相关论文: On some geometric aspects of the class of hv-conve…

Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…

We construct a splitting of the cohomology of configuration spaces of points on a smooth proper variety with a multiplicative Chow--K\"unneth decomposition. Applied to hyperelliptic curves, this shows that the hyperelliptic Torelli group…

Given a tiling of a 2D grid with several types of tiles, we can count for every row and column how many tiles of each type it intersects. These numbers are called the_projections_. We are interested in the problem of reconstructing a tiling…

Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…

We establish a system of PDE, called open WDVV, that constrains the bulk-deformed superpotential and associated open Gromov-Witten invariants of a Lagrangian submanifold $L \subset X$ with a bounding chain. Simultaneously, we define the…

To each oriented closed combinatorial manifold we assign the set (with repetitions) of isomorphism classes of links of its vertices. The obtained transformation L is the main object of study of the present paper. We pose a problem on the…

We consider a classical N. Steenrod's problem on realization of homology classes by images of the fundamental classes of manifolds. It is well-known that each integral homology class can be realized with some multiplicity as an image of the…

We define Gromov--Witten invariants of exploded manifolds. The technical heart of this paper is a construction of a virtual fundamental class $[\mathcal K]$ of any Kuranishi category $\mathcal K$ (which is a simplified, more general version…

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface.…

The Penrose triangle, staircase, and related ``impossible objects'' have long been understood as related to first cohomology $H^1$: the obstruction to extending locally consistent interpretations around a loop. This paper develops a…

The primary interest of this paper is to discuss the role of twisting cochains in the theory of characteristic classes. We begin with the homological description of monodromy map, associated with a connection on a trivial bundle over a…

We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…

These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the…

We construct H(curl) and H(div) conforming finite elements on convex polygons and polyhedra with minimal possible degrees of freedom, i.e., the number of degrees of freedom is equal to the number of edges or faces of the polygon/polyhedron.…

Finite convex geometries are combinatorial structures. It follows from a recent result of M.\ Richter and L.G.\ Rogers that there is an infinite set $T_{rr}$ of planar convex polygons such that $T_{rr}$ with respect to geometric convex…

We prove that the separated curve complex of a closed orientable surface of genus g is (g-3)-connected. We also obtain a connectivity property for a separated curve complex of the open surface that is obtained by removing a finite set from…

A closed formula is obtained for the integral $\int_{\mathcal{\bar{H}}_g^1}\kappa_{1}\psi^{2g-2}$ of tautological classes over the locus of hyperelliptic Weierstra\ss{} points in the moduli space of curves. As a corollary, a relation…

We introduce machinery to allow ``cut-and-paste''-style inductive arguments in the Torelli subgroup of the mapping class group. In the past these arguments have been problematic because restricting the Torelli group to subsurfaces gives…

For a graph $G$ and partition $\mathcal{U}$ of its vertex set, an independent transversal of $(G, \mathcal{U})$ is an independent set of $G$ that contains one vertex from each block of $\mathcal{U}$. Buys, Kang, and Ozeki studied when a…

We study the problem of classifying the irreducible projective varieties $X$ of dimension $n\ge 2$ in $\Bbb P^N$ which contain an algebraic family $\Cal F$ of dimension $h+1$ ($h<n$) of subvarieties $Y$ of dimension $n-h$, each one…