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BEGIN:VEVENT
SUMMARY:Wayne Lewis (University of Hawaiʻi)
DTSTART:20200414T160000Z
DTEND:20200414T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/1
DESCRIPTION:Title: Classification of Finite-Dimensional Periodic LCA Groups<
/a>\nby Wayne Lewis (University of Hawaiʻi) as part of Topological Groups
\n\nLecture held in Elysium.\n\nAbstract\nGeneralized resolutions of proto
ri have non-Archimedean component a periodic LCA group with finite non-Arc
himedean dimension. The previous session introduced the notion of non-Arch
imedean dimension of LCA groups. Applying published results by Dikranjan\,
Herfort\, Hofmann\, Lewis\, Loth\, Mader\, Morris\, Prodanov\, Ross\, and
Stoyanov\, we introduce new minimalist notation and accompanying definiti
ons to clarify the structure of these groups and their Pontryagin duals\,
enabling a parametrization of the spectrum of resolutions of finite-dimens
ional protori (the Grothendieck group is a moduli space).\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adolf Mader (University of Hawaiʻi)
DTSTART:20200421T160000Z
DTEND:20200421T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/2
DESCRIPTION:Title: Pontryagin Duals of Type Subgroups of Finite Rank Torsion
-Free Abelian Groups\nby Adolf Mader (University of Hawaiʻi) as part
of Topological Groups\n\nLecture held in Elysium.\n\nAbstract\nPontryagin
duals of type subgroups of finite rank torsion-free abelian groups are pre
sented. The interplay between the intrinsic study of compact abelian group
s\, respectively torsion-free abelian groups\, is discussed (how can resea
rchers better leverage the published results in each setting so there is a
dual impact?). A result definitively qualifying\, in the torsion-free cat
egory\, the uniqueness of decompositions involving maximal rank completely
decomposable summands is given\; the formulation of the result in the set
ting of protori is shown to optimally generalize a well-known result regar
ding the splitting of maximal tori from finite-dimensional protori.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dikran Dikranjan (University of Udine)
DTSTART:20200428T160000Z
DTEND:20200428T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/3
DESCRIPTION:Title: The Connection between the von Neumann Kernel and the Zar
iski Topology\nby Dikran Dikranjan (University of Udine) as part of To
pological Groups\n\nLecture held in Elysium.\n\nAbstract\nEvery group G ca
rries a natural topology Z_G defined by taking as a pre-base of the famil
y of all closed sets the solution sets of all one-variable equations in th
e 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 ex
plicitly introduced by Roger Bryant in 1978\, who named it the verbal topo
logy\, but the name Zariski topology was universally applied subsequently.
As a matter of fact\, this topology implicitly appeared in a series of pa
pers by Markov in the 1940’s in connection to his celebrated problem con
cerning unconditionally closed sets: sets which are closed in any Hausdorf
f group topology on G. These are the closed sets in the topology M_G obta
ined as the intersection of all Hausdorff group topologies on G\, which we
call the Markov topology\, although this topology did not explicitly appe
ar in Markov’s papers. Both Z_G and M_G are T1 topologies and M_G ≥
Z_G\, but they need not be group topologies. One can use these topologies
to formulate 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 eq
uality 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 t
o these topologies\, and to apply the theory to give a solution to the Com
fort-Protasov-Remus problem on minimally almost periodic topologies of abe
lian groups. This problem is associated to a more general problem of Gabri
yelyan concerning the realisation of the von Neumann kernel n(G) of a topo
logical group\; that is\, the intersection of the kernels of the continuou
s homomorphisms G → T into the circle group. More precisely\, given a pa
ir consisting of an abelian group G and a subgroup H\, one asks whether th
ere is a Hausdorff group topology τ on G such that n(G\,τ) = H. Since (G
\,τ) is 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://stable.researchseminars.org/talk/TopologicalGroups/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Giordano Bruno (University of Udine)
DTSTART:20200505T160000Z
DTEND:20200505T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/4
DESCRIPTION:Title: Topological Entropy and Algebraic Entropy on Locally Comp
act Abelian Groups\nby Anna Giordano Bruno (University of Udine) as pa
rt of Topological Groups\n\nLecture held in Elysium.\n\nAbstract\nSince it
s origin\, the algebraic entropy $h_{alg}$ was introduced in connection wi
th the topological entropy $h_{top}$ by means of Pontryagin duality. For a
continuous endomorphism $\\phi\\colon G\\to G$ of a locally compact abeli
an group $G$\, denoting by $\\widehat G$ the Pontryagin dual of $G$ and by
$\\widehat \\phi\\colon G\\to G$ the dual endomorphism of $\\phi$\, we pr
ove that $$h_{top}(\\phi)=h_{alg}(\\widehat\\phi)$$ under the assumption t
hat $G$ is compact or that $G$ is totally disconnected. It is known that t
his equality holds also when $\\phi$ is a topological automorphism.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wayne Lewis (University of Hawaiʻi)
DTSTART:20200407T160000Z
DTEND:20200407T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/5
DESCRIPTION:Title: Adelic Theory of Protori\nby Wayne Lewis (University
of Hawaiʻi) as part of Topological Groups\n\nLecture held in Elysium.\nAb
stract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolò Zava (University of Udine)
DTSTART:20200512T160000Z
DTEND:20200512T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/6
DESCRIPTION:Title: The Large-Scale Geometry of Locally Compact Abelian Group
s\nby Nicolò Zava (University of Udine) as part of Topological Groups
\n\nLecture held in Elysium.\n\nAbstract\nLarge-scale geometry\, also know
n as coarse geometry\, is the branch of mathematics that studies the globa
l\, large-scale properties of spaces. This theory is distinguished by its
applications which include the Novikov and coarse Baum-Connes conjectures.
