A seminar series.
Nadja Hempel
(UCLA)
Model theory of tame Groups and Fields: the $k$-dependent Hierarchy
Model theory can be view as a tool for organizing mathematics: one generally distinguishes between the "tame" and the "wild" structures. This division arises through combinatorial notions of tameness which Shelah identified in the 1970s. A central goal is to develop methods whereby model theory makes contributions to the mathematics of tame structures.
We will start with a gentle introduction of first order structures and definable sets. Afterwards, we shed light on "tame" structures focusing on stable and the hierarchy of $k$-dependent structures ($k$ a natural number). As the random $k$-hypergraph is $k$-dependent but not $(k-1)$-dependent, this hierarchy is strict. However, we are interested in finding and studying tame algebraic structures. Thus one might ask if there are any algebraic objects (groups, rings, fields) which are strictly $k$-dependent for every $k$? We will present the known results on $k$-dependent groups and (valued) fields. These were (more or less) inspired by the above question.
Santosh Kandel
Perturbative path integrals, Feynman diagrams, Cutting and Gluing
Path integrals are key tools in quantum physics. In mathematics, path integrals are to used to construct invariants of manifolds. A path integral is an integral defined over a space of "field configurations" which is, in general, an infinite dimensional manifold. For this reason, it is very challenging to rigorously define path integrals. In fact, only few examples are known to exist. In practice, one usually considers perturbative path integrals which is a notion pioneered by Feynman and based on the method of steepest descent and the method of stationary phase. A pertubative path integral gives rise to a power series which can be organized by Feymann diagrams.
In this talk, we will introduce perturbative path integrals and discuss an example associated to a "scalar field theory" on two dimensional closed oriented Riemannian manifolds. Time permitting, we will also examine interactions of path integrals when we cut a closed two dimensional manifold along a closed one dimensional submanifold.
Jack Fogliasso (CSUS)
Algebraic curves with complex multiplication
In this talk, I will describe a research project that I worked on as an undergraduate
student at CSU, Chico. First we provide some background information on algebraic
curves (e.g., elliptic curves). We de ne the Jacobian of an algebraic curve, and discuss
how to determine whether or not the Jacobian has a property called complex multiplication. Such Jacobians are important and rare objects in arithmetic geometry, and
generally it is difficult to determine whether or not a Jacobian has complex multiplication. To that end, we will briefly discuss a novel way to determine whether the Jacobian
of a curve has complex multiplication, based upon joint work with Prof. Chris Marks
(CSU, Chico), Prof. Luca Candelori (Wayne State University) and undergraduate Skip
Moses (CSU, Chico). This talk is aimed at an undergraduate and graduate student
audience.