# Calendar

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.

Jul
21
Tue
C.R.E. Raja (Indian Statistical Instititute) @ Lecture held in Elysium
Jul 21 @ 6:00 am – 8:00 am

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.

Jul
28
Tue
Dikran Dikranjan (University of Udine) @ Lecture held in Elysium
Jul 28 @ 6:00 am – 8:00 am

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.

Aug
4
Tue
George Willis (University of Newcastle) @ Lecture held in Elysium
Aug 4 @ 6:00 am – 8:00 am

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.