[Turkmath:10084] Joint seminar of Math. Dept. and Inst. App. Math. of METU

[Omer Kucuksakalli]

Dear Colleagues,

You are cordially invited to a joint seminar of Mathematics Department  
and Institute of Applied Mathematics of Middle East Technical  
University. As part of a Tubitak(112T011)-BMBF project, Stefan  
Hellbusch and Christian Neurohr from Universität Oldenburg will give  
talks on October 15, Wednesday between 15:40 and 17:30. The talks will  
be in Gunduz Ikeda room in Mathematics Department.

With best regards

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Stefan Hellbusch, October 15, Wednesday at 15:40 in Gunduz Ikeda Room.

Title: Riemann-Roch on graphs

Abstract: We all know the Riemann-Roch theorem. I will talk about an  
analogue on a finite graph by M. Baker and S. Norine in [2] and  
related results of F. Shokrieh [3] and myself [1]. As in the classic  
case, we get divisors, an equivalence relation and a (abelian) divisor  
class group, which is the quotient group of degree 0 divisors and  
principal divisors. When fixing a base vertex, in each equivalence  
class there is exactly one reduced divisor and the divisor reduction  
is related with an interesting, so called, unconstrained chip firing  
game. Using Dhar's Burning Algorithm, the reduction can be done fast  
and we get an efficient arithmetic in the divisor class group. We will  
see some examples and conclude, that for each finite abelian group,  
there is a graph with this group as divisor class group. We also give  
a short view on a cryptographic perspective and contrary to F.  
Shokrieh in [3], we conclude that there are graphs suitable for  
cryptography.

Literature:
[1] Stefan Hellbusch, Riemann-Roch Theorie auf Graphen und Anwendungen, 2013
[2] Matthew Baker, Serguei Norine, Riemann-Roch and Abel-Jacobi Theory  
on a finite Graph, 2007
[3] Farbod Shokrieh, The monodromy pairing and discrete logarithm on  
the Jacobian of finite graphs, 2010
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Christian Neurohr, October 15, Wednesday at 16:40 in Gunduz Ikeda Room.

Title: Integration on Riemann Surfaces: Homology

Abstract: The long-term goal of our project is to successfully  
implement algorithms in Magma in order to integrate differential form  
on Riemann Surfaces. In this talk, we will take a closer look at the  
first homology group of a given compact Riemann surface and the  
computation of a homology basis. Naturally, its elements, 1-cycles,  
are the objects along which differential forms are integrated. An  
important part of this is the computation of the monodromy group, for  
which we implemented an algorithm similar to the one used in the Maple  
package "algcurves".
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