A new simple proof for surface map degree inequality.
problem Degree of maps between closed surfaces.
method Elementary proof without additional techniques.
result A new proof of the inequality χ(M) ≤ d·χ(N).
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48 results for “geometric degree”
The paper geometrically characterizes graded manifolds and proves the Frobenius theorem.
problem Understanding and characterizing graded manifolds.
method Geometric characterization and Frobenius theorem proof.
result Frobenius theorem proven for graded distributions.
Prescribed Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degreemath.DG
Study on prescribing curvature on specific manifolds with negative Gauduchon degree.
problem Prescribing Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree.
method Analysis of geometric flow convergence to obtain existence results.
result Existence results for curvature functions that are nonzero and nonpositive, and sign-changing cases.
The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.
problem Geometric obstructions and volume bounds in singular spaces.
method Developed new degree theorem for Alexandrov spaces using integral currents.
result Entropy-volume minimization prevents metric singularities in Gromov-Hausdorff limits.
In this paper, it is shown that every orientable closed 3-manifold maps with nonzero degree onto at most finitely many homeomorphically distinct irreducible non-geometric orientable closed 3-manifolds. Moreover, given any nonzero integer, as a mapping degree up to sign, every orientable closed 3-manifold maps with that…
This paper determines the flexible exponent for non-geometric 3-manifolds.
problem Bounding the mapping degree in terms of the Lipschitz constant for non-geometric 3-manifolds.
method Analyzing the infimum of α such that the inequality holds for any Lipschitz map.
result The flexible exponent for non-geometric 3-manifolds is determined.
Maps between surfaces have degree constraints based on their Euler characteristics.
problem Constraints on the degree of maps between surfaces based on their Euler characteristics.
method Used the Kneser-Edmonds factorization theorem and provided a simple proof.
result Maps between surfaces have degree constraints based on their Euler characteristics.
A graph clustering method that moves nodes to highest-degree neighbors.
problem Graph clustering for data with Morse regularity.
method Max-degree hill-climbing on graph nodes.
result Asymptotically consistent for random geometric graphs.
On the Casson knot invariantmath.GT
We study the Vassiliev knot invariant v_2 of degree 2. We present it via the degrees of maps of various configuration spaces related to a knot to products of spheres. This gives rise to numerous geometrical and combinatorial formulas for this invariant.
Paper calculates stable cohomology of universal degree d hypersurfaces.
problem Computing stable cohomology of universal degree d hypersurfaces.
method Uses stable cohomology and geometric description of stable classes.
result Geometric description of stable classes of universal degree d hypersurfaces.
The study constructs a hyperbolic orbifold to show how certain Salem numbers can be realized geometrically.
problem Constructing a geometric model for Salem numbers of degree 4.
method Constructing an arithmetic hyperbolic 6-orbifold and proving its properties.
result Any square-rootable Salem number of degree at most 4 can be realized as the exponential of a closed geodesic length in the constructed orbifold.
We obtain an ordering of closed aspherical 4-manifolds that carry a non-hyperbolic Thurston geometry. As application, we derive that the Kodaira dimension of geometric 4-manifolds is monotone with respect to the existence of maps of non-zero degree.
Study shows no hidden symmetries in specific spacetime metrics.
problem Demonstrating the absence of Killing tensors in Koutras-McIntosh spacetimes.
method Geometric theory of overdetermined PDEs and Cartan prolongation-projection method.
result No Killing tensors of low degrees in Wils metrics and generic pp-waves.