Since the breakthrough work of Gromov\, large-scale geometry has played a
prominent role in geometric group theory\, in particular\, in the study o
f finitely generated groups and their word metrics. This large-scale appro
ach was successfully extended to all countable groups by Dranishnikov and
Smith. A further generalisation introduced by Cornulier and de la Harpe de
alt with locally compact σ-compact groups endowed with particular pseudo-
metrics. \nTo study the large-scale geometry of more general groups and to
pological groups\, coarse structures are required. These structures\, intr
oduced by Roe\, encode global properties of spaces. We also mention the eq
uivalent approach provided by Protasov and Banakh using balleans. Coarse s
tructures compatible with a group structure can be characterised by specia
l ideals of subsets\, called group ideals. While the coarse structure indu
ced 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 exemplified by the left coarse structure\, introduced by Rosendal\, and
the compact-group coarse structure\, induced by the group ideal of all re
latively compact subsets\, each suitable in disparate settings. \nWe prese
nt the large-scale geometry of groups via the historically iterative seque
nce of generalisations\, enlisting illustrative examples specific to disti
nct classes of groups and topological groups. We focus on locally compact
abelian groups endowed with compact-group coarse structures. In particular
\, we discuss the role of Pontryagin duality as a bridge between topologic
al properties and their large-scale counterparts. An overriding theme is a
n evidence-based tenet that the compact-group coarse structure is the righ
t choice for the category of locally compact abelian groups.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lydia Außenhofer (Universität Passau)
DTSTART:20200519T160000Z
DTEND:20200519T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/7
DESCRIPTION:Title: On the Mackey Topology of an Abelian Topological Group\nby Lydia Außenhofer (Universität Passau) as part of Topological Group
s\n\nLecture held in Elysium.\n\nAbstract\nFor a locally convex vector spa
ce $(V\,\\tau)$ there exists a finest locally convex vector space topolog
y $\\mu$ such that the topological dual spaces $(V\,\\tau)'$ and $(V\,\\mu
)'$ coincide algebraically. This topology is called the $Mackey$ $topology
$. If $(V\,\\tau)$ is a metrizable locally convex vector space\, then $\\t
au$ is the Mackey topology.\n\nIn 1995 Chasco\, Martín Peinador\, and T
arieladze asked\, "Given a locally quasi-convex group $(G\,\\tau)\,$ does
there exist a finest locally quasi-convex group topology $\\mu$ on $G$ suc
h that the character groups $(G\,\\tau)^\\wedge$ and $(G\,\\mu)^\\wedge$ c
oincide?"\n\nIn this talk we give examples of topological groups which\n\
n1. have a Mackey topology\,\n\n2. do not have a Mackey topology\,\n\nand
we characterize those abelian groups which have the property that every me
trizable locally quasi-convex group topology is Mackey (i.e.\, the finest
compatible locally quasi-convex group topology).\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Loth (Sacred Heart University)
DTSTART:20200526T160000Z
DTEND:20200526T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/8
DESCRIPTION:Title: Simply Given Compact Abelian Groups\nby Peter Loth (S
acred Heart University) as part of Topological Groups\n\nLecture held in E
lysium.\n\nAbstract\nA compact abelian group is called simply given if it
s Pontrjagin dual is simply presented. Warfield groups are defined to be d
irect summands of simply presented abelian groups. They were classified up
to isomorphism in terms of cardinal invariants by Warfield in the local c
ase\, and by Stanton and Hunter--Richman in the global case. In this talk\
, we classify up to topological isomorphism the duals of Warfield groups\,
dualizing Stanton's invariants. We exhibit an example of a simply given g
roup with nonsplitting identity component.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ajit Iqbal Singh (Indian National Science Academy)
DTSTART:20200609T160000Z
DTEND:20200609T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/9
DESCRIPTION:Title: Variants of Invariant Means of Amenability\nby Ajit I
qbal Singh (Indian National Science Academy) as part of Topological Groups
\n\nLecture held in Elysium.\n\nAbstract\nIt all started\, like many other
amazing theories\, in nineteen twenty-nine\,\nWith John von Neumann\, the
greatest 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 tra
nslation\, neither gaining nor losing any weight.\n\nMahlon M. Day\, in hi
s zest and jest\, giving double importance to semigroups\, too\,
\nTook up the study of conditions and properties\, and named it amenabi
lity.\nErling Folner followed it up\, more like a combinatorial maze to go
through\,\nWhether or not translated set meets the original in a sizeable
proportionality.\n\nHow could functional analysts sit quiet\, who measure
anything by their own norms\, \nLo and behold\, it kept coming back to th
e same concept over and over again.\nGroup algebras were just as good or b
ad\, approximate conditions did no harms\,\nWith the second duals of lofty
Richard Arens\, it became deeper\, but still a fun-game.\n\nEver since\,
with the whole alphabet names\, reputed experts or budding and slick\, \nC
onsidering several set-ups and numerous variants of the invariance.\nActio
ns on Manifolds or operators\, dynamical systems nimble or quick\,\nWe wil
l have a look at some old and some new\, closely or just from the fence.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Break (University of Hawaiʻi)
DTSTART:20200602T160000Z
DTEND:20200602T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/10
DESCRIPTION:Title: Topological Groups Seminar One-Week Hiatus\nby Break
(University of Hawaiʻi) as part of Topological Groups\n\nLecture held in
