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\title{LECTURE SERIES \\ 
Recent Results on
Algebraic Geometry over \\
Finite Fields and Its Applications\\
to Coding Theory and Cryptography
}
\author{by  H. L.   Janwa}

\date{}

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I plan to give a year long series  of lectures on 
Algebraic Geometry over Finite Fields and Its Applications
to Coding Theory and Cryptography.


We will meet once or twice a week as the pace demands.

{\em The first set of lectures will be on 
{\em Function Fields and Codes,}
as discussed by Stichtenoth in his book.}
I will provide additional material in the form of exercises and examples
from my book (under preparation.)

We will also discuss {\em Elliptic Curve Cryptosystems and Their
Generalizations.}

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{\bf BACKGROUND:} {\it Someone who has taken the basic graduate level 
courses in Abstract Algebra. Previous exposure to algebraic geometry,  coding 
theory, or cryptography is highly desirable.}

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{\bf If time permits, I will discuss some of my work (some with co-authors)  on algebraic geometry and its applications to
algebraic geometric codes, public key cryptosystems, one way 
functions in cryptography,  expander graphs, and sequence designs.}

Depending upon the interest of the participants, future topics may include:

\begin{itemize}
\item The work of Serre on ``rational points on curves over finite fields,"
and subsequent work by Oesterle and others.

\item Some New Results on Public Key Cryptosystems based on Algebraic Geometric Codes 

\item Low complexity Algorithms for the Constructions of 
Algebraic Geometric Codes Better than the Gilbert Varshamov Bounds 
%(After  Stichtenoth, Kumar, and others )

\item Generalizations of Sudan's Algorithm for  Decoding  
for Algebraic Geometric Codes 

%(after Hoholdt and others)


%(after Norm Elkies)

\item Some work of Norman Elkies:

\begin{enumerate}

\item  Curves and non(linear) codes 

\item Beyond Goppa Codes 

\end{enumerate}

\item Garcia-Stichtenoth Towers from  Drinfeld Modular Curves 

\item 
\begin{enumerate}
\item Carlitz Modules
\item Cyclotomic Function Fields
\item Explicit Class Field Theory for the Field of Rational Functions over Finite Fields
\item Explicit Cyclotomic Functional Fields with Many Rational Points

\item Drinfeld Modular Curves 

\item On zeta functions of curves over finite fields

\item Generalized Hamming weights and Algebraic Curves

 
\end{enumerate}

\item Codes from Curves Associated with  Deligne-Lusztig Varieties.

\item Asymptotically Good  Families of Curves from
Class Field Towers

%\item Explicit Cyclotomic Function Fields with Many Rational Points

\end{itemize}





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