New non-manifold monodromy spaces found for 3D branched covers.
problem Understanding monodromy spaces of branched covers between manifolds.
method Construction of a specific map to demonstrate non-manifold spaces.
result Found a self-map of the 3-sphere with non-manifold monodromy space.
We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type. For suitable open subspaces we can define the bifurcation braid monodromy takin…
Study shows hyperbolic knots' monodromy without fixed points.
problem Understanding fixed points in knot monodromy.
method Using Baldwin--Hu--Sivek argument and knot Floer homology.
result Monodromy of hyperbolic fibered knots is freely isotopic to a map with no fixed points.
Axis bundles in free-by-cyclic groups have non-generic monodromies.
problem Characterizing monodromies in free-by-cyclic groups.
method Analyzing monodromy actions on outer space and using properties of axis bundles.
result Monodromies having a 'lone axis' are non-generic and projectively discrete.
The article extends Thurston's Grafting Theorem to signed spaces and defines a framed monodromy map.
problem Extending Thurston's Grafting Theorem to signed spaces.
method Proves the analogue of Thurston's Grafting Theorem for signed spaces, defines a framed monodromy map.
result Characterizes PSL(2,C)-representations and shows the monodromy map is a local biholomorphism.
Reduces super Teichmüller space constraints for Ramond punctures.
problem Constraints on super Teichmüller space for Ramond punctures.
method Imposes constraints on odd coordinates of super Teichmüller space for monodromies around Ramond punctures.
result Reduces odd dimension to match moduli spaces of super Riemann surfaces.
The notion of local equivalence relation on a topological space is generalised to that of local subgroupoid. The main result is the construction of the holonomy and monodromy groupoids of certain Lie local subgroupoids, and the formulation of a monodromy principle on the extendibility of local Lie morphisms.
Authors compute monodromy groups for SL(n) and GL(n) Hitchin fibrations.
problem Computing monodromy groups for Higgs bundles on Riemann surfaces.
method Using spectral curves and Picard-Lefschetz transformations, they construct and classify vanishing lattices.
result They determine the structure of monodromy groups for SL(n) and GL(n) Hitchin fibrations.
Proves theorem about meromorphic projective structures with complex poles.
problem Proving a theorem about meromorphic projective structures with complex poles.
method Using coordinates on the moduli space of framed representations from Fock and Goncharov.
result Proves the analogue of a theorem of Gallo-Kapovich-Marden.
Holomorphic curves in moduli spaces are quasi-isometrically immersed.
problem Understanding the geometric properties of holomorphic curves in moduli spaces.
method Analyzing the quasi-isometric immersion of holomorphic maps from hyperbolic surfaces to moduli spaces.
result Holomorphic curves are quasi-isometrically immersed with parameters depending on surface and moduli space properties.
We compute Stokes matrices and monodromy for the quantum cohomology of projective spaces. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves.
Monodromy map from differential systems to character variety is generically immersive for complex G-representations.
problem Characterizing when the monodromy map is immersive for differential systems.
method Analyzing the space of g-differential systems on a compact Riemann surface and the character variety of G-representations. result The monodromy map is an immersion at the generic point when the complex dimension of G is at least three. We determine the Lagrangian monodromy group L(T) and the smooth monodromy group S(T) of a Clifford torus T in the symplectic 4-space. We show that L(T) is isomorphic to the infinite dihedral group, and S(T) is generated by three reflections. We give explicit formulas for both groups. We also show that if a Lagrangian t…
Study stabilizes components of Galois cover moduli spaces.
problem Decide equivalence and stable equivalence of monodromy maps.
method Develop algebraic framework to study equivalence classes of monodromy maps.
result Recover a homological invariant that distinguishes equivalence classes.
New manifold structures on Weyl group orbit spaces proven.
problem Constructing generalized Frobenius manifold structures.
method Construction on orbit spaces of affine Weyl groups.
result Monodromy groups are parabolic subgroups.
The paper discusses abelian integration and monodromy groups for Lie algebroids.
problem Existence and obstructions of smooth abelian integration of Lie algebroid abelianizations.
method Path-space construction and extended monodromy groups.
result Obstructions to abelian integration are related to extended monodromy groups.
We bound the index of a subgroup in iterated Kodaira fibrations.
problem Bounding the index of a subgroup in iterated Kodaira fibrations.
method Passing to a finite index subgroup of π_1(X) to achieve the desired structure.
result We provide a bound on the index of such a group.
