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For any genuinely ramified morphism f : Y ⟶ X between irreducible smooth projective curves we prove that (Y × X Y) ∖ Δ ‾ is connected, where Δ ⊂ Y × X Y is the diagonal. Using this result the following are proved: (1) If f is further Morse then the Galois closure is the symmetric group S d , where d = degree (f). (2) The Galois group of the general projection, to a line, of any smooth curve X ⊂ P n of degree d , which is not contained in a hyperplane and contains a non-flex point, is S d. [ABSTRACT FROM AUTHOR]

Each complex hyperplane arrangement gives rise to a Milnor fibration of its complement. Although the Betti numbers of the Milnor fiber F can be expressed in terms of the jump loci for rank 1 local systems on the complement, explicit formulas are still lacking in full generality, even for b1(F). We study here the "generic" case (in which b1(F) is as small as possible), and look deeper into the algebraic topology of such Milnor fibrations with trivial algebraic monodromy. Our main focus is on the cohomology jump loci and the lower central series quotients of π1(F). In the process, we produce a pair of arrangements for which the respective Milnor fibers have the same Betti numbers, yet non-isomorphic fundamental groups: the difference is picked by the higher-depth characteristic varieties and by the Schur multipliers of the second nilpotent quotients. [ABSTRACT FROM AUTHOR]

For a family of dynamical systems with k > 0 independent first integrals evolving in a compact region of a Euclidean space, we study the equilibrium locus. We show that under mild and generic conditions, it is a smooth manifold that can be viewed as the total space of a certain fiber bundle and that this bundle comes equipped with a natural connection. We then proceed to show parallel transport for this connection does exist and explore some of its properties. In particular, we elucidate how one can to some extent measure the variation of the system eigenvalues restricted to a given fiber. [ABSTRACT FROM AUTHOR]

Using square bridge position, Akbulut-Ozbagci and later Arikan gave algorithms both of which construct an explicit compatible open book decomposition on a closed contact 3-manifold which results from a contact (± 1) -surgery on a Legendrian link in the standard contact 3-sphere. In this article, we introduce the "generalized square bridge position" for a Legendrian link in the standard contact 5-sphere and partially generalize this result to the dimension five via an algorithm which constructs relative open book decompositions on relative contact pairs. [ABSTRACT FROM AUTHOR]

This paper is an expository one on necessary and sufficient conditions for the real Jacobian conjecture, which states that if F = f 1 , … , f n : R n → R n is a polynomial map such that det D F ≠ 0 , then F is a global injective. In Euclidean space R n , the Hadamard’s theorem asserts that the polynomial map F with det D F ≠ 0 is a global injective if and only if ‖ F x ‖ approaches to infinite as ‖ x ‖ → ∞ . The first part of this paper is to study the two-dimensional real Jacobian conjecture via Bendixson compactification, by which we provide a new version of Sabatini’s result. This version characterizes the global injectivity of polynomial map F by the local analysis of a singular point (at infinity) of a suitable vector field coming from the polynomial map F. Moreover, applying the above results we present a dynamical proof of the two-dimensional Hadamard’s theorem. In the second part, we give an alternative proof of the Cima’s result on the n-dimensional real Jacobian conjecture by the n-dimensional Hadamard’s theorem. [ABSTRACT FROM AUTHOR]

The paper considers the issue of constructing basises in the solution space of the system of partial differential equations, which is satisfied by the Lauricella hypergeometric function F D (N) (a ; b , c ; z) , depending on N complex variables (z 1 , ... , z N) =: z and having complex parameters (a 1 , ... , a N) =: a , b, c. For an arbitrary number N of variables, we have obtained explicit representations for such basis functions in the vicinity of points (0 , ... , 0) and (∞ , ... , ∞) in terms of the Horn type hypergeometric series in N variables. For some of these functions we have obtained formulas of analytic continuation. The found continuation formulas are important for calculating the solution of the Riemann – Hilbert problem with piecewise constant coefficients and studying its geometrical meaning. Besides, these formulas are effective for solving the parameters problem for the Schwarz – Christoffel integral and calculating conformal mapping of complex-shaped polygons. [ABSTRACT FROM AUTHOR]

The paper considers the issue of constructing basises in the solution space of the system of partial differential equations, which is satisfied by the Lauricella hypergeometric function F D (N) (a ; b , c ; z) , depending on N complex variables (z 1 , ... , z N) =: z and having complex parameters (a 1 , ... , a N) =: a , b, c. For an arbitrary number N of variables, we have obtained explicit representations for such basis functions in the vicinity of points (0 , ... , 0) and (∞ , ... , ∞) in terms of the Horn type hypergeometric series in N variables. For some of these functions we have obtained formulas of analytic continuation. The found continuation formulas are important for calculating the solution of the Riemann – Hilbert problem with piecewise constant coefficients and studying its geometrical meaning. Besides, these formulas are effective for solving the parameters problem for the Schwarz – Christoffel integral and calculating conformal mapping of complex-shaped polygons. [ABSTRACT FROM AUTHOR]

