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We study smoothness of generalized solutions of nonlocal elliptic problems in plane bounded domains with piecewise smooth boundary. The case where the support of nonlocal terms can intersect the boundary is considered. We announce…
We establish a criterion that ensures a bounded almost complex curve in a bounded almost complex 4-manifold minimizes genus amongst all smooth surfaces that share its homology class and the transverse link on its boundary. An immediate…
For a stationary sequence that is regularly varying and associated we give conditions which guarantee that partial sums of this sequence, under normalization related to the exponent of regular variation, converge in distribution to a…
Vertex similarity is a major problem in network science with a wide range of applications. In this work we provide novel perspectives on finding (dis)similar vertices within a network and across two networks with the same number of vertices…
For any elliptic normal surface singularity with rational homology sphere link we consider a new elliptic sequence, which differs from the one introduced by Laufer and S. S.-T. Yau. However, we show that their length coincide. Using the…
Consider a Hamiltonian action of a compact connected Lie group on a symplectic manifold $(M,\omega)$. Conjecturally, under suitable assumptions there exists a morphism of cohomological field theories from the equivariant Gromov-Witten…
We discuss an experimental approach to open problems in toric geometry: are smooth projective toric varieties (i) projectively normal and (ii) defined by degree 2 equations? We discuss the creation of lattice polytopes defining smooth toric…
We use Gromov-Witten theory to study rational curves in holomorphic symplectic varieties. We present a numerical criterion for the existence of uniruled divisors swept out by rational curves in the primitive curve class of a very general…
For real application and theoretical investigation of ordinary hypergraphs and non-ordinary hypergraphs, researchers need to establish standard rules and feasible operating methods. We propose a visualization tool for investigating…
A generalized-homology bordism-theory is constructed, such that for certain manifold homotopy stratified sets (MHSS; Quinn-spaces) homeomorphism-invariant geometric fundamental-classes exist. The construction combines three ideas: Firstly,…
In the present paper, we obtain real-analytic symplectic normal forms for integrable Hamiltonian systems with $n$ degrees of freedom near singular points having the type ``universal unfolding of $A_n$ singularity'', $n\ge1$ (local…
In this paper we exploit the geometric approach to the virtual fundamental class, due to Fukaya-Ono and Li-Tian, to compare the virtual fundamental classes of stable maps to a symplectic manifold and a symplectic submanifold whenever all…
We consider the number of crossings in a random embedding of a graph, $G$, with vertices in convex position. We give explicit formulas for the mean and variance of the number of crossings as a function of various subgraph counts of $G$.…
We introduce a parabolic flow of almost Kahler structures, providing an approach to constructing canonical geometric structures on symplectic manifolds. We exhibit this flow as one of a family of parabolic flows of almost Hermitian…
Many mechanical systems have configuration spaces that admit symmetries. Mathematically, such symmetries are modelled by the action of a group on a topological space. Several variations of topological complexity have emerged that take…
We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds in the spirit of topological field…
In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs,…
We initiate here the study of Gromov-Witten theory of locally conformally symplectic manifolds or $\lcs$ manifolds, $\lcsm$'s for short, which are a natural generalization of both contact and symplectic manifolds. We find that the main new…
Exploiting the affinity between stable generalized complex structures and symplectic structures, we explain how certain constructions coming from symplectic geometry can be performed in the generalized complex setting. We introduce…
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…