Related papers: A monomial basis for the holomorphic functions on …

We introduce new families of cylindric symmetric functions as subcoalgebras in the ring of symmetric functions $\Lambda$ (viewed as a Hopf algebra) which have non-negative structure constants. Combinatorially these cylindric symmetric…

We study representation stability in the sense of Church, Ellenberg, and Farb \cite{FI-module} through the lens of symmetric function theory and the different symmetric function bases. We show that a sequence, $(F_n)_n$, where $F_n$ is a…

Let $X$ be a Banach space with an unconditional finite-dimensional Schauder decomposition $(E_n)$. We consider the general problem of characterizing conditions under which one can construct an unconditional basis for $X$ by forming an…

Zagier introduced special bases for weakly holomorphic modular forms to give the new proof of Borcherds' theorem on the infinite product expansions of integer weight modular forms on $\SL_2(\ZZ)$ with a Heegner divisor. These good bases…

Let $X$ and $Y$ be complex Banach spaces with $B_X$ denoting the open unit ball of $X.$ This paper studies various aspects of the {\em holomorphic Lipschitz space} $\mathcal HL_0(B_X,Y)$, endowed with the Lipschitz norm. This space is the…

We introduce a $q$-deformation of the Fock space of holomorphic functions on $\mathbb{C}$, based on a geometric definition of $q$-analyticity. This definition is inspired by a standard construction in complex differential geometry. Within…

We construct a monomial basis of the positive part of the quantized enveloping algebra associated to a finite-dimensional simple Lie algebra. As an application we give a simple proof of the existence and uniqueness of the canonical basis of…

Let B be a unital Banach algebra. A projection in B which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal A in B. In this set-up we prove a theorem to the effect that the bounded Hochschild…

Given an arrangement of hyperplanes in $\P^n$, possibly with non-normal crossings, we give a vanishing lemma for the cohomology of the sheaf of $q$-forms with logarithmic poles along our arrangement. We give a basis for the ideal $\cal J$…

We use the formalism of the Bergman tau functions to study the geometry of moduli spaces of holomorphic quadratic differentials on complex algebraic curves. We introduce two natural tau functions and interpret them as holomorphic sections…

We study canonical bases for spaces of weakly holomorphic modular forms of level 4 and weights in $\mathbb{Z}+\frac{1}{2}$ and show that almost all modular forms in these bases have the property that many of their zeros in a fundamental…

An old theorem of Weil and Kodaira says that for a compact K\"ahler manifold $X$ there is a closed logarithmic $1$-form with residue divisor $D$ if and only if $D$ is homologous to zero in $H_{2n-2}(X,\mathbb C)$. In the first part of this…

This work deals with the topological classification of germs of singular foliations on $(\mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and…

We describe the strong dual space $({\mathcal O}^s (D))^*$ for the space ${\mathcal O}^s (D) = H^s (D) \cap {\mathcal O} (D)$ of holomorphic functions from the Sobolev space $H^s(D)$, $s \in \mathbb Z$, over a bounded simply connected plane…

Let $G$ be a discrete group. The topological category of finite dimensional unitary representations of $G$ is symmetric monoidal under direct sum and has an associated $\mathbb{E}_\infty$-space $\mathcal{K}^{\mathrm{def}}(G)$. We show that…

We consider weighted banach spaces of holomorphic functions on the upper half plane that are determined by $ \|f\|=\sup_{y>0,-\infty<x<\infty}p(y)|f(x+iy)|<\infty $ for a very large class of weight functions p. We completely solve the…

The Rabinowitz-Floer homology of a Liouville domain W is the Floer homology of the free period Hamiltonian action functional associated to a Hamiltonian whose zero energy level is the boundary of W. It has been introduced by K. Cieliebak…

The $Golomb$ $space$ $\mathbb N_\tau$ is the set $\mathbb N$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+bn:n\ge 0\}$ with coprime $a,b$. We prove that the Golomb…

The goal of this note is to show that the Riemann-Hilbert problem to find multivalued analytic functions with $SL(2,\mathbb{C})$-valued monodromy on Riemann surfaces of genus zero with $n$ punctures can be solved by taking suitable linear…

Let $\mathcal{M}\subset B(\mathcal{H})$ be a semifinite von Neumann algebra, where $B(\mathcal{H})$ denotes the algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$, and let $\tau$ be a fixed faithful normal semifinite…