Proves non-positivity of Hirzebruch form on stable weights and connects to flat logarithmic connections.
problem Non-positivity of Hirzebruch form on stable weights
method Kempf--Ness and frame-potential inequality
result Zero locus of Hirzebruch form on stable weights corresponds to flat logarithmic connections
Study logarithmic flat connections on principal bundles using Lie groupoids.
problem Classify flat connections on principal bundles with logarithmic singularities.
method Use tools from Lie groupoid theory to classify representations and establish van Kampen theorems.
result Obtain a functorial Riemann-Hilbert correspondence for logarithmic connections.
Study flat connections with logarithmic singularities on complex plane curves.
problem Modeling flat connections with logarithmic singularities.
method Explicit finite-dimensional model construction and detailed investigation of specific cases.
result Construction of shifted Poisson structure on moduli spaces.
Abstract: New geometric incarnation of isomonodromy functors.
problem Classical isomonodromic deformations.
method Functorial upgrade of isomonodromic deformations using Lie groupoids.
result Geometric incarnation of isomonodromy functors as Morita equivalences.
Normal forms and moduli stacks for flat connections on complex manifolds.
problem Understanding singular flat connections on complex manifolds.
method Introducing homogeneous Lie groupoids and studying their representation theory to prove normal form theorems and moduli space structures.
result Moduli spaces of singular flat connections admit the structure of algebraic quotient stacks.
The paper establishes a Poisson Poincaré-Dulac theorem for Poisson-flat connections.
problem Analyzing Poisson-flat connections with logarithmic poles.
method Defining an Euler-Poisson principal part and residue theory, establishing a Poisson Poincaré-Dulac theorem.
result Any logarithmic Poisson-flat connection is holomorphically gauge equivalent to a pure Euler-Poisson normal form.
In this paper, we investigate representations of At(N), the Atiyah algebroids of a holomorphic line bundles N over a complex manifold Y. In particular, we relate At(N)-modules with logarithmic connections through two functors. On the one hand, we use these functors to the define in…
Given a smooth manifold M equipped with a properly and discontinuous smooth action of a discrete group G, the nerve M∙G is a simplicial manifold and its vector space of differential forms TotN(ADR(M∙G)) carry a C∞-algebra structure m∙. We sh…
Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connectio…
In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology si…
We show that bi-flat F-manifolds can be interpreted as natural geometrical structures encoding the almost duality for Frobenius manifolds without metric. Using this framework, we extend Dubrovin's duality between orbit spaces of Coxeter groups and Veselov's ∨-systems, to the orbit spaces of exceptional well-gene…
Geodesics spiral around compact subsets in CAT(0) spaces.
problem Understanding geodesic spiraling in CAT(0) spaces.
method Logarithm law-type result for geodesics in quotients of rank one CAT(0) spaces.
result Proved logarithm law for geodesic spiraling in certain CAT(0) spaces.
We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodi…
The paper studies hybrid connections on Hessian manifolds and their properties.
problem Investigating hybrid connections on Hessian manifolds.
method Defining and analyzing hybrid connections as incompressible affine connections projective to a flat connection D. result The difference abla−D is determined by the logarithmic differential of a Hessian potential function. Logarithmic connections on principal bundles over normal varieties are studied.
problem Existence and properties of logarithmic connections on principal bundles over normal varieties.
method Introducing logarithmic connections, showing equivalence to covariant derivatives, and proving existence conditions.
result Existence of logarithmic connections on principal bundles over normal varieties is equivalent to certain conditions on the associated vector bundles and adjoint bundles.
Paper studies statistical manifolds with logarithmic divergences.
problem Understanding statistical manifolds induced by logarithmic divergences.
method Constructs dual foliation of the statistical manifold.
result Extends dual foliation of a dually flat manifold.
Study Bergman kernels on Kähler manifolds, answering Lu-Tian's question.
problem Understanding Bergman kernels on Kähler manifolds and their properties.
method Localization and expansion analysis of Bergman kernels.
result Answered Lu-Tian's question about Bergman kernels having no logarithmic singularity.
