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Let $k$ be a perfect field of positive characteristic and $Z$ an effective Cartier divisor in the projective line over $k$ with complement $U$. In this note, we establish some results about the formal deformation theory of overconvergent isocrystals on $U$ with fixed "local monodromy" along $Z$. En route, we show that a Hochschild cochain complex governs deformations of a module over an arbitrary associative algebra. We also relate this Hochschild cochain complex to a de Rham complex in order to understand the deformation theory of a differential module over a differential ring., 59 pages; fixed typos, improved exposition; comments welcome!

We show that on the generic degree-d hypersurface in P n + 1 , when d ≥ n + 2 , every closed point of degree d is separable, and has monodromy group S d . We study related Franchetta-type questions on generic objects, and give affirmative answers in specific cases.

We construct an irreducible holomorphic connection with SL ( 2 , R ) –monodromy on the trivial holomorphic vector bundle of rank two over a compact Riemann surface. This answers a question of Calsamiglia, Deroin, Heu and Loray in [5] .

We compute genus-0 Belyi maps with prescribed monodromy groups and verify the computed results. Among the computed examples are almost simple primitive groups that satisfy the well known rational rigidity criterion yielding polynomials with prescribed Galois groups over Q ( t ) . We also give an explicit version of a theorem of Magaard, which predicts all sporadic groups occurring as composition factors of monodromy groups of rational functions.

For a finite group G, the Hurwitz space H r , g i n ( G ) is the space of genus g covers of the Riemann sphere with r branch points and the monodromy group G. In this paper, we give a complete list of primitive genus one systems of affine type. That is, we assume that G is a primitive group of affine type. Under this assumption we determine the braid orbits on the suitable Nielsen classes, which is equivalent to finding connected components in H r , 1 i n ( G ) . Furthermore, we give a new algorithm for computing large braid orbits on Nielsen classes. This algorithm utilizes a correspondence between the components of H r , 1 i n ( G ) and H r , 1 i n ( M ) , where M is the point stabilizer in G.

Given a field K, a polynomial f ∈ K [ x ] of degree d, and a suitable element t ∈ K , the set of preimages of t under the iterates f ∘ n carries a natural structure of a d-ary tree. We study conditions under which the absolute Galois group of K acts on the tree by the full group of automorphisms. When d ≥ 20 is even and K = Q we exhibit examples of polynomials with maximal Galois action on the preimage tree, partially affirming a conjecture of Odoni. We also study the case of K = F ( t ) and f ∈ F [ x ] in which the corresponding Galois groups are the monodromy groups of the ramified covers f ∘ n : P F 1 → P F 1 .

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.

In this paper, the second in a series, we continue to study the generalized Lame equation with the Treibich-Verdier potential y ″ ( z ) = [ ∑ k = 0 3 n k ( n k + 1 ) ℘ ( z + ω k 2 | τ ) + B ] y ( z ) , n k ∈ Z ≥ 0 from the monodromy aspect. We prove the existence of a pre-modular form Z r , s n ( τ ) of weight 1 2 ∑ n k ( n k + 1 ) such that the monodromy data ( r , s ) is characterized by Z r , s n ( τ ) = 0 . This generalizes the result in [17] , where the Lame case (i.e. n 1 = n 2 = n 3 = 0 ) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δ u + e u = 16 π δ 0 and Δ u + e u = 8 π ∑ k = 1 3 δ ω k 2 on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.

Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials with constant coefficients. These algorithms rely on homotopy continuation, monodromy loops, and Cauchy integrals. Several examples are presented which are computed using an implementation that builds on the numerical algebraic geometry software Bertini., Comment: 23 pages, 5 figures

We present new examples of elliptic curves having the bounded rank in cyclotomic towers of function fields over C . Our key method is to utilize the monodromy of the Gaussian hypergeometric functions for the computation of the Mordell-Weil groups.

Let Γ be a triangulation of a connected closed 2-dimensional (not necessarily orientable) surface. Using zigzags (closed left–right walks), for every face of Γ we define the z -monodromy which acts on the oriented edges of this face. There are precisely 7 types of z -monodromies. We consider the following two cases: (M1) the z -monodromy is identity, (M2) the z -monodromy is the consecutive passing of the oriented edges. Our main result is the following: the subgraphs of the dual graph Γ ∗ formed by edges whose z -monodromies are of types (M1) and (M2), respectively, both are forests. We apply this statement to the connected sum of z -knotted triangulations.

