# Calendar

May
26
Tue
Peter Loth (Sacred Heart University) @ Lecture held in Elysium
May 26 @ 6:00 am – 8:00 am

Title: Simply Given Compact Abelian Groups
by Peter Loth (Sacred Heart University) as part of Topological Groups

Lecture held in Elysium.

Abstract
A compact abelian group is called simply given if its Pontrjagin dual is simply presented. Warfield groups are defined to be direct summands of simply presented abelian groups. They were classified up to isomorphism in terms of cardinal invariants by Warfield in the local case, and by Stanton and Hunter–Richman in the global case. In this talk, we classify up to topological isomorphism the duals of Warfield groups, dualizing Stanton’s invariants. We exhibit an example of a simply given group with nonsplitting identity component.

Jun
2
Tue
Break (University of Hawaiʻi) @ Lecture held in Elysium
Jun 2 @ 6:00 am – 8:00 am

Title: Topological Groups Seminar One-Week Hiatus
by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.
Abstract: TBA

Jun
9
Tue
Ajit Iqbal Singh (Indian National Science Academy) @ Lecture held in Elysium
Jun 9 @ 6:00 am – 8:00 am

Title: Variants of Invariant Means of Amenability
by Ajit Iqbal Singh (Indian National Science Academy) as part of Topological Groups

Lecture held in Elysium.

Abstract
It all started, like many other amazing theories, in nineteen twenty-nine,
With John von Neumann, the greatest of the great.
The question of existence of a finitely additive measure on a group, a mean of a kind,
That is invariant, under any translation, neither gaining nor losing any weight.

Mahlon M. Day, in his zest and jest, giving double importance to semigroups, too,
Took up the study of conditions and properties, and named it amenability.
Erling Folner followed it up, more like a combinatorial maze to go through,
Whether or not translated set meets the original in a sizeable proportionality.

How could functional analysts sit quiet, who measure anything by their own norms,
Lo and behold, it kept coming back to the same concept over and over again.
Group algebras were just as good or bad, approximate conditions did no harms,
With the second duals of lofty Richard Arens, it became deeper, but still a fun-game.

Ever since, with the whole alphabet names, reputed experts or budding and slick,
Considering several set-ups and numerous variants of the invariance.
Actions on Manifolds or operators, dynamical systems nimble or quick,
We will have a look at some old and some new, closely or just from the fence.

Jun
16
Tue
Riddhi Shah (Jawaharlal Nehru University, New Delhi, India) @ Lecture held in Elysium
Jun 16 @ 6:00 am – 8:00 am

Title: Dynamics of Distal Actions on Locally Compact Groups
by Riddhi Shah (Jawaharlal Nehru University, New Delhi, India) as part of Topological Groups

Lecture held in Elysium.

Abstract
Distal maps were introduced by David Hilbert on compact spaces to study non-ergodic maps. A homeomorphism T on a topological space X is said to be distal if the closure of every double T-orbit of (x, y) does not intersect the diagonal in X x X unless x=y. Similarly, a semigroup S of homeomorphisms of X is said to act distally on X if the closure of every S-orbit of (x,y) does not intersect the diagonal unless x=y. We discuss some properties of distal actions of automorphisms on locally compact groups and on homogeneous spaces given by quotients modulo closed invariant subgroups which are either compact or normal. We relate distality to the behaviour of orbits. We also characterise the behaviour of convolution powers of probability measures on the group in terms of the distality of inner automorphisms.

Jun
23
Tue
Karl Hofmann (Technische Universität Darmstadt) @ Lecture held in Elysium
Jun 23 @ 6:00 am – 8:00 am

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

Jun
30
Tue
Break (University of Hawaiʻi) @ Lecture held in Elysium
Jun 30 @ 6:00 am – 8:00 am

Title: Topological Groups Seminar One-Week Hiatus
by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.
Abstract: TBA

Jul
7
Tue
Indira Chatterji (Laboratoire J.A. Dieudonné de l’Université de Nice) @ Lecture held in Elysium
Jul 7 @ 6:00 am – 8:00 am

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.

Jul
14
Tue
Ajay Kumar (University of Delhi) @ Lecture held in Elysium
Jul 14 @ 6:00 am – 8:00 am

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.