Dedekind η\eta-function, Hauptmodul and invariant theory

We solve a long-standing open problem with its own long history dating back to the celebrated works of Klein and Ramanujan. This problem concerns the invariant decomposition formulas of the Hauptmodul for $\Gamma_0(p)$ under the action of finite simple groups $PSL(2, p)$ with $p=5, 7, 13$. The cases of $p=5$ and $7$ were solved by Klein and Ramanujan. Little was known about this problem for $p=13$. Using our invariant theory for $PSL(2, 13)$, we solve this problem. This leads to a new expression of the classical elliptic modular function of Klein: $j$-function in terms of theta constants associated with $\Gamma(13)$. Moreover, we find an exotic modular equation, i.e., it has the same form as Ramanujan's modular equation of degree $13$, but with different kinds of modular parametrizations, which gives the geometry of the classical modular curve $X(13)$.Comment: 46 pages. arXiv admin note: substantial text overlap with arXiv:1209.178

Equidistribution of expanding translates of curves in homogeneous spaces with the action of (SO(n,1))k(\mathrm{SO}(n,1))^k

Given a homogeneous space $X = G/\Gamma$ with $G$ containing the group $H = (\mathrm{SO}(n,1))^k$. Let $x\in X$ such that $Hx$ is dense in $X$. Given an analytic curve $\phi: I=[a,b] \rightarrow H$, we will show that if $\phi$ satisfies certain geometric condition, then for a typical diagonal subgroup $A =\{a(t): t \in \mathbb{R}\} \subset H$ the translates $\{a(t)\phi(I)x: t >0\}$ of the curve $\phi(I)x$ will tend to be equidistributed in $X$ as $t \rightarrow +\infty$. The proof is based on the study of linear representations of $\mathrm{SO}(n,1)$ and $H$.Comment: 19 page

Trace formulas for a class of compact complex surfaces

We give the trace formulas of weight $k$ for cocompact, torsion-free discrete subgroups of $SU(2, 1)$ and prove the analogue of the Riemann hypothesis on compact complex surfaces $M$ with $c_1^2(M)=3 c_2(M)$, where $c_i(M)$ is the $i$-th Chern class of $M$, $c_2(M)$ is a multiple of three and $c_2(M)>0$.Comment: 63 page

Poincar\'{e} series and modular functions for U(n, 1)

We construct infinitely many nonholomorphic automorphic forms and modular forms associated to a discrete subgroup of infinite covolume of $U(n, 1)$.Comment: 18 page

Modular curves, invariant theory and E8E_8

The $E_8$ root lattice can be constructed from the modular curve $X(13)$ by the invariant theory for the simple group $\text{PSL}(2, 13)$. This gives a different construction of the $E_8$ root lattice. It also gives an explicit construction of the modular curve $X(13)$.Comment: 39 pages. arXiv admin note: text overlap with arXiv:1511.0527

Icosahedron, exceptional singularities and modular forms

We find that the equation of $E_8$-singularity possesses two distinct symmetry groups and modular parametrizations. One is the classical icosahedral equation with icosahedral symmetry, the associated modular forms are theta constants of order five. The other is given by the group $\text{PSL}(2, 13)$, the associated modular forms are theta constants of order $13$. As a consequence, we show that $E_8$ is not uniquely determined by the icosahedron. This solves a problem of Brieskorn in his ICM 1970 talk on the mysterious relation between exotic spheres, the icosahedron and $E_8$. Simultaneously, it gives a counterexample to Arnold's $A, D, E$ problem, and this also solves the other related problem on the relation between simple Lie algebras and Platonic solids. Moreover, we give modular parametrizations for the exceptional singularities $Q_{18}$, $E_{20}$ and $x^7+x^2 y^3+z^2=0$ by theta constants of order $13$, the second singularity provides a new analytic construction of solutions for the Fermat-Catalan conjecture and gives an answer to a problem dating back to the works of Klein.Comment: 41 page

Finite Heat conduction in 2D Lattices

This paper gives a 2D hamonic lattices model with missing bond defects, when the capacity ratio of defects is enough large, the temperature gradient can be formed and the finite heat conduction is found in the model. The defects in the 2D harmonic lattices impede the energy carriers free propagation, by another words, the mean free paths of the energy carrier are relatively short. The microscopic dynamics leads to the finite conduction in the model

Complex version KdV equation and the periods solution

In this paper, the complex version KdV equation is discussed. The corresponding coupled equations is a integrable system in the sense of the bi-Hamiltonian structure, so the complex version KdV equation is integrable. A new spectral form is given, the periodic solution of the complex version KdV equation is obtained. It is showed that the periodic solution is the classical solution

Exact solutions of nonlinear PDE, nonlinear transformations and reduction nonlinear PDE to a quadrature

A method to construct the exact solution of the PDE is presents, which combines the two kind methods(the nonlinear transformation and RQ(Reduction the PDE to a Quadrature problem) method).The nonlinear diffusion equation is chosen to illustrate the method and the exact solutions are obtained

Higher spin holography with Galilean symmetry in general dimensions

We construct Schroedinger-like solutions of the Vasiliev higher spin theory in D>3 dimension. Symmetries of such solutions and the linearised equation of motion for the scalar on such backgrounds are analysed. We further propose Galilean invariant bosonic and fermionic field theories that could be dual to the two parity invariant higher spin theories on the Schroedinger-like background respectively. The discussion is phrased mainly in D=4 dimension, while similar constructions follow straightforwardly in higher dimensions.Comment: 31 page