Elysium.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Riddhi Shah (Jawaharlal Nehru University)
DTSTART:20200616T160000Z
DTEND:20200616T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/11
DESCRIPTION:Title: Dynamics of Distal Actions on Locally Compact Groups
\nby Riddhi Shah (Jawaharlal Nehru University) as part of Topological Grou
ps\n\nLecture held in Elysium.\n\nAbstract\nDistal maps were introduced by
David Hilbert on compact spaces to study non-ergodic maps. A homeomorphis
m 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 dista
lly 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 autom
orphisms on locally compact groups and on homogeneous spaces given by quot
ients 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 t
erms of the distality of inner automorphisms.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karl Hofmann (Technische Universität Darmstadt)
DTSTART:20200623T160000Z
DTEND:20200623T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/12
DESCRIPTION:Title: The group algebra of a compact group and Tannaka duality
for compact groups\nby Karl Hofmann (Technische Universität Darmstad
t) as part of Topological Groups\n\nLecture held in Elysium.\n\nAbstract\n
In the 4th edition of the text- and handbook "The Structure of Compact Gro
ups"\,\nde Gruyter\, Berlin-Boston\, having appeared June 8\, 2020\, Sidne
y A. Morris and\nI decided to include\, among material not contained in ea
rlier editions\, the Tannaka-Hochschild Duality Theorem which says that $t
he$ $category$ $of$ $compact$ $groups$ $is\\\,dual$\n$to$ $the$ $category\
\\,of\\\,real\\\,reductive$ $Hopf$ $algebras$. In the lecture I hope to ex
plain\nwhy this theorem was not featured in the preceding 3 editions and w
hy we decided\nto present it now. Our somewhat novel access led us into a
new theory of real\nand complex group algebras for compact groups which I
shall discuss. Some Hopf\nalgebra theory appears inevitable. Recent source
: K.H.Hofmann and L.Kramer\,\n$On$ $Weakly\\\,Complete\\\,Group\\\,Algebra
s$ $of$ $Compact$ $Groups$\, J. of Lie Theory $\\bold{30}$ (2020)\, 407-42
4.\n\n Karl H. Hofmann\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Break (University of Hawaiʻi)
DTSTART:20200630T160000Z
DTEND:20200630T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/13
DESCRIPTION:Title: Topological Groups Seminar One-Week Hiatus\nby Break
(University of Hawaiʻi) as part of Topological Groups\n\nLecture held in
Elysium.\nAbstract: TBA\n
LOCATION:https://stable.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:20260404T094531Z
UID:TopologicalGroups/14
DESCRIPTION:Title: Groups Admitting Proper Actions by Affine Isometries on
Lp Spaces\nby Indira Chatterji (Laboratoire J.A. Dieudonné de l'Unive
rsité de Nice) as part of Topological Groups\n\nLecture held in Elysium.\
n\nAbstract\nIntroduction\, known results\, and open questions regarding g
roups admitting a proper action by affine isometries on an $L_p$ space.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ajay Kumar (University of Delhi)
DTSTART:20200714T160000Z
DTEND:20200714T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/15
DESCRIPTION:Title: Uncertainty Principles on Locally Compact Groups\nby
Ajay Kumar (University of Delhi) as part of Topological Groups\n\nLecture
held in Elysium.\n\nAbstract\nSome of the uncertainty principles on $ \\m
athbb{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 meas
ures of the sets $\\{x: f(x)\\neq 0 \\}$ and $ \\{\\xi : \\widehat{f}(\\
xi) \\neq 0\\}$ cannot both be finite.\n\nHardy'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 \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
functions 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 $$\n \\l
eft( \\int_{\\mathbb{R}^n}\\|x-a\\|^2|f(x)|^2 dx \\right) \\left( \\int_{\
\mathbb{R}^n}\\|\\xi-b\\|^2|\\widehat{f}(\\xi)|^2 d\\xi \\right) \\geq \\f
rac{n^2\\|f\\|^4}{16\\pi^2}.\n $$\n Beurling's Theorem: Let $f \\in L^1(
\\mathbb{R}^n) $ and for some $ k(1\\leq k\\leq n) $ satisfies\n $$\n \\in
t_{\\mathbb{R}^{2n}} |f(x_1\, x_2\, \\dots \, x_n)||\\widehat{f}(\\xi_1\,
\\xi_2\, \\dots \, \\xi_n)|e^{2\\pi |x_k\\xi_k|} dx_1\\dots dx_n d\\xi_1\\
dots d\\xi_n< \\infty.\n $$\n Then $f = 0$ a.e.\n\nWe investigate these pr
inciples on locally compact groups\, in particular Type I\ngroups and nilp
otent Lie groups for Fourier transform and Gabor transform.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:C.R.E. Raja (Indian Statistical Instititute)
DTSTART:20200721T160000Z
DTEND:20200721T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/16
DESCRIPTION:Title: Probability Measures and Structure of Locally Compact Gr
oups\nby C.R.E. Raja (Indian Statistical Instititute) as part of Topol
ogical Groups\n\nLecture held in Elysium.\n\nAbstract\nWe will have an ove
rview of how existence of certain types of\nprobability measures forces lo
cally compact groups to have particular\nstructures and vice versa. Examp
les are Choquet-Deny measures\, recurrent\nmeasures etc.\, and groups of t
he kind amenable\, polynomial growth\, etc.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dikran Dikranjan (University of Udine)
DTSTART:20200728T160000Z
DTEND:20200728T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/17
DESCRIPTION:Title: On a Class of Profinite Groups Related to a Theorem of P
rodanov\nby Dikran Dikranjan (University of Udine) as part of Topologi
cal Groups\n\nLecture held in Elysium.\n\nAbstract\nA short history of min
imal groups is given\, featuring illustrative examples and leading to curr
ent research:$\\newline$\n$\\quad$ * non-compact minimal groups\,$\\newlin
e$\n$\\quad$ * equivalence between minimality and essentiality of dense s
ubgroups of compact groups\,$\\newline$\n$\\quad$ * equivalence between m
inimality and compactness in LCA\, $\\newline$\n$\\quad$ * hereditary for
mulations of minimality facilitate optimal statements of theorems\, $\\new
line$\n$\\quad$ * a locally compact hereditarily locally minimal infinite
