Related papers: Arithmetic Hirzebruch Zagier cycles
Minimal prime graphs (MPGs) are a special class of prime graphs (also known as Gruenberg-Kegel graphs) associated with finite solvable groups. A graph is an MPG if it has at least two vertices, is connected, its complement is triangle-free…
Every embedded surface $\mathcal{K}$ in the 4-sphere admits a bridge trisection, a decomposition of $(S^4,\mathcal{K})$ into three simple pieces. In this case, the surface $\mathcal{K}$ is determined by an embedded 1-complex, called the…
For a real affine hyperplane arrangement, we define an integer intersection matrix with a natural $q$-deformation related to the intersections of bounded chambers of the arrangement. By connecting the integer matrix to a bilinear form of…
We study the noncommutative geometry of algebras of Lipschitz continuous and H\"older continuous functions where non-classical and novel differential geometric invariants arise. Indeed, we introduce a new class of Hochschild and cyclic…
A Hilbert basis is a set of vectors X such that the integer cone (semigroup) generated by X is the intersection of the lattice generated by X with the cone generated by X. Define a graph to be (cut) Hilbert if its set of cuts forms a…
A type of directed multigraph called a W-digraph is introduced to model the structure of certain representations of Hecke algebras, including those constructed by Lusztig and Vogan from involutions in a Weyl group. Building on results of…
We study the algebraic structure of the mesonic moduli spaces of bipartite field theories by computing the Hilbert series. Bipartite field theories form a large family of 4d N=1 supersymmetric gauge theories that are defined by bipartite…
We study the algebraic hyperbolicity of the complement of very general degree $2n$ hypersurfaces in P^n. We prove the Algebraic Green-Griffiths-Lang Conjecture for these complements, and in the case of the complement of a quartic plane…
We propose an abelian categorification of $\hat{Z}$-invariants for Seifert $3$-manifolds. First, we give a recursive combinatorial derivation of these $\hat{Z}$-invariants using graphs with certain hypercubic structures. Next, we consider…
Near full-null degenerate singular points of analytic vector fields, asymptotic behaviors of orbits are not given by eigenvectors but totally decided by nonlinearities. Especially, in the case of high full-null degeneracy, i.e., the lowest…
The paper discusses the classification of surfaces of degree 10 and sectional genus 9 and 10. The surfaces of degree at most 9 are described through classical work dating from the last century up to recent years, while surfaces of degree 10…
We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann…
The matrix model of topological field theory for the moduli space of p-th spin curves is extended to the case of the Lie algebra of the orthogonal group. We derive a new duality relation for the expectation values of characteristic…
We consider the deformation of abelian integrals which arose from the study of SG form factors. Besides the known properties they are shown to satisfy Riemann bilinear identity. The deformation of intersection number of cycles on…
Motivated by problems arising in the complex analysis of perturbative quantum field theory, we investigate the homology of finite unions of certain non-degenerate quadratic affine hypersurfaces of complex dimension $n$ in general position.…
We first prove an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of hyperkaehler fourfolds of K3^{[2]}-type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is…
We propose the graph description of Teichm\"uller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe…
Arithmetical structures on a graph were introduced by Lorenzini as some intersection matrices that arise in the study of degenerating curves in algebraic geometry. In this article we study these arithmetical structures, in particular we are…
This article surveys some recent work of the author on Hilbert modular fourfolds X. After some preliminaries on the cohomology and special, codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate conjecture for X over…
This work is devoted to the study of the relationships between graph theory and the qualitative analysis of ordinary differential equations, with a special focus on two-dimensional systems. In particular, we reinterpret classical results…