Proves a stack of G-bundles with logarithmic connections is finite type.
problem Moduli of G-bundles with logarithmic connections over curves.
method Algebraic stack analysis and finite type proof.
result Proves the moduli stack is of finite type.
Logarithmic connections on complex manifolds with trivial tangent bundle.
problem Finding logarithmic connections on complex manifolds with specific properties.
method Analyzing holomorphic Cartan geometries and their connections.
result Logarithmic connections preserve holomorphic Cartan geometries.
Motivated by a remark and a question of Nicholas Katz, we characterize the tangent space of the space of Fuchsian equations with given generic exponents inside the corresponding moduli space of logarithmic connections: we construct a weight 1 Hodge structure on the tangent space of the moduli of logarithmic connections…
We prove a functorial correspondence between a category of logarithmic sl2-connections on a curve X with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover π:Σ→X. The proof is by constructing a pair of inverse functors $π^{\text{ab}}, π…
For a closed surface M with metric g, the Robin mass m(p) at the point p is the value of the Green function G(p,q) at p=q after the logarithmic singularity has been removed. The Laplacian-mass is the average value of the Robin mass, minus the value of the Robin mass for the round sphere of the same area. The Laplacian-…
Reconstructs fundamental groups from liquid local systems.
problem Reconstructing fundamental groups from category of liquid local systems.
method Theory of liquid vector spaces and liquid quasicoherent sheaves.
result Reconstructs topological fundamental group and twisted fundamental groupoids.
Study flat connections on Courant algebroids using Lie groups.
problem Flatness conditions on Courant algebroids.
method Metric generalized connections, flatness condition analysis.
result Existence of compact simple Lie groups as building blocks for flat transitive Courant algebroids.
Study of symplectically flat connections and their functionals on smooth manifolds.
problem Understanding symplectically flat connections and their functionals on smooth manifolds.
method Extend symplectically flat connections to ζ-flat connections, introduce functionals with zeroes as symplectically flat connections, study critical points of these functionals, describe characteristic classes of ζ-flat bundles. result Novel geometric flows and characteristic classes of ζ-flat bundles are described. Connected sum affects crossing numbers of flat virtual knots.
problem Understanding how connected sum impacts the crossing numbers of flat virtual knots.
method Analyzing minimal crossing diagrams and using super-additivity properties.
result Crossing number of flat virtual knots is super-additive under connected sum.
Introduces new info-geometric structure for dynamics on graphs and hypergraphs.
problem Modeling dynamics on discrete structures like graphs and hypergraphs.
method Introduces two dually flat structures: one on vertex space and another on edge space.
result Extends gradient flows to include nonequilibrium dynamics.
Study of conformal logarithmic Laplacian on sphere, connecting Yamabe problems and Sobolev spaces.
problem Yamabe-type problems and Sobolev spaces on the sphere.
method Detailed spectral analysis, conformal invariance, and Hilbert space introduction.
result Established precise connection between sphere and \(\mathbb{R}^N\) logarithmic Laplacian.
Let G be a Lie Group with a left invariant connection such that its connection function is skew-symmetric. Our main goal is to show a version of Pluzhnikov's Theorem for this kind of connection. To this end, we use the stochastic logarithm. More exactly, the stochastic logarithm gives characterizations for Brownian m…
Study connects G2-structures to flat connections on compact 3-manifolds.
problem Understanding moduli spaces of G2-structures and flat connections. method Equivalence between moduli spaces of G2-structures and flat connections on compact 3-manifolds. result Equivalence between moduli spaces of G2-structures and flat connections on compact 3-manifolds. Bi-flat F-structures link to differential bicomplexes and Gauss-Manin connections.
problem Understanding the geometric structure of bi-flat F-structures.
method Showed bi-flat F-structures define a differential bicomplex and relate to Gauss-Manin connections.
result Flat connections ablaGM associated with bi-flat structures can be identified with Levi-Civita connections of flat metrics. We show a correspondence between the set of all G-invariant projectively flat connections on a homogeneous apace M=G/K, and the one of all {G}^~-invariant flat connections on a homogeneous space {M}^~={G}^~/K, where {G}^~ is a central extension of G.