We give a proof of the monodromy conjecture relating the poles of motivic zeta functions with roots of b‐functions for isolated quasihomogeneous hypersurfaces, and more generally for semi‐quasihomogeneous hypersurfaces. We also give a strange generalization allowing a twist by certain differential forms. [ABSTRACT FROM AUTHOR]

Given a ribbon graph of arbitrary genus, we construct a spherical graph with the same two columns of the passport. And given a spherical graph, we present an algorithm for constructing a graph with the two columns of the same passport and any admissible genus. We also discuss possible applications to the theory of dessins d'enfants. [ABSTRACT FROM AUTHOR]

For a family of dynamical systems with k>0 independent first integrals evolving in a compact region of a Euclidean space, we study the equilibrium locus. We show that under mild and generic conditions, it is a smooth manifold that can be viewed as the total space of a certain fiber bundle and that this bundle comes equipped with a natural connection. We then proceed to show parallel transport for this connection does exist and explore some of its properties. In particular, we elucidate how one can to some extent measure the variation of the system eigenvalues restricted to a given fiber.

We study the geometry in the perturbations of principal submodules in the Drury-Arveson space. We show that the perturbations give rise to smooth vector bundles of Hilbert spaces which are equipped with natural Hermitian connections. We compute the associated parallel transport operators and explore properties of the monodromy. [ABSTRACT FROM AUTHOR]

An ordering of colours in an Adinkra leads to an embedding of this Adinkra into a Riemann surface X, and a branched covering map βX: X→CP¹. This paper shows how the dashing of edges in an Adinkra determines a signed permutation version of the monodromy group, and shows that it is isomorphic to a Salingaros vee group. [ABSTRACT FROM AUTHOR]

We formulate and prove Cutkosky’s Theorem regarding the discontinuity of Feynman integrals in the massive one-loop case up to the involved intersection index. This is done by applying the techniques to treat singular integrals developed in Fotiadi et al. (Topology 4(2):159–191, 1965). We write one-loop integrals as an integral of a holomorphic family of holomorphic forms over a compact cycle. Then, we determine at which points simple pinches occur and explicitly compute a representative of the corresponding vanishing sphere. This also yields an algorithm to compute the Landau surface of a one-loop graph without explicitly solving the Landau equations. We also discuss the bubble, triangle and box graph in detail. [ABSTRACT FROM AUTHOR]

Given a nilpotent singular point of a planar vector field, its monodromy is associated with its Andreev number n. The parity of n determines whether the existence of an inverse integrating factor implies that the singular point is a nilpotent center. For n odd, this is not always true. We give a characterization for a family of systems having Andreev number n such that the center problem cannot be solved by the inverse integrating factor method. Moreover, we study general properties of this family, determining necessary center conditions for every n and solving the center problem in the case n = 3 . [ABSTRACT FROM AUTHOR]

A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated quasihomogeneous singularity. The most important of them was a conjecture on the characteristic polynomial. Here, the conjecture is proved, and some of the other problems are solved, too. In the cases where also an old conjecture of Orlik on the integral monodromy holds, this has implications on the automorphism group of the Milnor lattice. The combinatorics used in the proof of the conjecture consists of tuples of orders on sets { 0 , 1 , ... , n } with special properties and may be of independent interest. [ABSTRACT FROM AUTHOR]

A well-known theorem of Kulikov, Persson and Pinkham states that a degeneration of a family of K3-surfaces with trivial monodromy can be completed to a smooth family. We give a simple example that an analogous statement does not hold for Calabi–Yau threefolds. [ABSTRACT FROM AUTHOR]

We classified finite orbits of monodromies of the Fuchsian system for five 2 × 2 matrices. The explicit proof of this result is given. We have proposed a conjecture for a similar classification for 6 or more 2 × 2 matrices. Cases in which all monodromy matrices have a common eigenvector are excluded from the consideration. To classify the finite monodromies of the Fuchsian system we combined two methods developed in this paper. The first is an induction method: using finite orbits of smaller number of monodromy matrices the method allows the construction of such orbits for bigger numbers of matrices. The second method is a formalism for representing the tuple of monodromy matrices in a way that is invariant under common conjugation way, this transforms the problem into a form that allows one to work with rational numbers only. The classification developed in this paper can be considered as the first step to a classification of algebraic solutions of the Garnier system. [ABSTRACT FROM AUTHOR]

