相关论文: A remark on Zagier's observation of the Mandelbrot…
The combinatorial Mandelbrot set is a continuum in the plane, whose boundary can be defined, up to a homeomorphism, as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by…
In this note, we prove a conjecture of Puder on an extension of the co-growth formula to any non-negative function defined on a bi-regular tree. A key component of our proof is the establishment of a resolvent identity, which serves as an…
Let $P_1,\dots,P_k \colon {\bf Z} \to {\bf Z}$ be polynomials of degree at most $d$ for some $d \geq 1$, with the degree $d$ coefficients all distinct, and admissible in the sense that for every prime $p$, there exists integers $n,m$ such…
We study regularized determinants of Laplacians acting on the space of Hilbert-Maass forms for the Hilbert modular group of a real quadratic field. We show that these determinants are described by Selberg type zeta functions introduced in…
We consider the action of Mandelbrot multiplicative cascades on probability measures supported on a symbolic space. For general probability measures, we obtain almost a sharp criterion of non-degeneracy of the limiting measure; it relies on…
In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of "Pacman Renormalization Theory" that combines features of quadratic-like and Siegel renormalizations.…
Let $\beta>1$ be a real number. In this paper, the Hausdorff dimension of sets consisting of pairs of numbers with prescribed quantitative waiting time indicators in $\beta$-expansions are determined. More precisely, let $I$ be the unit…
Several questions in CR geometry lead naturally to the study of bihomogeneous polynomials $r(z,\bar{z})$ on $\C^n \times \C^n$ for which $r(z,\bar{z})\norm{z}^{2d}=\norm{h(z)}^2$ for some natural number $d$ and a holomorphic polynomial…
Carrots for dessert is the title of a section of the paper `On polynomial-like mappings' by Douady and Hubbard. In that section the authors define a notion of dyadic carrot fields of the Mandelbrot set M and more generally for Mandelbrot…
In a series of papers we have been studying the geometric theta correspondence for non-compact arithmetic quotients of symmetric spaces associated to orthogonal groups. It is our overall goal to develop a general theory of geometric theta…
The higher Bruhat orders $\mathcal{B}(n,k)$ were introduced by Manin-Schechtman to study discriminantal hyperplane arrangements and subsequently studied by Ziegler, who connected $\mathcal{B}(n,k)$ to oriented matroids. In this paper, we…
We study the regularized determinants ${\rm det}\, \Delta$ of various self-adjoint extensions of symmetric Laplacians acting in spinor bundles over compact Riemann surfaces with flat singular metrics $|\omega|^2$, where $\omega$ is a…
In this paper, we present an application of mirror symmetry to arithmetic geometry. The main result is the computation of the period of a mixed Hodge structure, which lends evidence to its expected motivic origin. More precisely, given a…
In this expository introductory text we discuss the multiplier ideals in algebraic geometry. We state Kawamata-Viehweg's and Nadel's vanishing theorems, give a proof (following Ein and Lazarsfeld) of Koll\'ar's bound on the maximal…
This paper extends some results of [M5] and [M3], in particular, removing assumptions of positive lower density. We give conditions on a general family $P_{\lambda}:\mathbb{R}^{n}\to\mathbb{R}^{m}, \lambda \in \Lambda,$ of orthogonal…
We explore geometric properties of the Mandelbrot set M, and the corresponding Julia sets J_c, near the main cardioid. Namely, we establish that: a) M is locally connected at certain infinitely renormalizable parameters c of bounded…
In this paper, we prove a conjecture of Broadhurst and Zudilin \cite{BZ17} concerning a divisibility property of the Fourier coefficients of a meromorphic modular form using the generalization of the Shimura lift by Borcherds…
In this note, we study the arithmetic nature of values of modular functions, meromorphic modular forms and meromorphic quasi-modular forms with respect to arbitrary congruence subgroups, that have algebraic Fourier coefficients. This…
We show that the Julia set of quadratic maps with parameters in hyperbolic components of the Mandelbrot set is given by a transseries formula, rapidly convergent at any repelling periodic point. Up to conformal transformations, we obtain…
Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the…