Smooth classifying spaces for groups defined using diffeological spaces.
problem Classifying smooth principal bundles for a smooth group G. method Developed the theory of smooth principal bundles using diffeological spaces, defining D-numerable bundles and proving classification results. result Smooth structures on Milnor's spaces EG and BG classify all D-numerable principal bundles over any diffeological space. We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact Kähler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they…
Study numerically flat bundles on non-Kähler manifolds.
problem Characterize holomorphic vector bundles on non-Kähler manifolds.
method Analyze conditions for numerically flatness and effectiveness.
result Numerically flatness is equivalent to specific stability conditions.
Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.
problem Understanding foliations with numerically flat tangent bundles on Kähler manifolds.
method Analyzing the structure of foliations on compact Kähler manifolds, extending earlier results.
result Smooth foliations with numerically flat tangent bundles induce a decomposition of the ambient manifold's tangent bundle.
Study numerically flat bundles on Fujiki manifolds using algebraic groups.
problem Characterize numerically flat principal bundles on Fujiki manifolds.
method Analyzes holomorphic principal bundles and their quotient structures, proving equivalence of conditions involving numerically flat ad bundles and nef line bundles.
result Establishes equivalence among numerically flat ad bundles, nef line bundles, and degree inequalities for reductions of structure groups.
Study Higgs bundles and their reductions to prove stability and cohomology properties.
problem Analyzing Higgs bundles and their stability conditions.
method Introduced H-nflatness, proved stability conditions, and used cohomology rings.
result H-nflat Higgs bundles are either stable or reducible to a parabolic subgroup.
Calculates Laplacian spectra on Calabi-Yau hypersurfaces.
problem Computing the spectrum of the Laplacian on complex manifolds.
method Numerical computation of eigenvalues and eigenmodes for line bundles.
result Agreement with exact results for P3 and a torus, first numerical results for Fermat quintic. Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers o…
A new discrete calculus for bundle-valued forms is proposed and validated.
problem Discretization of exterior calculus for bundle-valued forms.
method Discretization of Cartan's exterior calculus for differential forms with values in vector bundles.
result The proposed discrete operator mimics the continuous exterior covariant derivative and ensures numerical convergence.
Study the Albanese map for Kähler manifolds with nef anticanonical bundle.
problem Characterize the structure of the Albanese map for Kähler manifolds with nef anticanonical bundle.
method Analyze two cases: general fiber is Calabi-Yau or projective space. Provide proofs for both cases.
result For the case where the general fiber is Calabi-Yau, the manifold itself must be Calabi-Yau.
Two types of nonvanishing results are presented for compact Kähler varieties.
problem Deriving geometric consequences from numerical information in Kähler geometry.
method Analyzing non-uniruled varieties and hyperkähler manifolds to establish nonvanishing results for adjoint and nef bundles.
result Strong abundance-type results are obtained in dimension 4.
Master's thesis reviews bundle gerbe theory and introduces new isomorphic gerbe.
problem Understanding bundle gerbe theory and its applications.
method Introduced cup product bundle gerbe and showed its isomorphism to the pullback of the basic bundle gerbe by the Weyl map.
result Stable isomorphism between cup product bundle gerbe and pullback of basic bundle gerbe by the Weyl map.
Generalizes nonabelian Hodge theory to klt singularities.
problem Applying nonabelian Hodge theory to spaces with klt singularities.
method Uses descent theorems for numerically flat vector bundles and a new restriction theorem for semistable Higgs sheaves.
result Establishes a new restriction theorem for semistable Higgs sheaves.
New integrators for mechanical systems on Lie groups simplify based on group properties.
problem Designing numerical integrators for mechanical systems on Lie groups.
method Leverage retraction maps and Lie group properties to design structure-preserving integrators.
result Simplified design of integrators for Euler-Poincare and Lie-Poisson equations.
Investigates J-equation on holomorphic vector bundles over Kähler manifolds.
problem Analyzes properties and solutions of J-equation on holomorphic vector bundles. method Introduces and studies J-equation, provides algebraic and numerical criteria. result Provides an algebraic condition (asymptotic J-stability) and a numerical criterion for vortex bundles. Ray Singer torsion is a numerical invariant associated with a compact Riemannian manifold equipped with a flat bundle and a Hermitian structure on this bundle. In this note we show how one can remove the dependence on the Riemannian metric and on the Hermitian structure with the help of a base point and of an Euler str…
Proves complex Finsler bundles with positive curvature are ample and biholomorphic to projective space.
problem Solving a 1975 problem posed by S. Kobayashi about complex Finsler vector bundles with positive Kobayashi curvature.
method Analyzes properties of complex Finsler vector bundles with positive Kobayashi curvature.
result Complex Finsler vector bundles with positive Kobayashi curvature are ample and biholomorphic to projective space.
