Seid Kassaw (University of Cape Town) @ Lecture held in Elysium
Nov 24 @ 6:00 am – 8:00 am

Title: The probability of commuting subgroups in arbitrary lattices of subgroups
by Seid Kassaw (University of Cape Town) as part of Topological Groups

Interactive livestream:
Lecture held in Elysium.

The subgroup commutativity degree $sd(G)$ of a finite group $G$ was introduced
almost ten years ago and deals with the number of commuting subgroups in the
subgroup lattice $L(G)$ of $G$. The extremal case $sd(G) = 1$ detects a class of groups
classified by Iwasawa in 1941 (in fact, $sd(G)$ represents a probabilistic measure which
allows us to understand how far $G$ is from the groups of Iwasawa). This means
$sd(G) = 1$ if and only if $G$ is the direct product of its Sylow $p$-subgroups and these
are all modular; or equivalently $G$ is a nilpotent modular group. Therefore, $sd(G)$ is
strongly related to structural properties of $L(G)$ and $G$.

In this talk, we introduce a new notion of probability $gsd(G)$ in which two arbitrary sublattices $S(G)$ and $T(G)$ of $L(G)$ are involved simultaneously. In case
$S(G) = T(G) = L(G)$, we find exactly $sd(G)$. Upper and lower bounds for $gsd(G)$
are shown and we study the behaviour of $gsd(G)$ with respect to subgroups and
quotients, showing new numerical restrictions. We present the commutativity
and subgroup commutativity degree for infinite groups and put some open problems
for further generalization.

Karen Lange (Wellesley College)
Nov 24 @ 11:00 am – 12:00 pm

Title: Complexity of root-taking in power series fields & related problems
by Karen Lange (Wellesley College) as part of Computability theory and applications

In earlier work with Knight and Solomon, we bounded the computational complexity of the root-taking process over Puiseux and Hahn series, two kinds of generalized power series. But it is open whether the bounds given are optimal. By looking at the most basic steps in the root-taking process for Hahn series, we together with Hall and Knight became interested in the complexity of problems associated with well-ordered subsets of a fixed ordered abelian group. Here we provide an overview of the results so far in both these settings.

Nov 26 – Nov 27 all-day
Linda Brown Westrick (Penn State)
Dec 8 @ 11:00 am – 12:00 pm

by Linda Brown Westrick (Penn State) as part of Computability theory and applications

Abstract: TBA

Last day of instruction
Dec 10 all-day
First day of instruction
Jan 11 all-day