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Tête-à-tête graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a generalization which define mixed tête-à-tête twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed tête-à-tête twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed Dehn twists around disjoint simple closed curves, including all boundary components. It follows that the class of tête-à-tête twists coincides with that of monodromies associated with reduced function germs on isolated complex surface singularities. [ABSTRACT FROM AUTHOR]

We give a criterion for the good reduction of semistable K3 surfaces over padic fields. We use neither p-adic Hodge theory nor transcendental methods as in the analogous proofs of criteria for good reduction of curves or K3 surfaces. We achieve our goal by realizing the special fiber Xs of a semistable model X of a K3 surface over the p-adic field K, as a special fiber of a log-family in characteristic p and use an arithmetic version of the Clemens-Schmid exact sequence in order to obtain a Kulikov-- Persson--Pinkham classification theorem in characteristic p. [ABSTRACT FROM AUTHOR]

Given a complex germ ƒ near the point x of the complex manifold X, equipped with a factorization ƒ = ƒ1 ⋅⋅⋅ ƒr, we consider the DX, x[s1, . . . , sr]-module generated by FS := ƒ1s1 ⋅⋅⋅ ƒrsr. We show for a large class of germs that the annihilator of FS is generated by derivations and this property does not depend on the chosen factorization of ƒ. We further study the relationship between the Bernstein-Sato variety attached to F and the cohomology support loci of ƒ, via the DX, x-map ∇A. This is related to multiplication by ƒ on certain quotient modules. We show that for our class of divisors the injectivity of ∇Aimplies its surjectivity. Restricting to reduced, free divisors, we also show the reverse, using the theory of Lie-Rinehart algebras. In particular, we analyze the dual of ∇Ausing techniques pioneered by Narváez-Macarro. As an application of our results we establish a conjecture of Budur in the tame case: if V(ƒ) is a central, essential, indecomposable, and tame hyperplane arrangement, then the Bernstein-Sato variety associated to F contains a certain hyperplane. By the work of Budur, this verifies the Topological Mulivariable Strong Monodromy Conjecture for tame arrangements. Finally, in the reduced and free case, we characterize local systems outside the cohomology support loci of ƒ near x in terms of the simplicity of modules derived from FS. [ABSTRACT FROM AUTHOR]

Abstract We give an algebraic description of screw dislocations in a crystal, especially simple cubic (SC) and body centered cubic (BCC) crystals, using free abelian groups and fibering structures. We also show that the strain energy of a screw dislocation based on the spring model is expressed by the Epstein-Hurwitz zeta function approximately.

This paper is about a generalization of celebrated Inoue’s surfaces. To each matrix M in SL(2n+1, Z) we associate a complex non-Kähler manifold TM of complex dimension n + 1. This manifold fibers over S¹ with the fiber T2n+1 and monodromy MT. Our construction is elementary and does not use algebraic number theory. We show that some of the OeljeklausToma manifolds are biholomorphic to the manifolds of type TM. We prove that if M is not diagonalizable, then TM does not admit a Kähler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds. [ABSTRACT FROM AUTHOR]

Algebraic hypergeometric functions can be compactly expressed as radical functions on pull-back curves where the monodromy group is simpler, say, a finite cyclic group. These so-called Darboux evaluations have already been considered for algebraic 2 F 1 -functions. This article presents Darboux evaluations for the classical case of 3 F 2 -functions with the projective monodromy group PSL (2 , F 7) . The pullback curves are of genus 0 (in the simplest case) or of genus 1. As an application of the genus 0 evaluations, appealing modular evaluations of the same 3 F 2 -functions are derived. [ABSTRACT FROM AUTHOR]

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fiber bundles. They are known to have positive signature divisible by 4. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent fibrations. Special attention is paid to ramified covers of product of curves which we analyze by studying the monodromy action for bundles of punctured curves. As a by-product, we obtain a classification of all fix-point-free automorphisms on curves of genus at most 9. [ABSTRACT FROM AUTHOR]

