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

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.

Farzana Nasrin (University of Hawaiʻi) @ Lecture held in Elysium
Dec 1 @ 6:00 am – 8:00 am

Title: Bayesian Statistics, Topology and Machine Learning for Complex Data Analysis
by Farzana Nasrin (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Analyzing and classifying large and complex datasets are generally challenging. Topological data analysis, that builds on techniques from topology, is a natural fit for this. Persistence diagram is a powerful tool that originated in topological data analysis that allows retrieval of important topological and geometrical features latent in a dataset. Data analysis and classification involving persistence diagrams have been applied in numerous applications. In this talk, I will provide a brief introduction of topological data analysis, focusing primarily on persistence diagrams, and a Bayesian framework for inference with persistence diagrams. The goal is to provide a supervised machine learning algorithm in the space of persistence diagrams. This framework is applicable to a wide variety of datasets. I will present applications in materials science, biology, and neuroscience.

Mikhail Tkachenko (Metropolitan Autonomous University) @ Lecture held in Elysium
Dec 8 @ 6:00 am – 8:00 am

Title: Pseudocompact Paratopological and Quasitopological Groups
by Mikhail Tkachenko (Metropolitan Autonomous University) as part of Topological Groups

Lecture held in Elysium.

Pseudocompactness is an interesting topological property which acquires very specific
features when applied to different algebrotopological objects. A celebrated theorem
of Comfort and Ross published in 1966 states that the Cartesian product of an arbitrary
family of pseudocompact topological groups is pseudocompact. We present a survey
of results related to the validity or failure of the Comfort-Ross’ theorem in the realm of
semitopological and paratopological groups and give some examples showing that
pseudocompactness fails to be stable when taking products of quasitopological groups.