BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Wayne Lewis (University of Hawaiʻi)
DTSTART:20200414T160000Z
DTEND:20200414T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/1
DESCRIPTION:Title: Classification of Finite-Dimensional Periodic LCA Groups\nby
Wayne Lewis (University of Hawaiʻi) as part of Topological Groups\n\nLec
ture held in Elysium.\n\nAbstract\nGeneralized resolutions of protori have
non-Archimedean component a periodic LCA group with finite non-Archimedea
n dimension. The previous session introduced the notion of non-Archimedean
dimension of LCA groups. Applying published results by Dikranjan\, Herfor
t\, Hofmann\, Lewis\, Loth\, Mader\, Morris\, Prodanov\, Ross\, and Stoyan
ov\, we introduce new minimalist notation and accompanying definitions to
clarify the structure of these groups and their Pontryagin duals\, enablin
g a parametrization of the spectrum of resolutions of finite-dimensional p
rotori (the Grothendieck group is a moduli space).\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adolf Mader (University of Hawaiʻi)
DTSTART:20200421T160000Z
DTEND:20200421T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/2
DESCRIPTION:Title: Pontryagin Duals of Type Subgroups of Finite Rank Torsion-Free A
belian Groups\nby Adolf Mader (University of Hawaiʻi) as part of Topo
logical Groups\n\nLecture held in Elysium.\n\nAbstract\nPontryagin duals o
f type subgroups of finite rank torsion-free abelian groups are presented.
The interplay between the intrinsic study of compact abelian groups\, res
pectively torsion-free abelian groups\, is discussed (how can researchers
better leverage the published results in each setting so there is a dual i
mpact?). A result definitively qualifying\, in the torsion-free category\,
the uniqueness of decompositions involving maximal rank completely decomp
osable summands is given\; the formulation of the result in the setting of
protori is shown to optimally generalize a well-known result regarding th
e splitting of maximal tori from finite-dimensional protori.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dikran Dikranjan (University of Udine)
DTSTART:20200428T160000Z
DTEND:20200428T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/3
DESCRIPTION:Title: The Connection between the von Neumann Kernel and the Zariski To
pology\nby Dikran Dikranjan (University of Udine) as part of Topologic
al Groups\n\nLecture held in Elysium.\n\nAbstract\nEvery group G carries a
natural topology Z_G defined by taking as a pre-base of the family of al
l closed sets the solution sets of all one-variable equations in the group
of the form (a_1)x^{ε_1}(a2)x^{ε_2}...(a_n)x^{ε_n} = 1\, where a_i ∈
G\, ε_i = ±1 for i = 1\,2\,...\,n\, n ∈ N. The topology was explicitl
y introduced by Roger Bryant in 1978\, who named it the verbal topology\,
but the name Zariski topology was universally applied subsequently. As a m
atter of fact\, this topology implicitly appeared in a series of papers by
Markov in the 1940’s in connection to his celebrated problem concerning
unconditionally closed sets: sets which are closed in any Hausdorff group
topology on G. These are the closed sets in the topology M_G obtained as
the intersection of all Hausdorff group topologies on G\, which we call t
he Markov topology\, although this topology did not explicitly appear in M
arkov’s papers. Both Z_G and M_G are T1 topologies and M_G ≥ Z_G\, b
ut they need not be group topologies. One can use these topologies to form
ulate Markov’s problem: does the equality M_G = Z_G hold? Markov proved
that M_G = Z_G if the group is countable and mentioned that the equality
holds also for arbitrary abelian groups (so one can speak about the Markov
-Zariski topology of an abelian group). The aim of the presentation is to
expose this history\, to describe some problems of Markov related to these
topologies\, and to apply the theory to give a solution to the Comfort-Pr
otasov-Remus problem on minimally almost periodic topologies of abelian gr
oups. This problem is associated to a more general problem of Gabriyelyan
concerning the realisation of the von Neumann kernel n(G) of a topological
group\; that is\, the intersection of the kernels of the continuous homom
orphisms G → T into the circle group. More precisely\, given a pair cons
isting of an abelian group G and a subgroup H\, one asks whether there is
a Hausdorff group topology τ on G such that n(G\,τ) = H. Since (G\,τ) i
s minimally almost periodic precisely when n(G) = G\, the solution of this
more general problem also gives a solution to the Comfort-Protasov-Remus
problem.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Giordano Bruno (University of Udine)
DTSTART:20200505T160000Z
DTEND:20200505T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/4
DESCRIPTION:Title: Topological Entropy and Algebraic Entropy on Locally Compact Abe
lian Groups\nby Anna Giordano Bruno (University of Udine) as part of T
opological Groups\n\nLecture held in Elysium.\n\nAbstract\nSince its origi
n\, the algebraic entropy $h_{alg}$ was introduced in connection with the
topological entropy $h_{top}$ by means of Pontryagin duality. For a contin
uous endomorphism $\\phi\\colon G\\to G$ of a locally compact abelian grou
p $G$\, denoting by $\\widehat G$ the Pontryagin dual of $G$ and by $\\wid
ehat \\phi\\colon G\\to G$ the dual endomorphism of $\\phi$\, we prove tha
t $$h_{top}(\\phi)=h_{alg}(\\widehat\\phi)$$ under the assumption that $G$
is compact or that $G$ is totally disconnected. It is known that this equ
ality holds also when $\\phi$ is a topological automorphism.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wayne Lewis (University of Hawaiʻi)
DTSTART:20200407T160000Z
DTEND:20200407T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/5
DESCRIPTION:Title: Adelic Theory of Protori\nby Wayne Lewis (University of Hawa
iʻi) as part of Topological Groups\n\nLecture held in Elysium.\nAbstract:
TBA\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolò Zava (University of Udine)
DTSTART:20200512T160000Z
DTEND:20200512T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/6
DESCRIPTION:Title: The Large-Scale Geometry of Locally Compact Abelian Groups\n
by Nicolò Zava (University of Udine) as part of Topological Groups\n\nLec
ture held in Elysium.\n\nAbstract\nLarge-scale geometry\, also known as co
arse geometry\, is the branch of mathematics that studies the global\, lar
ge-scale properties of spaces. This theory is distinguished by its applica
tions which include the Novikov and coarse Baum-Connes conjectures. Since
the breakthrough work of Gromov\, large-scale geometry has played a promin
ent role in geometric group theory\, in particular\, in the study of finit
ely generated groups and their word metrics. This large-scale approach was
successfully extended to all countable groups by Dranishnikov and Smith.