group $G$ is $\\newline$\n$\\quad$ $\\quad$ (a) $\\cong\\mathbb{Z}p$\, so
me prime $p$\, when $G$ is nilpotent\,$\\newline$\n$\\quad$ $\\quad$ (b) a
Lie group when $G$ is connected\,$\\newline$\n$\\quad$ * classification o
f hereditarily minimal locally compact solvable groups\,$\\newline$\n$\\qu
ad$ * existence of classes of hereditarily non-topologizable groups: $\\ne
wline$\n$\\quad$ $\\quad$ (a) bounded infinite finitely generated\,$\\newl
ine$\n$\\quad$ $\\quad$ (b) unbounded finitely generated\,$\\newline$\n$\\
quad$ $\\quad$ (c) countable not finitely generated\, $\\newline$\n$\\quad
$ $\\quad$ (d) uncountable.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Willis (University of Newcastle)
DTSTART:20200804T160000Z
DTEND:20200804T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/18
DESCRIPTION:Title: Totally disconnected locally compact groups and the scal
e\nby George Willis (University of Newcastle) as part of Topological G
roups\n\nLecture held in Elysium.\n\nAbstract\nThe scale is a positive\, i
nteger-valued function defined on any totally disconnected\, locally compa
ct (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. grou
ps\, the talk will introduce the scale and describe aspects of group struc
ture that it reveals. In particular\, the notions of tidy subgroup\, contr
action subgroup and flat subgroup of a t.d.l.c. will be explained and illu
strated with examples.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Helge Glöckner (Universität Paderborn)
DTSTART:20200811T160000Z
DTEND:20200811T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/19
DESCRIPTION:Title: Locally Compact Contraction Groups\nby Helge Glöckn
er (Universität Paderborn) as part of Topological Groups\n\nLecture held
in Elysium.\n\nAbstract\nConsider a locally compact group $G$\, together w
ith an automorphism $\\alpha$ which is $contractive$ in the sense that $\\
alpha^n\\rightarrow{\\rm id}_G$ pointwise as $n\\to\\infty$. Siebert showe
d 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 simply connected\, nilpotent real Lie group.\nI'll report on researc
h concerning the totally disconnected part\, obtained jointly with G. A. W
illis.\n\nFor each totally disconnected contraction group $(G\,\\alpha)$\,
the set ${\\rm tor} G$ of torsion elements is a closed subgroup of $G$. M
oreover\, $G$ is a direct product\n$$G=G_{p_1}\\times \\cdots\\times G_{p_
n}\\times {\\rm tor} G$$ of $\\alpha$-stable $p$-adic Lie groups $G_p$ for
certain primes $p_1\,\\ldots\, p_n$ and the torsion subgroup. The structu
re of $p$-adic contraction groups is known from the work of J. S. P. Wang\
; notably\, they are nilpotent. As shown with Willis\, ${\\rm tor} G$ admi
ts a composition series and there are countably many possible composition
factors\, parametrized by the finite simple groups. More recent research s
howed that there are uncountably many non-isomorphic torsion contraction g
roups\, but only countably many abelian ones. If a torsion contraction gro
up $G$ has a compact open subgroup which is a pro-$p$-group\, then $G$ is
nilpotent. Likewise if $G$ is locally pro-nilpotent.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elena Martín-Peinador (University of Madrid)
DTSTART:20200818T160000Z
DTEND:20200818T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/20
DESCRIPTION:Title: Group dualities: G-barrelled groups\nby Elena Martí
n-Peinador (University of Madrid) as part of Topological Groups\n\nLecture
held in Elysium.\n\nAbstract\nA natural notion in the framework of abelia
n groups are the group dualities. The most efficient definition of a grou
p duality is simply a pair $(G\, H)$\, where $G$ denotes an abstract abeli
an 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 eit
her 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
.\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 gro
ups. The fact that a locally convex topological vector space is in particu
lar a locally quasi-convex group serves as a nexus to emulate well-kno
wn results of Functional Analysis for the class of topological groups. \n\
nIn this talk we shall\ndeal with questions of the sort:\nUnder which co
nditions is there a locally compact topology in a fixed duality?\nThe sam
e question for a metrizable\, or a $k$-group topology.\nWe shall also intr
oduce the $g$-barrelled groups\, a class for which the Mackey-Arens Theo
rem admits an optimal counterpart. We study also the existence of $g$-barr
elled topologies in a group duality $(G\, H)$.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wayne Lewis (University of Hawaiʻi)
DTSTART:20200825T160000Z
DTEND:20200825T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/21
DESCRIPTION:Title: Classification of Periodic LCA Groups of Finite Non-Arch
imedean Dimension\nby Wayne Lewis (University of Hawaiʻi) as part of
Topological 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 c
ommon positive integer are classified via a complete set of topological is
omorphism invariants realized by an equivalence relation on pairs of exten
ded supernatural vectors.\n\nRemaining time will be devoted to a facilitat
ed discussion on how things are going this fall/winter academic semester i
n your part of the world as you see it.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dona Strauss (University of Leeds)
DTSTART:20200901T160000Z
DTEND:20200901T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/22
DESCRIPTION:Title: The Semigroup $\\beta S$\nby Dona Strauss (Universit
y of Leeds) as part of Topological Groups\n\nLecture held in Elysium.\n\nA
bstract\nIf $S$ is a discrete semigroup\, the semigroup operation on $S$ c
an be extended to a semigroup operation on its Stone–Čech compactificat
ion $\\beta S$. The properties of the semigroup $\\beta S$ have been a pow
erful tool in topological dynamics and combinatorics.\n \
nI shall give an introductory description of the semigroup $\\beta S$\, an