For certain powers q of odd primes p, and certain integers n ≥ 1 , we exhibit explicit rigid local systems on the affine line in characteristic p > 0 whose geometric and arithmetic monodromy groups are Sp ( 2 n , q ) .

We investigate a Curtis–Tits style group presentation based on the Dynkin diagram C g , in which the short root generators have order two while the long root generators have order four. We prove that this describes a finite group with an almost extraspecial normal subgroup of order 2 2 g + 2 and quotient isomorphic to the symplectic group Sp ( 2 g , 2 ) . Such an extension has to be non-split if g ⩾ 3 , and was proved to exist in papers of Bolt, Room and Wall from 1961/2 and Griess from 1973. Our presentation proves that it is a double cover of a finite quotient of Sp ( 2 g , Z ) . We investigate a 2 g dimensional complex representation on a suitable space of theta functions, and produce some consequences for the signatures of 4-manifolds described as surface bundles over surfaces. In particular, we prove that if the monodromy is contained in the theta subgroup Sp q ( 2 g , Z ) ⩽ Sp ( 2 g , Z ) then the signature of the 4-manifold is divisible by eight.

We introduce mod $l$ Milnor invariants of a Galois element associated to Ihara's Galois representation on the pro-$l$ fundamental group of a punctured projective line ($l$ being a prime number), as arithmetic analogues of Milnor invariants of a pure braid. We then show that triple quadratic (resp. cubic) residue symbols of primes in the rational (resp. Eisenstein) number field are expressed by mod $2$ (resp. mod $3$) triple Milnor invariants of Frobenius elements. For this, we introduce dilogarithmic mod $l$ Heisenberg ramified covering ${\cal D}^{(l)}$ of $\mathbb{P}^1$, which may be regarded as a higher analog of the dilogarithmic function, for the gerbe associated to the mod $l$ Heisenberg group, and we study the monodromy transformations of certain functions on ${\cal D}^{(l)}$ along the pro-$l$ longitudes of Frobenius elements for $l=2,3$., Comment: 33 pages, 1 figure

It was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a complete classification of all exceptional orthogonal polynomials. In some sense, this paper can be regarded as the extension of Bochner's result for classical orthogonal polynomials to the exceptional class. As a supplementary result, we derive a canonical form for exceptional operators based on a bilinear formalism, and prove that every exceptional operator has trivial monodromy at all primary poles., M.A.G.F. acknowledges the financial support of the Spanish MINECO through a Severo Ochoa FPI scholarship. The work of M.A.G.F. is supported in part by the ERC Starting Grant 633152 and the ICMAT-Severo Ochoa project SEV-2015-0554 . The research of D.G.U. has been supported in part by Spanish MINECO-FEDER Grants MTM2012-31714 and MTM2015-65888-C4-3 and by the ICMAT-Severo Ochoa project SEV-2015-0554 . The research of the third author (RM) was supported in part by NSERC grant RGPIN-228057-2009 . D.G.U. would like to thank Dalhousie University for their hospitality during his visit in the Spring semester of 2014 where many of the results in this paper were obtained.

Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R--matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in arXiv:1512.03041 to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of U_h(g). Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g., Expanded Introduction and Sec. 5 to discuss convolution product (5.11), cosimplicial structure on basis elements (5.13) and module structure on coinvariants (5.15). Minor revisions in Sec. 7.1 (gradings), 7.4 (deformation DY modules), 9.7 (exposition), 15.7 (Drinfeld double) and 15.15 (rigidity for diagrammatic KM algebras). Final version, to appear in Adv. Math. 81 pages

On the example of bending vibrational polyads of the acetylene molecule (C2H2) in the approximation of the 1 : 1 : 1 : 1 resonant oscillator with axial symmetry, whose geometry is similar to the n-shell approximation of the perturbed hydrogen atom, we show how remaining invariant tori of the underlying classical non-integrable system form a nontrivial continuous family with monodromy. We read this monodromy off the quantum energy spectrum which was observed experimentally by spectroscopists, and we uncover its origins through the particular topology, geometry, and symmetry. We explain how monodromy characterizes the chaotic region surrounded by the tori. We detail the explicit correspondence between the bending polyads of C2H2 and the n-shells of the hydrogen atom, and uncover the dynamical SO(3) symmetry of the bending polyads and the corresponding spherically localized vibrational states.

