Invited Speakers:
Lecture 1. Mike. Elliptic curves, modular forms and the modularity theorem.
We briefly review some modular forms terminology, and
explain the statement of the modularity theorem
due to Wiles, Breuil, Conrad, Diamond and Taylor.
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Lecture 2. Samir. Level lowering, Frey curves and Fermat's Last Theorem.
We state an explicit special case of Ribet's Levellowering Theorem
and explain how it is possible to use this together with
Frey curves to relate certain Diophantine equations to
modular forms. We show that Ribet's Theorem, the Modularity
Theorem and a theorem of Mazur give an immediate proof of Fermat's Last Theorem.
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Lecture 3. Samir. The equation x^{p}+y^{p}+L^{r}z^{p}=0
We apply the proof of Fermat's Last Theorem to the equation
of the title and see that it fails. But we can still
solve this equation for many values of L
using some ideas of Mazur.
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Lecture 4. Sander. The method of Kraus.
We consider the equation x^2+7=y^p, which is
known as the generalized RamanujanNagell equation.
We explain an extension of the modular method due
to Kraus that is capable of solving this equation
for one value of the exponent p at a time, and
so with the help of a computer
can solve the equation for a huge range of p.
The exponent p is bounded by Baker's Theory, so
this enables us to solve the equation completely.
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Lecture 5. Sander. Galois representations of elliptic curves, an introduction.
Galois representations are present behind the scenes in the previous lectures.
In this lecture we define Galois representations of elliptic curves,
and explain their relation to Galois representations of modular forms.
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Lecture 6. Mike. The generalized Fermat equation.
The generalized Fermat equation x^p+y^q+z^r=0
is the new 'holy grail of number theory', and
there is a $1M prize for solving it. We survey
what is known and the different methods that are
used for attacking it.
Contributed Talks:

Gökhan Soydan

Kazım Büyükboduk

Erhan Gürel

Burak Yıldız

Nikos Tzanakis

Ercan Altınışık