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Topological invariance of the vanishing holonomy group Ortiz-Bobadilla L. (Mexico) UNAM laura@math.unam.mx Rosales-Gonzlez E. (Mexico) UNAM ernesto@math.unam.mx Voronin S. M. (Russia) Chelyabinsk State University voron@csu.ru The problem about the topological invariance of vanishing holonomy groups was initially introduced by D. Cerveau and P. Sad in . Later, Yu.

Il’yashenko included it in the list of problems published in  (the problem 11.19). We give here its formulation as it was posed by Il’yashenko:

“Consider a germ of a holomorphic vector field at the singular point zero. Suppose that its (n - 1)-jet is zero, and that the Taylor expansion begins with the polynomial vector field (f, g) satisfying the following genericity assumption: the polynomial yf - xg has simple factors only. By one step of blowing-up, such a vector field is transformed into a complex line field having n+1 elementary singular points on the pasted-in divisor. This divisor with singular points deleted, is a leaf of the foliation obtained by the blow-up. The fundamental group of this leaf is free with n generators.

The corresponding monodromy group is called vanishing holonomy group.

Problem. Let two germs of the class described above be orbitally topologically equivalent. Is it true that their groups of vanishing holonomy are topologically equivalent as well This means that there exists a germ of a homeomorphism (C, 0) (C, 0) that conjugates all the corresponding generators of the two groups simultaneously.” In this work we give a positive answer to this question under the following genericity assumptions: all the singular points obtained after the desingularisation process (blow-up) are nondegenerated; exactly two separatrices pass through each singular point.

An analogous result was obtained before in the original work  under more strong assumptions of hyperbolicity of all the singular points (which means that the characteristic exponents of the singular points are non real numbers), and existence of a topologically trivial deformation from one germ of vector field to the other. Later, D. Marn, in , obtained the same result without using of the second assumption. In our work the first genericity assumption is weakened, by comparison with that on .

An analogous problem in the general case, where a “nice” blow-up of a vector field consists of several -process, was also formulated in  (general conjecture). This problem was solved, for typical nilpotent singular points, by F.Loray .

Our method basically coincides with the method used by F.Loray in , and it includes some constructions already used by D. Marn . The main tool of investigation is the so-called Extended Holonomy Group (see ).

References  Cerveau, D.; Sad,P. Problmes de modules pour les formes diffrentielles singulires dans le plan complexe (French), [Moduli problems for singular differential forms in the complex plane], Comment.

Math. Helv. 61 (1986), no. 2, 222–253.

 Ilyashenko,Y. Selected topics in differential equations with real and complex time. Normal forms, bifurcations and finiteness problems in differential equations, 317–354, NATO Sci. Ser. II Math. Phys.

Chem., 137, Kluwer Acad. Publ., Dordrecht, 2004.

 Marn,D. Moduli spaces of germs of holomorphic foliations in the plane. Comment. Math. Helv. 78 (2003), no. 3, 518–539.

 Loray,F. Rigidit topologique pour des singularits de feuilletages holomorphes. Ecuaciones Diferenciales y Singularidades (Colloque Medina 1995),J. Mozo-Fernandez (Ed.), Universidad de Valladolid (1997), p. 213-234.

 Voronin,S.M. Invariants for singular points of holomorphic vector fields on the complex plane. The Stokes phenomenon and Hilbert’s 16th problem (Groningen, 1995), 305–323, World Sci. Publishing, River Edge, NJ, 1996.

Hessian algebraic curves Ortiz Rodriguez Adriana (Mexico) IMATE aortiz@matem.unam.mx This talk is concerned with a global problem of the parabolic curve of an algebraic smooth surface in R3. A hessian curve in the real plane is a projection of the set of parabolic points of the graph of some real smooth function in two variables. In particular, when the function f is a polynomial function of degree n, its hessian polynomial has degree at most 2n - 4. By Harnack theorem, the number of compact connected components of the hessian curve, hessf(x, y) = 0, is at most (2n - 5)(n - 3) + 1.

It arises the natural question, Does this upper bound optimal We will discuss progress about this question and it will be ennonced open problems related to this subject.

First order local invariants of stable mappings from R3 to R3 with corank 1 singularities Oset R. (Spain) Universitat de Valencia raul.oset@uv.es Romero-Fuster M.C. (Spain) Universitat de Valencia carmen.romero@uv.es In  Vassiliev introduced a method to obtain topological invariants on function spaces. This method has proven to be very useful and has given interesting results in several cases:

i) Knots in R3 (Vassiliev in ).

ii) Inmmersed plane curves (Arnol’d in ).

iii) Stable mappings from surfaces to R3 (Goryunov in ).

iv) Stable mappings from the plane to the plane (Ohmoto and Aicardi in ).

v) Stable mappings form 3-manifolds to the plane (Yamamoto in ).