In this paper we give an affirmative answer to a conjecture proposed by Danny Neftin, that is, if the commutator of two permutations has at least n-4 fixed points where two permutations are in degree n symmetric group, then there exists a permutation in this symmetric inverting both of them., Comment: 16 pages,no figures

An extremal Kahler metric with finite singularities on a compact Riemann surface is often called an HCMU metric. It can be regarded as a generalization of CSC (constant scalar curvature) metric. In this paper we will prove that any HCMU metric is a pullback of an HCMU metric on S 2 with only two singularities (we call it football) by a multi-valued holomorphic function. Then we study the monodromy group of the multi-valued function.

This article presents a machinery based on polyhedral products that produces faithful representations of graph products of finite groups and direct products of finite groups into automorphisms of free groups $\rm Aut(F_n)$ and outer automorphisms of free groups $\rm Out(F_n)$, respectively, as well as faithful representations of products of finite groups into the linear groups $\rm SL(n,\mathbb Z)$ and $\rm GL(n,\mathbb Z)$. These faithful representations are realized as monodromy representations., 20 pages

We study tautological classes on the moduli space of stable $n$-pointed hyperelliptic curves of genus $g$ with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invariant classes in cohomology. The connection with recent conjectures by Pixton is also discussed., Comment: 38 pages, 1 figure

We investigate the occurrence of Shimura (special) subvarieties in the locus of Jacobians of abelian Galois covers of P1 in Ag and give classifications of families of such covers that give rise to Shimura subvarieties in the Torelli locus Tg inside Ag. Our methods are based on Moonen–Oort works as well as characteristic p techniques of Dwork and Ogus and Monodromy computations.

In this paper it is shown that the local topological zeta function of a germ associated with its Hessian differential form does not satisfy the monodromy conjecture, i.e., that the Hessian form is not an allowed differential 2-form.

We define a monodromy, directly from the spectrum of small non-selfadjoint perturbations of a selfadjoint semiclassical operator with two degrees of freedom, which is classically integrable. It is a combinatorial invariant that obstructs globally the existence of lattice structure of the spectrum, in the semiclassical limit. Moreover this spectral monodromy allows to recover a topological invariant (the classical monodromy) of the corresponding integrable system., 18 pages, 1 figure. arXiv admin note: text overlap with arXiv:1405.3930

For g ≥ 5 , we give a complete classification of the connected components of strata of abelian differentials over Teichmuller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building on work of the first author [2] , we find that the non-hyperelliptic components are classified by an invariant known as an r–spin structure. This is accomplished by computing a certain monodromy group valued in the mapping class group. To do this, we determine explicit finite generating sets for all r–spin stabilizer subgroups of the mapping class group, completing a project begun by the second author in [18] . Some corollaries in flat geometry and toric geometry are obtained from these results.

A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in ${\rm U(1)}$. By using the theory of indigenous bundles, we construct on a compact Riemann surface $X$ of genus $g_X \geq 1$ a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in $2 ��\mathbb{Z}_{>1}$, which is generically injective in the algebro-geometric sense as $g_X \geq 2$. As an application, we prove the following two results about irreducible metrics: $\bullet$ as $g_X \geq 2$ and $d$ is even and greater than $12g_X - 7$, the effective divisors of degree $d$ which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension $\geq 2(d+3-3g_X)$ in ${\rm Sym}^d(X)$; $\bullet$ as $g_X \geq 1$, for almost every effective divisor $D$ of degree odd and greater than $2g_X-2$ on $X$, there exist finitely many cone spherical metrics representing $D$., This manuscript supersedes arXiv:1808.04106. 34 pages

We study the Dynkin diagram associated to the monodromy of direct sums of polynomials. The monodromy problem asks under which conditions on a polynomial, the monodromy of a vanishing cycle generates the whole homology of a regular fiber. We consider the case $y^4+g(x)$, which is a generalization of the results of Christopher and Marde$\check{s}$i\'c about the monodromy problem for hyperelliptic curves. Moreover, We solve the monodromy problem for direct sums of fourth degree polynomials., Comment: 27 pages, 11 figures

In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and we prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. These properties allows us to obtain the main properties of the general o-minimal fundamental group, including: invariance and comparison results; existence of universal locally definable covering maps; monodromy equivalence for locally constant o-minimal sheaves - from which one obtains, as in algebraic topology, classification results for locally definable covering maps, o-minimal Hurewicz and Seifert - van Kampen theorems., Comment: 25 pages. arXiv admin note: substantial text overlap with arXiv:1507.03531