A further generalisation introduced by Cornulier and de la Harpe dealt wit
h locally compact σ-compact groups endowed with particular pseudo-metrics
. \nTo study the large-scale geometry of more general groups and topologic
al groups\, coarse structures are required. These structures\, introduced
by Roe\, encode global properties of spaces. We also mention the equivalen
t approach provided by Protasov and Banakh using balleans. Coarse structur
es compatible with a group structure can be characterised by special ideal
s of subsets\, called group ideals. While the coarse structure induced by
the family of all finite subsets is well-suited for abstract groups\, the
situation is less clear for groups endowed with group topologies\, as exem
plified by the left coarse structure\, introduced by Rosendal\, and the co
mpact-group coarse structure\, induced by the group ideal of all relativel
y compact subsets\, each suitable in disparate settings. \nWe present the
large-scale geometry of groups via the historically iterative sequence of
generalisations\, enlisting illustrative examples specific to distinct cla
sses of groups and topological groups. We focus on locally compact abelian
groups endowed with compact-group coarse structures. In particular\, we d
iscuss the role of Pontryagin duality as a bridge between topological prop
erties and their large-scale counterparts. An overriding theme is an evide
nce-based tenet that the compact-group coarse structure is the right choic
e for the category of locally compact abelian groups.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lydia Außenhofer (Universität Passau)
DTSTART:20200519T160000Z
DTEND:20200519T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/7
DESCRIPTION:Title: On the Mackey Topology of an Abelian Topological Group\nby L
ydia Außenhofer (Universität Passau) as part of Topological Groups\n\nLe
cture held in Elysium.\n\nAbstract\nFor a locally convex vector space $(V
\,\\tau)$ there exists a finest locally convex vector space topology $\\mu
$ such that the topological dual spaces $(V\,\\tau)'$ and $(V\,\\mu)'$ coi
ncide algebraically. This topology is called the $Mackey$ $topology$. If $
(V\,\\tau)$ is a metrizable locally convex vector space\, then $\\tau$ is
the Mackey topology.\n\nIn 1995 Chasco\, Martín Peinador\, and Tarielad
ze asked\, "Given a locally quasi-convex group $(G\,\\tau)\,$ does there e
xist a finest locally quasi-convex group topology $\\mu$ on $G$ such that
the character groups $(G\,\\tau)^\\wedge$ and $(G\,\\mu)^\\wedge$ coincide
?"\n\nIn this talk we give examples of topological groups which\n\n1. hav
e a Mackey topology\,\n\n2. do not have a Mackey topology\,\n\nand we char
acterize those abelian groups which have the property that every metrizabl
e locally quasi-convex group topology is Mackey (i.e.\, the finest compati
ble locally quasi-convex group topology).\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Loth (Sacred Heart University)
DTSTART:20200526T160000Z
DTEND:20200526T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/8
DESCRIPTION:Title: Simply Given Compact Abelian Groups\nby Peter Loth (Sacred H
eart University) as part of Topological Groups\n\nLecture held in Elysium.
\n\nAbstract\nA compact abelian group is called simply given if its Pontr
jagin dual is simply presented. Warfield groups are defined to be direct s
ummands of simply presented abelian groups. They were classified up to iso
morphism in terms of cardinal invariants by Warfield in the local case\, a
nd by Stanton and Hunter--Richman in the global case. In this talk\, we cl
assify up to topological isomorphism the duals of Warfield groups\, dualiz
ing Stanton's invariants. We exhibit an example of a simply given group wi
th nonsplitting identity component.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ajit Iqbal Singh (Indian National Science Academy)
DTSTART:20200609T160000Z
DTEND:20200609T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/9
DESCRIPTION:Title: Variants of Invariant Means of Amenability\nby Ajit Iqbal Si
ngh (Indian National Science Academy) as part of Topological Groups\n\nLec
ture held in Elysium.\n\nAbstract\nIt all started\, like many other amazin
g theories\, in nineteen twenty-nine\,\nWith John von Neumann\, the greate
st of the great.\nThe question of existence of a finitely additive measure
on a group\, a mean of a kind\,\nThat is invariant\, under any translatio
n\, neither gaining nor losing any weight.\n\nMahlon M. Day\, in his zest
and jest\, giving double importance to semigroups\, too\, \nTo
ok up the study of conditions and properties\, and named it amenability.\n
Erling Folner followed it up\, more like a combinatorial maze to go throug
h\,\nWhether or not translated set meets the original in a sizeable propor
tionality.\n\nHow could functional analysts sit quiet\, who measure anythi
ng by their own norms\, \nLo and behold\, it kept coming back to the same
concept over and over again.\nGroup algebras were just as good or bad\, ap
proximate conditions did no harms\,\nWith the second duals of lofty Richar
d Arens\, it became deeper\, but still a fun-game.\n\nEver since\, with th
e whole alphabet names\, reputed experts or budding and slick\, \nConsider
ing several set-ups and numerous variants of the invariance.\nActions on M
anifolds or operators\, dynamical systems nimble or quick\,\nWe will have
a look at some old and some new\, closely or just from the fence.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Break (University of Hawaiʻi)
DTSTART:20200602T160000Z
DTEND:20200602T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/10
DESCRIPTION:Title: Topological Groups Seminar One-Week Hiatus\nby Break (Unive
rsity of Hawaiʻi) as part of Topological Groups\n\nLecture held in Elysiu
m.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Riddhi Shah (Jawaharlal Nehru University)
DTSTART:20200616T160000Z
DTEND:20200616T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/11
DESCRIPTION:Title: Dynamics of Distal Actions on Locally Compact Groups\nby Ri
ddhi Shah (Jawaharlal Nehru University) as part of Topological Groups\n\nL
ecture held in Elysium.\n\nAbstract\nDistal 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. Si
milarly\, 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 diagona
l unless x=y. We discuss some properties of distal actions of automorphism
s on locally compact groups and on homogeneous spaces given by quotients m
odulo closed invariant subgroups which are either compact or normal. We re
late distality to the behaviour of orbits. We also characterise the behavi
