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In the opinion of Arnold, Enriques stands almost as high as Newton and Riemann in the hierarchy of all philosophers exercise recover the exact source. Sul gruppo di monodromia delle funzioni algebriche, appartenti ad una data superficie di Riemann, Rom. L Rend. Alcune osservazioni sulle superficie razionali reali, Acc. Scienze Ist. Bologna 2 16 , Enriques, O. Bieberbach [], Grunsky [] and for instance A. Bologna 23 , Some inequalities relating to conformal mapping upon canonical slitdomains, Bull. The kernel function and conformal invariants, J.

Math Mech. Eremenko, A. Rational functions with real critical points and B. Shapiro conjecture in real enumerative geometry, Ann. Introductio anlysis infinitorum, tom 2, Lausanae, Any Riemann surface may be considered as an algebraic curve defined over C.

### References

Sometimes this algebraic variety is already definable over the real numbers. This happens precisely if there exist an antiholomorphic involution on the surface, and these involutions correspond bijectively to the different real models of the curve. The real points of this real curve are the fixed-points of the involution, hence the points in the boundary of our original Riemann surface.

We shall see that all counterexample are of this form. To my knowledge not a single result of the sort is known and it is quite hard to speculate about any such topologicoarithmetical connection. Farkas, I. Riemann surfaces, Second Edition, Grad. Texts in Math. It has been proved in [3, p. Theta functions on Riemann surfaces, Lecture Notes in Math. Let z1, USSR Sb. Rohlin ca. So it seems that a more systematic method of prohibition still remains to be discovered and we tabulate on the usage of the Riemann map and the allied property of dextrogyration as sketched in Gabard B [].

Halle-Wittenberg, in press ? New congruences in the topology of real plane algebraic curves. New congruences in the topology of singular real plane algebraic curves. Additional inequalities in the topology of real plane algebraic curves. Emile Picard, Toulouse, If so, then we may expect to get new limitations on complex orientations and as a byproduct new restrictions in.

The main theorem claims that curves of degree 9 with 28 ovals the maximal possible number of ovals for degree 9 cannot realize oval arrangements among arrangements which have been in question before. The proof [ The aim of this paper is to expose these prohibitions. Orevkov, A. Algebraic Geom 11 , More precisely affine M-sextics are involved in the dissipation of which singularity [?

Sorry but we do not know the exact answer! London Math Soc. This classification is complete up to degree 7 and almost complete in degree 8 [yet in complete stagnation since Orevkov , with 6 casus irreducibilis resisting all attempts]. See e. Final version on arXiv March Theorem in v.

Hence it could still be of some interest to reconstruct a proof purely a priori assuming of course that there is a such. This looks quite likely, yet apparently quite elusive to implement. Field, On the circuit of a plane curve, Math. The topology of the complement of a real algebraic curve in CP2. Differential topology of quotients of complex surfaces by complex conjugation, Zap. Sem , According to V. It is well known and not difficult to see directly from the Hilbert and Gudkov constructions of nonsingular real sextics cf. On the topology of real plane algebraic curves with nondegenerate quadratic singularities.

Petersburg Math. Finashin, V. The resulting theory affords a refinement of earlier work by Mikhalkin Fischer, A. Riemannian - On a purely proof of the structure and dimension of the unramified moduli space of a compact Riemann surface, Math. Ahlfors [2]. In [4] Ahlfors shows that Lemma 1: Let z1, Then there is an inner function f i. Ahlfors-Royden extremal problem - Function theory on planar domains. A second course in complex analysis.

Pure and Applied Mathematics New York. A - Publication. Fisher, D. Murasugi-Tristram inequality for generalized signatures and application for real algebraic curves, Preprint Signatures of colored links with applications to real algebraic curves, J. Knot Th Ramif. Math 10 , This is also the smallest subset of the body permitting its complete reconstruction via the convex-hull operation. Indeed from the cited properties of convexity.

Fornaess, N. Question 3. Can one embed all Stein Riemann surfaces as closed complex submanifolds of C2? This may be restated as, p. Fraser, R. For higher eigenvalues, cf. Girouard-Polterovich [], and for Dirichlet-Neumann eingenvalues, cf. Gabard [].

## Analogy and Invention Some Remarks on Poincaré’s Analysis Situs Papers

Fricke, F. Reprinted by Johnson Reprint Corp. Il Principio di minmo i teoremi di esistenza per i problemi al contorno relativi alle equazioni alle derivate parziale di ordini pari, Rend. Palermo Sbornik N. S 16 58 The work of the writer is characterized by a low level of depth, trying rather first to chatter what is common knowledge, before entering into obscure behaviorism i.