d show how its properties can be used to prove some of the classical theor
ems of Ramsey Theory.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bharat Talwar (University of Dehli)
DTSTART:20200908T160000Z
DTEND:20200908T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/23
DESCRIPTION:Title: Closed Lie Ideals and Center of Generalized Group Algebr
as\nby Bharat Talwar (University of Dehli) as part of Topological Grou
ps\n\nLecture held in Elysium.\n\nAbstract\nThe closed Lie ideals of the g
eneralized group algebra $L^1(G\,A)$ are characterized in terms of element
s of the group $G$\, elements of the algebra $A$\, and the modular functio
n $\\Delta$ of the group $G$. Conditions under which for a given closed Li
e 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\, follo
wed by a discussion regarding a very special projection in $L^1(G\,A)$. Fi
nally\, a few restrictions are imposed on $G$ and $A$ under which $\\mathc
al{Z}(L^1(G\,A))\\cong\\mathcal{Z}(L^1(G))\\otimes^\\gamma\\mathcal{Z}(A)$
.\n\nThe presentation is based on joint work with Ved Prakash Gupta and Ra
njana Jain.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nico Spronk (University of Waterloo)
DTSTART:20200915T160000Z
DTEND:20200915T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/24
DESCRIPTION:Title: Topologies\, idempotents and ideals\nby Nico Spronk
(University 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
bijection 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 functi
ons. This has applications to decomposing weakly almost periodic represen
tations on Banach spaces\, generalizing results which go back to many auth
ors.\n\nMoving to unitary representations\, I will develop the Fourier-Sti
eltjes algebra $B(G)$ of $G$\, and give the analogous result there. As an
application\, I show that for a locally compact connected group\, operato
r amenability of $B(G)$ implies that $G$ is compact\, partially resolving
a problem of interest for 25 years.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mukund Madhav Mishra (Hansraj College)
DTSTART:20200922T160000Z
DTEND:20200922T180000Z
DTSTAMP:20260404T094531Z
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 subclas
s of the class of nilpotent Lie groups. The Lie algebra of a stratified Li
e group possesses a specific stratification (and hence the name)\, and an
interesting class of anisotropic dilations. Among the linear differential
operators of degree two\, there exists a family that is well behaved with
the automorphisms of the stratified Lie group\, especially with the anisot
ropic dilations. We shall see that one such family of operators mimics the
classical Laplacian in many aspects\, except for the regularity. More spe
cifically\, these Laplace-like operators are sub-elliptic\, and hence refe
rred to as the sub-Laplacians. We will review certain interesting properti
es of functions harmonic with respect to the sub-Laplacian on a stratified
Lie group\, and have a closer look at a particular class of stratified Li
e groups known as the class of Heisenberg type groups.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sutanu Roy (National Institute of Science Education and Research)
DTSTART:20200929T160000Z
DTEND:20200929T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/26
DESCRIPTION:Title: Compact Quantum Groups and their Semidirect Products
\nby Sutanu Roy (National Institute of Science Education and Research) as
part of Topological Groups\n\nLecture held in Elysium.\n\nAbstract\nCompac
t quantum groups are noncommutative analogs of compact groups in the realm
of noncommutative geometry introduced by S. L. Woronowicz back in the 80s
. Roughly\, they are unital C*-bialgebras in the monoidal category (given
by the minimal tensor product) of unital C*-algebras with some additional
properties. For real 0<|q|<1\, q-deformations of SU(2) group are the first
and well-studied examples of compact quantum groups. These examples were
constructed independently by Vaksman-Soibelman and Woronowicz also back i
n the 80s. In fact\, they are examples of a particular class of compact qu
antum groups namely\, compact matrix pseudogroups. The primary goal of thi
s 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 briefly discuss the semidirect product construction for compact quan
tum groups via an explicit example. The second part of this will be based
on a joint work with Paweł Kasprzak\, Ralf Meyer and Stanislaw Lech Woron
owicz.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolò Zava (University of Udine)
DTSTART:20201006T160000Z
DTEND:20201006T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/27
DESCRIPTION:Title: Towards a unifying approach to algebraic and coarse entr
opy\nby Nicolò Zava (University of Udine) as part of Topological Grou
ps\n\nLecture held in Elysium.\n\nAbstract\nIn each situation\, entropy as
sociates to a self-morphism a value that estimates the chaos created by th
e map application. In particular\, the algebraic entropy $h_{alg}$ can be
computed for (continuous) endomorphisms of (topological) groups\, while th
e coarse entropy $h_c$ is associated to bornologous self-maps of locally f
inite coarse spaces. Those two entropy notions can be compared because of
the following observation. If $f$ is a (continuous) homomorphism of a (top
ological) group $G$\, then $f$ becomes automatically bornologous provided
that $G$ is equipped with the compact-group coarse structure. For an endom
orphism $f$ of a discrete group\, $h_{alg}(f)=h_c(f)$ if $f$ is surjective
\, while\, in general\, $h_{alg}(f)\\neq h_c(f)$. That difference occurs b
ecause in many cases\, if $f$ is not surjective\, then $h_c(f)=0$. \n\nIn
the first part of the talk\, after briefly recalling the large-scale geome
try of topological groups\, we define the coarse entropy and discuss its r
elationship with the algebraic entropy. The second part is dedicated to th
e introduction of the algebraic entropy of endomorphisms of $G$-sets (i.e.