Flat Yang-Mills connections on pinched manifolds.
problem Stability of Yang-Mills connections on compact manifolds.
method Pinching conditions and weak stability criteria.
result No non-flat weakly stable Yang-Mills connections on δ(n)-pinched compact simply-connected Riemannian manifolds.
Complete classification of Hermitian manifolds with flat Gauduchon connections.
problem Classifying compact Hermitian manifolds with flat Gauduchon connections.
method Analyzing properties of Hermitian manifolds and using Gauduchon connections.
result Established a conjecture about Kähler-like conditions and flatness.
New invariant from non-acyclic flat connections.
problem Constructing a higher-loop perturbative invariant.
method Integral of a Chern-Simons volume form over moduli space of flat connections.
result Generalization of Chern-Simons invariant to non-acyclic connections.
We study the logarithmic L(α)-divergence which extrapolates the Bregman divergence and corresponds to solutions to novel optimal transport problems. We show that this logarithmic divergence is equivalent to a conformal transformation of the Bregman divergence, and, via an explicit affine immersion, is equivalent t…
New homogeneous manifolds with invariant Bismut Ricci flat connections are constructed.
problem Constructing homogeneous manifolds with invariant Bismut Ricci flat connections.
method Classification and construction of homogeneous spaces with specific properties.
result Examples of compact homogeneous Riemannian manifolds with invariant Bismut Ricci flat connections are provided.
Study jets of flat partial connections in foliations.
problem Characterize and understand flat partial connections in foliations.
method Define and apply jets to flat partial connections in smooth foliations and locally free sheaves, focusing on codimension one and arbitrary codimension foliations.
result Define and apply jets to characterize transversely affine and projective structures in foliations.
Introduces symplectic flatness for connections over symplectic manifolds.
problem Flatness conditions for connections over symplectic manifolds.
method Introduces symplectic flatness condition and twisting of differential complexes.
result Symplectic flat connections represent a subclass of Yang-Mills connections.
Study reveals connection between torus links and logarithmic VOAs.
problem Understanding the relationship between torus links and logarithmic VOAs.
method Proposed a geometric method to compute the singlet character of (s,t)-log VOA. result The singlet character of (s,t)-log VOA at the root of unity coincides with the Kashaev invariant and exhibits quantum modularity. New Poisson bracket connects to logarithmic manifolds.
problem Constructing a new Poisson bracket compatible with existing structures.
method Developed a new local Poisson bracket compatible with Adler-Gelfand-Dickey brackets, leading to a dispersionless limit.
result Leading term defines a logarithmic Dubrovin-Frobenius manifold.
A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.
Flatness of the loss curve is conjectured to be connected to the generalization ability of machine learning models, in particular neural networks. While it has been empirically observed that flatness measures consistently correlate strongly with generalization, it is still an open theoretical problem why and under whic…
In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita connection is flat. We show that no connected semisimple Lie group admits a left invarian…
Study on flat connections with controlled irregularity.
problem Boundedness of algebraic flat connections with limited irregularity.
method Analysis of families of algebraic flat connections and holonomic D-modules.
result Established boundedness of families of algebraic flat connections with controlled irregularity.
The paper characterizes surfaces in 4D space forms with flat normal connection.
problem Characterizing surfaces in 4D space forms with specific geometric properties.
method Analyzing linearly dependent conditions and using properties of sectional curvature.
result Characterizations of space-like and time-like surfaces with flat normal connection.
The parallel linear transports defined by flat linear connection are axiomatically described. On this basis a number of properties, some of which are new, of these transports and connections are derived.
SU(2) flat connection on 2D Riemann surface is shown to relate to the generalized twisted geometry in 3D space with cosmological constant. Various flat connection quantities on Riemann surface are mapped to the geometrical quantities in discrete 3D space. We propose that the moduli space of SU(2) flat connections on Ri…