Related papers: The Taylor expansion of Ruelle L-function at the o…

Borell's formula is a stochastic variational formula for the log-Laplace transform of a function of a Gaussian vector. We establish an extension of this to the Riemannian setting and give a couple of applications, including a new proof of a…

Starting from a hyperbolic toral automorphism, we obtain, for a small volume preserving perturbation, an exact and rigorous second order perturbation expansion of the Lyapunov exponents.

We find that an S-duality in SL(2) Chern-Simons theory for hyperbolic 3-manifolds emerges by the Borel resummation of a semiclassical expansion around a particular flat connection associated to the hyperbolic structure. We demonstrate it…

We refine the $L^p$ restriction estimates for Laplace eigenfunctions on a Riemannian surface, originally established by Burq, G\'erard, and Tzvetkov. First, we establish estimates for the restriction of eigenfunctions to arbitrary Borel…

We derive a local index theorem in Quillen's form for families of Cauchy-Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated…

For a fundamental solution of Laplace's equation on the $R$-radius $d$-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion…

We explore Liouville's theorem and the Strong Liouville Property (SLP) for harmonic functions on Riemannian cones and surfaces. Our approach recasts the classical Liouville property in terms of the growth of radial eigenfunctions (in the…

We propose an extension of the recently-proposed volume conjecture for closed hyperbolic 3-manifolds, to all orders in perturbative expansion. We first derive formulas for the perturbative expansion of the partition function of complex…

We show that part I of uniform Martin's conjecture follows from a local phenomenon, namely that if a non-constant Turing invariant function goes from the Turing degree $\boldsymbol x$ to the Turing degree $\boldsymbol y$, then $\boldsymbol…

We bound the $L^2$-norm of an $L^2$ harmonic $1$-form in an orientable cusped hyperbolic $3$-manifold $M$ by its topological complexity, measured by the Thurston norm, up to a constant depending on $M$. It generalizes two inequalities of…

The present paper is a continuation of our work [11], where we introduced a fractional operator calculus related to a fractional ${\psi}-$Fueter operator in the one-dimensional Riemann-Liouville derivative sense in each direction of the…

We analyse the role of vertex operator algebra and 2d amplitudes from the point of view of the representation theory of infinite dimensional Lie algebras, MacMahon and Ruelle functions. A p-dimensional MacMahon function is the generating…

We prove an upper bound for the volume-normalized second nonzero eigenvalue of the Laplace operator on closed Riemannian manifold, in terms of the conformal volume. This bound provides effective upper bound for a large class of manifolds,…

The object of this work is the spinor L-function of degree 3 and certain degeneration related to the functoriality principle. We study liftings of automorphic forms on the pair of symplectic groups $(\text{GSp}(2),\text{GSp}(4))$ to…

We study Ruelle's type zeta and $L$-functions for a torsion free abelian group $\G$ of rank $\n\ge 2$ defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when $\n=2,4$ and 8, and in…

We introduce a new operation, copolar addition, on unbounded convex subsets of the positive orthant of real euclidean space and establish convexity of the covolumes of the corresponding convex combinations. The proof is based on a technique…

A simple version for the extension of the Taylor theorem to the operator functions was found. The expansion was done with respect to a value given by a diagonal matrix for the non-commutative case, and the coefficients are given both by…

We obtain a new extension of Rogers-Shephard inequality providing an upper bound for the volume of the sum of two convex bodies $K$ and $L$. We also give lower bounds for the volume of the $k$-th limiting convolution body of two convex…

Let $L$ be a second-order elliptic operator with analytic coefficients defined in $B_1\subseteq\mathbb R^n$. We construct explicitly and canonically a fundamental solution for the operator, i.e., a function $u:B_{r_0}\to\mathbb R$ such that…

We show that L\"uck's conjecture on torsion growth in homology implies that two 3-manifolds have equal volume if the fundamental groups have the same set of finite quotients.