\, sets endowed with group actions). We show that it extends the usual al
gebraic entropy of group endomorphisms and we provide evidence that it can
represent a useful modification and generalisation of the coarse entropy
that overcome the non-surjectivity issue.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Break (University of Hawaiʻi)
DTSTART:20201013T160000Z
DTEND:20201013T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/28
DESCRIPTION:Title: Topological Groups Seminar Two-Week Hiatus\nby Break
(University of Hawaiʻi) as part of Topological Groups\n\nLecture held in
Elysium.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Break (University of Hawaiʻi)
DTSTART:20201020T160000Z
DTEND:20201020T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/29
DESCRIPTION:Title: Topological Groups Seminar Two-Week Hiatus\nby Break
(University of Hawaiʻi) as part of Topological Groups\n\nLecture held in
Elysium.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wayne Lewis (University of Hawaiʻi)
DTSTART:20201027T160000Z
DTEND:20201027T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/30
DESCRIPTION:Title: Abelian Varieties as Algebraic Protori?\nby Wayne Le
wis (University of Hawaiʻi) as part of Topological Groups\n\nLecture held
in Elysium.\n\nAbstract\nAn outcome of the structure theory of protori (c
ompact connected abelian groups) is their representability as quotients of
$\\mathbb{A}^n$ for the ring of adeles $\\mathbb{A}$. $\\mathbb{A}$ does
not contain zeros of rational polynomials\, but rather representations of
zeros. Investigating the relations between algebraicity of complex tori an
d algebraicity of protori leads one to the problem of computing the Pontry
agin dual of $\\mathbb{A}/\\mathbb{Z}$. Applying an approach by Lenstra in
the setting of profinite integers to the more general $\\mathbb{A}$ leads
to a definition of the closed maximal $Lenstra$ $ideal$ $E$ of $\\mathbb{
A}$\, whence the locally compact field of $adelic$ $numbers$ $\\mathbb{F}=
\\mathbb{A}/E$\, providing a long-sought connection to $\\mathbb{C}$ enabl
ing one to define a functor from the category of complex tori to the categ
ory of protori - is it possible to do so in a way that preserves algebraic
ity? While $\\mathbb{F}$ marks tentative progress\, much work remains...\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wayne Lewis (University of Hawaiʻi)
DTSTART:20201103T160000Z
DTEND:20201103T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/31
DESCRIPTION:Title: Accounting with $\\mathbb{QP}^\\infty$\nby Wayne Lew
is (University of Hawaiʻi) as part of Topological Groups\n\nLecture held
in Elysium.\n\nAbstract\nRational projective space provides a useful accou
nting tool in engineering decompositions of $\\mathbb{Q}[x]$ for desired e
ffect. The device is useful for defining a correspondence between summands
of such a decomposition and elements of a partition of $\\mathbb{A}$. Thi
s mechanism is applied to a decomposition of $\\mathbb{Q}[x]$ relative to
which the correspondence gives the $Lenstra$ $ideal$ $E$\, a closed maxima
l ideal yielding the $adelic$ $numbers$ $\\mathbb{F}=\\frac{\\mathbb{A}}{E
}$.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Monique Chyba (University of Hawaiʻi)
DTSTART:20201110T160000Z
DTEND:20201110T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/32
DESCRIPTION:Title: Journey in Hawaii's Challenges in the Fight Against COVI
D-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 from the first infectious disease that Hawaiʻi had to deal with. Duri
ng the 1918-1920 Influenza Pandemic\, the Hawaiian islands were not spared
as the disease ravaged through the whole world. Hawaiʻi and similar isla
nd populations can follow a different course of pandemic spread than large
cities/states/nations and are often neglected in major studies. It may be
too early to compare the 1918-1920 Influenza Pandemic and COVID-19 Pande
mic\, we do however note some similarities and differences between the two
pandemics.\n\nHawaiʻi and other US Islands have recently been noted by t
he media as COVID-19 hotspots after a relatively calm period of low case r
ates. U.S. Surgeon General Jerome Adams came in person on August 25 to Oah
u to address the alarming situation. We will discuss the peculiarity of th
e situation in Hawaiʻi and provide detailed modeling of current virus spr
ead patterns aligned with dates of lockdown and similar measures. We will
present a detailed epidemiological model of the spread of COVID-19 in Hawa
iʻi and explore effects of different intervention strategies in both a pr
ospective and retrospective fashion. Our simulations demonstrate that to c
ontrol the spread of COVID-19 both actions by the State in terms of testin
g\, contact tracing and quarantine facilities as well as individual action
s by the population in terms of behavioral compliance to wearing a mask an
d gathering in groups are vital. They also explain the turn for the worst
Oahu took after a very successful stay-at-home order back in March.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seid Kassaw (University of Cape Town)
DTSTART:20201124T160000Z
DTEND:20201124T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/33
DESCRIPTION:Title: The probability of commuting subgroups in arbitrary latt
ices of subgroups\nby Seid Kassaw (University of Cape Town) as part of
Topological Groups\n\nLecture held in Elysium.\n\nAbstract\nThe subgroup
commutativity degree $sd(G)$ of a finite group $G$ was introduced\nalmost
ten years ago and deals with the number of commuting subgroups in the\nsub
group lattice $L(G)$ of $G$. The extremal case $sd(G) = 1$ detects a class
of groups\nclassified by Iwasawa in 1941 (in fact\, $sd(G)$ represents a
probabilistic measure which\nallows us to understand how far $G$ is from t
he groups of Iwasawa). This means\n$sd(G) = 1$ if and only if $G$ is the d
irect product of its Sylow $p$-subgroups and these\nare all modular\; or e
quivalently $G$ is a nilpotent modular group. Therefore\, $sd(G)$ is\nstro
ngly related to structural properties of $L(G)$ and $G$.\n\nIn this talk\,
we introduce a new notion of probability $gsd(G)$ in which two arbitrary
sublattices $S(G)$ and $T(G)$ of $L(G)$ are involved simultaneously. In ca
se\n$S(G) = T(G) = L(G)$\, we find exactly $sd(G)$. Upper and lower bounds
for $gsd(G)$\nare shown and we study the behaviour of $gsd(G)$ with respe
ct to subgroups and\nquotients\, showing new numerical restrictions. We pr
esent the commutativity\nand subgroup commutativity degree for infinite gr
oups and put some open problems\nfor further generalization.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Farzana Nasrin (University of Hawaiʻi)
DTSTART:20201201T160000Z
DTEND:20201201T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/34
DESCRIPTION:Title: Bayesian Statistics\, Topology and Machine Learning for
Complex Data Analysis\nby Farzana Nasrin (University of Hawaiʻi) as p
art of Topological Groups\n\nLecture held in Elysium.\n\nAbstract\nAnalyzi
ng and classifying large and complex datasets are generally challenging. T
opological data analysis\, that builds on techniques from topology\, is a
natural fit for this. Persistence diagram is a powerful tool that origina
ted in topological data analysis that allows retrieval of important topolo
gical and geometrical features latent in a dataset. Data analysis and clas
sification involving persistence diagrams have been applied in numerous ap
plications. In this talk\, I will provide a brief introduction of topologi
cal data analysis\, focusing primarily on persistence diagrams\, and a Bay
esian framework for inference with persistence diagrams. The goal is to pr
ovide a supervised machine learning algorithm in the space of persistence
diagrams. This framework is applicable to a wide variety of datasets. I wi
ll present applications in materials science\, biology\, and neuroscience.
