Title: Towards a unifying approach to algebraic and coarse entropy

by Nicolò Zava (University of Udine) as part of Topological Groups

Lecture held in Elysium.

Abstract

In each situation, entropy associates to a self-morphism a value that estimates the chaos created by the map application. In particular, the algebraic entropy $h_{alg}$ can be computed for (continuous) endomorphisms of (topological) groups, while the coarse entropy $h_c$ is associated to bornologous self-maps of locally finite coarse spaces. Those two entropy notions can be compared because of the following observation. If $f$ is a (continuous) homomorphism of a (topological) group $G$, then $f$ becomes automatically bornologous provided that $G$ is equipped with the compact-group coarse structure. For an endomorphism $f$ of a discrete group, $h_{alg}(f)=h_c(f)$ if $f$ is surjective, while, in general, $h_{alg}(f)

eq h_c(f)$. That difference occurs because in many cases, if $f$ is not surjective, then $h_c(f)=0$.

In the first part of the talk, after briefly recalling the large-scale geometry of topological groups, we define the coarse entropy and discuss its relationship with the algebraic entropy. The second part is dedicated to the introduction of the algebraic entropy of endomorphisms of $G$-sets (i.e., sets endowed with group actions). We show that it extends the usual algebraic entropy of group endomorphisms and we provide evidence that it can represent a useful modification and generalisation of the coarse entropy that overcome the non-surjectivity issue.

Title: Topological Groups Seminar Two-Week Hiatus

by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract: TBA

Title: Topological Groups Seminar Two-Week Hiatus

by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract: TBA

Title: Abelian Varieties as Algebraic Protori?

by Wayne Lewis (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract

An outcome of the structure theory of protori (compact connected abelian groups) is their representability as quotients of $mathbb{A}^n$ for the ring of adeles $mathbb{A}$. $mathbb{A}$ does not contain zeros of rational polynomials, but rather representations of zeros. Investigating the relations between algebraicity of complex tori and algebraicity of protori leads one to the problem of computing the Pontryagin dual of $mathbb{A}/mathbb{Z}$. Applying an approach by Lenstra in the setting of profinite integers to the more general $mathbb{A}$ leads to a definition of the closed maximal $Lenstra$ $ideal$ $E$ of $mathbb{A}$, whence the locally compact field of $adelic$ $numbers$ $mathbb{F}=mathbb{A}/E$, providing a long-sought connection to $mathbb{C}$ enabling one to define a functor from the category of complex tori to the category of protori – is it possible to do so in a way that preserves algebraicity? While $mathbb{F}$ marks tentative progress, much work remains…

Title: Accounting with $mathbb{QP}^infty$

by Wayne Lewis (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract

Rational projective space provides a useful accounting tool in engineering decompositions of $mathbb{Q}[x]$ for desired effect. The device is useful for defining a correspondence between summands of such a decomposition and elements of a partition of $mathbb{A}$. This mechanism is applied to a decomposition of $mathbb{Q}[x]$ relative to which the correspondence gives the $Lenstra$ $ideal$ $E$, a closed maximal ideal yielding the $adelic$ $numbers$ $mathbb{F}=frac{mathbb{A}}{E}$.

Title: Journey in Hawaii’s Challenges in the Fight Against COVID-19

by Monique Chyba (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract

The COVID-19 pandemic is far from the first infectious disease that Hawaiʻi had to deal with. During the 1918-1920 Influenza Pandemic, the Hawaiian islands were not spared as the disease ravaged through the whole world. Hawaiʻi and similar island populations can follow a different course of pandemic spread than large cities/states/nations and are often neglected in major studies. It may be too early to compare the 1918-1920 Influenza Pandemic and COVID-19 Pandemic, we do however note some similarities and differences between the two pandemics.

Hawaiʻi and other US Islands have recently been noted by the media as COVID-19 hotspots after a relatively calm period of low case rates. U.S. Surgeon General Jerome Adams came in person on August 25 to Oahu to address the alarming situation. We will discuss the peculiarity of the situation in Hawaiʻi and provide detailed modeling of current virus spread patterns aligned with dates of lockdown and similar measures. We will present a detailed epidemiological model of the spread of COVID-19 in Hawaiʻi and explore effects of different intervention strategies in both a prospective and retrospective fashion. Our simulations demonstrate that to control the spread of COVID-19 both actions by the State in terms of testing, contact tracing and quarantine facilities as well as individual actions by the population in terms of behavioral compliance to wearing a mask and gathering in groups are vital. They also explain the turn for the worst Oahu took after a very successful stay-at-home order back in March.

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.

Abstract

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: 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.

Abstract

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.