Study integrability in Poisson and Dirac structures from quotients.
problem Integrability of quotient constructions in Poisson and Dirac geometry.
method Analysis of Poisson and Dirac structures from quotient constructions.
result Explicit constructions of Lie groupoids integrating specific geometric structures.
Researchers show how complex structures vary in symplectic quotients.
problem Understanding how complex structures change in symplectic quotients.
method Two approaches: complex geometry properties and variation of GIT quotients.
result Induced complex structure on symplectic quotients is locally invariant.
Reductive quotients preserve klt singularities in algebraic geometry.
problem Preserving klt singularities in quotients of klt singularities.
method Proving that the quotient of a klt type singularity by a reductive group is of klt type.
result The quotient of a klt variety by a reductive group results in a klt variety with a suitable boundary.
We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd …
Develops arithmetic PDE geometry using Fermat quotients.
problem Creating an arithmetic PDE analogue of Riemannian geometry.
method Using Fermat quotients and Frobenius elements in the absolute Galois group of a p-adic field. result Existence and uniqueness of geodesics and connections proved.
Study on curves on specific arithmetic quotients of hyperbolic 2-ball.
problem No complex curves of certain genus on these arithmetic quotients.
method Volume estimates and understanding special subvarieties.
result For large discriminants, no complex curves of fixed genus.
Polynomial density theorem for specific subgroup orbits in quotient spaces.
problem Effective density of orbits in arithmetic quotients of SL2(C) and SL2(R)imesSL2(R). method Use of Margulis function, incidence geometry tools, and spectral gap of ambient space.
result Proved effective density theorems with polynomial error rate.
Y. Nikonorov completes a proof in a geometry paper.
problem Completing a proof in a geometry paper.
method Completing an argument from a previous proof.
result Proof of Theorem 2.5 in JGA 27 (2017) is now complete.
Ray-marching method visualizes 8 Thurston geometries in real-time.
problem Accurately rendering and visualizing Thurston geometries in real-time.
method Ray-marching algorithms with theoretical framework for non-Euclidean geometries.
result Accurate interactive real-time views of Thurston geometries achieved.
The paper studies surface quotients of Fuchsian buildings.
problem Understanding group actions and symmetries in Fuchsian buildings.
method Developed theory of surface quotients, proved existence of discrete subgroups.
result Existence of discrete subgroups whose quotient is a compact surface.
We define and study noncommutative generalizations of submanifolds and quotient manifolds, for the derivation-based differential calculus introduced by M.~Dubois-Violette and P.~Michor. We give examples to illustrate these definitions.
In this article, we give a survey of Geometric Invariant Theory for Toric Varieties, and present an application to the Einstein-Weyl Geometry. We compute the image of the Minitwistor space of the Honda metrics as a categorical quotient according to the most efficient linearization. The result is the complex weighted pr…
Arithmetic Kleinian groups are distinguished by their finite quotients.
problem Distinguishing arithmetic Kleinian groups among all finitely generated residually finite groups.
method Constructing specific examples of arithmetic Kleinian groups and proving their profinite rigidity.
result Arithmetic Kleinian groups are uniquely identified by their finite quotients.
Study proves estimate for Hessian quotient equations on 2D Riemannian manifolds.
problem Problems posed by Delanoë and Urbas related to Hessian quotient equations.
method Maximum principle argument and new test function introduced to prove estimate.
result Unobstructed second order a priori estimate for real Hessian quotient equation in 2D.
The paper studies hyperbolic quotients of projection complexes and their actions.
problem Understanding the structure and properties of quotients of projection complexes.
method Analyzing the quotient of projection complexes by normal subgroups and studying the resulting actions.
result The quotient complex is δ-hyperbolic under certain conditions, and the quotient group is acylindrically hyperbolic.
A new Riemannian framework for robust covariance estimation.
problem Robust covariance estimation for elliptically distributed data with low-rank covariance structure.
method Original Riemannian geometry on quotient manifolds, new optimization framework, and divergence function.
result Derivation of intrinsic Cramér-Rao lower bounds for covariance and subspace estimation.