Study shows monodromy kernels are large, failing to prove commensurability in specific strata.
problem Proving commensurability of mapping class groups through monodromy kernels.
method Analyzing monodromy maps for specific strata in translation surfaces.
result Kernels of monodromy maps contain a non-abelian free group of rank 2.
Study homotopy groups of open books and their pages, pages, and bindings.
problem Homotopy groups of open books and their components.
method Homotopy theoretic conditions on monodromy, integral and rational loop space decompositions.
result Integral and rational loop space decompositions for open books under specific conditions.
This study examines arithmetic properties of GIB manifolds and their monodromy representations.
problem Arithmetic structure of generalized Inoue--Bombieri manifolds.
method Study of monodromy representations and their arithmetic properties.
result The image of the monodromy representation is a subgroup of a cocompact arithmetic lattice.
Unified understanding of integrability obstructions for Lie algebroids.
problem Equivalence of integrability obstructions for transitive Lie algebroids.
method Unified approach using cohomological and homotopical methods.
result Unified understanding of integrability obstructions.
New groups discovered with unique properties in a specific space.
problem Finding new discrete subgroups with special properties in a mathematical space.
method Proved by showing groups play ping-pong on cones, related to crooked surfaces.
result Infinite family of discrete subgroups with remarkable properties in Sp4(R). Characterizes monodromy groups for projective structures on surfaces with specified poles.
problem Characterizing monodromy groups for meromorphic projective structures on surfaces with specific singularities.
method Geometric interpretation of Fock-Goncharov coordinates and recent results on moduli spaces of representations.
result Proves the analogue of a theorem for closed surfaces and settles a long-standing question.
We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of L-twisted G-Higgs bundles, for the groups G=GL(2,C), SL(2,C) and PSL(2,C). We also determine the twisted Chern class of the regula…
Study perturbations of submodules in Drury-Arveson space, finding smooth vector bundles with Hermitian connections.
problem Geometry of perturbations in Drury-Arveson space.
method Analysis of smooth vector bundles with Hermitian connections and computation of parallel transport operators.
result Found natural Hermitian connections on perturbed submodules.
Study on constraints of monodromy group for smooth plane curves.
problem Understanding constraints on the monodromy group of smooth plane curves.
method Combining algebro-geometric and mapping class group theory.
result Invariant constrains the monodromy group, especially for degree 5.
We give a global version of the Bryant representation of surfaces of constant mean curvature one (cmc-1) in hyperbolic space. This allows to set the associated non-abelian period problem in the framework of flat unitary vector bundles on Riemann surfaces. We use this machinery to prove the existence of certain cmc-1 su…
Study monodromy relations in Seifert fibered spaces for 3-manifold fillings.
problem Understanding symplectic fillings of Seifert fibered 3-manifolds.
method Analyzes monodromy factorizations and their impact on fillings.
result Computes and bounds Euler characteristics of different fillings.
Let K be a fibered knot in the 3-sphere. We show that if the monodromy of K is sufficiently complicated, then Dehn surgery on K cannot yield a lens space. Work of Yi Ni shows that if K has a lens space surgery then it is fibered. Combining this with our result we see that if K has a lens space surgery then it is fibere…
We present a theorem on the unitarizability of loop group valued monodromy representations and apply this to show the existence of new families of constant mean curvature surfaces homeomorphic to a thrice-punctured sphere in the simply-connected 3-dimensional space forms R3, $\bbS^3 $ and $\bbH^3$. Additionally, we…
Monodromy groups help test Lie groupoids for faithful representations.
problem Testing Lie groupoids for faithful representations.
method Constructing monodromy groups and using them to test Lie groupoids.
result Monodromy groups can be used to determine if Lie groupoids lack faithful representations.
Explains solution space of Toda equations using Lie theory.
problem Parametrizing globally smooth solutions of tt*-Toda equations.
method Lie-theoretic explanation through monodromy data and algebraic group conjugacy classes.
result Expresses solutions as a convex subset of a Weyl alcove of SU_{n+1}.