We re-examine classical mechanics with both commuting and anticommuting degrees of freedom. We do this by defining the phase dynamics of a general Lagrangian system as an implicit differential equation in the spirit of Tulczyjew. Rather than parametrising our basic degrees of freedom by a specified Grassmann algebra, w…
We explain some interesting relations in the degree three bounded cohomology of surface groups. Specifically, we show that if two faithful Kleinian surface group representations are quasi-isometric, then their bounded fundamental classes are the same in bounded cohomology. This is novel in the setting that one end is d…
Transformers reduce redundancy by focusing on invariant relational quantities.
problem Substantial internal redundancy in Transformer models due to coordinate-dependent representations and continuous symmetries.
method Reformulate representations, attention mechanisms, and optimization dynamics in terms of invariant relational quantities, eliminating redundant degrees of freedom by construction.
result Architectures that operate directly on relational structures, providing a principled geometric framework for reducing parameter redundancy and analyzing optimization.
A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinator…
Kontsevich's formula for a deformation quantization of Poisson structures involves a Feynman series of graphs, with the weights given by some complicated integrals (using certain pullbacks of the standard angle form on a circe). We explain the geometric meaning of this series as degrees of maps of some grand configurat…
The paper studies the locus in the rank 2 Higgs bundle moduli space corresponding to points which are critical for d of the Poisson commuting functions. These correspond to the Higgs field vanishing on a divisor of degree D. The degree D critical locus has an induced integrable system related to K(-D)-twisted Higgs bun…
A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface is geometric, that is, it is combinatorially equivalent to a geodesic triangulation with respect to a constant curvature metric on , and we list the …
With the renewed and growing interest in geometric continuity in mind, this article gives a general definition of geometrically continuous polygonal surfaces and geometrically continuous spline functions on them. Polynomial splines defined by G1 gluing data in terms of rational functions are analyzed further. A general…
New bounds on mapping degrees for geometric 3-manifolds.
problem Bounding the mapping degree in terms of Lipschitz constant for geometric 3-manifolds.
method Constructing Legendrian maps to prove bounds on flexible exponent.
result Complete result for flexible exponent of geometric 3-manifolds.
Study geometric manifolds in arbitrary dimensions, focusing on maps and diffeomorphisms.
problem Existence and properties of maps and diffeomorphisms in geometric manifolds.
method Analysis of geometric structures and homotopy invariants in arbitrary dimensions.
result Existence of Anosov diffeomorphisms and monotonicity of homotopy invariants.
Study geometric flows with varying parameters and prove continuous dependence.
problem Continuous dependence of flows on parameters in geometric settings.
method Derived suitable topologies for vector fields and flows, proved new continuous dependence.
result Proved continuous dependence of flows on parameters in a general topological space.
Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if all interior edge degrees are even. Using a refinement process which cuts the in…
For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the local degrees at the preimages of the branching points. A very old problem asks whe…
Counting ends on shrinkersmath.DG
The study counts ends on shrinkers using geometric covering methods.
problem Counting the number of ends on shrinkers.
method Geometric covering method to study the number of ends.
result Proves that the number of ends on any complete non-compact shrinker is at most polynomial growth with fixed degree.
Maps between non-compact surfaces can have geometric kernels under certain conditions.
problem Understanding when maps between non-compact surfaces have geometric kernels.
method Using Brown's proper fundamental group to establish sufficient conditions for geometric kernels.
result Characterization of conjugacy classes in the proper fundamental group and sufficient conditions for geometric kernels.
We give estimates for the eigenvalues of multi-form modified Dirac operators which are constructed from a standard Dirac operator with the addition of a Clifford algebra element associated to a multi-degree form. In particular such estimates are presented for modified Dirac operators with a -degree form $0\leq k\leq…
Unified framework for observables in n-plectic geometry.
problem Quantization of extended objects in higher geometric contexts.
method Develops a semi-simplicial set model for observables, using a Grassmann variable to encode submanifold codimensions.
result Establishes a categorified pre-n-Hilbert space and a quantization scheme matching multisymplectic geometry.
We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable objects in terms of negative geometric intersection numbers between correspondin…
Geometry of supergravity and the Batalin--Vilkovisky formulation of the theory in ten dimensionshep-th
We simplify supergravity in 10D using geometric insights.
problem Formulating supergravity in 10D without Lorentz degrees of freedom.
method Using generalised geometry, we describe the fibred structure of field space.
result Our action satisfies the classical master equation without Lorentz terms.