our of convolution powers of probability measures on the group in terms of
the distality of inner automorphisms.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karl Hofmann (Technische Universität Darmstadt)
DTSTART:20200623T160000Z
DTEND:20200623T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/12
DESCRIPTION:Title: The group algebra of a compact group and Tannaka duality for co
mpact groups\nby Karl Hofmann (Technische Universität Darmstadt) as p
art of Topological Groups\n\nLecture held in Elysium.\n\nAbstract\nIn the
4th edition of the text- and handbook "The Structure of Compact Groups"\,\
nde Gruyter\, Berlin-Boston\, having appeared June 8\, 2020\, Sidney A. Mo
rris and\nI decided to include\, among material not contained in earlier e
ditions\, the Tannaka-Hochschild Duality Theorem which says that $the$ $ca
tegory$ $of$ $compact$ $groups$ $is\\\,dual$\n$to$ $the$ $category\\\,of\\
\,real\\\,reductive$ $Hopf$ $algebras$. In the lecture I hope to explain\n
why this theorem was not featured in the preceding 3 editions and why we d
ecided\nto present it now. Our somewhat novel access led us into a new the
ory of real\nand complex group algebras for compact groups which I shall d
iscuss. Some Hopf\nalgebra theory appears inevitable. Recent source: K.H.H
ofmann and L.Kramer\,\n$On$ $Weakly\\\,Complete\\\,Group\\\,Algebras$ $of$
$Compact$ $Groups$\, J. of Lie Theory $\\bold{30}$ (2020)\, 407-424.\n\n
Karl H. Hofmann\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Break (University of Hawaiʻi)
DTSTART:20200630T160000Z
DTEND:20200630T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/13
DESCRIPTION:Title: Topological Groups Seminar One-Week Hiatus\nby Break (Unive
rsity of Hawaiʻi) as part of Topological Groups\n\nLecture held in Elysiu
m.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Indira Chatterji (Laboratoire J.A. Dieudonné de l'Université de
Nice)
DTSTART:20200707T160000Z
DTEND:20200707T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/14
DESCRIPTION:Title: Groups Admitting Proper Actions by Affine Isometries on Lp Spac
es\nby Indira Chatterji (Laboratoire J.A. Dieudonné de l'Université
de Nice) as part of Topological Groups\n\nLecture held in Elysium.\n\nAbst
ract\nIntroduction\, known results\, and open questions regarding groups a
dmitting a proper action by affine isometries on an $L_p$ space.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ajay Kumar (University of Delhi)
DTSTART:20200714T160000Z
DTEND:20200714T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/15
DESCRIPTION:Title: Uncertainty Principles on Locally Compact Groups\nby Ajay K
umar (University of Delhi) as part of Topological Groups\n\nLecture held i
n Elysium.\n\nAbstract\nSome of the uncertainty principles on $ \\mathbb{R
}^n $ are as follows:\n\n Qualitative Uncertainty Principle: Let $f$ be a
non-zero function in $L^1(\\mathbb{R}^n)$. Then the Lebesgue measures of
the sets $\\{x: f(x)\\neq 0 \\}$ and $ \\{\\xi : \\widehat{f}(\\xi) \\n
eq 0\\}$ cannot both be finite.\n\nHardy's Theorem: Let $ a\,b\,c $ be th
ree real positive numbers and let $f: \\mathbb{R}^n \\to \\mathbb{C}$ be
a measurable function such that \n\n(i) $|f(x)| \\leq c\\exp{(-a\\pi \\
|x\\|^2)}$\, for all $ x \\in \\mathbb{R}^n$ \n(ii) $|\\widehat{f}(\\xi)|
\\leq c\\exp{(-b\\pi \\|\\xi\\|^2)}$\, for all $\\xi \\in \\mathbb{R}^n $
. \n\n Then following holds:\nIf $ab>1$\, then $f=0$ a.e.\n If $ab =1$\,
then $f(x)= \\alpha \\exp{(-a\\pi \\|x\\|^2)}$ for some constant $\\alpha$
.\n If $ab< 1$\, then there are infinitely many linear independent functio
ns satisfying above conditions.\n\n\n Heisenberg Inequality: If $f \\in L^
2(\\mathbb{R}^n)$ and $a\,b \\in \\mathbb{R}^n$\, then\n\n $$\n \\left( \\
int_{\\mathbb{R}^n}\\|x-a\\|^2|f(x)|^2 dx \\right) \\left( \\int_{\\mathbb
{R}^n}\\|\\xi-b\\|^2|\\widehat{f}(\\xi)|^2 d\\xi \\right) \\geq \\frac{n^2
\\|f\\|^4}{16\\pi^2}.\n $$\n Beurling's Theorem: Let $f \\in L^1(\\mathb
b{R}^n) $ and for some $ k(1\\leq k\\leq n) $ satisfies\n $$\n \\int_{\\ma
thbb{R}^{2n}} |f(x_1\, x_2\, \\dots \, x_n)||\\widehat{f}(\\xi_1\, \\xi_2\
, \\dots \, \\xi_n)|e^{2\\pi |x_k\\xi_k|} dx_1\\dots dx_n d\\xi_1\\dots d\
\xi_n< \\infty.\n $$\n Then $f = 0$ a.e.\n\nWe investigate these principle
s on locally compact groups\, in particular Type I\ngroups and nilpotent L
ie groups for Fourier transform and Gabor transform.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:C.R.E. Raja (Indian Statistical Instititute)
DTSTART:20200721T160000Z
DTEND:20200721T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/16
DESCRIPTION:Title: Probability Measures and Structure of Locally Compact Groups\nby C.R.E. Raja (Indian Statistical Instititute) as part of Topological
Groups\n\nLecture held in Elysium.\n\nAbstract\nWe will have an overview o
f how existence of certain types of\nprobability measures forces locally c
ompact groups to have particular\nstructures and vice versa. Examples are
Choquet-Deny measures\, recurrent\nmeasures etc.\, and groups of the kind
amenable\, polynomial growth\, etc.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dikran Dikranjan (University of Udine)
DTSTART:20200728T160000Z
DTEND:20200728T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/17
DESCRIPTION:Title: On a Class of Profinite Groups Related to a Theorem of Prodanov
\nby Dikran Dikranjan (University of Udine) as part of Topological Gro
ups\n\nLecture held in Elysium.\n\nAbstract\nA short history of minimal gr
oups is given\, featuring illustrative examples and leading to current res
earch:$\\newline$\n$\\quad$ * non-compact minimal groups\,$\\newline$\n$\\
quad$ * equivalence between minimality and essentiality of dense subgroup
s of compact groups\,$\\newline$\n$\\quad$ * equivalence between minimali
ty and compactness in LCA\, $\\newline$\n$\\quad$ * hereditary formulatio
ns of minimality facilitate optimal statements of theorems\, $\\newline$\n
$\\quad$ * a locally compact hereditarily locally minimal infinite group
$G$ is $\\newline$\n$\\quad$ $\\quad$ (a) $\\cong\\mathbb{Z}p$\, some prim
e $p$\, when $G$ is nilpotent\,$\\newline$\n$\\quad$ $\\quad$ (b) a Lie gr
oup when $G$ is connected\,$\\newline$\n$\\quad$ * classification of hered
itarily minimal locally compact solvable groups\,$\\newline$\n$\\quad$ * e
xistence of classes of hereditarily non-topologizable groups: $\\newline$\
n$\\quad$ $\\quad$ (a) bounded infinite finitely generated\,$\\newline$\n$
\\quad$ $\\quad$ (b) unbounded finitely generated\,$\\newline$\n$\\quad$ $
\\quad$ (c) countable not finitely generated\, $\\newline$\n$\\quad$ $\\qu