We find the exact values of complexity for an infinite series of 3-manifolds. Namely, by calculating hyperbolic volumes, we show that c(N_n)=2n, where c is the complexity of a 3-manifold and N_n is the total space of the punctured torus bundle over S^1 with monodromy 2&1 1&1 ^n$. We also apply a recent result of Matv…
The paper studies the monodromy of plane curves, finding a specific kernel.
problem Analyzing the monodromy of universal families of plane curves.
method Using algebraic geometry and Weil-Petersson geometry of Teichmüller space.
result The kernel of the monodromy homomorphism for quartic curves is isomorphic to F∞imesZ/3Z. Positive braids linked to knot invariants and geometric monodromy groups.
problem Understanding knot invariants and geometric monodromy groups for positive braids.
method Associate braid monodromy groups to positive braids, identify these groups with framed mapping class groups for knots, and use these to determine knot invariants.
result Geometric monodromy groups of irreducible singularities are determined by genus and Arf invariant of associated knots.
We show that a variety of monodromy phenomena arising in geometric topology and algebraic geometry are most conveniently described in terms of quandle homomorphisms from a knot quandle associated to the base to a quandle associated to a fiber. We consider the cases of the monodromy of a branched covering, braid monodro…
Paper studies Lagrangian submanifolds and their homological monodromy.
problem Understanding the homological monodromy of Lagrangian submanifolds.
method Proves triviality of homological Lagrangian monodromy under specific conditions.
result Homological Lagrangian monodromy is trivial if Hofer energy is less than minimum energy of J-holomorphic spheres and discs.
Study of origamis in minimal stratum with single cylinders, calculating spin parities and monodromy groups.
problem Understanding the structure and properties of origamis in the minimal stratum of moduli space.
method Construction and analysis of minimal [1,1]-origamis, calculation of spin parities, and investigation of monodromy groups. result All minimal [1,1]-origamis have monodromy groups that are almost always finite simple groups. Paper defines almost flat relative bundles and their correspondence with quasi-representations.
problem Understanding the equivalence of topological and smooth almost flatness.
method Introduces almost flatness for relative bundles, proves equivalence, and defines correspondence with quasi-representations.
result Equivalence of topological and smooth almost flatness and construction of almost flat bundles.
We study fillings of contact structures supported by planar open books by analyzing positive factorizations of their monodromy. Our method is based on Wendl's theorem on symplectic fillings of planar open books. We prove that every virtually overtwisted contact structure on L(p,1) has a unique filling, and describe fil…
This paper shows similarities in deformation spaces of Kleinian groups and anti-holomorphic maps.
problem Comparing deformation spaces of Kleinian reflection groups and anti-holomorphic rational maps.
method Established an analogue of Thurston's compactness theorem for critically fixed anti-rational maps and characterized deformation space interactions.
result Deformation spaces of Kleinian reflection groups and anti-holomorphic rational maps share striking similarities.
New conditions found for hyperbolic bicycle tracks.
problem Conditions for hyperbolic bicycle tracks.
method Hyperbolic development interpretation of bicycling monodromy.
result New necessary and sufficient conditions for hyperbolic bicycle tracks.
Introduces new spaces for configurations of points with specific monodromies.
problem Configurations of points with specific monodromies in complex plane subspaces.
method Introduces Hurwitz-Ran spaces for configurations of points in X∖Y and Y with monodromies in Q and G. result Proves homeomorphism between Hurwitz-Ran spaces and simplicial Hurwitz spaces.
Study plane curve singularities to determine vanishing cycles and monodromy groups.
problem Understanding vanishing cycles and monodromy groups for plane curve singularities.
method Intrinsic description of geometric monodromy group, easy criterion for vanishing cycles, canonical framing.
result Monodromy groups are injective for singularities with Milnor fiber of genus at least 7.
Study on surface bundles, CAT(0) groups, and mapping class groups.
problem Properties of surface bundles and their mapping class groups.
method Geometric and group theoretical analysis of surface bundles and their monodromies.
result Constraints on Kodaira fibrations and properties of surface bundles over surfaces.
Noninjective monodromy found in polynomial critical point tracking.
problem Tracking critical points in polynomials leads to noninjective monodromy.
method Monic squarefree complex polynomials with prescribed critical point multiplicities.
result Monodromy map is noninjective for polynomials with exactly two critical points.
The study finds arithmetic groups often in square-tiled surface monodromies.
problem Understanding arithmetic properties of square-tiled surfaces.
method Analyzing variations of Hodge structures and Kontsevich-Zorich monodromies.
result Arithmetic groups are frequent in low genus square-tiled surfaces.
Characterizes conical angles for metrics with dihedral symmetry.
problem Understanding metrics with specific symmetry properties.
method Using recent results on local invariants of quadratic differentials.
result Complete characterization of conical angles for dihedral spherical metrics.