We provide a complete classification and an explicit representation of rational solutions to the fourth Painleve equation P IV and its higher order generalizations known as the A 2 n -Painleve or Noumi-Yamada systems. The construction of solutions makes use of the theory of cyclic dressing chains of Schrodinger operators. Studying the local expansions of the solutions around their singularities we find that some coefficients in their Laurent expansion must vanish, which express precisely the conditions of trivial monodromy of the associated potentials. The characterization of trivial monodromy potentials with quadratic growth implies that all rational solutions can be expressed as Wronskian determinants of suitably chosen sequences of Hermite polynomials. The main classification result states that every rational solution to the A 2 n -Painleve system corresponds to a cycle of Maya diagrams, which can be indexed by an oddly coloured integer sequence. Finally, we establish the link with the standard approach to building rational solutions, based on applying Backlund transformations on seed solutions, by providing a representation for the symmetry group action on coloured sequences and Maya cycles.

The partial (Poincare) dual with respect to a subset A of edges of a ribbon graph G was introduced by Chmutov in connection with the Jones–Kauffman and Bollobas–Riordan polynomials. In developing the theory of maps, Wilson and others have composed Poincare duality ∗ with Petrie duality × to give Wilson duality ∗ × ∗ and two trialities ∗ × and × ∗ . In further expanding the theory, Abrams and Ellis-Monaghan have called the five operators twualities. Part I of this investigation (Gross et al., 2020) introduced as a partial- ∗ polynomial of G , the generating function enumerating partial Poincare duals by Euler-genus. In this sequel, we introduce the corresponding partial- × , - ∗ × , - × ∗ , and - ∗ × ∗ polynomials for their respective twualities. For purposes of computation, we express each partial twuality in terms of the monodromy of permutations of the flags of a map. We analyze how single-edge partial twualities affect the three types (proper, untwisted, twisted) of edges. Various possible properties of partial-twuality polynomials are studied, including interpolation and log-concavity; machine-computed unimodal counterexamples to some log-concavity conjectures from Gross et al. (2020) are given. It is shown that the partial- ∗ × ∗ polynomial for a ribbon graph G equals the partial- × polynomial for G ∗ . Formulas or recursions are given for various families of graphs, including ladders and, for Wilson duality, a large subfamily of series–parallel networks. All of these polynomials are shown to be log-concave.

We describe a counter-example which shows that ( 2 ) of theorem ( 11 ) in Algaba et al. (2012) is not correct. This part of the theorem, pinpoints whether the origin of quasi-homogeneous system ( 15 ) in Algaba et al. (2012) is a center or not. It is shown in this note that the given necessary and sufficient conditions of theorem ( 11 ), in Algaba et al. (2012) are not complete.

We find some new one-parameter families of exponential sums in every odd characteristic whose geometric and arithmetic monodromy groups are G 2 .

We consider general n -dimensional systems of differential equations having an ( n − 2 ) -dimensional, locally invariant manifold on which there exist equilibria connected by heteroclinic orbits for n ≥ 3 . The system may be non-Hamiltonian and have no saddle-centers, and the equilibria are allowed to be the same and connected by a homoclinic orbit. Under additional assumptions, we prove that the monodromy group for the normal variational equation , which is represented by components of the variational equation normal to the locally invariant manifold and defined on a Riemann surface, is diagonalizable or infinitely cyclic if the system is real-meromorphically integrable in the meaning of Bogoyavlenski. We apply our theory to a three-dimensional volume-preserving system describing the streamline of a steady incompressible flow with two parameters, and show that it is real-meromorphically nonintegrable for almost all values of the two parameters.

We give an elementary description of the relationship between the classical and quantum monodromy of a completely integrable system, from the point of view of geometric quantization, as a consequence of the construction of action–angle variables. We also describe the relation to Symington’s notion of “affine monodromy”.

The paper concerns a simple model of bicycle kinematics: a bicycle is represented by an oriented segment of constant length in n -dimensional space that can move in such a way that the velocity of its rear end is aligned with the segment (the rear wheel is fixed on the bicycle frame). Starting with a closed trajectory of the rear end, one obtains the two respective trajectories of the front end, corresponding to the opposite directions of motion. These two curves are said to be in the bicycle correspondence. Conjecturally, this correspondence is completely integrable; we present a number of results substantiating this conjecture. In dimension three, the bicycle correspondence is the Backlund transformation for the filament equation; we discuss bi-Hamiltonian features of the bicycle correspondence and its integrals.