ad$ (d) uncountable.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Willis (University of Newcastle)
DTSTART:20200804T160000Z
DTEND:20200804T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/18
DESCRIPTION:Title: Totally disconnected locally compact groups and the scale\n
by George Willis (University of Newcastle) as part of Topological Groups\n
\nLecture held in Elysium.\n\nAbstract\nThe 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 o
verview of the main directions of current research on t.d.l.c. groups\, th
e talk will introduce the scale and describe aspects of group structure th
at 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.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Helge Glöckner (Universität Paderborn)
DTSTART:20200811T160000Z
DTEND:20200811T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/19
DESCRIPTION:Title: Locally Compact Contraction Groups\nby Helge Glöckner (Uni
versität Paderborn) as part of Topological Groups\n\nLecture held in Elys
ium.\n\nAbstract\nConsider a locally compact group $G$\, together with an
automorphism $\\alpha$ which is $contractive$ in the sense that $\\alpha^n
\\rightarrow{\\rm id}_G$ pointwise as $n\\to\\infty$. Siebert showed that
$G$ is the direct product of its connected component $G_e$ and an $\\alpha
$-stable\, totally disconnected closed subgroup\;\nmoreover\, $G_e$ is a s
imply connected\, nilpotent real Lie group.\nI'll report on research conce
rning the totally disconnected part\, obtained jointly with G. A. Willis.\
n\nFor each totally disconnected contraction group $(G\,\\alpha)$\, the se
t ${\\rm tor} G$ of torsion elements is a closed subgroup of $G$. Moreover
\, $G$ is a direct product\n$$G=G_{p_1}\\times \\cdots\\times G_{p_n}\\tim
es {\\rm tor} G$$ of $\\alpha$-stable $p$-adic Lie groups $G_p$ for certai
n 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\; notab
ly\, they are nilpotent. As shown with Willis\, ${\\rm tor} G$ admits a co
mposition series and there are countably many possible composition factors
\, parametrized by the finite simple groups. More recent research showed t
hat 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 nilpote
nt. Likewise if $G$ is locally pro-nilpotent.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elena Martín-Peinador (University of Madrid)
DTSTART:20200818T160000Z
DTEND:20200818T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/20
DESCRIPTION:Title: Group dualities: G-barrelled groups\nby Elena Martín-Peina
dor (University of Madrid) as part of Topological Groups\n\nLecture held i
n Elysium.\n\nAbstract\nA natural notion in the framework of abelian group
s are the group dualities. The most efficient definition of a group duali
ty is simply a pair $(G\, H)$\, where $G$ denotes an abstract abelian grou
p 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 thi
s sense we shall define the term "compatible topology" and loosely speakin
g we consider the compatible topologies as members of the duality.\n\n
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. Th
e fact that a locally convex topological vector space is in particular a l
ocally quasi-convex group serves as a nexus to emulate well-known resu
lts of Functional Analysis for the class of topological groups. \n\nIn thi
s talk we shall\ndeal with questions of the sort:\nUnder which condition
s is there a locally compact topology in a fixed duality?\nThe same quest
ion for a metrizable\, or a $k$-group topology.\nWe shall also introduce
the $g$-barrelled groups\, a class for which the Mackey-Arens Theorem adm
its an optimal counterpart. We study also the existence of $g$-barrelled t
opologies in a group duality $(G\, H)$.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wayne Lewis (University of Hawaiʻi)
DTSTART:20200825T160000Z
DTEND:20200825T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/21
DESCRIPTION:Title: Classification of Periodic LCA Groups of Finite Non-Archimedean
Dimension\nby Wayne Lewis (University of Hawaiʻi) as part of Topolog
ical Groups\n\nLecture held in Elysium.\n\nAbstract\nA periodic LCA group
such that the $p$-components all have $p$-rank bounded above by a common p
ositive integer are classified via a complete set of topological isomorphi
sm invariants realized by an equivalence relation on pairs of extended sup
ernatural vectors.\n\nRemaining time will be devoted to a facilitated disc
ussion on how things are going this fall/winter academic semester in your
part of the world as you see it.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dona Strauss (University of Leeds)
DTSTART:20200901T160000Z
DTEND:20200901T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/22
DESCRIPTION:Title: The Semigroup $\\beta S$\nby Dona Strauss (University of Le
eds) as part of Topological Groups\n\nLecture held in Elysium.\n\nAbstract
\nIf $S$ is a discrete semigroup\, the semigroup operation on $S$ can be e
xtended to a semigroup operation on its Stone–Čech compactification $\\
beta S$. The properties of the semigroup $\\beta S$ have been a powerful t
ool in topological dynamics and combinatorics.\n \nI shal
l 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.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bharat Talwar (University of Dehli)
DTSTART:20200908T160000Z
DTEND:20200908T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/23
DESCRIPTION:Title: Closed Lie Ideals and Center of Generalized Group Algebras\
nby Bharat Talwar (University of Dehli) as part of Topological Groups\n\nL
ecture held in Elysium.\n\nAbstract\nThe closed Lie ideals of the generali
zed group algebra $L^1(G\,A)$ are characterized in terms of elements of th
e group $G$\, elements of the algebra $A$\, and the modular function $\\De
lta$ of the group $G$. Conditions under which for a given closed Lie ideal
$L\\subseteq A$ the subspace $L^1(G\,L)$ is a Lie ideal\, and vice versa\
, are discussed. The center of $L^1(G\,A)$ is characterized\, followed by
a discussion regarding a very special projection in $L^1(G\,A)$. Finally\,
a few restrictions are imposed on $G$ and $A$ under which $\\mathcal{Z}(L
^1(G\,A))\\cong\\mathcal{Z}(L^1(G))\\otimes^\\gamma\\mathcal{Z}(A)$.\n\nTh
e presentation is based on joint work with Ved Prakash Gupta and Ranjana J
ain.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nico Spronk (University of Waterloo)
DTSTART:20200915T160000Z
DTEND:20200915T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/24
DESCRIPTION:Title: Topologies\, idempotents and ideals\nby Nico Spronk (Univer
sity of Waterloo) as part of Topological Groups\n\nLecture held in Elysium
