An autohomeomorphism of the Čech--Stone remainder $\beta N\setminus N$ is called trivial if it has a continuous extension to a map from $\beta N$ into itself. Such map is determined by an almost permutation, which is a bijection between cofinite subsets of $N$. By results of W. Rudin and S. Shelah, the question whether nontrivial autohomeomorphisms of $\beta N\setminus N$ exist is independent from ZFC. We will be considering the so-called rotary autohomeomorphisms. An autohomeomorphism is called rotary if it corresponds to a permutation of $N$ all of whose cycles are finite. If all autohomeomorphisms are trivial, then the problem of their conjugacy is also trivial (in the usual sense of the word). However the Continuum Hypothesis makes the conjugacy relation nontrivial. While our results are somewhat incomplete, they suffice to decide whether for example the rotary autohomeomorphisms whose cycles have lengths $2^{2n}$, for $n\in N$, and $2^{2n+1}$, for $n\in N$, are conjugate. This is a joint work with Will Brian.

A field K is "large" if every smooth curve over K with at least one K-rational point has infinitely many K-rational points. In this talk, I'll discuss what we know about the relations between the arithmetic condition of largeness and the model-theoretic conditions of stability and NIP. Stable large fields are separably closed. For NIP large fields, we know something much weaker: there is a canonical field topology satisfying a weak form of the implicit function theorem for polynomials. Conjecturally, any stable or NIP infinite field should be large. I will discuss these results, as well as the following conjecture: if K is a field and p is a prime and every separable extension of K has degree prime to p, then K is large. This conjecture would imply that NIP fields of positive characteristic are large, and would classify stable fields of positive characteristic. I will present some (very weak) evidence for this conjecture.