In this paper, we give algorithm for calculating Lp-partial indices for some classes of the piecewise constant matrix functions. We reduce this problem to calculation of partial indices of a rational matrix function, which leads Fuchsian system of differential equations to a system of a standard form. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove that a large class of quaternionic slice regular functions result to be (ramified) covering maps. By means of the topological implications of this fact and by providing further topological structures, we are able to give suitable natural conditions for the existence of k-th ⋆ -roots of a slice regular function. Moreover, we are also able to compute all the solutions which, quite surprisingly, in the most general case, are in number of k 2 . The last part is devoted to compute the monodromy and to present a technique to compute all the k 2 roots starting from one of them. [ABSTRACT FROM AUTHOR]

We introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting the roots of a complex polynomial f (x) = c 0 + c 1 x d 1 + ⋯ + c k x d k by varying its coefficients. If the GCD of the exponents is d, then the polynomial admits the change of variable y = x d , and its roots split into necklaces of length d. At best we can expect to permute these necklaces, i.e. the Galois group of f equals the wreath product of the symmetric group over d k / d elements and Z / d Z . We study the multidimensional generalization of this equality: the Galois group of a general system of polynomial equations equals the expected wreath product for a large class of systems, but in general this expected equality fails, making the problem of describing such Galois groups unexpectedly rich. [ABSTRACT FROM AUTHOR]

Designing and analyzing large cable-driven parallel robots (CDPRs) for precision tasks can be challenging, as the position kinematics are governed by kineto-statics and cable sag equations. Our aim is to find all equilibria for a given set of unstrained cable lengths using numerical continuation techniques. The Irvine sagging cable model contains both non-algebraic and multi-valued functions. The former removes the guarantee of finiteness on the number of isolated solutions, making homotopy start system construction less clear. The latter introduces branch cuts, which could lead to failures during path tracking. We reformulate the Irvine model to eliminate multi-valued functions and propose a heuristic numerical continuation method based on monodromy, removing the reliance on a start system. We demonstrate this method on an eight-cable spatial CDPR, resulting in a well-constrained non-algebraic system with 31 equations. The method is applied to four examples from literature that were previously solved in bounded regions. Our method computes the previously reported solutions along with new solutions outside those bounds much faster, showing that this numerical method enhances existing approaches for comprehensively analyzing CDPR kineto-statics. • We introduce a numerical continuation method for solving the kinetostatics of CDPRs. • The Irvine sagging cable model is reformulated to eliminate multi-valued functions. • We propose a new heuristic monodromy algorithm to solve non-algebraic systems. • We solve an eight-cable spatial CDPR, a well-constrained system with 31 equations. • The new approach finds solutions more quickly than previously published methods. [ABSTRACT FROM AUTHOR]

Let X be an irreducible, reduced complex projective hypersurface of degree d. A point P not contained in X is called uniform if the monodromy group of the projection of X from P is isomorphic to the symmetric group S d . We prove that the locus of non-uniform points is finite when X is smooth or a general projection of a smooth variety. In general, it is contained in a finite union of linear spaces of codimension at least 2, except possibly for a special class of hypersurfaces with singular locus linear in codimension 1. Moreover, we generalise a result of Fukasawa and Takahashi on the finiteness of Galois points. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada's classification of connected components of the moduli of elliptically fibred K3 surfaces and is closely related to the root lattices of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positivedimensional ambi-typical strata, that is, strata which are both Shimada root strata and monodromy strata. We also discuss the relationship with moduli spaces of latticepolarised K3 surfaces. The appendix by M. Kirschmer contains computational results about the 1-dimensional ambi-typical strata. [ABSTRACT FROM AUTHOR]

We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well-known result of Gabriélov, Lazzeri and Lê for hypersurfaces. We use A'Campo's theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety. [ABSTRACT FROM AUTHOR]

We study the analog of the classical infinitesimal center problem in the plane, but for zero cycles. We define the displacement function in this context and prove that it is identically zero if and only if the deformation has a composition factor. That is, we prove that here the composition conjecture is true, in contrast with the tangential center problem on zero cycles. Finally, we give examples of applications of our results. [ABSTRACT FROM AUTHOR]

We express Darboux transformations of discrete polarised curves as parallel sections of discrete connections in the quaternionic formalism. This immediately leads to the linearisation of the monodromy of the transformation. We also consider the integrable reduction to the case of discrete bicycle correspondence. Applying our method to the case of discrete circles, we obtain closed-form discrete parametrisations of all (closed) Darboux transforms and (closed) bicycle correspondences. [ABSTRACT FROM AUTHOR]