\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikhail Tkachenko (Metropolitan Autonomous University)
DTSTART:20201208T160000Z
DTEND:20201208T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/35
DESCRIPTION:Title: Pseudocompact Paratopological and Quasitopological Group
s\nby Mikhail Tkachenko (Metropolitan Autonomous University) as part o
f Topological Groups\n\nLecture held in Elysium.\n\nAbstract\nPseudocompac
tness is an interesting topological property which acquires very specific
\nfeatures when applied to different algebrotopological objects. A celebra
ted theorem\nof Comfort and Ross published in 1966 states that the Cartesi
an product of an arbitrary\nfamily of pseudocompact topological groups is
pseudocompact. We present a survey \nof results related to the validity or
failure of the Comfort-Ross' theorem in the realm of \nsemitopological an
d paratopological groups and give some examples showing that \npseudocompa
ctness fails to be stable when taking products of quasitopological groups.
\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph A. Wolf (University of California)
DTSTART:20210126T160000Z
DTEND:20210126T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/36
DESCRIPTION:Title: Gelfand Pairs\nby Joseph A. Wolf (University of Cali
fornia) as part of Topological Groups\n\nLecture held in Elysium.\n\nAbstr
act\nGelfand pairs are at the intersection of locally compact group theory
\, differential geometry and harmonic analysis. In this talk I'll give so
me examples and sketch how the Plancherel Theorem \nfor Gelfand pairs is p
retty much the same as Pontryagin Duality for locally compact abelian grou
ps.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Fournier-Facio (ETH Zürich)
DTSTART:20210202T160000Z
DTEND:20210202T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/37
DESCRIPTION:Title: Normed p-adic amenability and bounded cohomology\nby
Francesco Fournier-Facio (ETH Zürich) as part of Topological Groups\n\nL
ecture held in Elysium.\n\nAbstract\nAmenability is a property of a locall
y compact group that allows to average bounded real-valued functions. It i
s closely related to continuous bounded cohomology\, a fundamental tool in
rigidity theory. Both notions are very Archimedean in nature\, in that th
ey 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 r
eplace these by normed vector spaces over the p-adics\, which are ultramet
ric. We will see that the corresponding notion of amenability (which needs
to be defined a little differently) is much more restrictive than the usu
al one\, and that bounded cohomology is often close to ordinary continuous
cohomology.\n\nNo previous knowledge of bounded cohomology is assumed.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eniola Kazeem (University of Cape Town)
DTSTART:20210209T160000Z
DTEND:20210209T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/38
DESCRIPTION:Title: Subgroup Commutativity Degree of Profinite Groups\nb
y Eniola Kazeem (University of Cape Town) as part of Topological Groups\n\
nLecture held in Elysium.\n\nAbstract\nWe find a probability measure which
counts the pairs of closed commuting subgroups in infinite groups. This m
easure turns out to be an extension of what was known in the finite case a
s subgroup commutativity degree. The extremal case of probability one desc
ribes the so-called topologically quasihamiltonian groups and is a useful
tool in describing the distance of a profinite group from this special cla
ss. We have been inspired by an idea of Heyer in the context of our proble
m.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Russo (University of Cape Town)
DTSTART:20210216T160000Z
DTEND:20210216T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/39
DESCRIPTION:Title: A probabilistic measure for the number of commuting subg
roups in locally compact groups\nby Francesco Russo (University of Cap
e Town) as part of Topological Groups\n\nLecture held in Elysium.\n\nAbstr
act\nIn the present talk I will construct a probability measure which coun
ts the pairs of closed commuting subgroups in locally compact groups\, ext
ending in this way the so-called ``subgroup commutativity degree'' of pr
ofinite groups\, and in particular\, of finite groups. The extremal case o
f probability one characterizes topologically quasihamiltonian groups\, in
troduced by K. Iwasawa in 1941. We use a general approach\, involving mea
sure theory and\, in particular\, we study the number of closed commuting
subgroups in locally compact groups and pro-Lie groups with a correspondin
g description of their Chabauty spaces. \n\nTime permitting I will discuss
a classical example of Andersen and Jessen\, which shows some natural lim
itations in the techniques of construction when we involve the notion of p
resheaves of measure spaces.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tea N. Coffee (University of Hawaiʻi)
DTSTART:20210223T160000Z
DTEND:20210223T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/40
DESCRIPTION:Title: Two Years Forward\, One Year Back\nby Tea N. Coffee
(University of Hawaiʻi) as part of Topological Groups\n\nLecture held in
Elysium.\n\nAbstract\nTea & Coffee:\nIn lieu of our usual presentation...s
hifting the tectonic plates of mathematics...we take a guided look back at
the last \nyear of presentation topics and speakers. Come and go as you p
lease as we informally meet and pass the ball around the table.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph A. Wolf (University of California)
DTSTART:20210302T160000Z