.\n\nAbstract\nLet $G$ be a topological group. I wish to exhibit a bijecti
on between (i) a certain class of weakly almost periodic topologies\, (ii)
idempotents in the weakly almost periodic compactification of $G$\, and (
iii) certain ideals of the algebra of weakly almost periodic functions. T
his has applications to decomposing weakly almost periodic representations
on Banach spaces\, generalizing results which go back to many authors.\n\
nMoving to unitary representations\, I will develop the Fourier-Stieltjes
algebra $B(G)$ of $G$\, and give the analogous result there. As an applic
ation\, I show that for a locally compact connected group\, operator amena
bility of $B(G)$ implies that $G$ is compact\, partially resolving a probl
em of interest for 25 years.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mukund Madhav Mishra (Hansraj College)
DTSTART:20200922T160000Z
DTEND:20200922T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/25
DESCRIPTION:Title: Potential Theory on Stratified Lie Groups\nby Mukund Madhav
Mishra (Hansraj College) as part of Topological Groups\n\nLecture held in
Elysium.\n\nAbstract\nStratified Lie groups form a special subclass of th
e class of nilpotent Lie groups. The Lie algebra of a stratified Lie group
possesses a specific stratification (and hence the name)\, and an interes
ting class of anisotropic dilations. Among the linear differential operato
rs of degree two\, there exists a family that is well behaved with the aut
omorphisms of the stratified Lie group\, especially with the anisotropic d
ilations. We shall see that one such family of operators mimics the classi
cal Laplacian in many aspects\, except for the regularity. More specifical
ly\, these Laplace-like operators are sub-elliptic\, and hence referred to
as the sub-Laplacians. We will review certain interesting properties of f
unctions harmonic with respect to the sub-Laplacian on a stratified Lie gr
oup\, and have a closer look at a particular class of stratified Lie group
s known as the class of Heisenberg type groups.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sutanu Roy (National Institute of Science Education and Research)
DTSTART:20200929T160000Z
DTEND:20200929T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/26
DESCRIPTION:Title: Compact Quantum Groups and their Semidirect Products\nby Su
tanu Roy (National Institute of Science Education and Research) as part of
Topological Groups\n\nLecture held in Elysium.\n\nAbstract\nCompact quant
um groups are noncommutative analogs of compact groups in the realm of non
commutative geometry introduced by S. L. Woronowicz back in the 80s. Rough
ly\, they are unital C*-bialgebras in the monoidal category (given by the
minimal tensor product) of unital C*-algebras with some additional propert
ies. For real 0<|q|<1\, q-deformations of SU(2) group are the first and we
ll-studied examples of compact quantum groups. These examples were constru
cted independently by Vaksman-Soibelman and Woronowicz also back in the 8
0s. In fact\, they are examples of a particular class of compact quantum g
roups namely\, compact matrix pseudogroups. The primary goal of this talk
is to motivate and discuss some of the interesting aspects of this theory
from the perspective of the compact groups. In the second part\, I shall b
riefly discuss the semidirect product construction for compact quantum gro
ups via an explicit example. The second part of this will be based on a jo
int work with Paweł Kasprzak\, Ralf Meyer and Stanislaw Lech Woronowicz.\
n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolò Zava (University of Udine)
DTSTART:20201006T160000Z
DTEND:20201006T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/27
DESCRIPTION:Title: Towards a unifying approach to algebraic and coarse entropy
\nby Nicolò Zava (University of Udine) as part of Topological Groups\n\nL
ecture held in Elysium.\n\nAbstract\nIn each situation\, entropy associate
s to a self-morphism a value that estimates the chaos created by the map a
pplication. In particular\, the algebraic entropy $h_{alg}$ can be compute
d for (continuous) endomorphisms of (topological) groups\, while the coars
e entropy $h_c$ is associated to bornologous self-maps of locally finite c
oarse spaces. Those two entropy notions can be compared because of the fol
lowing observation. If $f$ is a (continuous) homomorphism of a (topologica
l) group $G$\, then $f$ becomes automatically bornologous provided that $G
$ is equipped with the compact-group coarse structure. For an endomorphism
$f$ of a discrete group\, $h_{alg}(f)=h_c(f)$ if $f$ is surjective\, whil
e\, in general\, $h_{alg}(f)\\neq h_c(f)$. That difference occurs because
in many cases\, if $f$ is not surjective\, then $h_c(f)=0$. \n\nIn the fir
st part of the talk\, after briefly recalling the large-scale geometry of
topological groups\, we define the coarse entropy and discuss its relation
ship with the algebraic entropy. The second part is dedicated to the intro
duction of the algebraic entropy of endomorphisms of $G$-sets (i.e.\, sets
endowed with group actions). We show that it extends the usual algebraic
entropy of group endomorphisms and we provide evidence that it can repres
ent a useful modification and generalisation of the coarse entropy that ov
ercome the non-surjectivity issue.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Break (University of Hawaiʻi)
DTSTART:20201013T160000Z
DTEND:20201013T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/28
DESCRIPTION:Title: Topological Groups Seminar Two-Week Hiatus\nby Break (Unive
rsity of Hawaiʻi) as part of Topological Groups\n\nLecture held in Elysiu
m.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Break (University of Hawaiʻi)
DTSTART:20201020T160000Z
DTEND:20201020T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/29
DESCRIPTION:Title: Topological Groups Seminar Two-Week Hiatus\nby Break (Unive
rsity of Hawaiʻi) as part of Topological Groups\n\nLecture held in Elysiu
m.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wayne Lewis (University of Hawaiʻi)
DTSTART:20201027T160000Z
DTEND:20201027T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/30
DESCRIPTION:Title: Abelian Varieties as Algebraic Protori?\nby Wayne Lewis (Un
iversity of Hawaiʻi) as part of Topological Groups\n\nLecture held in Ely
sium.\n\nAbstract\nAn outcome of the structure theory of protori (compact
connected abelian groups) is their representability as quotients of $\\mat
hbb{A}^n$ for the ring of adeles $\\mathbb{A}$. $\\mathbb{A}$ does not con
tain zeros of rational polynomials\, but rather representations of zeros.
