Abstract of Blumenthal Lectures 2014

Lecture 1:

Deterministic limit shapes in statistical mechanics

In statistical mechanics states of the system are

distributed randomly according to the Boltzmann law.

However when the size of the system increases,

randomness may manifest itself differently at different

scales. At the macroscopic scale it may develop

deterministic limit shape. Such limit shapes are

usually minimizers of certain functionals which are

determined by the microscopic nature of the system.

The lecture will be focused on the nature of such limit

shapes in dimer models and questions of universality of

fluctuations around various regions of limit shapes.

Lecture 2:

Ice and 6-vertex models in statistical mechanics:

mathematical perspective

The talk will start with a historical overview of

how the model appeared in statistical mechanics.

This will be followed by an overview of some related

combinatorial problems. After this the focus will shift to

the "exact solution" of the model

with periodic boundary conditions followed by the

discussion of algebraic structures involved in the exact

solution (quantum groups with braiding). Then there will

be a brief discussion of other boundary conditions and of

limit shape phenomenon. If time permit, we will see how

the ASEP (Asymmetric Exclusion Process) fits into the

picture.