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,$

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$quad$ * equivalence between minimality and essentiality of dense subgroups of compact groups,$

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$quad$ * equivalence between minimality and compactness in LCA, $

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$quad$ * hereditary formulations of minimality facilitate optimal statements of theorems, $

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$quad$ * a locally compact hereditarily locally minimal infinite group $G$ is $

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$quad$ $quad$ (a) $congmathbb{Z}p$, some prime $p$, when $G$ is nilpotent,$

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$quad$ $quad$ (b) a Lie group when $G$ is connected,$

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$quad$ * classification of hereditarily minimal locally compact solvable groups,$

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$quad$ * existence of classes of hereditarily non-topologizable groups: $

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$quad$ $quad$ (a) bounded infinite finitely generated,$

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$quad$ $quad$ (b) unbounded finitely generated,$

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$quad$ $quad$ (c) countable not finitely generated, $

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$quad$ $quad$ (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.

Title: Group dualities: G-barrelled groups

by Elena Martín-Peinador (University of Madrid) as part of Topological 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 $(G, H)$.

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.

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.