arXiv research

The paper generalizes Hodge theory to semisimple local systems and proves a geometric Decomposition theorem.

New proof of Alesker's Irreducibility Theorem using localization techniques.

We reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two $K$-equivalent varieties are the same, which implies that several conjectures of algebraic cycl…

We prove a global algebraic version of the Lie-Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.

Counterexample disproves Yashiro's theorem on surface knots.

A geometric model for twisted $K$-homology is introduced. It is modeled after the Mathai-Melrose-Singer fractional analytic index theorem in the same way as the Baum-Douglas model of $K$-homology was modeled after the Atiyah-Singer index theorem. A natural transformation from twisted geometric $K$-homology to the new g…

This paper extends the Good Covering Theorem and Jordan Curve Theorem for proximal Alexandrov spaces.

Study Agol cycles for pseudo-Anosov 3-braids.

We extend the edge version of the classical Menger's Theorem for undirected graphs to $n$-dimensional simplicial complexes with chains over the field $\mathbb{F}_2$. The classical Menger's Theorem states that two different vertices in an undirected graph can be connected by $k$ pairwise edge-disjoint paths if, and only…

Brooks and Makover introduced an approach to studying the global geometric quantities (in particular, the first eigenvalue of the Laplacian, injectivity radius and diameter) of a ``typical'' compact Riemann surface of large genus based on compactifying finite-area Riemann surfaces associated with random cubic graphs; b…

Paper extends theorem on covering spaces and Jordan curves.

Proves existence of manifolds with Kervaire invariant one in specific dimensions.

Study Agol cycles in pseudo-Anosov 3-braids.

New Vassiliev invariant of order three derived from Fox-Hatcher cycles.

We introduce an invariant linked to some foundational questions in geometric measure theory and provide bounds on this invariant by decomposing an arbitrary cycle into uniformly rectifiable pieces. Our invariant measures the difficulty of cutting a nonorientable closed manifold or mod-2 cycle in $\mathbb{R}^n$ into ori…

Euler's theorem extended to complex structures.

The paper proves global existence and convergence of Möbius-invariant Willmore flow in 3-sphere.

New parities defined on virtual knots linked to crossing indices.

Starting from the candidate Bloch-Beilinson filtration on Chow groups of 0-cycles constructed by J. Lewis, we develop and describe geometrically a series of Hodge-theoretic invariants defined on the graded pieces. Explicit formulas (in terms of currents and membrane integrals) are given for certain quotients of the inv…

We prove that the group-homological version of the generalized Goncharov invariant of finite-volume locally rank one symmetric spaces determines their generalized Neumann-Yang invariant, which is defined using ideal fundamental cycles.

Intelligence emerges from stabilizing invariant cycles in memory.

The paper extends sphere theorems to higher-order mean curvature functions on specific hypersurfaces.

New method uses cycle consistency to enforce invariance in latent space.

The index theorem connects anomalies on a domain wall to global integrals.

New concept of k-holes in simple drawings and convex drawings.

We consider the classical problem of a position of n-dimensional manifold M in R^{n+2}. We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of a knotting M to R^{n+2}. In particular, we show that for any fixed quandle, quandle coloring, and shadow quan…

Smooth approximation of integral cycles mod 2 in Riemannian manifolds.

We give a new proof of the Alexander-Wermer Theorem that characterizes the oriented curves in C^n which bound positive holomorphic chains, in terms of the linking numbers of the curve with algebraic cycles in the complement. In fact, we establish a slightly stronger version which applies to a wider class of boundary 1-…

Global homotopies upgrade classical map in differential geometry.

By Gromov's mapping theorem for bounded cohomology, the projection of a group to the quotient by an amenable normal subgroup is isometric on group homology with respect to the $\ell^1$-semi-norm. Gromov's description of the diffusion of cycles also implicitly produces efficient cycles in this situation. We present an e…

Proves existence of longest paths in sub-Lorentzian problems.

The accurate characterization of the business cycles in the nonlinear dynamic financial and economic systems in the time of globalization represents a formidable research problem. The central banks and other financial institutions make their decisions on the minimum capital requirements, countercyclical capital buffer …

New approach to proving Chen-Donaldson-Sun theorem with examples.

New theorem bounds link volume using surface coefficients.

We study the Harvey-Lawson spark characters of level p on complex manifolds. Presenting Deligne cohomology classes by sparks of level $p$, we give an explicit analytic product formula for Deligne cohomology. We also define refined Chern classes in Deligne cohomology for holomorphic vector bundles over complex manifolds…

We propose to consider ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. "to be orthogonal", "to be tangent", etc.), as new objects in an extended Moebius--Lie geometry. It was recently demonstrated in several related papers, that such ensembles of cycles nat…

New graph invariant measures embeddability in 3D.

The simplest patterns of qualitative changes on the configurations of lines of principal curvature} around umbilic points on surfaces whose immersions into $\mathbb R^3$ depend smoothly on a real parameter (codimension one umbilic bifurcations) are described in this paper. Global effects, due to umbilic bifurcations, o…

The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In …

PerCDL learns personalized dictionaries for physiological signals combining global and local structures.

A singularity theorem based on asymptotic volume growth

In this paper we present new proofs of the Conway-Gordon-Sachs and Sachs Theorems on the linked cycles in graphs embedded in $\R^3$. We reduce these theorems to certain property of graphs mapped to the plane.

The paper extends a theorem about momentum maps to singular symplectic spaces.

Let $π$ be a finitely presented group. If h is a non trivial homology class in Hn($π$; Z), a theorem of Gromov (see [Gro83], 6) asserts the existence of regular geometric cycles which represent h, whose relative systolic volume is as close as desired to the systolic volume of h, in which we can control the volume of ba…

Properties of general Legendrian cycles $T$ acting in ${\mathbb R}^d\times S^{d-1}$ are studied. In particular, we give short proofs for certain uniqueness theorems with respect to the projections on the first and second component of such currents: In general, $T$ is determined by its restriction to the Gauss curvature…

The paper finds solutions for specific curvature conditions on 5D Lie groups.

New filling functions for groups with coefficients show different asymptotic behavior.

We investigate triangulations of the two-dimensional sphere and torus with the faces properly colored white and black. We focus on matchings between white triangles and incident vertices. On the torus our objects are perfect pairings, whereas on the sphere this is only true after removing one triangle and its vertices.…