By Alexander I. Bobenko (auth.), Alexander I. Bobenko, Christian Klein (eds.)

This quantity bargains a well-structured evaluate of existent computational methods to Riemann surfaces and people at present in improvement. The authors of the contributions signify the teams supplying publically on hand numerical codes during this box. hence this quantity illustrates which software program instruments can be found and the way they are often utilized in perform. furthermore examples for suggestions to partial differential equations and in floor thought are provided. The meant viewers of this booklet is twofold. it may be used as a textbook for a graduate path in numerics of Riemann surfaces, within which case the normal undergraduate historical past, i.e., calculus and linear algebra, is needed. specifically, no wisdom of the speculation of Riemann surfaces is predicted; the mandatory heritage during this concept is inside the creation bankruptcy. while, this ebook can be meant for experts in geometry and mathematical physics utilizing the speculation of Riemann surfaces of their study. it's the first publication on numerics of Riemann surfaces that displays the development made during this box over the past decade, and it comprises unique effects. There are increasingly more purposes that contain the assessment of concrete features of types analytically defined when it comes to Riemann surfaces. Many challenge settings and computations during this quantity are inspired via such concrete purposes in geometry and mathematical physics.

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E. ∃D ∈ C2 : δD = γ1 − γ2 . Deﬁnition 13. The factor group H1 (R, ZZ) = Z/B is called the ﬁrst homology group of R. Freely homotopic closed curves are homologous. However, the converse is false in general, as one can see from the example in Fig. 16. I. Bobenko Fig. 16. A cycle homologous to zero but not homotopic to a point The ﬁrst homology group is the fundamental group “made commutative”. , the subgroup of π(R) generated by all elements of the form ABA−1 B −1 , A, B ∈ π(R). To introduce intersection numbers of elements of the ﬁrst homology group it is convenient to represent them by smooth cycles.

The following two characterizations of a point e ∈ Jac(R) are equivalent: • The theta function and all its partial derivatives up to order s− 1 vanish in e and at least one partial derivative of order s does not vanish at e. • e = A(D) + K where D is a positive divisor of degree g and i(D) = s. 7 Holomorphic Line Bundles In this section some results of the previous sections are formulated in the language of holomorphic line bundles. This language is useful for generalizations to manifolds of higher dimension, where one does not have concrete tools as in the case of Riemann surfaces, and where one has to rely on more abstract geometric constructions.

71) i=1 Qi Theorem 17 (Abel’s theorem). The divisor D ∈ Div (R) is principal if and only if: (1) deg D = 0 (2) A(D) ≡ 0 The necessity of the ﬁrst condition is already shown in Corollary 2. Let f be a meromorphic function with the divisor (f ) = P1 + . . + PN − Q1 − . . − QN 1 Riemann Surfaces 39 (these points need not be diﬀerent). Then Ω= df = d(log f ) f is an Abelian diﬀerential of the third kind. Since periods of Ω are integer multiples of 2πi, it can be represented as g N Ω= ΩPi Qi + i=1 n k ωk k=1 with nk ∈ ZZ.