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.

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.

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,$

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$quad$ * 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.