Brigham Young University

A proper simply connected one-ended metric space is call semi-stable if any two proper rays are properly homotopic.  A finitely presented group is called semi-stable if the universal cover of its presentation 2-complex is semi-stable.  
It is conjectured that every finitely presented group is semi-stable.  We will examine the known results for the cases where the group in question is relatively hyperbolic or CAT(0). 
 

For $k \geq 3$ we give new values of $\rho_k$ such that
$$ \| \alpha p^k + \beta \| < p^{-\rho_k} $$
has infinitely many solutions in primes whenever $\alpha$ is irrational and $\beta$ is real. The mean
value results of Bourgain, Demeter, and Guth are useful for $k \geq 6$; for all $k$, the results also
depend on bounding the number of solutions of a congruence of the form

$$ \left\| \frac{sy^k}{q} \right\| < \frac{1}{Z} \ \ (1 \leq y \leq Y < q) $$

where $q$ is a given large natural number.

I describe joint work with Alastair Irving in which we improve a result of
D.H.J. Polymath on the length of intervals in $[N,2N]$ that can be shown to
contain $m$ primes. Here $m$ should be thought of as large but fixed, while $N$
tends to infinity.
The Harman sieve is the key to the improvement. The preprint will be
available on the Math ArXiv before the date of the talk.