2015년 6월에 1주간 개최할 예정인 제 3회 Summer school의 주제는 Interacting particle systems and random matrices이다. 제 1회 주제는 Stochastic analysis and its potential theory였고 제 2회 주제는 Stochastic partial differential equation였다. 3회에 다룰 내용은 한층 더 확률론의 응용을 다룬다고 볼 수 있다. 확률론 이론이 어떻게 구체적으로 자연현상을 설명하는 데 쓰이는지 토론하게 될 것이다.
좀 더 구체적으로 여름학교에서 다룰 주제를 소개하겠다.

1.Phase Transitions, Critical Phenomena, and the Renormalisation Group a course by David Brydges and Gordon Slade

The subject of phase transitions and critical phenomena in physics has had a major influence on mathematics for over half a century, especially in probability theory but also in other disciplines. In fact the influence on mathematics is now greater than ever before.

The subject is mainly focussed on the study of various specific models.
This course of 16-20 hours will include an introduction to some of the models of greatest interest: percolation, the Ising and $|varphi|^4$ spin models, and models of self-avoiding walks. One of the fascinating features of these models is their dependence on the spatial dimension, and the course will provide a survey of what is rigorously known, and of what is predicted but not yet rigorously known,in the different dimensions. Emphasis will be placed on the existence of phase transitions and the accompanying universal critical behaviour, characterised by universal critical exponents. Exact solutions and conformal invariance are predominant themes in dimension $d=2$,and mean-field behaviour and the lace expansion are important in high dimensions.

Recent joint work with R.~Bauerschmidt for self-avoiding walk and spin models in dimension $d=4$, which uses a rigorous implementation of Wilson's renormalisation group method to prove the existence of logarithmic corrections to mean-field scaling, will be introduced in the course. The application of the method to the self-avoiding walk makes use of a functional integral representation involving anti-commuting(fermionic) variables. This representation will be discussed in the course;such representations are becoming increasingly useful in probability theory.

The level of the course will be suitable for
graduate students in probability without previous background in statistical mechanics;the course will provide the necessary background and will not assume specialised knowledge.

2. Lectures by Tadahisa FUNAKI (President of the Mathematical Society of Japan)

(1) Effective interface model: Assuming that the interface is described as a height function measured from a fixed reference plane disretized in space, the system is described by the energy called Hamiltonian of the height function. Static and dynamic theories are developed for this model. One possible topic to be discussed in my course is the scaling limit for Gaussian random fields with a weak pinning effect under the situation that the rate functional of the large deviation principle corresponding to this scaling limit admits non-unique minimizers.

(2) Dynamics of two-dimensional Young diagrams: If the value of the height function is also discrete and monotone, then it forms a Young diagram. We can define dynamics of Young diagrams in a natural way and study its hydrodynamic behavior and fluctuations.

(3) Kardar-Parisi-Zhang (KPZ) equation: KPZ equation describes a fluctuation of interfaces, and recently attracts a lot of attentions. This is a kind of stochastic partial differential equation which involves a divergent term. I will discuss the invariant measures of this equation.

(4) Sharp interface limit for stochastic Allen-Cahn equations: Sharp interface limit for the Allen-Cahn equation, that is a reaction-diffusion equation with bistable reaction term, leads to a motion of mean curvature for the interface. I will discuss its stochastic perturbation, especially the effect of the stochastic term in the limit.