Related papers: Generalized Tonnetz and discrete Abel-Jacobi map

We show how to construct an ideal triangulation of a mapping torus of a pseudo-Anosov map punctured along the singular fibers. This gives rise to a new conjugacy invariant of mapping classes, and a new proof of a theorem of…

This article represents a major step in the unification of the theory of algebraic, topological and singular transition matrices by introducing a definition which is a generalization that encompasses all of the previous three. When this…

The Bitableax correspondence isomorphism/Koszul map Theorem (BCK Theorem, for short, Theorem 6.5 below) describes a relevant pair of mutually inverse vector space isomorphisms, the Koszul map K : U(gl(n))-> Sym(gl(n)) and the bitableaux…

Let $B$ be a Borel subgroup of $\mathrm{GL}_n(\mathbb{C})$ and $\mathbb{T}$ a maximal torus contained in $B$. Then $\mathbb{T}$ acts on $\mathrm{GL}_{n}(\mathbb{C})/B$ and every Schubert variety is $\mathbb{T}$-invariant. We say that a…

We discuss a generalization of Clifford algebras known as generalized Clifford algebras (in particular, ternary Clifford algebras). In these objects, we have a fixed higher-degree form (in particular, a ternary form) instead of a quadratic…

Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. In this paper, differential geometric properties of these metrics are…

Let $X \subset \mathbb{C}^n$ be an algebraic variety, and let $\Lambda \subset \mathbb{C}^n$ be a discrete subgroup whose real and complex spans agree. We describe the topological closure of the image of $X$ in $\mathbb{C}^n / \Lambda$,…

In this paper we evaluate some Toeplitz-type determinants. Let $n>1$ be an integer. We prove the following two basic identities: \begin{align*} \det{[j-k+\delta_{jk}]_{1\leq j,k\leq n}}&=1+\frac{n^2(n^2-1)}{12}, \\…

Let $\Theta_{k_1,\cdots,k_\ell}$ denote the generalized theta graph, which consists of $\ell$ internally disjoint paths with lengths $k_1,\cdots, k_{\ell}$, connecting two fixed vertices. We estimate the corresponding extremal number…

Building on works of Boulanger and Goto, we show that Goto's scalar curvature is the moment map for an action of generalized Hamiltonian automorphisms of the associated Courant algebroid, constrained by the choice of an adapted volume form.…

In this paper, we define a generalized elliptic genus of an almost complex manifold with an extra complex bundle which generalize the elliptic genus in [10]. This generalized elliptic genus is a generalized Jacobi form. By this generalized…

Let $G$ be a connected reductive complex algebraic group with a maximal torus $T$. We denote by $\Lambda$ the cocharacter lattice of $(T,G)$. Let $\Lambda^+ \subset \Lambda$ be the submonoid of dominant coweights. For $\lambda \in…

If (Q,A) is a marked polygon with one interior point, then a general polynomial f in K[x,y] with support A defines an elliptic curve C on the toric surface X_A. If K has a non-archimedean valuation into the real numbers we can tropicalize C…

Let E be a transitive Courant algebroid with scalar product of neutral signature. A generalized almost complex structure \mathcal J on E is a skew-symmetric smooth field of endomorphisms of E which squares to minus the identity. We say that…

A generalized topology in a set $X$ is a collection $\text{Cov}_X$ of families of subsets of $X$ such that the triple $(X,\bigcup \text{Cov}_X,\text{Cov}_X)$ is a generalized topological space in the sense of Delfs and Knebusch. In this…

Let $(\mathfrak{g},[p])$ be a finite-dimensional restricted Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $p>0$, and $G$ be the adjoint group of $\mathfrak{g}$. We say that $\mathfrak{g}$ satisfying the {\sl…

Given a double cover $\pi: \mathcal{G} \rightarrow \hat{\mathcal{G}}$ of finite groupoids, we explicitly construct twisted loop transgression maps, $\tau_{\pi}$ and $\tau_{\pi}^{ref}$, thereby associating to a Jandl $n$-gerbe…

A compact solvmanifold of completely solvable type, i.e. a compact quotient of a completely solvable Lie group by a lattice, has a K\"ahler structure if and only if it is a complex torus. We show more in general that a compact solvmanifold…

This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the…

We give a geometric Langlands type proof of the geometrization conjecture of the local Langlands correspondence introduced by the author for GL_1. For this we study an Abel-Jacobi morphism. We prove that this morphism is a pro-\'etale…