This article reviews ∞-bundles and their applications in geometry and physics.
problem Understanding higher bundles in geometry and physics.
method An ∞-categorical formulation of higher bundles. result Identification of higher bundles in various contexts.
We produce new non-Kähler complete steady gradient Ricci solitons whose asymptotics combine those of the Bryant solitons and the Hamilton cigar. We also obtain a family of complete Ricci-flat metrics with asymptotically locally conical asymptotics. Finally, we obtain numerical evidence for complete steady soliton struc…
The paper proves conditions for minimal compact Kähler manifolds with vanishing second Chern class.
problem Conditions for minimal compact Kähler manifolds with vanishing second Chern class.
method Study of the abundance conjecture and associated Iitaka fibrations.
result For a minimal compact Kähler manifold, the second Chern class vanishes if and only if the cotangent bundle is nef and the canonical bundle has numerical dimension 0 or 1.
The study of Chern numbers on vector bundles uses combinatorial methods to establish bounds and ordering.
problem Establishing bounds and ordering for Chern numbers on vector bundles.
method Combinatorial ideas to study Chern numbers on ample and numerically effective vector bundles.
result An effective lower bound for Chern numbers of ample vector bundles and reverse dominance ordering for nef vector bundles.
We produce new non-Kähler, non-Einstein, complete expanding gradient Ricci solitons with conical asymptotics and underlying manifold of the form R2×M2×⋯×Mr, where r≥2 and Mi are arbitrary closed Einstein spaces with positive scalar curvature. We also find numerical evidence for…
Study on hermitian Yang-Mills connections on blown-up manifolds.
problem Analyzing hermitian Yang-Mills connections on blown-up Kähler manifolds.
method Investigates connections for pullback vector bundles under specific conditions.
result Provides numerical criterion for convergence of hermitian Yang-Mills connections.
In this paper we study numerical properties of quotients of holomorphic log-tensors.
New Einstein metrics constructed on complex line bundle over CP1.
problem Constructing SU(2)-invariant negative Einstein metrics on complex line bundles. method Rigorous numerics to approximate, then fixed-point methods to perturb to genuine Einstein metrics.
result Complete, asymptotically hyperbolic Einstein metrics constructed.
On compact Kähler manifolds, we classify regular holomorphic foliations of codimension 1 whose canonical bundle is numerically trivial.
Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character χof P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle …
The first part of this paper discusses general procedures for finding numerical approximations to distinguished Kahler metrics, such as Calabi-Yau metrics, on complex projective manifolds. These procedures are closely related to ideas from Geometric Invariant Theory, and to the asymptotics of high powers of positive li…
The study connects monopole chains to Higgs bundles and classifies symmetric chains.
problem Classifying symmetric monopole chains invariant under cyclic actions.
method Formulation of a correspondence between monopole chains and spectral data, using the Nahm transform.
result Classification of symmetric monopole chains of charge k.
Let M be a once-punctured torus bundle over S1 with monodromy h. We show that, under certain hypotheses on h, "most" Dehn-fillings of M (in some cases all but finitely many) are virtually Z-representable. We apply our results to show that surgeries on the figure-eight knot with even numerator are …
Variant of Seiberg-Witten equations for multiple-spinors connects to stability of holomorphic bundles.
problem Detecting stability of holomorphic vector bundles using Seiberg-Witten equations.
method Abelian gauge-theoretic variant of Seiberg-Witten equations for multiple-spinors.
result Constructs a numerical invariant related to φ−stability of SU(n)−holomorphic vector bundles. A new method solves convex optimization on curved spaces.
problem Optimization on curved spaces with non-smooth functions.
method Convex bundle method on Riemannian manifolds.
result The method converges to a minimizer under mild conditions.
