Related papers: A discrete Morse perspective on knot projections a…
The perfect matching complex of a simple graph $G$ is a simplicial complex having facets (maximal faces) as the perfect matchings of $G$. This article discusses the perfect matching complex of polygonal line tilings and the $\left(2 \times…
We study the $k$-th nearest neighbor distance function from a finite point-set in $\mathbb{R}^d$. We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric…
Atiyah's formulation of what is nowadays called the convexity theorem of Atiyah-Guillemin-Sternberg has two parts: (a) the image of the moment map arising from a Hamiltonian action of a torus on a symplectic manifold is a convex polytope,…
We consider a finite group $G$ acting on a manifold $M$. For any equivariant Morse function, which is a generic condition, there does not always exist an equivariant metric $g$ on $M$ such that the pair $(f,g)$ is Morse-Smale. Here, the…
Motivated by the Landau-Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function $f$ near infinity. We prove that the…
Kirchhoff's matrix tree theorem is a well-known result that gives a formula for the number of spanning trees in a finite, connected graph in terms of the graph Laplacian matrix. A closely related result is Wilson's algorithm for putting the…
A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain…
In this paper we introduce linear mapping D from WnF\subset Ln into Lm\times Rm, induced by linear differential equation d/dt Fx(t)-C(t)x(t)=f(t),Fx(t_0)=f_0. We prove that D is closed dense defined mapping for any m\times n-matrix F. Also…
We prove that any perturbation of the symplectic part of the derivative of a Poisson diffeomorphism can be realized as the derivative of a $C^1$-close Poisson diffeomorphism. We also show that a similar property holds for the Poincar\'e map…
The cohomology theory known as Tmf, for "topological modular forms," is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to…
We present an alternative application of discrete Morse theory for two-particle graph configuration spaces. In contrast to previous constructions, which are based on discrete Morse vector fields, our approach is through Morse functions,…
We investigate the interplay of generalized global symmetries in 2+1 dimensions in a lattice model that couples a $\mathbb{Z}_N$ clock model to a $\mathbb{Z}_N$ gauge theory via a topological interaction. This coupling binds the charges of…
We study a certain discrete differentiation of piecewise-constant functions on the adjoint of the braid hyperplane arrangement, defined by taking finite-differences across hyperplanes. In terms of Aguiar-Mahajan's Lie theory of hyperplane…
Given a compact symplectic toric manifold $(M,\omega, \mathbb{T})$, we identify a class $DGK_{\omega}^{\mathbb{T}}(M)$ of $\mathbb{T}$-invariant generalized K\"ahler structures for which a generalisation the Abreu-Guillemin theory of toric…
We study Morse theory on noncompact manifolds equipped with exhaustions by compact pieces, defining the Morse homology of a pair which consists of the manifold and related geometric/homotopy data. We construct a collection of Morse data…
We derive an explicit formula for the fundamental solution $K_{T_{q+1}}(x,x_{0};t)$ to the discrete-time diffusion equation on the $(q+1)$-regular tree $T_{q+1}$ in terms of the discrete $I$-Bessel function. We then use the formula to…
The 't Hooft expansion of SU(N) Chern-Simons theory on $S^3$ is proposed to be exactly dual to the topological closed string theory on the $S^2$ blow up of the conifold geometry. The $B$-field on the $S^2$ has magnitude $Ng_s=\lambda$, the…
We study the Borel summation of the Gromov-Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson-Thomas invariants of the resolved conifold, having a direct…
A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar map models (e.g., uniform triangulations…
The necessity of an introduction of discrete physical objects in physics conception is analysed taking into consideration an optimum stage for postulating of some like objects in microworld as well as in macroworld including the new…