SR1

I will speak about the Zilber trichotomy for weakly minimal difference varieties, and the definable structure on them.

A difference field is a field with a distinguished automorphism $\sigma$. Solution sets of systems of polynomial difference equations like

$3 x \sigma(x) +4x +\sigma^2(x) +17 =0$ are the quantifier-free definable subsets of difference fields. These \emph{difference varieties} are similar to varieties in algebraic geometry, except uglier, both from an algebraic and from a model-theoretic point of view.

ACFA, the model-companion of the theory of difference fields, is a supersimple theory whose minimal (i.e. U-rank $1$) types satisfy the Zilber's Trichotomy Conjecture that any non-trivial definable structure on the set of realizations of a minimal type $p$ must come from a definable one-based group or from a definable field. Every minimal type $p$ in ACFA contains a (weakly) minimal quantifier-free formula $\phi_p$, and often the difference variety defined by $\phi_p$ determines which case of the Zilber Trichotomy $p$ belongs to.

Defined in terms of $\zeta(\frac{1}{2} +it)$ are the Riemann-Siegel functions, $\theta(t)$ and $Z(t)$. A zero of $\zeta(s)$ on the critical line corresponds to a sign change in $Z(t)$, since $Z$ is a real function. Points where $\theta(t) = n\pi$ are called Gram points, and the so called Gram's Law states between each Gram point there is a zero of $Z(t)$, and hence of $\zeta(\frac{1}{2} +it)$. This is known to be false in general and work will be presented to attempt to quantify how frequently this fails.

We give different criteria for transience of branching Markov chains. These conditions enable us to give a classification of branching random walks in random environment (BRWRE) on Cayley graphs in recurrence and transience. This classification is stated explicitly for BRWRE on $\Z^d.$ Furthermore, we emphasize the interplay between branching Markov chains, the spectral radius, and some generating functions.

Given an algebraic variety $X$ over the finite field ${\bf F}_{q}$, it is known that the zeta function of $X$,

$$ Z(X,T):=\mbox{exp}\left( \sum_{k=1}^{\infty} \frac{#X({\bf F}_{q^{k}})T^{k}}{k} \right) $$

is a rational function of $T$. It is an ongoing topic of research to efficiently compute $Z(X,T)$ given the defining equation of $X$.

I will summarize how we can use Berthelot's rigid cohomology (sparing you the actual construction) to compute $Z(X,T)$, first done for hyperelliptic curves by Kedlaya. I will go on to describe Lauder's deformation algorithm, and the promising fibration algorithm, outlining the present drawbacks.

I will review the construction of algebraic de Rham cohomology, relative de Rham cohomology, and the Gauss-Manin connection. I will then show how we can find a basis for the cohomology and the matrix for the connection with respect to this basis for certain families of curves sitting in weighted projective spaces.

In this talk I will introduce hyperbolic 3-manifolds, state some major conjectures about them, and discuss some group-theoretic properties of their fundamental groups.