We consider minimal compact complex surfaces S with Betti numbers b_1=1 and n=b_2>0. A theorem of Donaldson gives n exceptional line bundles. We prove that if in a deformation, these line bundles have sections, S is a degeneration of blown-up Hopf surfaces. Besides, if there exists an integer m>0 and a flat line bundle…
Study Higgs bundles and flat connections on quasi-regular Sasakian manifolds.
problem Investigate Higgs bundles and flat connections on compact Sasakian manifolds.
method Introduce quasi-regularity and regularity of vector bundles, relate to orbibundles, extend non-abelian Hodge correspondence.
result Extend non-abelian Hodge correspondence to quasi-regular Sasakian manifolds.
This work proposes a geometric approach to equivariant message passing on Riemannian manifolds.
problem Efficiently processing data on Riemannian manifolds with equivariance.
method Geometric insight into equivariant message passing on Riemannian manifolds, using an equivariant embedding and diffusion process.
result A new class of equivariant GNNs on Riemannian manifolds.
Paper proposes a method to estimate consumer valuations from bundle sales data.
problem Estimating consumer valuations from bundle sales data using classical methods is challenging.
method Proposes an approach using EM algorithm and Monte Carlo simulation to estimate consumer valuations from bundle sales data.
result The approach can recover the distribution of consumers' valuations and is robust to unobserved no-purchases and clustered market segments.
Let G be a Lie supergroup and H a closed subsupergroup. We study the unimodularity of the homogeneous supermanifold G/H, i.e. the existence of G-invariant sections of its Berezinian line bundle. To that end, we express this line bundle as a G-equivariant associated bundle of the principal H-bundle G over G/H. We also s…
Study shows non-existence of almost complex structures on certain sphere bundles over complex projective spaces.
problem Existence of almost complex structures on sphere bundles over complex projective spaces.
method Chern class computations and divisibility properties of characteristic classes.
result Establishes a necessary condition for the non-existence of almost complex structures on certain bundles.
Newton's method tackles nonlinear mappings into vector bundles with connections and retractions.
problem Finding zeros of mappings from a manifold into a vector bundle.
method Local convergence using differentiability concepts, Banach space Riemannian distance, and affine covariant damping strategy.
result Illustrated application to generalized non-symmetric eigenvalue problems.
The paper studies foliations on smooth projective varieties and their properties.
problem Characterizing and understanding foliations on smooth projective varieties.
method Develops a structure theorem for smooth projective varieties with almost nef regular foliations, using a smooth morphism and MRC fibration.
result An almost nef regular foliation on a smooth projective variety can be decomposed into a numerically flat regular foliation and a smooth morphism.
In \cite{D3}, Donaldson defines a dynamical system on the space of Fubini-Study metrics on a polarized compact Kähler manifold. Sano proved that if there exists a balanced metric for the polarization, then this dynamical system always converges to the balanced metric (\cite{S}). In \cite{DKLR}, Douglas, et. al., conjec…
Proves strong version of Hopf problem for hyperbolic circle bundles.
problem Strong version of Hopf problem for circle bundles over aspherical manifolds with hyperbolic fundamental groups.
method Analyzes circle bundles over aspherical manifolds with hyperbolic fundamental groups.
result Every self-map of non-zero degree is either homotopic to a homeomorphism or a non-trivial covering and the bundle is virtually trivial.
The paper explores Parseval frames on vector bundles, proving their existence for certain cases.
problem Existence of Parseval frames on vector bundles.
method Using G-bundles and algebraic topology, the authors prove the existence of Parseval frames for orientable vector bundles and provide conditions for smaller size frames. result The existence of Parseval frames for orientable vector bundles and conditions for smaller size frames.
New stability criteria for vector bundles linked to Hermite-Einstein geometry.
problem Stability of higher-rank vector bundles and their moduli spaces.
method Introducing m-positivity and a smooth function for coherent subbundles, linking to Hermite-Einstein geometry. result Hermite-Einstein bundles are uniformly semi-stable, and new stability conditions are established.
Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other discretisation schemes. There are, however, several theoretical approaches to systems…
Solves a long-standing problem in Kähler geometry.
problem Existence of constant scalar curvature Kähler metrics on projectivized vector bundles.
method Introduces adiabatic slope stability, a weaker version of K-stability, and uses test configurations from subsheaves.
result Equivalence between adiabatic slope stability and existence of cscK metrics for simple vector bundles.
Novel framework improves wind power forecasts by bundling assets and using machine learning.
problem Inaccurate forecasts of intermittent renewable generation, especially wind power.
method Bundle-Predict-Reconcile (BPR) framework integrating asset bundling, machine learning, and forecast reconciliation.
result Significant improvement in forecast accuracy, especially at the fleet level.
Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons a…