Previous unpublished work on the same question due to Kharlamov-Viro in the Leningrad seminar of topology supervised by V. Rohlin Confirms incidentally a desideratum of Gross-Harris []. Up to minor redactional change this is the same as the next entry Gabard Thesis of the author published in , cf. Notwithstanding, more mature knowledge of mine ca. Gabard, D. Jordan and Schoenflies in non-metrical analysis situs, arXiv Seeking extreme generality always brings to an equivalence, which is just a logical tautology, not really worth emphasizing it.

Dynamics of non-metric manifolds, arXiv This would maybe be a spectacular application of the Ahlfors map, or maybe some allied conformal maps, e. Ebullition in foliated surfaces vs. Ahlfors circle maps: historical ramblings, arXiv Also instrumental for this expansion of material is the Rohlin conjecture or at least the vestige thereof post Shustin that a scheme of type I is maximal.

Konstruktive Methoden der konformen Abbildung, Ergebnisse d. Do we recover an Ahlfors circle map? Lincei 8 1 , e IV 27 , Gamelin, M. The space Nc is finite-dimensional. Uniform algebras Gamelin, J. J Math. The Ahlfors function tries hard to be unimodular on the boundary of an arbitrary domain. In fact, he proves that the Ahlfors function for U is unimodular. In fact, an example is given of a reasonably well-behaved domain U with the following property: If f is any.

Garnett, L. Rubel, A. Despite this contrast, badly approximable functions are shown to have constant modulus along the boundary Theorem 1. An expository account is given of T. The Corona Theorem answers affirmatively a question raised by S. Kakutani as to whether the open unit disc in the complex plane is dense in the Problem 1. Problem 2. Is the Ahlfors function unique? Gander, G. Garabedian, M. Problem I.. A new proof of the Riemann mapping theorem. The Dirichlet problem, Math. Analytic capacity and measures, Lecture Notes in Math. Bounded analytic functions, Pure and Appl.

## Notes on Hilbert's 16th: experiencing Viro's theory - INSPIRE-HEP

Garnett, J. Imbeddeding of Riemann surfaces by canal surfaces, Rend. Palermo 2 9 , Garsia, E. It has been an open question for some time whether or not the classical i. C2 surfaces of Euclidean space 3-dimensional exhaust all possible conformal types. In the non compact case the question is still open. It will be shown that for any given compact Riemann surface conformally equivalent models can be found among the algebraic surfaces of ordinary space.

Here by - On the conformal type of algebraic surfaces in euclidean space The braid group and other groups, Quart. Oxford Ser. Disquisitiones arithmeticae Also in: Werke, Bd. Disquisitiones generales circa superficies curvas, Werke vol. Gauthier, M. From local to global properties of subharmonic functions on Green spaces, J. Now the authors shows the converse statement. Gelfand, M. Kapranov, A. Tagung Math. Oberwolfach, , Geyer, G. Gilmer a student of P. Emery Thomas well-known for his knottheoretical approach to real curves. Kharlamov et al. Floppy curves with applications to real algebraic curves, arXiv Arf invariant of real algebraic curves, arXiv , v.

Girouard, I. Steklov eigenvalues, arXiv J Globevnik, B. The problem is reduced to the solution of a finite system of functional equations which are solved by successive approximations. Russian original published in Goryacheva, G. Zh Mat. The author studies the set of Ahlfors functions on R. These functions are the extremal functions obtained by maximizing the derivative f' P in some local parameter at P in the class of holomorphic functions on R bounded by one.

Each Ahlfors function has modulus 1 on the boundary of R and gives a complete30 covering of the unit disk.

It is known that the degree N of any Ahlfors function satisfies q - Ahlfors functions on non-planar Riemann surfaces whose double are hyperelliptic A50 [ detailed study of the degrees of the Ahlfors map in the hyperelliptic case a complement tour de force is to be found in Yamada [] for an application to proper holomorphic embeddings in C2, cf. Cerne-Forstneric [] Koditz's summary MathReviews : 5.

Conformal mapping of abstract Riemann surfaces, Published by the author, Univ. In , Ahlfors showed that a number of extremal problems on compact subregions of open Riemann surfaces could be solved by studying the class of Schottky differentials [Ahlfors [19]; errata, MR 13, p. I the present paper, the authors study the class of SchottkyAhlfors forms with singularities. V Dumkin, An extremal problem for forms with singularities on Riemannian manifolds, Perm. On the history of the Riemann mapping theorem, Rend. Palermo 2 34 , Gray, M. A very important contribution to the development of the manifold concept was made by H.

