The braid monodromy factorization of the branch curve of a surface of general type is known to be an invariant that completely determines the diffeomorphism type of the surface (see [2]). Calculating this factorization is of high technical complexity; computing the braid monodromy factorization of branch curves of surfaces uncovers new facts and invariants of the surfaces. Since finding the bra...

We are solving the classical Riemann-Hilbert problem of rank N > 1 on the extended complex plane punctured in 2m + 2 points, for N × N quasi-permutation monodromy matrices with 2m(N − 1) parameters. Our approach is based on the finite gap integration method applied to study the Riemann-Hilbert by Deift, Its, Kapaev and Zhou [1] and Kitaev and Korotkin [2, 3]. This permits us to solve the Rieman...

A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in ...

In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is able to check conjugacy of a given braid to one of E. Artin’s generators in any power, and compute i...

One of the consequences of the classification of finite simple groups is the fact that non-rigid polynomials (those with more than two finite critical values), considered as branched coverings of the sphere, have exactly three exceptional monodromy groups (one in degree 7, one in degree 13 and one in degree 15). By exceptional here we mean primitive and not equal to Sn or An, where n is the deg...

We are solving the classical Riemann-Hilbert problem of rank N > 1 on the extended complex plane punctured in 2m + 2 points, for N × N quasi-permutation monodromy matrices. Our approach is based on the finite gap integration method applied to study the Riemann-Hilbert by Kitaev and Korotkin [1], Deift, Its, Kapaev and Zhou [2] and Korotkin, [3]. This permits us to solve the Riemann-Hilbert prob...

We show that a variety of monodromy phenomena arising in geometric topology and algebraic geometry are most conveniently described in terms of homomorphisms from a(n augmented) knot quandle associated with the base to a suitable (augmented) quandle associated with the fiber. We consider the cases of the monodromy of a branched covering, braid monodromy and the monodromy of a Lefschetz fibration...

A comprehensive analysis of the Euler-Jacobi problem of motion in the field of two fixed attracting centers is given, first classically and then quantum mechanically in semiclassical approximation. The system was originally studied in the context of celestial mechanics but, starting with Pauli’s dissertation, became a model for one-electron molecules such as H2 (symmetric case of equal centers)...

Consider a polynomial function f : C → C with generic fiber F . Let Bf be the bifurcation set of f; hence f induces a smooth locally trivial fibration over C \ Bf . Then, for any integer q ≥ 0 and any coefficient ring R, there is an associated monodromy representation ρ(f )q : π1 ( C \ Bf , pt ) −→ Aut H̃q(F,R) ) in (reduced) homology. Going around a circle in C large enough to contain all of th...