New proof of Khovanov-Rozansky homology base point independence in finite characteristic.
problem Base point independence of Khovanov-Rozansky homology in characteristic p. method New proof using infinitesimal symmetries of gl(p)-homology. result Proof of base point independence in characteristic p. Study characterizes points on projective surfaces using a cubic form.
problem Characterize points on projective surfaces and their impact on Euler characteristic.
method Define local indices for umbilics and godrons, use fundamental cubic form.
result Formulas relating indices to Euler characteristic determine coexistences of points.
Integrability of mean curvature near degenerate points in Heisenberg group.
problem Integrability of sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group.
method Introduction of mildly degenerate characteristic points and use of perimeter measure.
result The sub-Riemannian mean curvature is integrable in a neighborhood of these points.
Unified interpretation of sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.
problem Proving a sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.
method Measure-theoretic perspective, focusing on singular measures and characteristic points.
result Unified interpretation of previous results and natural geometric conditions for the theorem.
Study investigates induced geometry on surfaces in 3D contact manifolds.
problem Understanding the metric structure on surfaces embedded in 3D contact sub-Riemannian manifolds.
method Defined a coefficient to characterize characteristic points and identified global conditions for finite induced distance.
result Proved induced distance finite for certain surfaces with isolated characteristic points.
Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable equations. In this paper we discuss characteristic integrals in 3D and demonstrate that…
Study on knot 74 surgeries reveals infinite residue characteristics and infinite order points.
problem Arithmetic properties of Dehn surgery points on knot 74. method Analyzing the canonical component of the SL2(C)-character variety. result Infinite set of ramified places and infinite order points in the Mordell-Weil group.
The paper defines and calculates Euler characteristics for quandles.
problem Defining and calculating Euler characteristics for quandles.
method Definition and calculation of Euler characteristics for quandles.
result The quandle Euler characteristic of a compact connected Riemannian symmetric space coincides with the topological Euler characteristic.
Expands Euler-Poincare characteristic to supergeometry.
problem No new problem introduced.
method Introduced Euler-Poincare characteristic pair in supergeometry.
result Examined transversality in Π-symmetric supermanifolds. This paper studies fixed points of graph selfmaps and iwip endomorphisms of free groups.
problem Fixed points and fixed subgroups of selfmaps on graphs and surfaces.
method Analyzes iwip outer endomorphisms of free groups acting on stable trees.
result A sufficient condition for equality in a bound on a fixed point quantity is found.
Given an S1-manifold with isolated fixed points, some recent papers are concerned with the relationship between the least number of fixed points and the characteristic numbers of this manifold, and their proofs have some similar features. The main purpose of this short survey article is, by using the language of equ…
The paper proves group actions on spheres with odd fixed points.
problem Finite group actions on homology six-spheres with odd Euler characteristics.
method Analyzes smooth actions and fixed point sets of finite groups.
result The group is one of three specific types, and the fixed point set is a single point.
Affirmatively answers Kosniowski conjecture for unitary S^1-manifolds.
problem Proving Kosniowski conjecture for specific types of manifolds.
method Analyzing unitary S^1-manifolds with isolated fixed points.
result 4χ(M) > dim(M) for the given manifolds, affirming the conjecture.
Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincaré--Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related with corres…
The study counts critical points of Steklov eigenfunctions on manifolds.
problem Counting critical points of Steklov eigenfunctions on manifolds.
method Established an identity relating indexes of eigenfunctions and their restrictions to the boundary, and used it to count critical points.
result A precise count of interior critical points of Steklov eigenfunctions in terms of manifold's Euler characteristic and boundary sign changes.
The study finds rational points on specific types of hypersurfaces.
problem Identifying rational points on generic marked hypersurfaces.
method Analyzing hypersurfaces in projective spaces over various fields.
result Conditions for the existence of rational points on hypersurfaces.
