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.
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.
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.
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.
The geometric monodromy of a plane curve singularity is a quasi-finite diffeomorphism. In this paper we locate the reduction curves of the geometric monodromy and the quadratic vanishing cycles of the singularity. An application to the geometric monodromy group is given.
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.
Study of monodromy and vanishing cycles for complete intersection curves.
problem Computing topological monodromy of complete intersection curves.
method Innovative tools for studying monodromy of tensor products of very ample line bundles, induction on multi-degree.
result Answer given by the r-spin mapping class group associated to the maximal root of the adjoint line bundle.
In this work we describe a method to reconstruct the braid monodromy of the preimage of a curve by a Kummer cover. This method is interesting, since it combines two techniques, namely, the reconstruction of a highly non-generic braid monodromy with a systematic method to go from a non-generic to a generic braid monodro…
The paper studies the index of a specific monodromy for origamis in a particular stratum.
problem Determining the index of a Kontsevich-Zorich monodromy for origamis in H(2). method Analyzing the action of the Veech group on the non-tautological part of the homology.
result The index of the Kontsevich-Zorich monodromy is either 1 or 3 for origamis in H(2). We solve a long-standing question about the monodromy of certain complex surfaces.
problem When is the monodromy group of an algebraic family of complex varieties arithmetic?
method Topological analysis of the 'geometric' monodromy, valued in the mapping class group of the fiber.
result We resolve the question affirmatively for Atiyah-Kodaira manifolds.
Polynomials with distinct critical values have braid monodromy groups equal to braid groups.
problem Understanding the structure of braid monodromy groups of polynomials.
method Analyzing the critical values of polynomials to determine their braid monodromy groups.
result The braid monodromy group of a polynomial equals the braid group if the polynomial has distinct critical values.
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…
The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot; however, in full generality it is proven only for zeta functions associated to a polynomial in two variables. In this article we consider zeta…
Arithmetic Kontsevich-Zorich monodromy found in a specific origami surface.
problem Exploring the monodromy of a symmetric origami in genus 4.
method Analyzing the Veech group and symplectic group properties of the origami.
result Existence of arithmetic Kontsevich-Zorich monodromy in a specific origami.
Holomorphic connections found on Riemann surfaces with Fuchsian monodromy.
problem Existence of holomorphic connections with Fuchsian monodromy on Riemann surfaces.
method Construction of compact Riemann surfaces with specific holomorphic vector bundles and connections.
result Existence of holomorphic connections with maximal Euler class and Fuchsian monodromy.
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.
We outline the construction of the holonomy groupoid of a locally Lie groupoid and the monodromy groupoid of a Lie groupoid. These specialise to the well known holonomy and monodromy groupoids of a foliation, when the groupoid is just an equivalence relation.
This work classifies monodromy in vineyards using singularity theory.
problem Understanding and predicting monodromy in vineyards for topological data analysis.
method Using a connection with singularity theory, the study classifies monodromy in vineyards of 1-manifolds in R^2.
result Monodromy in vineyards occurs only if they contain a specific singularity of the distance function.
This paper generalizes monodromy maps for projective structures with poles.
problem Understanding the monodromy of projective structures with poles.
method Generalizing the monodromy map from compact curves to those with poles.
result The monodromy map for projective structures with poles is a local biholomorphism.
LA-MOKA computes braid monodromy for line arrangements over complex projective plane.
problem Computing braid monodromy for algebraic curves in P2. method Discretization of continuous geometry into a discrete combinatorial structure.
result Explicit conditions ensure topological correctness of computed braid monodromy.
New lattices are linked to higher hypergeometric functions.
problem Understanding non-arithmetic lattices in PU(2,1).
method Showed all known non-arithmetic lattices are monodromy groups of higher hypergeometric functions.
result Non-arithmetic lattices in PU(2,1) are linked to higher hypergeometric functions.
In this paper we consider completed coverings that are branched coverings in the sense of Fox. For completed coverings between PL manifolds we give a characterization of the existence of a monodromy representation and the existence of a locally compact monodromy representation. These results stem from a characterizatio…
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. Characterizes monodromies on orbifolds using antitwists.
problem Understanding monodromies on orientable 2-orbifolds.
method Characterizes monodromies as products of antitwists with respect to a cylinder decomposition.
result Characterizes many divide monodromies as pseudo-Anosov with stretch factors close to one.
