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# Mathematica Summer School on Theoretical Physics

This school is focused on applications of Mathematica to advanced topics in Theoretical Physics. It is targeted at physics Master and Ph.D students from all over Europe and the world. The school aims at developing competences in the use of Mathematica, which probably became the most powerful tool for a theoretical physicist. Moreover, while developing the student’s efficient use of Mathematica, the school also intends to teach an advanced, exciting and actual research topic in theoretical physics.

The school first edition
will be dedicated to the theme of Integrability and the Gauge/String
duality. This duality, commonly referred as the AdS/CFT duality, is an
equivalence between a string theory in a curved space and a gauge
theory. The study of these gauge theories is of particular importance
both to theoretical and high-energy physics, since our current
understanding of the strong interactions is based on a gauge theory
known as QCD. Quite remarkably, on the other side of the equivalence
there is a theory with gravity, so that the AdS/CFT duality became in
the last decade an incredibly rich field for the interplay between
experts in gauge theory, string theory and gravitational physics. Very
recently this subject became even more fascinating, since experimental
results at particle accelerators (RHIC and the forthcoming LHC) require
the knowledge of the dynamical behavior of QCD that is not within reach
of standard computational techniques. However, such dynamical regime
can be analyzed by using the duality, which is now also seen as an
important computational tool. Moreover in recent years it was
understood that the most known examples of the AdS/CFT duality relate
integrable theories, thus amenable of exact solution. Since this
discovery this research field has observed a huge growth and has
already allowed to access strongly coupled regimes of particles physics
and quantum gravity which had never been explored before.

In
sum, the subject of Integrability in AdS/CFT is actually one of the
hottest and most interesting topics in theoretical physics.This
research program envisages solving, for the very first time, a
nontrivial four dimensional gauge theory. If such goal is accomplished,
it would also teach us a lot about the nature of Quantum Gravity.
Moreover Integrability is an extremely beautiful subject per se, with
many connections to other areas of physics such as Condensed Matter and
Mathematical Physics.

Finally, Integrability is an excellent
subject where the power and usefulness of Mathematica can be mostly
appreciated. This subject is a very rich and technical one, with many
examples where the Mathematica tool is of great importance. To name a
few:

• Algebraic manipulations involving the N=4 spin chain
S-matrix such as checking its symmetries, the triangle Yang-Baxter
relations and understanding the scattering of bound states can take
huge advantage of Mathematica.

• The computation of effective
spin chain Hamiltonians, such as that following from the Hubbard model
at half-filling (an example where this subject touches the condensed
matter literature), can also benefit a lot from Mathematica.

•
Higher loop computations, such as the remarkable recent four-loop
computation of the anomalous dimension of the Konishi operator, could
never be done without the use of computer symbolic manipulation and
Mathematica is an excellent choice to do that.

• The numeric solution to Bethe equations involving often hundreds of variables can be found with the help of this software.

•
High precision expansions checking the transcendentally nature of the
N=4 spectrum from the known Bethe equations can be done using
Mathematica.

• The computation of classical string solutions and
the computation of their semiclassical spectrum using algebraic curve
techniques is another example where Mathematica can be of great use.

•
The computation of the scaling function, a first example of a non-BPS
quantity whose interpolation from weak to strong coupling in this gauge
theory is believed to be under control, can be also optimized using
Mathematica.

This list could easily go on. The examples covered
in the school will certainly be among those listed above but they will
reflect the current state of the art as of June 2009. In the mean time,
it is certain that many interesting problems where Mathematica can be
of great utility will appear and those will be covered in the school.