Paper introduces Chern minimal surfaces in Hermitian surfaces and establishes identities related to their points and bundles.
problem Understanding the properties of Chern minimal surfaces in Hermitian surfaces.
method Using the Chern connection, the paper introduces Chern minimal surfaces and establishes identities related to their points and bundles.
result Established two identities relating the orders of complex and anticomplex points, the cap product of pull-back of first Chern class, and the Euler characteristics of tangent and normal bundles.
Unified framework detects changes in complex system models.
problem Accurate identification of dynamic changes in simulation models.
method Combines machine learning and process-driven simulation modeling.
result Significantly improves change point detection accuracy.
Study on symplectic semi-characteristic using cohomology and vector fields.
problem Defining and calculating the symplectic semi-characteristic of symplectic manifolds.
method Defined using even-degree primitive cohomology and proved a counting formula using vector fields.
result Established a counting formula for symplectic semi-characteristic and derived vanishing properties.
The Grove-Searle theorem on 2d manifolds with 8 or less symmetry groups has positive Euler characteristic.
problem Proving positive Euler characteristic for 2d manifolds with specific symmetry groups.
method Direct proof and analysis of fixed point components N with geodesic properties.
result Fixed point components N have amazing geodesic properties and can be S^2, RP^2, CP^d, HP^d, etc.
New method assesses prediction intervals across different operating points.
problem Difficulty in comparing prediction intervals across studies.
method Operating characteristics curves and gain over a simple reference.
result A novel operating point agnostic assessment methodology for prediction intervals.
In this paper we will explore a way to prove the hundred years old Gronwall's conjecture: if two plane linear 3-webs with non-zero curvature are locally isomorphic, then the isomorphism is a homography. Using recent results of S. I. Agafonov, we exhibit an invariant, the {\sl characteristic}, attached to each generic p…
Study fixed point indices and words at infinity for graph selfmaps.
problem Estimate indices of fixed point classes for graph selfmaps.
method Extend attracting fixed words at infinity, use relative train track technique, algebraic approach.
result Upper bound for attracting fixed words of injective endomorphisms of free groups.
Study SL(2,C) connections on Seifert-fibered spaces using gauge theory.
problem Counting SL(2,C) connections on Seifert-fibered spaces. method Introduced perturbations of the SL(2,C) Chern--Simons functional and proved a localisation result. result Formulae for the Euler characteristic and Poincaré polynomial of the stable locus of the SL(2,C) character variety of a Seifert-fibered homology 3-sphere. Study of intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
problem Characterizing the intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
method Construction and analysis of the intrinsic sub-Laplacian using Riemannian approximations and stochastic processes.
result The intrinsic sub-Laplacian is stochastically complete, ensuring the process does not hit characteristic points.
Formula conjectured for rational cuspidal curves in projective plane.
problem Counting rational cuspidal curves in projective plane.
method Extending Kontsevich's recursion formula and using geometric input about tangency of curves at nodal points.
result Conjectural formula agrees with earlier computations and extends to rational quartics with E6 singularity.
Suppose one is given a discrete group G, a cocompact proper G-manifold M, and a G-self-map f of M. Then we introduce the equivariant Lefschetz class of f, which is globally defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f, which is locally defined in terms of fixed point data…
Culler-Shalen theory extended to arbitrary characteristic.
problem Understanding essential surfaces in 3-manifolds over arbitrary fields.
method Using character varieties and group actions on trees over arbitrary algebraically closed fields.
result Examples show changes in the A-polynomial and essential surfaces in characteristic 2. The paper derives Gauss-Bonnet formulas for mappings between surfaces with boundary.
problem Calculating topological invariants for mappings between surfaces with boundaries.
method Defining singular points, constructing coherent tangent bundles, and applying Gauss-Bonnet formulas.
result Derives two Gauss-Bonnet type formulas for mappings between surfaces with boundaries.
The indicatrix or curvature ellipse and the characteristic curve of a surface in R4 are presented, as well as the projective duality connecting them. The characterisation of points in the surfaces as elliptic, parabolic and hyperbolic points, and the inflection points, are also discussed.
