Related papers: Generating Functions in $\mathbb{R}^{2n}$ and the …

We study the integrable structure of the 2D Laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in the transverse direction. Similar to the Laplacian growth in radial geometry, this…

Given a square matrix with elements in the group-ring of a group, one can consider the sequence formed by the trace (in the sense of the group-ring) of its powers. We prove that the corresponding generating series is an algebraic…

We show how to construct, for a Lagrangian of arbitrary order, a Lepage equivalent satisfying the closure property: that the Lepage equivalent vanishes precisely when the Lagrangian is null. The construction uses a homotopy operator for the…

This is an overview of recent results on the use of 2D Toda $\tau$-functions as generating functions for multiparametric families of weighted Hurwitz numbers. The Bose-Fermi equivalence composed with the characteristic map provides an…

We construct exact Lagrangian fillings of Legendrian torus links $\Lambda(k, n-k)$ that are fixed by a Legendrian loop that acts by $2\pi\ell/n$ rotation. Using these rotationally symmetric fillings, we produce fillings of the corresponding…

Starting with two copies of the random energy model coupled with independent magnetic fields, the generating function for the connected correlator of the magnetization is exactly derived. Without use of the replica trick, it is shown that…

Suppose a Lagrangian is constructed from its fields and their derivatives. When the field configuration is a distribution, it is unambiguously defined as the limit of a sequence of smooth fields. The Lagrangian may or may not be a…

Given an operad P with a finite Groebner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function…

Let $T$ be an underlying space with a non-atomic measure $\sigma$ on it. In [{\it Comm.\ Math.\ Phys.}\ {\bf 292} (2009), 99--129] the Meixner class of non-commutative generalized stochastic processes with freely independent values,…

We use spectral invariants in Lagrangian Floer theory in order to show that there exist \emph{isometric} embeddings of normed linear spaces (finite or infinite dimensional, depending on the case) into the space of Hamiltonian deformations…

We establish the following Hadamard--Stoker type theorem: Let $f:M^n\rightarrow\mathscr{H}^n\times\mathbb R$ be a complete connected hypersurface with positive definite second fundamental form, where $\mathscr H^n$ is a Hadamard manifold.…

We investigate the Lagrangian formulation of the double-copy correspondence between gauge theories and gravity, up to the cubic order. Building on the definition of the double-copy field as a convolution of two vectors, we obtain free…

We evaluate the quantum gravity partition function that counts the dimension of the Hilbert space of a simply connected spatial region of fixed proper volume in the context of Lovelock gravity, generalizing the results for Einstein gravity…

We propose a new class of gravity-matter-gauge theories in terms of two different non-Riemannian volume-forms independent of the Riemannian metric. The nonlinear gauge field system contains a square-root $\sqrt{-F^2}$ of the standard…

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on $\mathbb{C}^{2n+1}$ for any $n\in\mathbb{N}$. We provide several approximation and desingularization results which enable us to prove…

The purpose of this paper is to establish several new results about the Hodge theory of Lagrangian fibrations on (not necessarily compact) holomorphic symplectic manifolds. Let $M$ be a holomorphic symplectic manifold of dimension $2n$ that…

Let $\OO$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $(X,\omega).$ We define a functional $\CC:\OO \to \R$ for each differential form $\beta$ of middle…

We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit…

This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a…

Let $G$ be $2$-generated group. The generating graph of $\Gamma(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G=\langle g,h\rangle$. This graph encodes the combinatorial…