# Calendar

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.

Aug
11
Tue
Helge Glöckner (Universität Paderborn) @ Lecture held in Elysium
Aug 11 @ 6:00 am – 8:00 am

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.

Aug
18
Tue
Aug 18 @ 6:00 am – 8:00 am

Title: Group dualities: G-barrelled groups

Lecture held in Elysium.

Abstract
A natural notion in the framework of abelian groups are the group dualities. The most efficient definition of a group duality is simply a pair $(G, H)$, where $G$ denotes an abstract abelian group and $H$ a subgroup of characters of $G$, that is $H leq {rm Hom}(G, mathbb T)$. Two group topologies for $G$ and $H$ appear from scratch in a group duality $(G, H)$: the weak topologies $sigma(G, H)$ and $sigma (H, G)$ respectively. Are there more group topologies either in $G$ or $H$ that can be strictly related with the duality $(G, H)$? In this sense we shall define the term “compatible topology” and loosely speaking we consider the compatible topologies as members of the duality.

The locally quasi-convex topologies defined by Vilenkin in the 50′s form a significant class for the construction of a duality theory for groups. The fact that a locally convex topological vector space is in particular a locally quasi-convex group serves as a nexus to emulate well-known results of Functional Analysis for the class of topological groups.

In this talk we shall
deal with questions of the sort:
Under which conditions is there a locally compact topology in a fixed duality?
The same question for a metrizable, or a $k$-group topology.
We shall also introduce the $g$-barrelled groups, a class for which the Mackey-Arens Theorem admits an optimal counterpart. We study also the existence of $g$-barrelled topologies in a group duality $(G, H)$.

Aug
25
Tue
Wayne Lewis (University of Hawaiʻi) @ Lecture held in Elysium
Aug 25 @ 6:00 am – 8:00 am

Title: Classification of Periodic LCA Groups of Finite Non-Archimedean Dimension
by Wayne Lewis (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract
A periodic LCA group such that the $p$-components all have $p$-rank bounded above by a common positive integer are classified via a complete set of topological isomorphism invariants realized by an equivalence relation on pairs of extended supernatural vectors.

Remaining time will be devoted to a facilitated discussion on how things are going this fall/winter academic semester in your part of the world as you see it.

Sep
1
Tue
Dona Strauss (University of Leeds) @ Lecture held in Elysium
Sep 1 @ 6:00 am – 8:00 am

Title: The Semigroup $beta S$
by Dona Strauss (University of Leeds) as part of Topological Groups

Lecture held in Elysium.

Abstract
If $S$ is a discrete semigroup, the semigroup operation on $S$ can be extended to a semigroup operation on its Stone–Čech compactification $beta S$. The properties of the semigroup $beta S$ have been a powerful tool in topological dynamics and combinatorics.

I shall give an introductory description of the semigroup $beta S$, and show how its properties can be used to prove some of the classical theorems of Ramsey Theory.