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# Category Archives: MA 59800

## Homework Assignment 3

Recall that the upper half plane consists of complex numbers with . Let denote the closed unit disk as the union of the unit disk with its boundary . This assignment is meant to explain the pictures at the Wolfram … Continue reading

## Lecture 15: Monday, September 23, 2013

Today we discuss how some familiar objects — namely the circle, the sphere, the torus, and even elliptic curves — are each examples of non-singular algebraic curves which are also compact, connected Riemann surfaces.

## Lecture 14: Friday, September 20, 2013

Eventually, we wish to show that every compact, connected Riemann surface is a nonsingular algebraic curve. Today, we discuss what is means to be a “nonsingular algebraic curve” using Dedekind Domains.

Posted in MA 59800
Tagged Algebraic Curve, Algebraic Variety, Coordinate ring, Dedekind domain, Discrete Valuation, Discrete Valuation Ring, Integral domain, Jacobian matrix, Localization, Maximal ideal, Oscar Zariski, Polynomial ring, Prime ideal, Principal Ideal, Radical of an Ideal, Tangent space, Zariski cotangent space
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## Lecture 13: Wednesday, September 18, 2013

In Lecture 11 and Lecture 12, we discussed examples of Riemann Surfaces and their relation with the Riemann Sphere . Today, we put things into perspective by considering the larger picture: we discuss tesselations of the upper half plane, covering … Continue reading

## Lecture 12: Monday, September 16, 2013

Last time, we introduced the Fundamental Domain in terms of the extended complex plane . Felix Klein showed the existence of a function , invariant under , giving these isomorphisms. Today, we use Richard Dedekind‘s function to create similar functions … Continue reading

## Lecture 11: Friday, September 13, 2013

Over the past several lectures, we have focused on triangle groups and ways to tesselate both the plane and the sphere. In order to generalize this, we will focus on ways to tesselate compact, connected Riemann surfaces. Today we’ll begin … Continue reading

## Homework Assignment 2

Recall that the collection of Möbius transformations is the group while the (ordinary) triangle group is the set This homework set is meant to discuss the relationship with these two sets. This assignment is due Friday, September 20, 2013 at … Continue reading

Posted in MA 59800
Tagged Euler Characteristic, Genus, Hecke Group, Mobius Transformation, Modular Group, Principal Congruence Subgroup
1 Comment