We show how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus g curves with n labeled points. This produces a polynomial with coefficients that include the Euler characteristic of the moduli space, and tautological intersection numbers on the compactified moduli space.
This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas …
Systemic risks characterizing the Russian overnight interbank market from the network point of view are analyzed.
We define a "circle Euler characteristic" of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group of ZG where G is the fundamental group of X. It is analogous in many ways to the ordinary Euler characteristic. One application is an intuitively satisfying formula for …
Motivated by the prediction of cell loads in cellular networks, we formulate the following new, fundamental problem of statistical learning of geometric marks of point processes: An unknown marking function, depending on the geometry of point patterns, produces characteristics (marks) of the points. One aims at learnin…
The study shows that nonpositively curved 4-manifolds with zero Euler characteristic have degenerating Ricci curvature.
problem Characterizing nonpositively curved 4-manifolds with zero Euler characteristic.
method Analyzing the Ricci curvature and foliations in neighborhoods of points.
result Nonpositively curved 4-manifolds with zero Euler characteristic have degenerating Ricci curvature.
Bound critical points for minimal Radó functions.
problem Counting interior critical points for minimal Radó functions.
method Bounding critical points in terms of boundary data and domain Euler characteristic.
result Bound the number of interior critical points.
We consider a C1 smooth surface with prescribed p(or H)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed p-mean curvature H∈C0, we show that any characteristic curve is C2 smooth and its (line) curvature equals −H in the nonsingular domain. By introducing ch…
Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs play an important role both in Mathematics …
Study stochastic processes on surfaces in contact sub-Riemannian manifolds using Riemannian approximations.
problem Analyzing stochastic processes on surfaces in contact sub-Riemannian manifolds.
method Employing Riemannian approximations, a second order partial differential operator is derived on the surface. The stochastic process moves along the characteristic foliation induced by the contact distribution.
result Elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from separatrices.
Suppose (Xn) is a sequence of positive-dimensional smooth projective complete intersections over Fq with dimensions bounded from above and with characteristic zero lifts (X~n) to smooth projective geometrically connected varieties. Suppose each complex variety X~nan has (underlying…
PUMA augments models to remove unique data points without performance loss.
problem Preserving model performance while removing unique training data points.
method Explicitly models data influence, reweights remaining data optimally.
result PUMA effectively removes unique data points without performance degradation.
Counting lattice points in moduli space of Klein surfaces.
problem Count lattice points in moduli space of Klein surfaces.
method Introduced metric Möbius graphs, counted lattice points weighted by non-orientability measure, deduced recursion for volumes.
result Proved refined version of Norbury's recursion and computed refined Euler characteristic.
Study heat content in sub-Riemannian structures, proving asymptotic series existence and coefficients.
problem Analyzing heat content in sub-Riemannian manifolds.
method Adapting Savo's technique to sub-Riemannian structures, computing coefficients up to order 5.
result Existence of full asymptotic series and explicit computation of coefficients up to order 5.
We describe the structure of the asymptotic lines near an inflection point of a Lagrangean surface, proving that in the generic situation it corresponds to two of the three possible cases when the discriminant curve has a cusp singularity. Besides being stable in general, inflection points are proved to exist on a comp…
We show that the Cappell-Shaneson version of Pick's theorem for simple lattice polytopes is a consequence of a general relation between characteristic numbers of virtual submanifolds dual to the characteristic classes of a stably almost complex manifold. This relation is analogous to the miraculous cancellation formula…
Proposes a new method to describe graph vertex features using characteristic functions.
problem Describing the distribution of vertex features at multiple scales on graphs.
method Introduces FEATHER, a computationally efficient algorithm to calculate characteristic functions based on random walk transition probabilities.
result Demonstrates that the proposed method creates high-quality graph representations and is robust to data corruption.
We discuss analogies between number theory and the theory of dynamical systems on spaces with a one-codimensional foliation. The emphasis is on comparing the "explicit formulas" of analytic number theory with certain dynamical Lefschetz trace formulas. We also point out a possible relation between an Arakelov-Euler cha…