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.
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.
For any topological groupoid G and any homomorphism from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group. We prove that Morita equivalent topological groupoids have the same monodromy groups. We show how the monodromy groups can be used to test if a Lie groupoid lacks fa…
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.
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.
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.
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. 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.
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 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).
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.
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). For large genus, precise monodromy groups are calculated for surface covers.
problem Calculating precise monodromy groups for large genus surface covers.
method Hodge-theoretic methods, including a generic Torelli theorem with coefficients.
result Precise connected monodromy groups are calculated for large genus surface covers.
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). 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.
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…
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.
Study shows infinite kernels in topological monodromy for curve families.
problem Understanding kernels of topological monodromy representations.
method Extending Kuno's arguments and using Carlson-Toledo techniques.
result Kernels are infinite for certain linear systems on surfaces.
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…
Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.
problem Characterizing curves that can be vanishing cycles in degenerations of linear systems.
method Computing mapping class group-valued monodromy and identifying it with r-spin mapping class groups.
result Identifies simple closed curves as vanishing cycles and provides characterizations of discriminants and Lefschetz fibrations.
This paper investigates the relationship between strata of abelian differentials and various mapping class groups afforded by means of the topological monodromy representation. Building off of prior work of the authors, we show that the fundamental group of a stratum surjects onto the subgroup of the mapping class grou…
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.
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.
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.
Given an ample line bundle on a toric surface, a question of Donaldson asks which simple closed curves can be vanishing cycles for nodal degenerations of smooth curves in the complete linear system. This paper provides a complete answer. This is accomplished by reformulating the problem in terms of the mapping class gr…
The study proposes a conjecture about the monodromy group of singular hyperbolic metrics and provides evidence and confirmations.
problem Understanding the monodromy group of singular hyperbolic metrics on Riemann surfaces.
method Using meromorphic differentials and affine connections, the study examines the monodromy group and confirms the conjecture for specific Riemann surfaces.
result The monodromy group of the singular hyperbolic metric is Zariski dense in PSL(2, R) and cannot be contained in certain Lie subgroups.
A question of Griffiths-Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for the class of algebraic surfaces known as Atiyah-Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in …
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.
The study provides obstructions and unusual subgroup properties in mapping class groups.
problem Obstructing finite index monodromy and unusual subgroup properties in mapping class groups.
method Analyzes geometric monodromy groups and Johnson filtrations to provide obstructions and unusual subgroup properties.
result Construction of subgroups with unusual properties in mapping class groups.
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.
We show that all the currently known non-arithmetic lattices in PU(2,1) are monodromy groups of higher hypergeometric functions.
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.
We study several geometric and group theoretical problems related to Kodaira fibrations, to more general families of Riemann surfaces, and to surface-by-surface groups. First we provide constraints on Kodaira fibrations that fiber in more than two distinct ways, addressing a question by Catanese and Salter about their …
Study monodromy factorizations for lines on del Pezzo surfaces.
problem Understanding monodromy factorizations for lines on del Pezzo surfaces.
method Listed monodromy factorizations in the mapping class group of a torus.
result Explicit correspondence between factorizations and roots of E8.
Researchers compute monodromy groups of surface families over quartic curves.
problem Computing monodromy groups of surface families over smooth quartic curves.
method Analyzing cyclic branched covers of P2 over smooth quartic curves, computing monodromy groups for del Pezzo and K3 surfaces. result Obtained monodromy groups for del Pezzo and K3 surfaces, including Weyl group $W\left(E_{7}
ight)$ and arithmetic lattice $U\left(h_{L_{-}}
ight)$.
We compute the monodromy of the Hitchin fibration for the moduli space of L-twisted SL(n,C) and GL(n,C)-Higgs bundles for any n, on a compact Riemann surface of genus g>1. We require the line bundle L to either be the canonical bundle or satisfy deg(L)>2g−2. The monodromy group is gene…
This paper is the first part of a series of three papers about the fundamental groups of conic-line arrangements consist of two tangented conics and up to two additional tangented lines. In this part, we compute the local braid monodromies and the local fundamental groups of the singularities which appear in such arran…
Let G be a connected reductive affine algebraic group defined over C and g its Lie algebra. We study the monodromy map from the space of g-differential systems on a compact connected Riemann surface Σ of genus g≥2 to the character variety of G-representations of the fu…
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. In this paper, we introduce two new classes of representations of the framed braid groups. One is the homological representation constructed as the action of a mapping class group on a certain homology group. The other is the monodromy representation of the confluent KZ equation, which is a generalization of the KZ equ…
Study Alexander invariants and cohomology jump loci in group extensions with trivial monodromy.
problem Understanding Alexander invariants and cohomology jump loci in group extensions with specific conditions.
method Analyzing integral, rational, and modular Alexander invariants and cohomology jump loci of groups as extensions with trivial algebraic monodromy.
result Established a tight relationship between Alexander invariants, characteristic varieties, and resonance varieties, leading to an inequality between Chen ranks.
We consider the space of smooth complex projective plane curves of degree d. Defined over this is the tautological family of plane curves, and hence there is a monodromy representation into the mapping class group of the fiber. We show two results concerning this monodromy group. First, we show that the presence of an …
Study a specific line arrangement and compute its fundamental group via braid monodromy.
problem Compute the fundamental group of a specific line arrangement's complement.
method Use braid monodromy to compute the fundamental group.
result The resulting presentation of the fundamental group coincides with the modified Artin presentation.
Study of polynomial strata using braid groups and translation surfaces.
problem Understanding the monodromy of polynomial strata.
method Using infinite-area translation surfaces and braid groups.
result Determine the monodromy of polynomial strata in the braid group.
In this paper it is proven that there is at most one way, up to isotopy, in which a connected, hyperbolic, orientable 3-manifold can fiber over the circle with monodromy in the Torelli group.
We show that the Hamiltonian Lagrangian monodromy group, in its homological version, is trivial for any weakly exact Lagrangian submanifold of a symplectic manifold. The proof relies on a sheaf approach to Floer homology given by a relative Seidel morphism.
A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group B_n. In their papers, Bigelow and Krammer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this pape…
For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided…