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In the author's paper ''Poincar\'{e} series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity'' a relation is proved between the Poincar\'{e} series of the coordinate algebra of a two-dimensional quasihomogeneous…
We prove that the coefficients of certain mock theta functions possess no linear congruences modulo 3. We prove similar results for the moduli 2 and 3 for a wide class of weakly holomorphic modular forms and discuss applications. This…
In this paper we deal with semilinear problems at resonance. We present a sufficient condition for the existence of a weak solution in terms of the asymptotic properties of nonlinearity. Our condition generalizes the classical…
In this paper, we discuss the Poincar\'{e} series of Kac-Moody Lie algebras, especially for indefinite type. Firstly, we compute the Poincar\'{e} series of certain indefinite Kac-Moody Lie algebras whose Cartan matrices have the same type…
We discuss mock automorphic forms from the point of view of representation theory, that is, obtained from weak harmonic Maass forms give rise to nontrivial $(\mathfrak{g},K)$-cohomology. We consider the possibility of replacing the…
We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $\rho: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that…
We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterised naturally by the divisors of twelve. The Mathieu group…
We construct infinitely many nonholomorphic automorphic forms and modular forms associated to a discrete subgroup of infinite covolume of $U(n, 1)$.
We study the zeros of Poincar\'e series $P_{k,m}$ for the full modular group. We consider the case where $m \sim \alpha k$ for some constant $\alpha > 0$. We show that in this case a positive proportion of the zeros lie on the line…
Moduli spaces of stable coherent sheaves on a surface are of much interest for both mathematics and physics. Yoshioka computed generating functions of Poincare polynomials of such moduli spaces if the surface is the projective plane P2 and…
We study the Poincar\'e series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of $2\times2$ matrices; a denominator, which we…
We survey divisibility properties of the Fourier coefficients of modular functions inspired by Ramanujan. Then using recent results of the generalized Hecke operator on harmonic Maass functions and known divisibility of Fourier coefficients…
We study the coefficients of a natural basis for the space of weak harmonic Maass forms of weight $5/2$ on the full modular group. The non-holomorphic part of the first element of this infinite basis encodes the values of the partition…
In this paper we investigate Poincar\'e-type integral inequalities in the functional Musielak structure. We extend the ones already well known in Sobolev, Orlicz and variable exponent Sobolev spaces. We introduce conditions on the Musielak…
In this survey one discusses the notion of the Poincar\'e series of multi-index filtrations, an alternative approach to the definition, a method of computation of the Poincar\'e series based on the notion of integration with respect to the…
This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In two previous papers, we develop the basic theory of formal manifolds,…
This paper explores affine Weyl groups and their associated Hecke algebras, concentrating on the Poincar\'e series with coefficients in Hecke algebra. We investigate its relationship with zeta functions on complexes and extend existing…
The action of origin-preserving diffeomorphisms on a space of jets of symmetric connections is considered. Dimensions of moduli spaces of generic connections are calculated. Poincar\'e series of the geometric structure of symmetric…
A version of the twisted Poincar\'{e} duality is proved between the Poisson homology and cohomology of a polynomial Poisson algebra with values in an arbitrary Poisson module. The duality is achieved by twisting the Poisson module structure…
The purpose of this article is to give a simple and explicit construction of mock modular forms whose shadows are Eisenstein series of arbitrary integral weight, level, and character. As application, we construct forms whose shadows are…