), Discrete complex analysis: convergence results

Various discretizations of complex analysis have been actively studied since the 1920s because of applications to numerical analysis, statistical physics, and integrable systems. This talk concerns complex analysis on quadrilateral lattices tracing back to the works of J. Ferrand, R. Isaacs, R. Duffin.

We solve a problem by S.K. Smirnov from 2010 on the convergence of discrete harmonic functions on planar nonrhombic lattices to their continuous counterparts under lattice refinement. This generalizes the results of R.Courant-K.Friedrichs-H.Lewy, L.Lusternik, D.S.Chelkak-S.K.Smirnov, P.G.Ciarlet-P.-A.Raviart.

We also prove convergence of discrete period matrices and discrete Abelian integrals to their continuous counterparts (this is a joint work with A.I. Bobenko). The proofs are based on energy estimates inspired by electrical network theory.


1. Ю.Ю. Кочетков, Мера и аксиома выбора;

2. А.Р. Моляков, О парах Белого рода 1 степени 5;

3. Разное.


1. Г.Б. Шабат, О семействах торических рисунков;

2. Разное.


1. Ф.Б. Пакович (Ben Gurion University of Negev), Об уравнении A(X)=A(Y)

2. Н.М. Адрианов, О реализуемости паспортов и группах монодромии (продолжение)


Н.М. Адрианов, О реализуемости паспортов и группах монодромии (продолжение)


Н.М. Адрианов, О реализуемости паспортов и группах монодромии


1. Г.Б. Шабат, О парах Белого рода 2.

2. Н.Я. Амбург, Производящая функция для числа раскрасок графа Кочеткова


1. Г.Б. Шабат, Эллипсы и эллиптические кривые.

2. Как мы провели зимние каникулы.

3. Разное.


Anton Zorich, University of Paris, Square-tiled surfaces, Masur-Veech volumes and count of meanders.

(joint work with V. Delecroix, E. Goujard and P. Zograf)

I will introduce square-tiled surfaces and explain why they represent integer points in the moduli spaces of Abelian and quadratic differentials. I will also explain why count of square-tiled surfaces provides Masur-Veech volumes of these moduli spaces.

To justify my interest to count of square-tiled surfaces I will show how it allows to count meanders. The results are obtained jointly with V. Delecroix, E. Goujard, P. Zograf.


1. А. Моляков, Г.Б. Шабат, Об одном семействе Фрида степени 5

2. Н.М. Адрианов, Введение в теорию Галуа-Тайхмюллера (продолжение)

3. Разное


Andrei Bogatyrev (Institute of Numerical Mathematics), Degeneration of graphs and moduli of curves

Abstract: We consider the cell decomposition of the moduli space of real genus two curves with a marked point on the only real oval. The cells are enumerated by certain graphs with their weights describing the complex structure on a curve. We show that collapse of an edge of the graph results in a root like singularity of the natural mapping from the graph weights to the moduli space of curves.


1.  Елизавета Бриль (НИУ ВШЭ), Многочлен Татта – вовсе  не Галуа-инвариант!

2. Н.М. Адрианов, Введение в теорию Галуа-Тайхмюллера (продолжение).

3. Разное.


George Shabat, Dessins d'enfants and piecewise-euclidean metrics on Riemann surfaces

Abstract: There are several ways to introduce such metrics: using Strebel differentials, by metrized ribbon graphs, etc. The talk will contain an overview of these structures and their relation with desssins d'enfants.

According to an old result of the speaker and Voevodsky, a Riemann surface admits a conformal structure, defined by an equilateral triangulation, if and only if the corresponding algebraic curve can be defined over the field of the algebraic numbers. The similar result where the equilateral triangles are replaced by squares, will be presented. As the corresponding dessins d'enfants the square-tiled surfaces (origamis) arise.

Hopefully, soon we'll have a talk by Anton Zorich concerning the statistics of the square-tiled surfaces. It is possible that the relations with the distribution of sizes of their Galois orbits will be found.


1. Г.Б. Шабат, Применение производной Шварца для вычислений функций Белого (продолжение)

2. Н.М. Адрианов, Введение в теорию Галуа-Тайхмюллера (продолжение)


1. Н.Я. Амбург, О матричной модели, связанной с сокрестиями (окончание)

2. Г.Б. Шабат, Применение производной Шварца для вычислений функций Белого

3. Разное.


1. Н.А. Амбург, О матричной модели, связанной с сокрестиями

2. Н.М. Адрианов, Введение в теорию Галуа-Тайхмюллера (продолжение)


Natalia Amburg (ITEP, HSE), Elena Kreines (MSU, ITEP), Belyi pair of the cell decomposition of oriented covering of the Deligne-Mumford compactification for M^R_{0,5}

Abstract: We consider the Deligne-Mamford compactification of the moduli space of genus 0 real algebraic curves with 5 marked and numbered points and its orienting covering. The second one is a surface. So, standard cell decomposition of the original moduli space provides a dessin d'enfant on this surface. We compute the Belyi pair for this dessin. In particular, it appears that the corresponding curve is a celebrated Bring curve.


1. Г.Б. Шабат, Семейства Фрида и география алгебраических поверхностей

2. Н.М. Адрианов, Введение в теорию Галуа-Тайхмюллера (продолжение)

3. Разное.


1. П. Мартынюк, Г.Б. Шабат, Сокрестия

2. Н.М. Адрианов, Введение в теорию Галуа-Тайхмюллера (продолжение)


Petr Dunin-Barkowski (HSE), Review of topological recursion, quasi-polynomiality and rationality results for various types of Hurwitz numbers

Abstract: We review results on spectral curve, topological recursion, quasi-polynomiality and rationality for various types of Hurwitz numbers, including simple, monotone, and r-spin Hurwitz numbers, Bousquet-Mélou–Schaeffer numbers, coefficients of Ooguri-Vafa partition functions for colored HOMFLY polynomials of knots, etc.


1. К.Т. Гадахабадзе (НИУ ВШЭ), Изображение алгебраических чисел вершинами плоских деревьев

2. Ю.Ю. Кочетков (НИУ ВШЭ), О геометрии многогранников

3. Н.М. Адрианов (МГУ), Введение в теорию Галуа-Тайхмюллера


Ilya Mednykh and Liliya Grunwald (Sobolev Institute of Mathematics, Novosibirsk State University, Russia), Enumerating rooted spanning forests in circulant graphs

Abstract: In this talk, we develop a new method to produce explicit formulas for the number fG(n) of rooted spanning forests in the circulant graphs G = Cn(s1, s2, . . . , sk) and G = C2n(s1, s2, . . . , sk, n). These formulas are expressed through Chebyshev polynomials. We prove that in both cases the number of rooted spanning forests can be represented in the form fG(n) = p a(n)2, where a(n) is an integer sequence and p is a prescribed natural number depending on the parity of n. Finally, we find an asymptotic formula for fG(n) through the Mahler measure of the associated Laurent polynomial.