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Content archived on 2024-04-16

Computation problems in the theory of Riemann surfaces and algebraic curves


Period matrices of symmetric compact Riemann surfaces have been computed numerically. An algorithm for general compact Riemann surfaces has been developed but suffers numerical stability problems. A program that computes periods of differentials for symmetric Riemann surfaces of low genus has been written.

A program for computing period matrices of hyperelliptic curves has been written and tested for stability and accuracy. It has been used to study the behaviour of period matrices as the boundary of the moduli space of smooth curves is approached and has provided optimistic results towards a general program for computing period matrices of normalization of singular hyperelliptic curves.

For nonhyperelliptic curves, a strategy has been developed for computing period matrices of smooth plane quartics. The algorithm is not entirely automatic as yet, except for a family of curves which includes most of the bielliptic curves.

An algorithm has been developed to compute species of symmetrics of Riemann surfaces. A program has been written (KLEIN) to compute all orders and ramification indices of cyclic actions on compact surfaces (automorphism). Antiholomorphic automorphisms have been studied, and the case of involutions on surfaces with or without boundary, orientable or not.


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15,Yliopistonkatu 5

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