Stochastic Population
Systems - Don
Dawson

Historically,
the modelling of biological populations has been an important stimulus
for the development of stochastic processes. The revolutionary changes
in the biological sciences over the past 50 years have created many new
challenges and open problems. At the same time probabilists have
developed new classes of stochastic processes such as interacting
particle systems and measure-valued processes and made advances in
stochastic analysis that make possible the modelling and analysis of
populations having complex structures and dynamics. This course will
focus on these developments. In particular stochastic processes that
model populations distributed in space as well as their genealogies and
interactions will be considered. This will include branching particle
systems, interacting Wright-Fisher diffusions, Fleming-Viot processes
and superprocesses. Basic methodologies including martingale problems,
diffusion approximations, dual representations, coupling methods,
random measures and particle representations will be involved. A
principal objective is to describe the dynamics and structure of
populations in large and small space and time scales using dual
processes asymptotics, mean-feld methods and multiscale analysis. Some
recent developments based on the use of these methods and models to
approach some challenging problems in evolutionary biology, genetics,
ecology and epidemiology will be described. Finally, we will discuss
some open problems in stochastic population processes and their
applications to modelling biological populations.

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Statistical
Mechanics and the Renormalisation Group - David
Brydges

Course
Outline
* Some canonical models in equilibrium statistical
mechanics and connections between them

o ideal gas =
Poisson field

o lattice
Gaussian field

o hard core gas

o Ising model

o mean field
models

o self-avoiding
walk and random walk

* Gibbs measures, correlations

o program
to classify scaling limits

o relation
to CLT and the Newman-Wright theorem

* Central role of the lattice Gaussian field

o graphical
expansions

o Hermite
polynomials

* Generalisations of Gaussian field

o Grassmann
variables versus differential forms

o supersymmetry

o self-avoiding
walk as a Gaussian integral

o matrix tree
theorems

* Symmetry breaking and phase transitions

o the basic
phenomenon at the lattice Gaussian field level

o proof of
symmetry breaking by infra-red bounds

o role of the
transfer matrix and Osterwalder-Schrader positivity

* Hierarchical lattice

o
Renormalisation Group (RG) for models on the hierarchical lattice

o Relevant,
Irrelevant interactions

o critical
models and tuning the initial mass

o Why four
dimensions is special

* RG for models on the Euclidean lattice

o space of
interactions defined in analogy to hierarchical case

o theorems on
local existence of RG flow

o global
existence for critical models

o When scaling
limits are Gaussian

References
(incomplete)
* Supersymmetry/differential forms:

o Differential Forms with Applications to the Physical Sciences, Harley
Flanders

o
Advanced Calculus: A Differential Forms Approach, Harold M.
Edwards

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