In this paper we apply this method to stable mappings with corank singularities from 3-manifolds to R3.

Starting from the classification of germs obtained by Marar and Tari in  we determine a complete list of germs and multigerms up to codimension 2. The analysis of the different unfoldings allows us to determine the structure of the discriminant subset (non-stable mappings) in a neighbourhood of each of the codimension 2 strata as well as to provide suitable coorientations to the codimension 1 strata. In this way we obtain 5 cocycles that form a complete set of generators for the cohomology ring H0(E1(R3, R3), Z) (first order Vassiliev type invariants), where E1 stands for stable corank 1 mappings.

Besides the obvious invariants (number of triple points, number of swallowtails and number of intersections between cuspidal edges and fold planes) we provide a geometrical interpretation for the other two invariants.

References  Arnold, V. I., Topological invariants of plane curves and caustics.

Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, New Jersey. University Lecture Series, 5. American Mathematical Society, Providence, RI, 1994. viii+pp. ISBN: 0–8218–0308– Goryunov, V.V., Local invariants of mappings of surfaces into three-space, in: V.I. Arnold (Ed.), The Arnold-Gelfand Math Seminars: Geometry and Singularity Theory, Birkhauser, Basal, 1997, pp. 223–255.

 Marar, W. L.; Tari, F., On the geometry of simple germs of corank 1 maps from R3 to R3. Math. Proc. Cambridge Philos. Soc.

119 (1996), no. 3, 469–481.

 Ohmoto, Toru; Aicardi, Francesca; First order local invariants of apparent contours. Topology 45 (2006), no. 1, 27–45.

 Vassiliev, V.A., Cohomology of knot spaces, Adv. Soviet.Math.

21 (1990) 23—69.

 Yamamoto, M., First order semi-local invariants of stable mappings of 3-manifolds into the plane, Thesis, Department of Math., Kyushu University, 2004.

The hydrodynamic-statistical model of forecast of the catastrophic phenomena like squalls, tornadoes, floods, landslides and mudflows Perekhodtseva E. V. (Russia) Hydrometeorological Center of Russia perekhod@mecom.ru This report is devoted to the development of the hydrodynamic-statistical model of forecast of catastrophic phenomena of dangerous wind, including squalls and tornadoes, and study of correlations between landslides, mudflows and floods with heavy rainfalls and their hydrodynamicstatistical forecast. The probability of forecast of landslides and mudflows is the function of intensity and duration of heavy rainfalls in the previous two - three days where those events are probable due to the soil structure of soil and the height over the sea level.

The model of the forecast of summer-season half-day precipitation exceeding 50mm/12h is developed using data of the objective analysis on the basis of statistical interpretation of output data of the hemispheric hydrodynamic model on the full equation (the author – Berkowich L.V.). Before that the problem was solved for selection of the most informative vectorpredictor thus reducing the dimension of the space of parameters without noticeable losses of information. For this purpose the sample correlation matrix R for all potential predictors is calculated. The correlation matrix of predictors R may be reduced to a diagonal form in which the blocks with strongly dependent predictors are located near the diagonal. For diagonalization of the matrix R we put it into one-to-one correspondence with a connected graph G whose sides correspond to couple correlation coefficients rij of predictors. Depending on the given threshold r of connectedness rij we remove the sides of the graph whose rij > r. So the connected graph G decays into several non-connected subgraphs Gij and isolated vertices. The most informative vector-predictor includes representatives of the blocks and the predictors corresponding to isolated vertices (the criteria of the informativity are the criterion of the Makhalanobis distance and the criterion of the minimum entropy by Vapnik V.N). The optimum number of predictors in the vector-predictor is usually determined by the number 6-8. This number is connected with quantity of the eigen values of the matrix R. The number of initial potential predictors for the forecast of dangerous precipitation was about 40 predictors. As the results of the said selection we have chosen the vector-predictor for recognition of dangerous precipitation with the numbers of predictors seven.

For the given predictors the discriminant function F was calculate on the data of objective analysis and was used for the forecast of this phenomena to 12, 24 and 36 h ahead.

For the forecast of storm wind (V more 24 m/c) to 12 - 36 h were placed such method of diagonalization of new matrix R1 and selection of the new informative vector-predictor with other discriminant functions U(X). For the forecast of squalls and tornadoes at the territory of Russia we usually use in the beginning the hydrodynamic-statistical forecast and then we use the expert systems of empirical rules of these events. At the report are given the examples of the forecasts of the flood on 2002 year and of landslides at the North Caucasus and the examples of tornadoes and dangerous squalls at the territory of Russia and Europe.