Investigating the relations between algebraicity of complex tori and algeb
raicity of protori leads one to the problem of computing the Pontryagin du
al of $\\mathbb{A}/\\mathbb{Z}$. Applying an approach by Lenstra in the se
tting of profinite integers to the more general $\\mathbb{A}$ leads to a d
efinition of the closed maximal $Lenstra$ $ideal$ $E$ of $\\mathbb{A}$\, w
hence the locally compact field of $adelic$ $numbers$ $\\mathbb{F}=\\mathb
b{A}/E$\, providing a long-sought connection to $\\mathbb{C}$ enabling one
to define a functor from the category of complex tori to the category of
protori - is it possible to do so in a way that preserves algebraicity? Wh
ile $\\mathbb{F}$ marks tentative progress\, much work remains...\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wayne Lewis (University of Hawaiʻi)
DTSTART:20201103T160000Z
DTEND:20201103T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/31
DESCRIPTION:Title: Accounting with $\\mathbb{QP}^\\infty$\nby Wayne Lewis (Uni
versity of Hawaiʻi) as part of Topological Groups\n\nLecture held in Elys
ium.\n\nAbstract\nRational projective space provides a useful accounting t
ool in engineering decompositions of $\\mathbb{Q}[x]$ for desired effect.
The device is useful for defining a correspondence between summands of suc
h a decomposition and elements of a partition of $\\mathbb{A}$. This mecha
nism is applied to a decomposition of $\\mathbb{Q}[x]$ relative to which t
he correspondence gives the $Lenstra$ $ideal$ $E$\, a closed maximal ideal
yielding the $adelic$ $numbers$ $\\mathbb{F}=\\frac{\\mathbb{A}}{E}$.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Monique Chyba (University of Hawaiʻi)
DTSTART:20201110T160000Z
DTEND:20201110T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/32
DESCRIPTION:Title: Journey in Hawaii's Challenges in the Fight Against COVID-19\nby Monique Chyba (University of Hawaiʻi) as part of Topological Groups
\n\nLecture held in Elysium.\n\nAbstract\nThe COVID-19 pandemic is far fro
m the first infectious disease that Hawaiʻi had to deal with. During the
1918-1920 Influenza Pandemic\, the Hawaiian islands were not spared as the
disease ravaged through the whole world. Hawaiʻi and similar island popu
lations can follow a different course of pandemic spread than large cities
/states/nations and are often neglected in major studies. It may be too ea
rly to compare the 1918-1920 Influenza Pandemic and COVID-19 Pandemic\, w
e do however note some similarities and differences between the two pandem
ics.\n\nHawaiʻi and other US Islands have recently been noted by the medi
a as COVID-19 hotspots after a relatively calm period of low case rates. U
.S. Surgeon General Jerome Adams came in person on August 25 to Oahu to ad
dress the alarming situation. We will discuss the peculiarity of the situa
tion in Hawaiʻi and provide detailed modeling of current virus spread pat
terns aligned with dates of lockdown and similar measures. We will present
a detailed epidemiological model of the spread of COVID-19 in Hawaiʻi an
d explore effects of different intervention strategies in both a prospecti
ve and retrospective fashion. Our simulations demonstrate that to control
the spread of COVID-19 both actions by the State in terms of testing\, con
tact tracing and quarantine facilities as well as individual actions by th
e population in terms of behavioral compliance to wearing a mask and gathe
ring in groups are vital. They also explain the turn for the worst Oahu to
ok after a very successful stay-at-home order back in March.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seid Kassaw (University of Cape Town)
DTSTART:20201124T160000Z
DTEND:20201124T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/33
DESCRIPTION:Title: The probability of commuting subgroups in arbitrary lattices of
subgroups\nby Seid Kassaw (University of Cape Town) as part of Topolo
gical Groups\n\nLecture held in Elysium.\n\nAbstract\nThe subgroup commuta
tivity degree $sd(G)$ of a finite group $G$ was introduced\nalmost ten yea
rs ago and deals with the number of commuting subgroups in the\nsubgroup l
attice $L(G)$ of $G$. The extremal case $sd(G) = 1$ detects a class of gro
ups\nclassified by Iwasawa in 1941 (in fact\, $sd(G)$ represents a probabi
listic measure which\nallows us to understand how far $G$ is from the grou
ps of Iwasawa). This means\n$sd(G) = 1$ if and only if $G$ is the direct p
roduct of its Sylow $p$-subgroups and these\nare all modular\; or equivale
ntly $G$ is a nilpotent modular group. Therefore\, $sd(G)$ is\nstrongly re
lated to structural properties of $L(G)$ and $G$.\n\nIn this talk\, we int
roduce a new notion of probability $gsd(G)$ in which two arbitrary sublatt
ices $S(G)$ and $T(G)$ of $L(G)$ are involved simultaneously. In case\n$S(
G) = T(G) = L(G)$\, we find exactly $sd(G)$. Upper and lower bounds for $g
sd(G)$\nare shown and we study the behaviour of $gsd(G)$ with respect to s
ubgroups and\nquotients\, showing new numerical restrictions. We present t
he commutativity\nand subgroup commutativity degree for infinite groups an
d put some open problems\nfor further generalization.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Farzana Nasrin (University of Hawaiʻi)
DTSTART:20201201T160000Z
DTEND:20201201T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/34
DESCRIPTION:Title: Bayesian Statistics\, Topology and Machine Learning for Complex
Data Analysis\nby Farzana Nasrin (University of Hawaiʻi) as part of
Topological Groups\n\nLecture held in Elysium.\n\nAbstract\nAnalyzing and
classifying large and complex datasets are generally challenging. Topologi
cal data analysis\, that builds on techniques from topology\, is a natural
fit for this. Persistence diagram is a powerful tool that originated in
topological data analysis that allows retrieval of important topological a
nd geometrical features latent in a dataset. Data analysis and classificat
ion involving persistence diagrams have been applied in numerous applicati
ons. In this talk\, I will provide a brief introduction of topological dat
a analysis\, focusing primarily on persistence diagrams\, and a Bayesian f
ramework for inference with persistence diagrams. The goal is to provide a
supervised machine learning algorithm in the space of persistence diagram
s. This framework is applicable to a wide variety of datasets. I will pres
ent applications in materials science\, biology\, and neuroscience.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikhail Tkachenko (Metropolitan Autonomous University)
DTSTART:20201208T160000Z
DTEND:20201208T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/35
DESCRIPTION:Title: Pseudocompact Paratopological and Quasitopological Groups\n
by Mikhail Tkachenko (Metropolitan Autonomous University) as part of Topol
ogical Groups\n\nLecture held in Elysium.\n\nAbstract\nPseudocompactness i
s an interesting topological property which acquires very specific \nfeatu
res when applied to different algebrotopological objects. A celebrated the
orem\nof Comfort and Ross published in 1966 states that the Cartesian prod
uct of an arbitrary\nfamily of pseudocompact topological groups is pseudoc
ompact. We present a survey \nof results related to the validity or failur
e of the Comfort-Ross' theorem in the realm of \nsemitopological and parat
opological groups and give some examples showing that \npseudocompactness
fails to be stable when taking products of quasitopological groups.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph A. Wolf (University of California)
DTSTART:20210126T160000Z
DTEND:20210126T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/36
