The hyperbolic structure on a 3-dimensional cone-manifold with a knot as singularity can often be deformed into a limiting Euclidean structure. In the present work, we show that the respective normalised Euclidean volume is always an algebraic number, which is reminiscent of Sabitov’s theorem (the Bellows Conjecture). This fact also stands in contrast to hyperbolic volumes whose number–theoretic nature is usually quite complicated. This is our joint work with Alexander Kolpakov and Alexander Mednykh.

Though the series of the inverses of integers diverges, a nice,now classical, result of Kempner (in 1914) states that the sum of the inverses of the integers whose expansion in base 10 contains no occurrence of a given digit converges. Several generalizations or extensions of this result have been given. In particular a paper of Farhi (in 2008) proves the somehow unexpected result that

where s10(n) is the sum of the digits of n in base 10.

We show how to extend this result, first by replacing s10(n) with s2(n), the sum of the binary digits of the integer n; then by replacing s2(n) by any binary pattern counting sequence (i.e., any sequence aw(n) that counts the number of occurrences of a fixed block w of 0’s and 1’s in the binary expansion of the integer n).

This presentation is composed of two parts which represent my current research. In the first part, I will discuss large deviations of volatility estimators and some limit theorems for bifurcating Markov chains and in the second part I will present some results on global sensitivity analysis applied to some models. This is a synthetic presentation, we will point out the difficulties and the interest of the work.

这个报告的主题是马氏过程观察量的经验均值趋于真实均值的集中不等式，或者说随机算法的错误概率的绝对估计。我将介绍与集中不等式等价的“运费vs Fisher信息量”不等式，以及怎样使用Poincare或logSobolev不等式，与最优传输，熵这些工具去证明“运费vs Fisher信息量”不等式。作为应用例子，我将介绍高维分布Gibbs算法的错误概率绝对估计。

The notion of a cofinitary groups was named by Peter Cameron in the 1980ies, as a dual to the notion of finitary permutation groups. Cameron asked about such groups which are maximal in the sense that they are not proper subgroups of another cofinitary group. 

Presumably due to a superficial similarity with so-called MAD families, it was long thought that such maximal cofinitary groups are necessarily extremely complicated objects, akin to monsters like the paradoxical decomposition of the sphere. It turns out that on the contrary, they can be constructed in a very, very concrete manner.

In this talk, I will sketch some ideas leading to this construction and give a some context to explain why it surprised everyone. There are also some possible connections to the world of operator algebras which I will discuss if time permits.

I will talk about automatic sequences and their link with combinatorics, number theory, and theoretical computer science.

In this report, I will introduce some fully nonlinear partial differential equations arising in geometric problems. I will also report some results which I (with my collaborators) got recently.

In this talk, I will briefly review recent developments of the derivation problem for noncommutative symmetric spaces. This talk is based on a joint work with Fedor Sukochev.

Given q between 0 and 1, one considers the following problems about the q-difference-differential equation y′(x) = ay(qx) + by(x) + f (x),where a and b are two complex numbers and where f is a rational function:

(1) The pantograph equations following Kato and McLeod;

(2) The indexes of the associated operator d/dx − aσ_q − b;

(3) The connection formulas between zero and infinity;

(4) The asymptotic behavior of the solutions at infinity.

This talk is partially based on a joint work with H. Dai and G. Chen (HITSZ).

This is a joint work with Junyi Xie. In 1987, McMullen proved a remarkable rigidity theorem which asserts that aside from the flexible Lattès family, the multiplier spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. The proof relies on Thurston's rigidity theorem for post-critically finite maps, in where Teichmüller theory is an essential tool. In this talk we give a new proof of McMullen's theorem without using quasiconformal maps or Teichmüller theory. We also show that aside from the flexible Lattès family, the length spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. This generalize the aforementioned McMullen's theorem.

The talk is based on a joint work with Charles Favre and Tuyen Trung Truong. We prove that the topological entropy of any dominant rational self-map of a projective variety defined over a complete non-Archimedean field is bounded from above by the maximum of its dynamical degrees, thereby extending a theorem of Gromov and Dinh-Sibony from the complex to the non-Archimedean setting. We proceed by proving that any regular self-map which admits a regular extension to a projective model defined over the valuation ring has necessarily zero entropy. To this end we introduce the -reduction of a Berkovich analytic space, a notion of independent interest.

We are concerned here with Sobolev-type space of functions valued in Banach space or in metric space. We review two ways of defining Sobolev maps valued in metric space: Reshetnyak’s approach vs deffinition via postcomposition with the Kuratowski embedding. In particular, we show that Sobolev map with values in dual-to-separable Banach space can be described in terms of classical weak derivatives in a weak* sense.

1937年，Kolmogorov, Petrovskii和Piskunov在研究反应扩散方程的柯西问题时，提出一类传播问题。从上世纪70年代起，该问题吸引了包括偏微分方程，动力系统和生物数学等多个领域学者的关注。这个报告主要从动力系统的角度来介绍近期关于这类问题的一些研究进展。

Modular form is one of the central objects in modern number theory. In this introductory talk, we shall focus on the very basics of modular forms: some history, some basic theories, a few important conjectures and quite a few examples. This talk is for general audience.

In this talk, I will briefly review Sharifi's conjectures which relate the K groups of cylotomic fields to the homology groups of modular curves. Then I will talk about some generalizations of these conjectures.  This is a joint work in progress with Emmanuel Lecouturier, Romyar Sharifi and Sheng-Chi Shih.