In this paper we describe a feasible construction of universal set of quantum gates using monodromy matrices of Fuchsian system. Fuchsian systems are considered as Schrödinger type equations and it is shown that such quantum systems are exactly solvable. We also show that dynamics of trapped cold ions may be described by a Fuchsian system which also describes the critical points of logarithmic potential associated with equilibrium positions of trapped ions in line geometry. Two different approaches to the inverse problem are also discussed. [ABSTRACT FROM AUTHOR]

We extend the middle convolution for monodromy of Fuchsian ordinary differential equations due to Katz to an operation for monodromy of KZ equations. This operation, also called the middle convolution, gives a monodromy of KZ equation obtained by the additive middle convolution. Since the fundamental group of the domain of definition of KZ equation is the pure braid group, our middle convolution also gives a recursive way of constructing irreducible representations of the pure braid group. Several applications are considered. [ABSTRACT FROM AUTHOR]

Here we investigate the Minkowski box dimension of complex integral curves of the vector fields near resonant saddles in C 2 . The results provide the geometrical explanation of the order of the saddle points and a quantitative description for the non-integrability via monodromy. [ABSTRACT FROM AUTHOR]

We first develop some basic facts about certain sorts of rigid local systems on the affine line in characteristic p > 0. We then apply them to exhibit a number of rigid local systems of rank 23 on the affine line in characteristic p = 3 whose arithmetic and geometric monodromy groups are the Conway group Co 3 in its orthogonal irreducible representation of degree 23. [ABSTRACT FROM AUTHOR]

This paper proposes seven combinatorial problems around formulas for the characteristic polynomial and the spectral numbers of an isolated quasihomogeneous hypersurface singularity. One of them is a new conjecture on the characteristic polynomial. It is an amendment to an old conjecture of Orlik on the integral monodromy of an isolated quasihomogeneous singularity. The search for a combinatorial proof of the new conjecture led us to the seven purely combinatorial problems. [ABSTRACT FROM AUTHOR]

This paper deals with both integrability and linearizability problems of three dimensional Lotka–Volterra systems at the origin with rank-2. We give necessary conditions of a system with quadratic nonlinearities with (1 : 3 : - 1) -resonance. We proved the sufficiency of these conditions by showing the existence of two functionally independent first integrals via the Darboux method with inverse Jacobi multiplier together with some other techniques such as two equations define a linearizable node and the third equation is linearizable using the power series argument as well as the monodromy method. [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]

We study a problem related to Kontsevich's homological mirror symmetry conjecture for the case of a generic curve Y with bi-degree (2,2) in a product of projective lines ℙ1 × ℙ1. We calculate two differenent monodromy representations of period integrals for the affine variety X(2,2) obtained by the dual polyhedron mirror variety construction from Y. The first method that gives a full representation of the fundamental group of the complement to singular loci relies on the generalised Picard-Lefschetz theorem. The second method uses the analytic continuation of the Mellin-Barnes integrals that gives us a proper subgroup of the monodromy group. It turns out both representations admit a Hermitian quadratic invariant form that is given by a Gram matrix of a split generator of the derived category of coherent sheaves on on Y with respect to the Euler form.

We consider three linear operators determining automorphisms of the solution space of a special double confluent Heun equation of positive integer order (L-operators). We propose a new method for describing properties of the solution space of this equation based on using eigenfunctions of one of the L-operators, called the universal L-operator. We construct composition laws for L-operators and establish their relation to the monodromy transformation of the solution space of the special double confluent Heun equation. We find four functionals quadratic in eigenfunctions of the universal automorphism; they have a property with respect to the considered equation analogous to the property of the first integral. Based on them, we construct matrix representations of the L-operators and also the monodromy operator. We give a method for extending solutions of the special double confluent Heun equation from the subset Re z > 0 of a complex plane to a maximum domain on which the solution exists. As an example of its application to the RSJ model theory of overdamped Josephson junctions, we give the explicit form of the transformation of the phase difference function induced by the monodromy of the solution space of the special double confluent Heun equation and propose a way to continue this function from a half-period interval to any given interval in the domain of the function using only algebraic transformations. [ABSTRACT FROM AUTHOR]

