We shall follow the
growth of probability theory and applications from the 1650s onwards, in
parallel with the development of statistical inference. Bayesian,
Neyman-Pearson hypothesis testing and Fisherian likelihood methods will all be
covered, with an emphasis on relating theory to a wide range of
applications. Practical sessions will use SciPy and feature
closed-form solutions, iterative and Monte Carlo simulation methods.
We shall follow the
growth of probability theory and applications from the 1650s onwards, in
parallel with the development of statistical inference. Bayesian,
Neyman-Pearson hypothesis testing and Fisherian likelihood methods will all be
covered, with an emphasis on relating theory to a wide range of
applications. Practical sessions will use SciPy and feature
closed-form solutions, iterative and Monte Carlo simulation methods.
It has been known for some time that
a system with a filled band will have an integer quantum Hall conductance equal
to its Chern number, a toplogical index associated with the band. While this is
true for a system in a magnetic field with filled Landau Levels, even a system
in zero external field can exhibit the QHE if its band has a Chern number. I
review this issue and discuss a more recent question of whether a partially
filled Chern band can exhibit the Fractional QHE. I describe the work done with
Ganpathy Murthy in which we show how composite fermions, which were so useful
in explaining the usual FQHE, can be introduced here and with equal success by
adapting our Hamiltonian Theory of CFs developed for the FQHE in the continuum.
The
combinatorial problems associated with the counting of black hole states in
loop quantum gravity can be analyzed by using suitable generating functions.
These not only provide very useful tools for exact computations, but can also
be used to perform an statistical analysis of the black hole degeneracy
spectrum, study the interesting substructure found in the entropy of
microscopic black holes and its asymptotic behavior for large horizon areas.
The methods that will be described are relevant for the discussion of the
thermodynamic limit for black holes in the area canonical ensemble. This is an
important issue in order to understand sub-leading corrections to the
Bekenstein-Hawking law.