We consider the (finite-dimensional) restricted quantum group U ‾ q s l ( 2 ) at q = i . We show that U ‾ i s l ( 2 ) does not allow for a universal R-matrix, even though U ⊗ V ≅ V ⊗ U holds for all finite-dimensional representations U , V of U ‾ i s l ( 2 ) . We then give an explicit coassociator Φ and a universal R-matrix R such that U ‾ i s l ( 2 ) becomes a quasi-triangular quasi-Hopf algebra. Our construction is motivated by the two-dimensional chiral conformal field theory of symplectic fermions with central charge c = − 2 . There, a braided monoidal category, SF , has been computed from the factorisation and monodromy properties of conformal blocks, and we prove that Rep ( U ‾ i s l ( 2 ) , Φ , R ) is braided monoidally equivalent to SF .

We consider a generalization of Riemann–Hilbert problem on elliptic curves. For a given elliptic curve and irreducible representation of free group with two generators we construct explicitly a semistable vector bundle of degree zero obeying a logarithmic connection such that its monodromy over fundamental parallelogram is equivalent to given free group representation, monodromy along -cycle is trivial and monodromy along -cycle belong to certain orbit. This work has been funded by the Russian Academic Excellence Project ‘5-100’. The results of Section 2 have been obtained under support of the RSF grant 16-11-10160. Research has also been supported by the RFBR grant 17-01-00585.

We study the massive ODE/IM correspondence for modified B 2 ( 1 ) affine Toda field equation. Based on the ψ-system for the solutions of the associated linear problem, we obtain the Bethe ansatz equations. We also discuss the T–Q relations, the T-system and the Y-system, which are shown to be related to those of the A 3 / Z 2 integrable system. We consider the case that the solution of the linear problem has a monodromy around the origin, which imposes nontrivial boundary conditions for the T-/Y-system. The high-temperature limit of the T- and Y-system and their monodromy dependence are studied numerically.

Inspired by the motivic monodromy conjecture, Halle and Nicaise defined the global monodromy property for Calabi–Yau varieties over a discretely valued field. In this note, we discuss this property for K 3 surfaces allowing a strict normal crossings model where no three components in the special fiber have a common intersection. The main result is that the global monodromy property holds for a K 3 surface with a so-called flowerpot degeneration. It also holds for K 3 surfaces with a chain degeneration under certain conditions.

We determine the Bernstein–Sato polynomials for the ideal of maximal minors of a generic m × n matrix, as well as for that of sub-maximal Pfaffians of a generic skew-symmetric matrix of odd size. As a corollary, we obtain that the Strong Monodromy Conjecture holds in these two cases.

In this paper, we identify the moduli space for germs of generic unfoldings of nonresonant linear differential systems with an irregular singularity of Poincar\'e rank $k$ at the origin, under analytic equivalence. The modulus of a given family was determined in \cite{HLR}: it comprises a formal part depending analytically on the parameters, and an analytic part given by unfoldings of the Stokes matrices. These unfoldings are given on "Douady-Sentenac" (DS) domains in the parameter space covering the generic values of the parameters corresponding to Fuchsian singular points. Here we identify exactly which moduli can be realized. A necessary condition on the analytic part, called compatibility condition, is saying that the unfoldings define the same monodromy group (up to conjugacy) for the different presentations of the modulus on the intersections of DS domains. With the additional requirement that the corresponding cocycle is trivial and good limit behavior at some boundary points of the DS domains, this condition becomes sufficient. In particular we show that any modulus can be realized by a $k$-parameter family of systems of rational linear differential equations over $\mathbb C\mathbb P^1$ with $k+1$, $k+2$ or $k+3$ singular points (with multiplicities). Under the generic condition of irreducibility, there are precisely $k+2$ singular points which are Fuchsian as soon as simple. This in turn implies that any unfolding of an irregular singularity of Poincar\'e rank $k$ is analytically equivalent to a rational system of the form $y'=\frac{A(x)}{p_\epsilon(x)}\cdot y$, with $A(x)$ polynomial of degree at most $k$ and $p_\epsilon(x)$ is the generic unfolding of the polynomial $x^{k+1}$., Comment: 45 pages, 19 figures

We prove that the cohomology jump loci of rank one local systems on the complement in a small ball of a germ of a complex analytic set are finite unions of torsion translates of subtori. This is a generalization of the classical Monodromy Theorem stating that the eigenvalues of the monodromy on the cohomology of the Milnor fiber of a germ of a holomorphic function are roots of unity., Comment: v4: references adjusted. Final version

Correlators based on $s\ell_2$ Yangian symmetry and its quantum deformation are studied. Symmetric integral operators can be defined with such correlators as kernels. Yang-Baxter operators can be represented in this way. Particular Yangian symmetric correlators are related to the kernels of QCD parton evolution. The solution of the eigenvalue problem of Yangian symmetric operators is described., Comment: 39 pages, comments and references added, published version