We prove geometric and cohomological stabilization results for the universal smooth degree d hypersurface section of a fixed smooth projective variety as d goes to infinity. We show that relative configuration spaces of the universal smooth hypersurface section stabilize in the completed Grothendieck ring of varieties, and deduce from this the stabilization of the Hodge Euler characteristic of natural families of local systems constructed from the vanishing cohomology. We prove explicit formulas for the stable values using a probabilistic interpretation, along with the natural analogs in point counting over finite fields. We explain how these results provide new geometric examples of a weak version of representation stability for symmetric, symplectic, and orthogonal groups. This interpretation of representation stability was studied in the prequel [20] for configuration spaces. [ABSTRACT FROM AUTHOR]

We consider germs of holomorphic vector fields in (ℂ n , 0) , n ≥ 3, with non-isolated singularities. We assume that the set of singular points forms a submanifold of codimension 2, and the sum of the nonzero eigenvalues of the linearization of the germs at each singular point is zero. We give the orbital analytic classification of generic germs of such type. It happens that, unlike the formal classification (which is trivial), the analytic one has functional moduli. The same result is obtained in the real-analytic case (the smooth normalization was obtained earlier in Roussarie Astérisque. 1975;30:1–181). [ABSTRACT FROM AUTHOR]

In 1973, B. Josephson received a Nobel Prize for discovering a new fundamental effect concerning a Josephson junction,—a system of two superconductors separated by a very narrow dielectric: there could exist a supercurrent tunneling through this junction. We will discuss the model of the overdamped Josephson junction, which is given by a family of first-order nonlinear ordinary differential equations on two-torus depending on three parameters: a fixed parameter ω (the frequency); a pair of variable parameters (B, A) that are called respectively the abscissa, and the ordinate. It is important to study the rotation number of the system as a function ρ = ρ(B, A) and to describe the phase-lock areas: its level sets Lr = {ρ = r} with non-empty interiors. They were studied by V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi, who observed in their joint paper in 2010 that the phase-lock areas exist only for integer values of the rotation number. It is known that each phase-lock area is a garland of infinitely many bounded domains going to infinity in the vertical direction; each two subsequent domains are separated by one point, which is called constriction (provided that it does not lie in the abscissa axis). Those points of intersection of the boundary ∂Lr of the phase-lock area Lr with the line Λr = {B = rω} (which is called its axis) that are not constrictions are called simple intersections. It is known that our family of dynamical systems is related to appropriate family of double–confluent Heun equations with the same parameters via Buchtaber–Tertychnyi construction. Simple intersections correspond to some of those parameter values for which the corresponding "conjugate" double-confluent Heun equation has a polynomial solution (follows from results of a joint paper of V.M. Buchstaber and S.I.Tertychnyi and a joint paper of V.M. Buchstaber and the author). There is a conjecture stating that all the constrictions of every phase-lock areaLrlie in its axis Λr. This conjecture was studied and partially proved in a joint paper of the author with V.A.Kleptsyn, D.A.Filimonov, and I.V.Schurov. Another conjecture states that for any two subsequent constrictions in Lr with positive ordinates, the interval between them also lies in Lr. In this paper, we present new results partially confirming both conjectures. The main result states that for every r ∈ ℤ ∖ { 0 } the phase-lock area Lr contains the infinite interval of the axis Λr issued upwards from the simple intersection in ∂Lr ∩Λr with the biggest possible ordinate. The proof is done by studying the complexification of the system under question, which is the projectivization of a family of systems of second-order linear equations with two irregular non-resonant singular points at zero and at infinity. We obtain new results on the transition matrix between appropriate canonical solution bases of the linear system; on its behavior as a function of parameters. A key result, which implies the main result of the paper, states that the off-diagonal terms of the transition matrix are both non-zero at each constriction. We also show that their ratio is real at the constrictions. We reduce the above conjectures on constrictions to the conjecture on negativity of the ratio of the latter off-diagonal terms at each constriction. [ABSTRACT FROM AUTHOR]

We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy-based solvers in terms of decorated graphs. Under the theoretical. that monodromy actions are generated uniformly, we show that the expected number of homotopy paths tracked by an algorithm following this framework is linear in the number of solutions. We demonstrate that our software implementation is competitive with the existing state-of-the-art methods implemented in other software packages. [ABSTRACT FROM AUTHOR]