Geography of 3-folds Persson Ulf (Sweden) Department of Mathematics, Chalmers University of Technology ulfp@math.chalmers.se That every even number e 2 can occur as the euler number of a compact complex curve has been known since antiquity (i.e., since the introduction of the concept) and can easily be demonstrated by exhibiting hyper-elliptic curves The corresponding problem of determining the possible chern invariants c2, c2 for (minimal) surfaces is not yet completly settled, the case of surfaces with positive index c2 - c2 > 0 is particular subtle, but it is expected that no restictions appear except those given by the standard inequalities. What is known though are the possible ratios c2/c2 which can occur Those consist of a ‘continous’ part and a ‘discrete’.

The natural question is now to consider the case of compact complex 3-folds the notion of ‘minimal models’ having been clarified by Mori. Now three invari ants come into play c3, c1c2, c3 and instead of considering the fine question, we are concerned with ratios, which can be plotted on a sphere. (This incidentally gives the notion of ‘Geography’ a very literal meaning.) Also here there are some a priori inequalities, but it is far from clear that those are the only ones My task is to systematically consider standard constructions and see what areas those cover. In this way one can identify interesting invariants, and pose the challenge to find new constructions of 3-folds in order to actually exhibit those. In my talk I will report on progress so far, which has mostly been concentrated on smooth 3-folds (i.e., not allowing non-smooth terminal singularities) where one incidentally notes that c3 = 0(2) for those (note that in general the chern-invariants will not be integers). One reasonable conjecture is to show that the ‘continous’ part of the invariants forms a ‘convex’ set. The content of this conjecture depends on what meaning you attribute to the terms, and I will pre sent and prove based on a very natural notion of the continous part K that the closure of the latter is indeed convex in the usual sense.

Specific features of Hamiltonian system Petrova L. I. (Russia) Moscow State University ptr@cs.msu.su In present work, in addition to skew-symmetric exterior differential forms, skew-symmetric differential forms, which differ in their properties from exterior forms, are used under investigation of Hamiltonian systems.

These are skew-symmetric differential forms defined on manifolds that are nondifferentiable ones. Such manifolds result, for example, under describing physical processes by differential equations. This approach to investigation of Hamiltonian systems enables one to see peculiarities of Hamiltonian systems and relevant phase spaces.

The specific features of Hamiltonian system are analyzed in the case when the Lagrangian manifold is not a differentiable manifold. This can take place, for example, for mechanical systems with nonholonomic constraints. In this case the tangent and cotangent manifolds are not mutually connected. The Legendre transformation, which converts the Lagrangian function defined on tangent manifold into the Hamilton function defined on cotangent manifold, is a degenerate transformation, and hence, a correspondence between the Lagrange equation and the Hamiltonian system will be fulfilled only discretely - on pseudostructures (sections of cotangent bundle). The phase space will be formed by discrete structures.

The Lagrange equation has been obtained from the condition of maximum of the action functional S. This condition is one of conditions needed for existence of the invariant, namely, closed exterior form. But for existence of invariant it is necessary that the closure condition of dual form (determing the manifold or structure, on which the skew-symmetric form is specified) be fulfilled.

As such a condition it just serves the first relation of the Hamiltonian system which is not fulfilled for the Lagrange equation in general case (for nonholonomic constraints).

The transition from the Lagrange equation to Hamiltonian system is achieved with the help of the Legendre transformation, which transforms the tangent manifold into cotangent one. When tangent manifold is a differentiable one , such transition is a nondegenerate transformation. The transition from tangent manifold to cotangent one is one-to-one mapping, and Hamiltonian system and the Lagrange equation are identical.

In the case on nonholonomic constraints the tangent manifold of Lagrangian equation will be not a differentiable manifold. In this case the transition from tangent manifold to cotangent one, that is, the transition from the Lagrange equation to Hamiltonian system, is possible only as a degenerate transformation. This means that the transition to cotangent manifold composed of pseudostructures (sections of cotangent bundles) is only possible. That is, Hamiltonian system can be realized only discretely, namely, on pseudostructures. The first relation of Hamiltonian system (not connected with the Lagrange equation) is a condition of degenerate transformation and defines pseudostructure, on which the Lagrange equation proves to be integrable and is equivalent to the Hamiltonian system.

In this case as the phase space it can serve only cotangent bundle sections of Lagrangian manifold.

It is known that in the case when the tangent manifold is differentiable and hence when the transition from tangent space to cotangent space is one-to one mapping, in the extended phase space {t, q, p} there exists the Poincare invariant ds = - Hdt + pdq (the differential form - Hdt + pdq is a closed exterior form, that is, the differential of this form vanishes).

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