Related papers: $C^{\infty}$ Stability, Canonical Maps, and Discre…
After a brief review of the historical role of analyticity in the study of critical phenomena, an account is given of recent discoveries of discretely holomorphic observables in critical two-dimensional lattice models. These are objects…
In this paper, I give a new construction of a K\"{a}hler-Einstein metrics on a smooth projective variety with ample canonical bundle. This result can be generalized to the construction of a singular K\"{a}hler-Einstein metric on a smooth…
Motivated by the Chern-Weil theory, we prove that for a given vector bundle $E$ on a smooth scheme $X$ over a field $k$ of any characteristic, the Chern classes of $E$ in the Hodge cohomology can be recovered from the Atiyah class. Although…
Let W be the germ of a smooth complex surface around an exceptional curve and let E be a rank 2 vector bundle on W. We study the cohomological properties of a finite sequence $E_i, 1 \leq i \leq t$ of rank 2 vector bundles canonically…
A general canonical formalism for discrete systems is developed which can handle varying phase space dimensions and constraints. The central ingredient is Hamilton's principal function which generates canonical time evolution and ensures…
Given two discrete Morse functions on a simplicial complex, we introduce the {\em connectedness homomorphism} between the corresponding discrete Morse complexes. This concept leads to a novel framework for studying the connectedness in…
We consider the relationship between the tight-binding Hamiltonian of the two-dimensional honeycomb lattice of carbon atoms with nearest neighbor hopping only and the 2+1 dimensional Hamiltonian of quantum electrodynamics which follows in…
For $S$ a closed surface of genus $g\geq2$, we construct a canonical diffeomorphism from the degree $3$ Fock-Thomas space $\mathcal{T}^3(S)$ of higher complex structures to the $\text{SL}(3,\mathbb{R})$ Hitchin component. Our construction…
Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete…
We present a complete theory of higher-order autonomous contact mechanics, which allows us to describe higher-order dynamical systems with dissipation. The essential tools for the theory are the extended higher-order tangent bundles, ${\rm…
On the basis of shell model simulations, it is conjectured that the Lanczos construction at fixed quantum numbers defines---within fluctuations and behaviour very near the origin---smooth canonical matrices whose forms depend on the rank of…
In the present chapter, we discuss an approach for transition from discrete to continuum description of thermomechanical behavior of solids. The transition is carried out for several anharmonic systems: one-dimensional crystal,…
A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce $C^r$ open sets ($r=1, 2, ..., \infty$) of symplectic diffeomorphisms and Hamiltonian systems, exhibiting…
We study the geometric motion of sets in the plane derived from the homogenization of discrete ferromagnetic energies with weak inclusions. We show that the discrete sets are composed by a `bulky' part and an external `mushy region'…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
A covariant Hamiltonian description was introduced in the dynamics of charges and electromagnetic interaction. By a canonical transformation this Hamiltonian formalism was transformed to obtain the Dirac generators for any form of…
In this survey paper we review classical results and recent progress about a certain topic in the spectral theory of two-dimensional canonical systems. Namely, we consider the questions whether the spectrum $\sigma$ is discrete, and if it…
In this paper, we study a geometric counterpart of the cyclic vector which allow us to put a rank 2 meromorphic connection on a curve into a ``companion'' normal form. This allow us to naturally identify an open set of the moduli space of…
We consider smooth vector bundles over smooth manifolds equipped with non-smooth geometric data. For nilpotent differential operators acting on these bundles, we show that the kernels of induced Hodge-Dirac-type operators remain isomorphic…
Gauge invariance in discrete dynamical systems and its connection with quantization are considered. For a complete description of gauge symmetries of a system we construct explicitly a class of groups unifying in a natural way the space and…