Title: The group algebra of a compact group and Tannaka duality for compact groups
by Karl Hofmann (Technische Universität Darmstadt) as part of Topological Groups
Lecture held in Elysium.
Abstract
In the 4th edition of the text- and handbook “The Structure of Compact Groups”,
de Gruyter, Berlin-Boston, having appeared June 8, 2020, Sidney A. Morris and
I decided to include, among material not contained in earlier editions, the Tannaka-Hochschild Duality Theorem which says that $the$ $category$ $of$ $compact$ $groups$ $is,dual$
$to$ $the$ $category,of,real,reductive$ $Hopf$ $algebras$. In the lecture I hope to explain
why this theorem was not featured in the preceding 3 editions and why we decided
to present it now. Our somewhat novel access led us into a new theory of real
and complex group algebras for compact groups which I shall discuss. Some Hopf
algebra theory appears inevitable. Recent source: K.H.Hofmann and L.Kramer,
$On$ $Weakly,Complete,Group,Algebras$ $of$ $Compact$ $Groups$, J. of Lie Theory $bold{30}$ (2020), 407-424.
Karl H. Hofmann
Title: Topological Groups Seminar One-Week Hiatus
by Break (University of Hawaiʻi) as part of Topological Groups
Lecture held in Elysium.
Abstract: TBA
Title: Groups Admitting Proper Actions by Affine Isometries on Lp Spaces
by Indira Chatterji (Laboratoire J.A. Dieudonné de l’Université de Nice) as part of Topological Groups
Lecture held in Elysium.
Abstract
Introduction, known results, and open questions regarding groups admitting a proper action by affine isometries on an $L_p$ space.
Title: Uncertainty Principles on Locally Compact Groups
by Ajay Kumar (University of Delhi) as part of Topological Groups
Lecture held in Elysium.
Abstract
Some of the uncertainty principles on $ mathbb{R}^n $ are as follows:
Qualitative Uncertainty Principle: Let $f$ be a non-zero function in $L^1(mathbb{R}^n)$. Then the Lebesgue measures of the sets ${x: f(x)
eq 0 }$ and $ {xi : widehat{f}(xi)
eq 0}$ cannot both be finite.
Hardy’s Theorem: Let $ a,b,c $ be three real positive numbers and let $f: mathbb{R}^n to mathbb{C}$ be a measurable function such that
(i) $|f(x)| leq cexp{(-api |x|^2)}$, for all $ x in mathbb{R}^n$
(ii) $|widehat{f}(xi)| leq cexp{(-bpi |xi|^2)}$, for all $xi in mathbb{R}^n $.
Then following holds:
If $ab>1$, then $f=0$ a.e.
If $ab =1$, then $f(x)= alpha exp{(-api |x|^2)}$ for some constant $alpha$.
If $ab< 1$, then there are infinitely many linear independent functions satisfying above conditions.
Heisenberg Inequality: If $f in L^2(mathbb{R}^n)$ and $a,b in mathbb{R}^n$, then
$$
left( int_{mathbb{R}^n}|x-a|^2|f(x)|^2 dx right) left( int_{mathbb{R}^n}|xi-b|^2|widehat{f}(xi)|^2 dxi right) geq frac{n^2|f|^4}{16pi^2}.
$$
Beurling's Theorem: Let $f in L^1(mathbb{R}^n) $ and for some $ k(1leq kleq n) $ satisfies
$$
int_{mathbb{R}^{2n}} |f(x_1, x_2, dots , x_n)||widehat{f}(xi_1, xi_2, dots , xi_n)|e^{2pi |x_kxi_k|} dx_1dots dx_n dxi_1dots dxi_n< infty.
$$
Then $f = 0$ a.e.
We investigate these principles on locally compact groups, in particular Type I
groups and nilpotent Lie groups for Fourier transform and Gabor transform.
Title: Probability Measures and Structure of Locally Compact Groups
by C.R.E. Raja (Indian Statistical Instititute) as part of Topological Groups
Lecture held in Elysium.
Abstract
We will have an overview of how existence of certain types of
probability measures forces locally compact groups to have particular
structures and vice versa. Examples are Choquet-Deny measures, recurrent
measures etc., and groups of the kind amenable, polynomial growth, etc.
Title: On a Class of Profinite Groups Related to a Theorem of Prodanov
by Dikran Dikranjan (University of Udine) as part of Topological Groups
Lecture held in Elysium.
Abstract
A short history of minimal groups is given, featuring illustrative examples and leading to current research:$
ewline$
$quad$ * non-compact minimal groups,$
ewline$
$quad$ * equivalence between minimality and essentiality of dense subgroups of compact groups,$
ewline$
$quad$ * equivalence between minimality and compactness in LCA, $
ewline$
$quad$ * hereditary formulations of minimality facilitate optimal statements of theorems, $
ewline$
$quad$ * a locally compact hereditarily locally minimal infinite group $G$ is $
ewline$
$quad$ $quad$ (a) $congmathbb{Z}p$, some prime $p$, when $G$ is nilpotent,$
ewline$
$quad$ $quad$ (b) a Lie group when $G$ is connected,$
ewline$
$quad$ * classification of hereditarily minimal locally compact solvable groups,$
ewline$
$quad$ * existence of classes of hereditarily non-topologizable groups: $
ewline$
$quad$ $quad$ (a) bounded infinite finitely generated,$
ewline$
$quad$ $quad$ (b) unbounded finitely generated,$
ewline$
$quad$ $quad$ (c) countable not finitely generated, $
ewline$
$quad$ $quad$ (d) uncountable.
Title: Totally disconnected locally compact groups and the scale
by George Willis (University of Newcastle) as part of Topological Groups
Lecture held in Elysium.
Abstract
The scale is a positive, integer-valued function defined on any totally disconnected, locally compact (t.d.l.c.) group that reflects the structure of the group. Following a brief overview of the main directions of current research on t.d.l.c. groups, the talk will introduce the scale and describe aspects of group structure that it reveals. In particular, the notions of tidy subgroup, contraction subgroup and flat subgroup of a t.d.l.c. will be explained and illustrated with examples.
Title: Locally Compact Contraction Groups
by Helge Glöckner (Universität Paderborn) as part of Topological Groups
Lecture held in Elysium.
Abstract
Consider a locally compact group $G$, together with an automorphism $alpha$ which is $contractive$ in the sense that $alpha^nrightarrow{rm id}_G$ pointwise as $ntoinfty$. Siebert showed that $G$ is the direct product of its connected component $G_e$ and an $alpha$-stable, totally disconnected closed subgroup;
moreover, $G_e$ is a simply connected, nilpotent real Lie group.
I’ll report on research concerning the totally disconnected part, obtained jointly with G. A. Willis.
For each totally disconnected contraction group $(G,alpha)$, the set ${rm tor} G$ of torsion elements is a closed subgroup of $G$. Moreover, $G$ is a direct product
$$G=G_{p_1}times cdotstimes G_{p_n}times {rm tor} G$$ of $alpha$-stable $p$-adic Lie groups $G_p$ for certain primes $p_1,ldots, p_n$ and the torsion subgroup. The structure of $p$-adic contraction groups is known from the work of J. S. P. Wang; notably, they are nilpotent. As shown with Willis, ${rm tor} G$ admits a composition series and there are countably many possible composition factors, parametrized by the finite simple groups. More recent research showed that there are uncountably many non-isomorphic torsion contraction groups, but only countably many abelian ones. If a torsion contraction group $G$ has a compact open subgroup which is a pro-$p$-group, then $G$ is nilpotent. Likewise if $G$ is locally pro-nilpotent.