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.
Title: Stably free cancellation for orders
Abstract: A ring has the stably free cancellation property if every stably free module is free. We will discuss this property for orders in semisimple algebras and show how the question of whether an order has this property can be answered algorithmically. As an application we obtain new cancellation results for integral groups rings. This is joint work…
Title: Density of R-free sets
Abstract: We will discuss a number of new results about the density of certain translates in R-free numbers. We present a solution to Problem 486 in the Erdős' Problems website, and a partial resolution of Problem 25. We will also discuss a new result from van Doorn and Tao.
Title: Counting $D_4$ extensions by multiple invariants
Abstract: Malle’s conjecture is a central question in arithmetic statistics, predicting an asymptotic formula for the number of number fields with a prescribed Galois group and bounded discriminant. Recently, such counting problems have received considerable attention. In 2022 Gundlach introduced the notion of ordering number fields by…
