Past

  We present a link between polymer pinning by a columnar defect in a random medium and a particular model of interacting particles on the line, related to polynuclear growth. While the question of whether an arbitrarily small intensity for the defect always results in pinning is still open, in a 'randomized' version of the model, which is closely related to the zero-temperature Glauber dynamics of the Ising model, we are able to obtain explicit results and a complete understanding of the process. This is joint work with Vladas Sidoravicius and Maria Eulalia Vares.  

  Model theory typically looks at classical mathematical structures in novel ways. The guiding principle is to understand what relations are definable, and there are usually related questions of effectivity. In the case of Lie theory, there are two current lines of research, both of which I will describe, but with more emphasis on the first. The most advanced work concerns exponentials and logarithms, in both real and complex situations. To understand the definable relations, and to show various natural problems are decidable, one uses a mixture of analytic geometry with number-theoretic conjectures related to Schanuel's Conjecture. More recent work, not yet closely connected to the preceding, concerns the limit behaviour (model-theoretically), of finite -dimensional modules over semisimple Lie algebras, and here again, for decidability, one seems obliged to consider number-theoretic decision problems, around Siegel's Theorem.

 

We consider as a starting point a problem raised by Kunen and Tall as to whether

the real continuum can have non-homeomorphic versions in different submodels of

the universe of all sets. Its resolution depends on modest large cardinals.

In general Junqueira and Tall have made a study of such "substructure spaces"

where the topology of a subspace can be different from the usual relative

topology.

  We will shortly review the known bounds on the angular momentum J in terms of mass m assuming a negative cosmological constant, and describe in more detail Brill's proof of the axisymmetric positive energy theorem, and Dain's upper bound on angular momentum J for vacuum initial data sets with an axial Killing vector and with two asymptotically flat regions.

  We start from the stochastic Ising model(or Glauber Dynamics) and have a short review of some important topics in Particle Systems such as ergodicity, convergence rates and so on. Then an abstract nonlinear model will be introduced by an evolution differential equation. We will build the existence and uniqueness theorem, and give some nice properties such as convergence exponentially and monotonicity for the abstract systems. To apply our abstract theory, we will study a family of nonlinear interacting particle systems generalized from Glauber Type Dynamics(we call them nonlinear Glauber Type Dynamics) and prove that such generalization can be done in infinitely many ways. For nonlinear Glauber Type Dynamics, we have two interesting inequalities related to Gibbs measures. Finally, we will concentrate on one specific nonlinear dynamics, and provide the relation between nonlinear system and the linear one, and that between Gibbs measures and tangent functionals to a nonlinear transfer operator.