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We will study links between integrable systems in statistical mechanics, combinatorial problems and special functions in mathematics. This area of research has attracted many scientist's attention during the last decade and revealed unexpected links to other areas of mathematics like enumeration problems and differential equations.
The aim of this project is to introduce quantum integrable systems which play a very important role in modern theoretical physics. Such systems provide one of very few ways to analyze nonlinear effects in continuous and discrete quantum systems.
A separation of variables is a standard technique in classical mechanics which allows to reduce a complicated dynamics with many degrees of freedom to a set of one-dimensional problems. Surprisingly this method finds its natural generalization in the theory of quantum integrable systems. This project aims to study such systems and apply results to the theory of special functions in one and several variables.
There are many interesting physical statistical systems which never reach thermal equilibrium. Examples include surface growth, diffusion processes or traffic flow. In the absence of general theory of such systems a study of particular models plays a very important role. Integrable systems provide examples of such systems where one can analyze time dynamics using analytic methods.
In recent years there was a large boost in development of advanced variational methods which play an important role in analytic and numerical studies of 1D and 2D quantum spin systems. Such methods are based on the ideas coming from the renormalization group theory which states that physical properties of spin systems become scale invariant near criticality. One of the most powerful variational algorithms is the corner-transfer matrices (CTM) method which allows to predict properties of large systems based on a simple iterative algorithm.
Conformal Field Theory (CFT) in two-dimensions describes physics of the second order transitions in statistical mechanics and also plays important role in string theory, which is expected to unify the theory of strong interaction with quantum gravity. The project aims to explore and further develop mathematical techniques of CFT.
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