We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space R^{4d} by the action of a compact Abelian group. These 4n-dimensional quotients carry a multi-Hamilitonian action of an n-torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration o…
Polynomial error equidistribution for SL2 groups.
problem Equidistribution of orbits in arithmetic quotients.
method Margulis function, incidence geometry, spectral gap.
result Polynomial error rate for equidistribution.
SU(n)-structures derived from Kähler manifolds with torus actions.
problem Understanding SU(n)-structures on quotient spaces.
method Using Kähler manifolds with Hamiltonian actions of tori.
result Symplectic quotients inherit SU(n)-structures under certain conditions.
Embeds CR manifolds into complex spaces using equivariant actions.
problem Embedding strongly pseudoconvex CR manifolds into complex spaces.
method Equivariant CR maps and quotient maps.
result Universal quotient map property for CR manifolds.
Constructs Ricci-flat K3 metrics using D-geometry.
problem Computing the BPS index of a heterotic string theory.
method Hyper-Kähler quotient and D-geometry technology.
result Contains solution to BPS state counting problem.
The study of infinite groups through their finite quotients in geometry.
problem Understanding properties of infinite groups from their finite images.
method Analyzing infinite groups through their finite quotients and using low-dimensional topology.
result Recent results show how finite images can determine the group completely in some cases.
New Spin(7) metrics found from Kähler quotients.
problem Finding Spin(7) metrics from Kähler quotients. method Kähler reduction and PDEs on quotient manifolds.
result Infinitely many new explicit examples of Spin(7) metrics. We show that certain submanifolds of generalized complex manifolds ("weak branes") admit a natural quotient which inherits a generalized complex structure. This is analog to quotienting coisotropic submanifolds of symplectic manifolds. In particular Gualtieri's generalized complex submanifolds ("branes") quotient to sp…
Construct special Lagrangian fibrations on abelian varieties using retraction techniques.
problem Constructing special Lagrangian fibrations on abelian varieties.
method Explicit construction using special techniques in non-Archimedean geometry.
result Solved a conjecture of Kontsevich-Soibelman for finite quotients of abelian varieties.
New method uses quotient predictor space for better PAC-Bayes bounds, reducing KL divergence and improving model performance.
problem Overparameterized models with continuous symmetries can lead to biased predictions.
method Perform PAC-Bayesian analysis on quotient predictor space, constructing a canonical prior that reflects model's implicit bias.
result The new prior reduces KL divergence and improves model performance in experiments.
We obtain a generalization of the Kodaira-Morrow stability theorem for cosymplectic structures. We investigate cosymplectic geometry on Lie groups and on their compact quotients by uniform discrete subgroups. In this way we show that a compact solvmanifold admits a cosymplectic structure if and only if it is a finite q…
A new presentation of a quotient of braid groups leads to a new type of Burnside group.
problem Understanding the structure of quotient groups of braid groups.
method Purely group-theoretic methods, including presentations and finiteness results.
result A new presentation for the kernel of a truncated quotient map of braid groups.
Diffeology extends differential geometry to complex spaces.
problem Handling singular and infinite-dimensional settings in differential geometry.
method Introduces diffeology as a new framework.
result Diffeology provides a natural and effective framework for complex spaces.
The paper examines conditions for Gromov-Hausdorff convergence of metric quotients and provides examples of conic-flat surfaces.
problem Conditions for Gromov-Hausdorff convergence of metric quotients.
method Analyzes sufficient conditions for Gromov-Hausdorff convergence of metric quotients of a metric space.
result Concrete examples of sequences of two-dimensional conic-flat spheres converging to spheres with singularities.
Smooth resolutions found for quotient of R^2 by infinite discrete groups.
problem Symplectic resolutions of quotient spaces by infinite discrete subgroups.
method Constructing smooth symplectic resolutions for R^2 under infinite discrete subgroups of GL_2(R).
result Minimal resolutions of Du Val singular varieties are symplectic resolutions of R^2/G.