The notion of a higher bundle gerbe is introduced to give a geometric realization of the higher degree integral cohomology of certain manifolds. We consider examples using the infinite dimensional spaces arising in gauge theories.
Paper connects free-energy and low-degree hardness in high-dimensional statistics.
problem High-dimensional statistical inference problems are computationally hard.
method Defines a free-energy criterion and connects it to low-degree hardness.
result Establishes connection between free-energy and low-degree hardness for Gaussian models.
New geometric metrics improve Bayesian optimization performance evaluation.
problem Current metrics lack geometric insights and cannot compare algorithms effectively.
method Proposed four geometric metrics: precision, recall, average degree, and average distance.
result Proposed metrics provide more detailed evaluation of Bayesian optimization.
We show that uniformly finite homology of products of trees vanishes in all degrees except degree , where it is infinite dimensional. Our method is geometric and applies to several large scale homology theories, including almost equivariant homology and controlled coarse homology. As an application we determine …
Unpublished results of S Straus and W Browder state that two notions of homotopy equivalence for manifolds with smooth group actions - isovariant and equivariant - often coincide under a condition called the Gap Hypothesis; the proofs use deep results in geometric topology. This paper analyzes the difference between th…
The study explores which sets of integers can be realized as the degrees of maps between manifolds.
problem Which sets of integers can be realized as the degrees of maps between manifolds?
method Analyzes the set of degrees of maps between closed oriented manifolds of the same dimension.
result Finite arithmetic progressions and geometric progressions starting from 1 can be realized as degrees of maps between manifolds.
We prove that the Gromov-Hausdorff compactification of the moduli space of Kahler-Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian's theorem on the existence of Kahler-Einstein metrics on smooth Del Pezzo surfaces and classifies the deg…
New algebraic structures extend Courant algebroids to higher multi-Courant algebroids.
problem Extending Courant algebroid structures to higher multi-Courant algebroids.
method Constructing higher geometric versions of algebraic structures defined by Keller and Waldmann.
result Higher multi-Courant algebroids form a Poisson algebra.
Study shows non-trivial homology classes in certain symmetric manifolds.
problem Understanding bounded cohomology of symmetric manifolds.
method Analyzing totally geodesic submanifolds and their homology.
result Comparison map is not surjective in specific cohomology degrees.
We define a new class of irreducible groups, called groups not infinite-index presentable by products or not IIPP. We prove that certain aspherical manifolds with fundamental groups not IIPP do not admit maps of non-zero degree from direct products. This extends previous results of Kotschick and Loeh, providing new cla…
Symplectifying Biarcsmath.SG
A new method joins two arcs with a degree of freedom.
problem Joining two arcs with a precise point.
method Geometric approach using tangent vectors and points.
result A novel method to determine the join point.
The paper studies geometric constants under modified Ricci flows with variable parameters.
problem Understanding geometric constants under variable coupling parameters in Ricci flows.
method Introduced modified Ricci flows with variable coefficients, derived evolution formulas, and proved monotonicity conditions.
result Conditions for maintaining monotonicity of geometric constants under modified Ricci flows.
Study the expressivity and training complexity of polynomial neural networks.
problem Understanding the expressivity and training complexity of polynomial neural networks.
method Use algebraic geometry to describe neuromanifolds and neurovarieties, analyzing their dimension and learning degree.
result Characterized the dimension and learning degree of neuromanifolds, providing geometric and complexity measures.
The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the to…
The aim of this paper is to extend basic understanding of Engel structures through developing geometric constructions which are canonical to a certain degree and the dynamics of Cauchy characteristics in the transverse spaces which may exhibit elliptic, parabolic, or hyperbolic natures in typical cases.
In this paper we compute the Leray Schauder degree for a fourth order elliptic boundary value problem with exponential nonlinearity and Navier boundary condition. This will be made by proving a Poincare'-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of ba…