Many problems involving phase transitions have a variational formulation.
Treatment of these problems with the aid of tools from PDE , dynamical systems, and geometry as well as the calculus of variations leads to many interesting results and open questions. We will survey the results and methods and also mention several open questions.

Composite materials can have properties unlike any found in nature, and in this case they are known as metamaterials. Recent attention has been focussed on obtaining metamaterials which have an interesting dynamic behavior. Their effective mass density can be anisotropic, negative, or even complex. Even the eigenvectors of the effective mass density tensor can vary with frequency. Within the framework of linear elasticity, internal masses can cause the effective elasticity tensor to be frequency dependent, yet not contribute at all to the effective mass density at any frequency. One may use coordinate transformations of the elastodynamic equations to get novel unexpected behavior. A classical propagating wave can have a strange behavior in the new abstract coordinate system. However the problem becomes to find metamaterials which realize the behavior in the new coordinate system. This can be solved at a discrete level, by replacing the original elastic material with a network of masses and springs and then applying transformations to this network. The realization of the transformed network requires a new type of spring, which we call a torque spring. The forces at the end of the torque spring are equal and opposite but not aligned with the line joining the spring ends. We show how torque springs can theoretically be realized.

 The aim of this talk is to introduce some theory of algebraic geometry to Commutative Ring Theory and to translate some properties of singularities to the language of Commutative Ring Theory over fields of positive characteristic.
 The contents includes the following topics.
   (1) Resolution of singularities and rational singularities.
   (2) Positive characteristic counterpart of rational singularities and log terminal singularities.
   (3) Construction of normal graded rings from projective varieties and Q- divisors.
   (4) Ideal theory of integrally closed ideals and cycles on the resolution.