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.
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.
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.
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.
Title: Luzin’s (N) and randomness reflection
by Linda Brown Westrick (Penn State) as part of Computability theory and applications
We show that a computable real-valued function f has Luzin’s property (N) if and only if it reflects Pi^1_1-randomness, if and only if it reflects Delta^1_1-randomness relative to Kleene’s O, and if and only if it reflects Kurtz randomness relative to Kleene’s O. Here a function f is said to reflect a randomness notion R if whenever f(x) is R-random, then x is R-random as well. If additionally f is known to have bounded variation, then we show f has Luzin’s (N) if and only if it reflects weak-2-randomness, and if and only if it reflects Kurtz randomness relative to 0′. This links classical real analysis with algorithmic randomness. Joint with Arno Pauly and Liang Yu.
Title: Automorphism argument and reverse mathematics
by Keita Yokoyama (Japan Advanced Institute of Science and Technology) as part of Computability theory and applications
In the study of models of Peano (or first-order) arithmetic, there are
many results on recursively saturated models and their automorphisms.
Here, we apply such an argument to models of second-order arithmetic
and see that any countable recursively saturated model (M,S) of WKL_0*
is isomorphic to its countable coded omega-submodel if
Sigma_1-induction fails in (M,S). From this result, we see some
interesting but weird properties of WKL_0* with the absence of
Sigma_1-induction such as the collapse of analytic hierarchy. This
argument can also be applied to the reverse mathematical study of
Ramsey’s theorem for pairs (RT22), and we see some new relations
between the computability-theoretic characterizations of RT22 and the
famous open question on the first-order part of RT22+RCA_0.
This work is a part of a larger project joint with Marta Fiori
Carones, Leszek Kolodziejczyk, Katarzyna Kowalik and Tin Lok Wong.