Grassmann in [t]his work, in which he spoke of n-tuply extended manifolds for the first time and developed, among other things, modern n-dimensional analytic geometry and linear algebra, in which the mathematical theory structure is worked out in a coordinated-free way, allowing the simplest treatment of problems in n-dimensional geometry, and in other fields as well. An essay on the application of mathematical analysis to the theories of electricity and magnetism.

Printed for the Author by Whellhouse T Nottingham, , 72 pp. Greuel, U. Principles of Algebraic Geometry New York, , pp. Russian transl. Gromov, V. In Appendix 4 we show that the real projective plane with a metric of positive curvature, in particular, the elliptic plane, cannot be isometrically C2 -embedded in R4. Of course the uniformization required for Loewner torus and Pu projective plane are of a simpler nature, Abel and Riemann, Schwarz resp. Spaces and questions, Preprint Some remarks on conformal representation, Ann.

Real algebraic curves and their Jacobians, preprint, Gross, J. Schneps and P. Lochak eds. Armeisen - reproduced in: Geometric Galois Actions I. Lecture Note Ser. Leipzig 80 , Leipzig , Das Kreisbogenschlitztheorem der konformen Abbildung schlichter Bereiche, Ber.

Schriften Math. Berlin 1 , This periodicity modulo 8 incarnates a whole new era, and Viro often wondered why it was not already conjectured by V. Ragsdale ca. Gudkov also contributed to the construction of 2 new M-schemes of degree 8 in [] reported on our Fig. Establishing all existing types of non-singular plane algebraic curves of the sixth order with real coefficients, Ph. Dissertation Gorki, Gudkov, G. Utkin, M. A complete classification of indecomposable curves of the fourth order, Mat.

Construction of a new series of M-curves.

Construction of a curve of degree 6 of type 5 1 5, Izv. VUZ Gudkov, A. On the periodicity of the Euler characteristic of real algebraic M - 1 -manifolds. The topology of real projective algebraic varieties. On the topology of algebraic curves on a hyperboloid.

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Algebraic curves on a hyperboloid of one sheet have been studied for a long time. Generalization of a theorem of Brusotti for curves on a surface of second order If all the singular points of a curve F in the complex projective plane CP2 x0 : x1 : x2 are simple double points, then the simplifications of these singular points are independent. Gudkov, E. Classification of non-singular curves of degree 8 on an ellipsoid, in: Method in Qualitative Theory of Differential Equations, Gorky, , Stratification of a space of fourth-order curves. Special forms of fourth order curves, Part 1,2,3,4,5, Deposited in Vinity, , resp.

Gudkov, N. Biographical enigmas, Nizhnii Novgorod, monograph in print. Guillemin, S. Guillou, A. Acad Sci Paris Ser. A , Lectures on Riemann surfaces. Princeton Acad. Press, Princeton, Gunning, R. A piece de resistance is served in the appendix in the form of a proof of the existence of functions of order - Lectures on Riemann surfaces: Jacobi varieties, Princeton Univ.

Press, Princeton, N. J , pp. Martens's review in MathReviews is capital for it brings the hope to gain a Teichmuller theoretic approach to the existence of circle maps with the best possible bounds hence hinting how to recover Ahlfors and even Gabard [] by an analytic or rather geometric! Acta Appl. Basel ca. Also often quoted for a proof of Ragsdale within the context of maximal T -curves, so giving some additional evidence toward the truth of the still open Ragsdale conjecture for M-curves.

If so is the case this would be a good piece of experimental evidence toward the truth of Ragsdale. Alas, like any small perturbation method nothing seems to tell us that all curves arise through patchwork, and so we are forced to a theoretical attack. At the time of writing we have 2 vague strategies.

Gabard B [] and the second an attempt to make the Arnold surface always orientable by a surgery cf. Gabard B, v.

## On the Foundations of Plane Analysis Situs.

Real algebraic curves and combinatorial constructions, Ph. Ambient surfaces and T -fillings of T -curves, arXiv Sur le principe de Dirichlet, Bull. Soc Math. France Contemporary Mathematics 71, Amer. Hara, M. J 37 , It is reported somewhere that Klein did not believed first in this phenomenon discovered by Harnack. Ueber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Die Grundlagen der Theorie des logarithmischen Potentiales, und der eindeutigen Potentialfunktionen in der Ebene Leipzig, Hardt, D.