DESCRIPTION:Title: Gelfand Pairs\nby Joseph A. Wolf (University of California)
as part of Topological Groups\n\nLecture held in Elysium.\n\nAbstract\nGe
lfand pairs are at the intersection of locally compact group theory\, diff
erential geometry and harmonic analysis. In this talk I'll give some exam
ples and sketch how the Plancherel Theorem \nfor Gelfand pairs is pretty m
uch the same as Pontryagin Duality for locally compact abelian groups.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Fournier-Facio (ETH Zürich)
DTSTART:20210202T160000Z
DTEND:20210202T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/37
DESCRIPTION:Title: Normed p-adic amenability and bounded cohomology\nby France
sco Fournier-Facio (ETH Zürich) as part of Topological Groups\n\nLecture
held in Elysium.\n\nAbstract\nAmenability is a property of a locally compa
ct group that allows to average bounded real-valued functions. It is close
ly related to continuous bounded cohomology\, a fundamental tool in rigidi
ty theory. Both notions are very Archimedean in nature\, in that they deal
with normed real vector spaces and "boundedness" is intended with respect
to such norms. In this talk we will explore what happens when we replace
these by normed vector spaces over the p-adics\, which are ultrametric. We
will see that the corresponding notion of amenability (which needs to be
defined a little differently) is much more restrictive than the usual one\
, and that bounded cohomology is often close to ordinary continuous cohomo
logy.\n\nNo previous knowledge of bounded cohomology is assumed.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eniola Kazeem (University of Cape Town)
DTSTART:20210209T160000Z
DTEND:20210209T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/38
DESCRIPTION:Title: Subgroup Commutativity Degree of Profinite Groups\nby Eniol
a Kazeem (University of Cape Town) as part of Topological Groups\n\nLectur
e held in Elysium.\n\nAbstract\nWe find a probability measure which counts
the pairs of closed commuting subgroups in infinite groups. This measure
turns out to be an extension of what was known in the finite case as subgr
oup commutativity degree. The extremal case of probability one describes t
he so-called topologically quasihamiltonian groups and is a useful tool in
describing the distance of a profinite group from this special class. We
have been inspired by an idea of Heyer in the context of our problem.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Russo (University of Cape Town)
DTSTART:20210216T160000Z
DTEND:20210216T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/39
DESCRIPTION:Title: A probabilistic measure for the number of commuting subgroups i
n locally compact groups\nby Francesco Russo (University of Cape Town)
as part of Topological Groups\n\nLecture held in Elysium.\n\nAbstract\nIn
the present talk I will construct a probability measure which counts the
pairs of closed commuting subgroups in locally compact groups\, extending
in this way the so-called ``subgroup commutativity degree'' of profinite
groups\, and in particular\, of finite groups. The extremal case of proba
bility one characterizes topologically quasihamiltonian groups\, introduce
d by K. Iwasawa in 1941. We use a general approach\, involving measure th
eory and\, in particular\, we study the number of closed commuting subgrou
ps in locally compact groups and pro-Lie groups with a corresponding descr
iption of their Chabauty spaces. \n\nTime permitting I will discuss a clas
sical example of Andersen and Jessen\, which shows some natural limitation
s in the techniques of construction when we involve the notion of presheav
es of measure spaces.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tea N. Coffee (University of Hawaiʻi)
DTSTART:20210223T160000Z
DTEND:20210223T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/40
DESCRIPTION:Title: Two Years Forward\, One Year Back\nby Tea N. Coffee (Univer
sity of Hawaiʻi) as part of Topological Groups\n\nLecture held in Elysium
.\n\nAbstract\nTea & Coffee:\nIn lieu of our usual presentation...shifting
the tectonic plates of mathematics...we take a guided look back at the la
st \nyear of presentation topics and speakers. Come and go as you please a
s we informally meet and pass the ball around the table.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph A. Wolf (University of California)
DTSTART:20210302T160000Z
DTEND:20210302T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/41
DESCRIPTION:Title: Uncertainty Principles for Gelfand Pairs\nby Joseph A. Wolf
(University of California) as part of Topological Groups\n\nLecture held
in Elysium.\n\nAbstract\nI'll sketch an extension of the classical Uncerta
inty Principle to the context\nof Gelfand pairs. The Gelfand pair setting
includes Riemannian symmetric\nspaces\, compact topological groups\, loca
lly compact abelian groups\, and\nhomogeneous graphs. In the case of the
locally compact group R^n one\nrecovers a sharp form of the signal process
ing version of the classical\nHeisenberg uncertainty principle. If time p
ermits I'll briefly indicate \nsome applications to spherical functions on
Riemannian symmetric spaces\,\nto Cayley complexes\, and to hypergroups.\
n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manoj Kummini (Chennai Mathematical Institute)
DTSTART:20210309T160000Z
DTEND:20210309T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/42
DESCRIPTION:Title: Polynomial invariant rings\nby Manoj Kummini (Chennai Mathe
matical Institute) as part of Topological Groups\n\nLecture held in Elysiu
m.\n\nAbstract\nLet $G$ be a finite group acting linearly on a\nfinite-dim
ensional vector-space $V$ over a field $K$. Let $R$ denote\nthe symmetric
algebra on $V^*$\; then $G$ acts as graded $K$-algebra\nautomorphisms on $
R$. If $R^G$ is a polynomial ring\, then $G$ is\ngenerated by elements tha
t act as pseudo-reflections on $V$. The\nconverse holds when $|G|$ is inve
rtible in $K$. The above results are\nShephard-Todd-Chevalley-Serre theore
m. If $\\mathrm{char}(K) = p>0$ and\n$G$ is a $p$-group\, then a conjectur
e of Shank-Wehlau-Broer asserts that\n$R^G$ is a polynomial ring if $R^G$
is a direct summand of $R$ as an\n$R^G$-module. We show that this is true
for a class of groups called\ngeneralized Nakajima groups. The key step in
proving this is showing\nthat the Hilbert ideal (i.e. the ideal of $R$ ge
nerated by positive\ndegree elements in $R^G$) is a complete intersection
ideal. This is\njoint work with Mandira Mondal.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adolf Mader (University of Hawaiʻi)
DTSTART:20210316T160000Z
DTEND:20210316T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/43
DESCRIPTION:Title: Free Subgroups with Torsion Quotients and Profinite Subgroups w
ith Torus Quotients: Arbitrary Rank\nby Adolf Mader (University of Haw
aiʻi) as part of Topological Groups\n\nLecture held in Elysium.\n\nAbstra
ct\nCompact connected groups contain $\\delta$-subgroups\, that is\, compa
ct\ntotally disconnected subgroups with torus quotients\, which are essent
ial ingredients in the important Resolution Theorem\, a description of com
pact groups.\nDually\, free subgroups with torsion quotient of discrete to
rsion-free groups\nare studied in order to obtain a comprehensive picture
of the abundance of \n$\\delta$-subgroups of arbitrary compact connected a
belian groups. Previously we\nconsidered finite dimensional groups $G$ onl
y but even for these we obtain new\nresults on the canonical subgroup $\\b
oldsymbol{\\Delta}_G$ which is the sum of all \n$\\delta$-subgroups of G.