We obtain an explicit formula for the characteristic polynomial of the local monodromy of A-hypergeometric functions with respect to small loops around a coordinate hyperplane xi = 0. This formula is similar to the one obtained by Ando, Esterov and Takeuchi for the local monodromy at infinity. Our proof is combinatorial and can be adapted to provide an alternative proof for the latter formula as well. On the other hand, we also prove that the solution space at a nonsingular point of certain irregular and irreducible A-hypergeometric D-modules has a nontrivial global monodromy invariant subspace. [ABSTRACT FROM AUTHOR]

Abstract We exhibit a rigid local system of rank six on the affine line in characteristic p = 5 whose arithmetic and geometric monodromy groups are the finite group 2. J 2 (J 2 the Hall-Janko sporadic group) in one of its two (Galois-conjugate) irreducible representation of degree six. [ABSTRACT FROM AUTHOR]

Algebraic hypergeometric functions can be compactly expressed as radical or dihedral functions on pull-back curves where the monodromy group ismuchsimpler. This article considers the classical 3F2- functions with the projective monodromy group PSL(2, F7) and their pull-back transformations of degree 21 that reduce the projective monodromy to the dihedral group D4 of 8 elements. [ABSTRACT FROM AUTHOR]

Abstract Given a prime p and an integer d > 1 , we give a numerical criterion to decide whether the ℓ -adic sheaf associated to the one-parameter exponential sums t ↦ ∑ x ψ (x d + t x) over F p has finite monodromy or not, and work out some explicit cases where this is computable. [ABSTRACT FROM AUTHOR]

We construct a new kind of elliptic fibration in terms of group action and group rep- resentation: a dihedral group acts on a regular polygon - thickening its edges yields a torus on which the dihedral group acts. We give a complex structure to the torus such that this action is holomorphic. Then from representations of the dihedral group, we con- struct fibrations of elliptic curves (dihedral quotient families). We classify them - their singular loci/fibers, as well as the dynamic motion of the singular fibers, are completely described. Moreover, we construct binary dihedral quotient families and classify them. Although the representations of the dihedral and binary dihedral groups are analogous, it turns out that their associated quotient families are totally different. [ABSTRACT FROM AUTHOR]

Abstract We describe an algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of any reduced projective plane curve C. The relation to the zero set of Bernstein–Sato polynomial of the defining homogeneous polynomial for C is also discussed. When C has some non-weighted homogeneous singularities, then we have to assume that a conjecture holds in order to get some of our results. In all the examples computed so far this conjecture holds. [ABSTRACT FROM AUTHOR]

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables. [ABSTRACT FROM AUTHOR]

The double-dimer model consists in superimposing two independent, identically distributed perfect matchings on a planar graph, which produces an ensemble of non-intersecting loops. In [20], Kenyon established conformal invariance in the small mesh limit by considering topological observables of the model parameterized by SL2(C) representations of the fundamental group of the punctured domain. The scaling limit is conjectured to be CLE4, the Conformal Loop Ensemble at k = 4 [36]. In support of this conjecture, we prove that a large subclass of these topological correlators converge to their putative CLE4 limit. Both the small mesh limit of the double-dimer correlators and the corresponding CLE4 correlators are identified in terms of the τ-functions introduced by Jimbo, Miwa and Ueno [14] in the context of isomonodromic deformations. [ABSTRACT FROM AUTHOR]

This paper establishes certain existence and classification results for solutions to SU(n) Toda systems with three singular sources at 0, 1, and8. First, we determine the necessary conditions for such an SU(n) Toda system to be related to an nth order hypergeometric equation. Then, we construct solutions for SU(n) Toda systems that satisfy the necessary conditions and also the interlacing conditions from Beukers and Heckman. Finally, for SU(3) Toda systems satisfying the necessary conditions, we classify, under a natural reality assumption, that all the solutions are related to hypergeometric equations. This proof uses the Pohozaev identity. [ABSTRACT FROM AUTHOR]