Title: Generalizing Erdős B-free systems: Erdős Sieves
Abstract: Given a set B = {b_1,b_2,...} of pairwise coprime numbers, we say it forms a B free system, if sum_i 1/b_i is finite. If for each i we instead consider a set of congruence classes mod b_i, we get Erdős sieves. For a sieve R we investigate the dynamical systems associated to the R-free numbers, those integers not contained in any…
Title: Local-global principle for p-extensions in characteristic p
Abstract: One of the main themes of number theory is the description of extensions of a fixed (local or global) field. For abelian extensions, this is accomplished by class field theory, which has the distinctive property that local and global extensions are tightly connected. When restricted to abelian p-extensions in…
Title: Frobenius and Octopus
Abstract: We count semisimple nxn-matrices over a finite field F_q that commute with their (p-th power) Frobenius conjugate (for fixed n and for q going to infinity). This involves some linear algebra, algebraic geometry, and graph theory.
