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\centerline{MATH 400 -- Directed Readings in Complex Functions and Riemann Surfaces}
\centerline{Spring 2014}
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{\it Course Format}
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The course will be closely based on topics from the book {\it Complex Functions, an algebraic
and geometric viewpoint}, by G. Jones and D. Singerman (I have a copy for you).

We will meet for about 2 hours each Monday afternoon during the semester in my office, 
Swords 331.  The assignment for each meeting will include some reading from the Jones 
and Singerman book (to be done {\it before} our meeting), 
plus some problems (starting with the second meeting).  I'm hoping to spend about a half hour each week 
after the first on questions that might have come up from the reading, and having you 
present some parts of solutions to the problems orally. Then I'll do a ``lecture'' on 
some new material.  But we will not prove every theorem in the book in detail in
these lectures, as you might expect in a more formal course; some of your work
each week should be figuring out some of these arguments individually.

For formal assessments, I will give a larger midterm problem set around the 
middle of the semester, and then either a similar final problem set or a 
project paper on a topic related to the mathematics we study (we will discuss
this, and I will let you choose which option fits your goals and interests 
better).  

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{\it (Very) Tentative Schedule}
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Note:  This might be too ambitious, but it would be very nice to cover
this much of the book.  There's no real pressure, though, and we won't rush to 
fit this much in if it turns out we need to go more slowly.  We'll see that even covering
the material in Chapters 1 - 3 will give a nice follow-up to the Complex
Analysis course and tie things you have seen in other courses together in 
a beautiful way as a ``capstone'' course.  
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\item{$\bullet$}  Week 1 -- January 27:  The Riemann sphere $\Sigma$ and mermorphic functions.  
Reading:  Appendix 1 and \S\S 1.1 - 1.4; no problems due this first meeting.
\item{$\bullet$}  Week 2 -- February 3:  Meromorphic functions as mappings $f : \Sigma \to \Sigma$, 
M\"obius transformations.  Reading: \S 1.5, \S\S 2.1 - 2.9.  Problems: to be assigned.
\item{$\bullet$}  Week 3 -- February 10:  Geometric classification of M\"obius transformations, 
finite subgroups of the group of M\"obius transformations.  Reading: \S\S 2.10 - 2.15.  Problems: to be assigned.
\item{$\bullet$}  Week 4 -- February 17:  Discrete subgroups of ${\bf C}, +$, periodic functions.  
Reading: \S\S 3.1 - 3.6.  Problems: to be assigned.
\item{$\bullet$}  Week 5 -- February 24:  Sum and product expansions, the Weierstrass $\zeta, \sigma, 
\wp$ functions.  Reading: \S\S 3.7 - 3.9.  Problems: to be assigned.
\item{$\bullet$}  Spring Break -- week of March 3
\item{$\bullet$}  Week 6 -- March 10:  The differential equation of the $\wp$-function,
the field of elliptic functions.  Reading: \S\S 3.10 - 3.12.  Problems: to be assigned.
\item{$\bullet$}  Week 7 -- March 17:  Elliptic functions with given zeroes and poles; with given 
principal parts.  Reading: \S\S 3.13 - 3.15.  Problems: Midterm problem set.
\item{$\bullet$}  Week 8 -- March 24:  Elliptic curves, the group law, applications.  Reading: \S\S 3.16- 3.17.  Problems: to be assigned.
\item{$\bullet$}  Week 9 -- March 31:  Analytic/meromorphic continuation.  Reading \S\S 4.1 - 4.4.
Problems:  to be assigned.
\item{$\bullet$}  Week 10 -- April 7:  Homotopy of curves; the monodromy theorem.  Reading \S\S 4.5 - 4.6.
Problems:  to be assigned.
\item{$\bullet$}  Week 11 -- April 14:  Riemann surfaces of $\log z, z^{1/q}, \sqrt{p(z)}$.  Reading:
\S\S 4.7 - 4.9.  Problems:  to be assigned.
\item{$\bullet$}  Easter break includes Monday, April 21
\item{$\bullet$}  Week 12 -- April 28:  Abstract Riemann surfaces. Reading: \S\S 4.10 - 4.11.  Problems:  to be 
assigned.
\item{$\bullet$}  Week 13 -- May 5:  Function theory on Riemann surfaces, course wrap-up.  Reading: 
\S\S 4.12 - 4.16.  Problems:  Final problem set, or project.
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