We first construct a genus zero positive allowable Lefschetz fibration over the disk (a genus zero PALF for short) on the Akbulut cork and describe the monodromy as a positive factorization in the mapping class group of a surface of genus zero with five boundary components. We then construct genus zero PALFs on infinitely many exotic pairs of compact Stein surfaces such that one is a cork twist of the other along an Akbulut cork. The difference of smooth structures on each of exotic pairs of compact Stein surface is interpreted as the difference of the corresponding positive factorizations in the mapping class group of a common surface of genus zero., 14 pages, 15 figures

We prove John Hubbard’s conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex Henon map. In fact, we show that there exist several non-trivial loops in the locus which generate infinitely many mutually different monodromies. Furthermore, we prove that the dynamics of the real Henon map is completely determined by the monodromy of the complex Henon map, providing the parameter of the map is contained in the hyperbolic horseshoe locus.

We thank useful discussion with Profs. Francesco Iachello, Renato Lemus, and Georg Mellau. This project has received fund-ing from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 872081 and from the Spanish National Research, Development, and Innovation plan (RDI plan) under the project PID2019-104002GB-C21. This work has also been partially supported by the Consejeria de Conocimiento, Investigacion y Universidad, Junta de Andalucia and European Regional Development Fund (ERDF), ref. SOMM17/6105/UGR, by the Ministerio de Ciencia, Innovacion y Universidades (ref.COOPB20364), and by the Centro de Estudios Avanzados en Fisica, Matematicas y Computacion (CEAFMC) of the University of Huelva. JKR thanks support from the Youth Employment Initiative and the Youth Guarantee program supported by the European Social Fund. FPB and JKR thank the obtained support from the project UHU-1262561. Computing resources supporting this work were provided by the CEAFMC and Universidad de Huelva High Performance Computer (HPC@UHU) located in the Campus Universitario El Carmen and funded by FEDER/MINECO project UNHU-15CE-2848., We present an extension of the Hamiltonian of the two dimensional limit of the vibron model to encompass all possible interactions up to four-body operators. We apply this Hamiltonian to the modeling of the bending spectrum of four molecules: HNC, H2S, Si2C, and NCNCS. The selected molecular species include linear, bent, and nonrigid equilibrium structures, proving the versatility of the algebraic approach which allows for the consideration of utterly different physical cases within a single Hamiltonian and a general formalism. For each case we compute predicted bending energies and wave functions, that we use to depict the associated quantum monodromy diagram, Birge-Sponer plot, and participation ratio. In nonrigid cases, we also show the bending energy functional obtained using the coherent -or intrinsicstate formalism., European Commission 872081, Spanish National Research, Development, and Innovation plan (RDI plan) PID2019-104002GB-C21, Junta de Andalucia, European Commission SOMM17/6105/UGR, Ministerio de Ciencia, Innovacion y Universidades COOPB20364, Centro de Estudios Avanzados en Fisica, Matematicas y Computacion (CEAFMC) of the University of Huelva, European Social Fund (ESF), FEDER/MINECO UNHU-15CE-2848, UHU-1262561

In this paper we study quantum del Pezzo surfaces belonging to a certain class. In particular we introduce the generalised Sklyanin-Painleve algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painleve equations.

In literature, it is known that any solution of Painleve VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE on CP 1 . In this paper, we extend this isomonodromy theory on CP 1 to the moduli space of elliptic curves by studying the isomonodromic deformation of the generalized Lame equation. Among other things, we prove that the isomonodromic equation is a new Hamiltonian system, which is equivalent to the elliptic form of Painleve VI equation for generic parameters. For Painleve VI equation with some special parameters, the isomonodromy theory of the generalized Lame equation greatly simplifies the computation of the monodromy group in CP 1 . This is one of the advantages of the elliptic form.

An LCK manifold with potential is a compact quotient of a Kahler manifold $X$ equipped with a positive Kahler potential $f$, such that the monodromy group acts on $X$ by holomorphic homotheties and multiplies $f$ by a character. The LCK rank is the rank of the image of this character, considered as a function from the monodromy group to real numbers. We prove that an LCK manifold with potential can have any rank between 1 and $b_1(M)$. Moreover, LCK manifolds with proper potential (ones with rank 1) are dense. Two errata to our previous work are given in the last Section., 14 pages. Supersedes arXiv:1512.00968. Contains errata to arXiv:math/0305259