We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group G. These manifolds are obtained as quotients of the domains of discontinuity in generalized flag varieties G/P constructed by Kapovich-…
In this thesis we study the topology and geometry of hyperkähler quotients, as well as some related non-compact Kähler quotients, from the point of view of Hamiltonian group actions. The main technical tool we employ is Morse theory with moment maps. We prove a Lojasiewicz inequality which permits the use of Morse theo…
The paper studies binary icosahedral representations of hyperbolic 3-manifolds.
problem Understanding the representations of hyperbolic integral homology spheres into the binary icosahedral group.
method Relating 2I representations to quotient dimension and analyzing finite covers. result Hyperbolic 3-manifolds have quotient dimension 2 or 3, with specific cases obtained infinitely many times.
Constructing exponential families from statistical manifolds.
problem The central problem of constructing exponential families from statistical manifolds.
method Constructive approach proving every compact statistical manifold admits a foliation of Hessian manifolds.
result Compact orientable leaves are either finite quotients of flat torus or mapping torus with periodic monodromy.
We classify the 5-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 3 of 3) classifies those in which the linear isotropy representation is nontrivial but reducible. Most of the resulting geometries are products. Some interesting examples include a countably infinite family of inequiv…
Study nearly Kähler 6-manifolds with 2-torus symmetry, proving geometric properties and constructing new manifolds.
problem Characterize nearly Kähler 6-manifolds with 2-torus symmetry.
method Use multi-moment map and Laplace operator eigenfunctions, analyze quotient geometry, and construct new manifolds.
result Prove T2-action is free on level sets and determine the geometry of quotients. In this survey article we describe the geometry of toric hyperkähler varieties, which are hyperkähler quotients of the quaternionic vector spaces by tori. In particular, we discuss the Betti numbers, the cohomology ring, and variation of hyperkähler structures of these spaces with many improved results and proofs.
Several representations of geometric shapes involve quotients of mapping spaces. The projection onto the quotient space defines two sub-bundles of the tangent bundle, called the horizontal and vertical bundle. We investigate in these notes the sub-Riemannian geometries of these bundles. In particular, we show for a sel…
Given a compact Lie group, endowed with a bi-invariant Riemannian metric, its complexification inherits a Kaehler structure having twice the kinetic energy of the metric as its potential, and Kaehler reduction with reference to the adjoint action yields a stratified Kaehler structure on the resulting adjoint quotient. …
Near isospectrality forces full isospectrality for compact quotients of symmetric spaces.
problem Inverse spectral problem for Riemannian manifolds
method Proving near isospectrality implies full isospectrality
result Compact quotients of symmetric spaces have full isospectrality
Positive mass theorem and Yamabe equation on CR manifolds
problem Positive mass theorem and Yamabe equation on CR manifolds
method Positive mass theorem and Yamabe equation on CR manifolds
result Positive mass theorem in 3-dimensional CR geometry
Formula calculates Riemann-Roch number for singular symplectic quotients.
problem Computing Riemann-Roch number for singular symplectic quotients.
method Complete singular stationary phase expansion of Witten integral.
result New explicit local invariant of singularities in symplectic quotients.
Study of foliations' geometric and topological structures.
problem Analyzing the geometric and topological properties of transversely affine foliations.
method Attach holonomy group and quotient stack, identify reparametrisations, classify them, and study the Kato-Nakayama space.
result Holonomy group controls the geometric part, while the Kato-Nakayama space captures the topological and dynamical aspects.
Recently V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry. In this note we generalize this argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtai…
New sub-Riemannian structures fail synthetic curvature bounds.
problem Failure of synthetic curvature bounds in sub-Riemannian geometry.
method New stability results for local MCP under quotients, applied to specific sub-Riemannian structures.
result Ideal sub-Riemannian structures can fail the MCP, generically for high dimensions and rank > 3.
The study examines connections between Coxeter groups and their alternating quotients.
problem Characterizing connections between Coxeter groups and their alternating quotients.
method Analyzes the structure of right-angled Coxeter groups and their quasiconvex subgroups.
result Establishes conditions for the connectivity of alternating quotients of Coxeter groups.
When a complex semisimple group G acts holomorphically on a Kähler manifold (X,ω) such that a maximal compact subgroup K⊂G preserves the symplectic form ω, a basic result of symplectic geometry says that the corresponding categorical quotient X/G can be identified with quotient of the zero-set of the m…