We also study a topology on torsion-free abelian groups \nfor which the fr
ee subgroups with torsion quotient form a neighborhood basis at 0.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wolfgang Herfort (TU Wien)
DTSTART:20210323T160000Z
DTEND:20210323T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/44
DESCRIPTION:Title: LCA p-groups in which every maximal monothetic subgroup splits<
/a>\nby Wolfgang Herfort (TU Wien) as part of Topological Groups\n\nLectur
e held in Elysium.\n\nAbstract\nA well-known result about any discrete ab
elian $p$-group $G$\nby Kulikov states that every element of finite $p$-he
ight \nis contained in a maximal cyclic subgroup\, a direct summand of $G$
.\nMoreover\, every bounded pure subgroup is a direct summand of $G$.\nThe
se facts motivated us to pose the question which LCA $p$-groups\n$G$ have
the following two properties: \n\n(N.1) Every monothetic subgroup is conta
ined in some maximal monothetic\nsubgroup.\n\n(N.2) Every maximal monothet
ic subgroup is a direct summand\, algebraically\nand topologically.\n\nLCA
$p$-groups with these properties we term $\\it neat$.\nWe say that an LCA
$p$-group $G$ is $p^e$-homocyclic provided every maximal\nmonothetic subg
roup of $G$ has order $p^e$.\nWe will discuss some examples first and then
present a classification result:\n\n$\\bf Theorem 1.$ \nA LCA $p$-group $
G$ is neat if and only if one of the following exclusive \nstatements hold
:\n\n$\\rm (i)$ $G$ is torsion-free. \n\n$\\rm (ii)$ $G$ is torsion. Then
there is $e\\ge 1$ and closed subgroups $A$ and\n$B$ such that $G=A\\oplus
B$\, $A$ is $p^e$-homocyclic and $B$ is $p^{e-1}$-homocyclic.\n\nAbout th
e structure in (i) we conjecture that \n$G=\\Z_p^I$ for some set $I$ and w
e let $C$ be any $p$-adically\nclosed subgroup of $G$ with $G/C$ torsion.
\nThen the topology on $G$ is redefined\nby letting $C$ be an open compact
subgroup.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rafael Dahmen (Karlsruher Institut für Technologie)
DTSTART:20210330T160000Z
DTEND:20210330T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/45
DESCRIPTION:Title: Long Direct Limits of Topological Groups\nby Rafael Dahmen
(Karlsruher Institut für Technologie) as part of Topological Groups\n\nLe
cture held in Elysium.\n\nAbstract\nGiven a directed system of topological
groups\, one can consider the direct limit (colimit) in the category of t
opological spaces. Unfortunately\, sometimes this topology may fail to be
a group topology due to discontinuity of the multiplication map. In these
cases the topology underlying the colimit in the category of topological g
roups is different from the colimit in the category of topological spaces.
In this talk\, I want to present some well-known results on when this pat
hology occurs in the case of countable directed systems -- as well as some
newer results on certain uncountable systems (called "long directed syste
ms") which behave very differently than countable ones. This will be illus
trated by some (hopefully) motivating examples. This is joint work with G
ábor Lukács.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pratulananda Das (Jadavpur University)
DTSTART:20210413T160000Z
DTEND:20210413T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/46
DESCRIPTION:Title: On certain "differently" characterized subgroups of the circle
group\nby Pratulananda Das (Jadavpur University) as part of Topologica
l Groups\n\nLecture held in Elysium.\n\nAbstract\nThe talk is based on rec
ent joint works with Prof. Dikran Dikranjan\, Prof. Wei He\, my Ph.D. stud
ents Kumardipta Bose and Ayan Ghosh. In this talk\, we will discuss certai
n new versions of characterized subgroups of the circle group $\\mathbb{T}
$ which we name as statistically characterized subgroups and $\\alpha$-sta
tistically characterized subgroups. Our investigations show that these sub
groups (for arithmetic sequences of integers) are essentially different a
nd strictly larger in size than the much investigated class of characteriz
ed subgroups\, having cardinality $\\mathfrak{c}$ but remaining nontrivial
(i.e. different from $\\mathbb{T}$) though remaining topologically nice.
As a natural consequence\, we consider an extended version of Armacost's p
roblem of ``description of topologically torsion elements" of the circle g
roup and describe topologically $s$-torsion and topologically $\\alpha$-to
rsion elements (which form the statistically characterized subgroups and $
\\alpha$-statistically characterized subgroups respectively) in terms of t
he support and provide a complete solution.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linus Kramer (Universität Münster)
DTSTART:20210427T160000Z
DTEND:20210427T180000Z
DTSTAMP:20260422T212857Z
UID:TopologicalGroups/47
DESCRIPTION:Title: Automatic continuity for topological groups\nby Linus Krame
r (Universität Münster) as part of Topological Groups\n\nLecture held in
Elysium.\n\nAbstract\nAn abstract homomorphism $f\\colon G\\rightarrow H$
between topological groups \nis a group homomorphism which is not assumed
to be continuous. I will\ndiscuss old and new results which say that unde
r certain assumptions\non $G$ or $H$\, such an abstract homomorphism is au
tomatically continuous.\n\nThis is based on joint work with Karl H. Hofman
n and Olga Varghese.\n
LOCATION:https://researchseminars.org/talk/TopologicalGroups/47/
END:VEVENT
END:VCALENDAR