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.
New conditions ensure surface group extensions are non-positively curved.
problem Conditions for surface group extensions to be CAT(0).
method Generalized necessary conditions for surface-by-surface groups.
result If G is CAT(0) with infinite monodromy, the monodromy representation has a finite kernel. New knot found with unique property.
problem Existence of hyperbolic fibered slice knots with specific monodromy.
method Constructed a specific type of hyperbolic fibered slice knot.
result Negative answer to a question posed by Hubbard et al.
Researchers use Mellin-Barnes integrals to study trinomial equations and their braids.
problem Analyzing the roots of trinomial algebraic equations.
method Global analytic continuation and Mellin-Barnes integral representations.
result Precise description of the Galois group of trinomial equations.
Paper computes braid monodromy for special curves using a new method.
problem Computing invariants of completely reducible n-gonal curves. method Rectangular braid diagram method and Burau representations.
result Alexander polynomial of curve complements computed successfully.
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.
This paper solves a question about curves on toric surfaces.
problem Which curves can be vanishing cycles for degenerations?
method Reformulated as a mapping class group problem and determined the monodromy group.
result A complete answer to the question of which curves can be vanishing cycles.
Identifies filtration in Lagrangian fibrations to monodromy weight filtration in degenerations.
problem Understanding the relationship between Lagrangian fibrations and degenerations of hyper-Kähler manifolds.
method Identifies and compares perverse filtration with monodromy weight filtration.
result Identifies the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a degeneration.
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.
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.
We exhibit low-dilatation families of surface homeomorphisms among monodromies of Lorenz knots.
Study fibrations over S2 with same singularities, showing monodromies are equivalent up to direct sums.
problem Classifying torus fibrations over S2 up to fibre sum stabilisation. method Analyzing monodromies and using direct sums with certain torus Lefschetz fibrations.
result Global monodromies of fibrations with same singularities are Hurwitz equivalent after direct sums.
Study projective structures with poles, mapping them to local systems.
problem Understanding projective structures with poles on surfaces.
method Define a monodromy map from parameter space to framed local systems, proving containment in cluster chart domains.
result The image of the monodromy map is contained in the union of cluster chart domains.
Authors construct symplectic Lefschetz pencils on complex projective plane.
problem Construct symplectic Lefschetz pencils on complex projective plane.
method Differential topological construction, analogous to holomorphic pencils.
result Explicit monodromy factorization and topological construction for d=4. Study the monodromy and center-focus problems for rational maps defined by products of generic lines.
problem Monodromy and center-focus problems for rational maps defined by products of generic lines.
method Analyze the 1-homology group and meromorphic 1-forms to characterize vanishing Abelian integrals.
result Characterize meromorphic 1-forms whose Abelian integrals vanish on cycles around a center singularity.
In this paper we construct effective invariants for braid monodromy of affine curves. We also prove that, for some curves, braid monodromy determines their topology. We apply this result to find a pair of curves with conjugate equations in a number field but which do not admit any orientation-preserving homeomorphism.
Study establishes monodromy equivalence for Lamé-type equations and constructs cone spherical metrics.
problem Investigating monodromy equivalence and finite-gap structures of Lamé-type equations.
method Analyzing finite-gap structures and constructing cone spherical metrics.
result Established monodromy equivalence between classical and generalized Lamé-type equations, derived finite-gap structures, and constructed cone spherical metrics.
Study of translation covers of platonic solids reveals monodromy group structures.
problem Understanding monodromy groups of translation covers of platonic solids.
method Computed Zariski closures using generators, constraints, and Lyapunov spectrum analysis.
result Zariski closures of monodromy groups are powers of SL(2, R).
Monodromy acts on character varieties for Lefschetz pencils, answering a Conjecture.
problem Understanding the monodromy action on character varieties for Lefschetz pencils.
method Analyzing the monodromy subgroup of a family of complex curves arising from a Lefschetz pencil.
result The group acts with Zariski dense orbits or ergodically on certain character varieties.
For isolated complex hypersurface singularities with real defining equation we show the existence of a monodromy vector field such that complex conjugation intertwines the local monodromy diffeomorphism with its inverse. In particular, it follows that the geometric monodromy is the composition of the involution induced…