Transition from Classical Mechanics to Quantum Mechanics involves replacing a certain commutative ring by its "non commutative deformation". We will discuss some algebraic aspects of this transition, and recent developments in algebra and algebraic geometry which are motivated by applications to quantum field theory

Advisor:  Anton Gorodetski

As is well known, a class of one-dimensional lattice models, such as Ising models, Jacobi and CMV operators and others, are susceptible to renormalization analysis that can be carried out via the transfer matrix formalism. As a result, models on the one-dimensional lattice of a certain quasi-periodic type (namely, those generated by primitive substitutions) can be studied via dynamics of so-called trace maps, which are polynomial maps acting on the real (or complex) Euclidean space of appropriate dimension. A prominent example is the widely studied Fibonacci model. Much work has been done in this direction. At some point it became apparent that a model-independent framework, based on the dynamics of trace maps, can be built, that would cover essentially all models the relevant information of which is encapsulated in the traces of the associated transfer matrices (and, as experience has shown, this information is very difficult if not impossible to obtain via techniques other than the trace map). The purpose of this talk is to give a broad overview of past and very recent results on the dynamics of the Fibonacci trace map in a model-independent fashion, motivated by a class of models from physics, and with a view towards applications to those models. We shall cover hyperbolicity and partial hyperbolicity of the trace map and implications in spectral theory of Jacobi operators; some applications to Ising models; recent advances in understanding invariant measures on the invariant hyperbolic sets and implications for the density of states measures for the Jacobi operators; the Newhouse phenomenon and mixed behavior with large (in the sense of Hausdorff dimension) chaotic sea, and some connections with kicked two-level systems. Time permitting, we shall also state some open problems.

Talk Abstract:
Over finite fields, if the image of a polynomial (a.k.a. the value set) is not the entire field, its cardinality is often quite a bit smaller. Earlier results bound this cardinality using the degree of the polynomial. We improve upon these bounds for multivariable polynomial vectors over a finite field using the polytope of the polynomial.
Advisor:  Daqing Wan