Hautes Etudes Sci. On the mean modulus of an analytic function, Proc. Conformal mappings onto domains with arbitrarily specified boundary shapes, J. Algebraic Geometry, Grad. Riesz ca. Then the route is paved for Veblen-Whitehead, Whitney, etc. Havinson, On extremal properties of functions mapping a region on a multi-sheeted circle, Doklady Akad. SSSR N. Havinson, Extremal problems for certain classes of analytic functions in finitely connected regions, Mat. Havinson, G.

Havinson, Analytic capacity of sets, joint nontriviality of various classes of analytic functions and the Schwarz lemma in arbitrary domains. Havinson, Factorization theory for single-valued analytic functions on compact Riemann surfaces with boundary, Uspekhi Math. Havinson, Duality relations in the theory of analytic capacity Russian version published in Hawley, M. Precisely, for a planar membrane with hyperelliptic double, all W-points are located on the boundary.

The author gives an example, derived from a real quartic with 4 ovals, whose W-points are not confined to the boundary. Such questions make good sense over positive genus membranes and are perhaps worth investigating further. Probably updates are already known, and one would like to explicit any possible relation between Wpoints and the degree of the Ahlfors function.

He, O. Fixed points, Koebe uniformization and circle packings, Ann. Ueber trigonometrische Reihen, J Reine Angew. A lemma on positive harmonic functions, Ann. Hardy classes on Riemann surfaces, Lecture Notes in Mathematics 98 As a pessimistic remark it seems that there is a wide variety of extremal problems, somehow reflecting. It is appropriate to cite instances of convexity considerations related to the present paper. Extreme points are present in the fundamental work of R.

Such Riemann surfaces will be termed finite Riemann surfaces henceforth. Math Kyoto Univ. Linear extremal problems for analytic functions with interior side conditions, Ann. Hensel, W. Leipzig 93 , Riesz and cf. Acad Sc. Paris , Hersch, L. Payne, M. Schiffer Some. Mech Anal. Hidalgo, A. Annales Acad. Schottky uniformization of stable symmetric Riemann surfaces, Notas de la Soc. Hidalgo, B. On the retrosection theorem, Proyecciones 27 , All these attempts where judged unconvincing, and from the Russian rating agency not judged as sufficiently rigorous until the intervention of Petrovskii [] and Gudkov , cf.

Grundlagen der Geometrie, Mathematische Probleme, Arch. III, p. Was die Kurven 6. Ordnung des dreidimensionalen Raumes im Maximum wirklich besitzt [solution of this problem at most 10 components, cf. Ich bediene mich im folgenden dieses klassischen Beispiels zur Darlegung meines strengen Beweisverfahrens. Galafassi [] , yet probably just a French translation of the German original. Kahn und K. Loebenstein Vgl.

Hilbert, S. Anschauliche Geometrie Berlin, We show with a new variational approach that any Riemannian metric on a multiply connected schlicht domain in R2 can be represented by globally conformal parameters. This yields a - Math , On the circuits of a plane sextic curve, Rend. Circolo Mat. Palermo 60 , Topological methods in algebraic geometry The signature of ramified coverings, in: Global Analysis Papers in honor of K. Kodaira University Press, Tokyo, , The signature theorem: reminiscences and recreations, in: Prospects in Mathematics Annals of Math.

Studies 70 , Princeton University Press, Princeton, , III, pp. Tome I. VII, pp. VI, pp. Analysis situs.

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Tome III. Bulletin astronomique , 16 , — Sur les nombres de Betti. VI, p. Rendiconti del Circolo matematico di Palermo , 13 , — Proceedings of the London Mathematical Society , 32 , — Philosophical Transactions , A , — V, pp. Rendiconti del Circolo matematico di Palermo , 18 , 45— Science and Hypothesis W. Larmor, Trans. The Walter Scott Publishing Co. Paris: Flammarion. Rivista di scienza , 2nd year, 3 , pp. Rendiconti del Circolo matematico di Palermo , 33 , — The value of science, G.

Halsted Trans. Dover, New York orig. Science and Method Francis Maitland, with a preface by B. Russell Trans. Thoemmes Press, Bristol Analyse de ses travaux scientifiques faite par H. Acta Mathematica , 38 , 36— Gray and S. Walters Eds. Pont, J. Paris: Presses Universitaires de France. Riemann, B. Also in Gesammelte Mathematische Werke…, pp. Rourke, C. Bartocci Ed. La matematica. Problemi e teoremi pp. Einaudi, Torino. Sakaria, K. Saviliev, N. Invariants for homology 3-spheres.

Encyclopedia of mathematical sciences , vol. Berlin-Heidelberg: Springer.