DTEND:20210302T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/41
DESCRIPTION:Title: Uncertainty Principles for Gelfand Pairs\nby Joseph
A. Wolf (University of California) as part of Topological Groups\n\nLectur
e held in Elysium.\n\nAbstract\nI'll sketch an extension of the classical
Uncertainty Principle to the context\nof Gelfand pairs. The Gelfand pair
setting includes Riemannian symmetric\nspaces\, compact topological groups
\, locally compact abelian groups\, and\nhomogeneous graphs. In the case
of the locally compact group R^n one\nrecovers a sharp form of the signal
processing version of the classical\nHeisenberg uncertainty principle. If
time permits I'll briefly indicate \nsome applications to spherical funct
ions on Riemannian symmetric spaces\,\nto Cayley complexes\, and to hyperg
roups.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manoj Kummini (Chennai Mathematical Institute)
DTSTART:20210309T160000Z
DTEND:20210309T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/42
DESCRIPTION:Title: Polynomial invariant rings\nby Manoj Kummini (Chenna
i Mathematical Institute) as part of Topological Groups\n\nLecture held in
Elysium.\n\nAbstract\nLet $G$ be a finite group acting linearly on a\nfin
ite-dimensional vector-space $V$ over a field $K$. Let $R$ denote\nthe sym
metric algebra on $V^*$\; then $G$ acts as graded $K$-algebra\nautomorphis
ms on $R$. If $R^G$ is a polynomial ring\, then $G$ is\ngenerated by eleme
nts that act as pseudo-reflections on $V$. The\nconverse holds when $|G|$
is invertible in $K$. The above results are\nShephard-Todd-Chevalley-Serre
theorem. If $\\mathrm{char}(K) = p>0$ and\n$G$ is a $p$-group\, then a co
njecture 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 i
s 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
$R$ generated by positive\ndegree elements in $R^G$) is a complete inters
ection ideal. This is\njoint work with Mandira Mondal.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adolf Mader (University of Hawaiʻi)
DTSTART:20210316T160000Z
DTEND:20210316T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/43
DESCRIPTION:Title: Free Subgroups with Torsion Quotients and Profinite Subg
roups with Torus Quotients: Arbitrary Rank\nby Adolf Mader (University
of Hawaiʻi) as part of Topological Groups\n\nLecture held in Elysium.\n\
nAbstract\nCompact connected groups contain $\\delta$-subgroups\, that is\
, compact\ntotally disconnected subgroups with torus quotients\, which are
essential ingredients in the important Resolution Theorem\, a description
of compact groups.\nDually\, free subgroups with torsion quotient of disc
rete torsion-free groups\nare studied in order to obtain a comprehensive p
icture of the abundance of \n$\\delta$-subgroups of arbitrary compact conn
ected abelian groups. Previously we\nconsidered finite dimensional groups
$G$ only but even for these we obtain new\nresults on the canonical subgro
up $\\boldsymbol{\\Delta}_G$ which is the sum of all \n$\\delta$-subgroups
of G. We also study a topology on torsion-free abelian groups \nfor which
the free subgroups with torsion quotient form a neighborhood basis at 0.\
n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wolfgang Herfort (TU Wien)
DTSTART:20210323T160000Z
DTEND:20210323T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/44
DESCRIPTION:Title: LCA p-groups in which every maximal monothetic subgroup
splits\nby Wolfgang Herfort (TU Wien) as part of Topological Groups\n\
nLecture held in Elysium.\n\nAbstract\nA well-known result about any disc
rete abelian $p$-group $G$\nby Kulikov states that every element of finite
$p$-height \nis contained in a maximal cyclic subgroup\, a direct summand
of $G$.\nMoreover\, every bounded pure subgroup is a direct summand of $G
$.\nThese 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 i
s contained in some maximal monothetic\nsubgroup.\n\n(N.2) Every maximal m
onothetic 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\nmonothet
ic subgroup of $G$ has order $p^e$.\nWe will discuss some examples first a
nd 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 \nstatemen
ts 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\nA
bout the structure in (i) we conjecture that \n$G=\\Z_p^I$ for some set $I
$ and we let $C$ be any $p$-adically\nclosed subgroup of $G$ with $G/C$ to
rsion. \nThen the topology on $G$ is redefined\nby letting $C$ be an open
compact subgroup.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rafael Dahmen (Karlsruher Institut für Technologie)
DTSTART:20210330T160000Z
DTEND:20210330T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/45
DESCRIPTION:Title: Long Direct Limits of Topological Groups\nby Rafael
Dahmen (Karlsruher Institut für Technologie) as part of Topological Group
s\n\nLecture held in Elysium.\n\nAbstract\nGiven a directed system of topo
logical groups\, one can consider the direct limit (colimit) in the catego
ry of topological 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 topolo
gical groups is different from the colimit in the category of topological
spaces. In this talk\, I want to present some well-known results on when t
his pathology occurs in the case of countable directed systems -- as well
as some newer results on certain uncountable systems (called "long directe
d systems") which behave very differently than countable ones. This will b
e illustrated by some (hopefully) motivating examples. This is joint work
with Gábor Lukács.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pratulananda Das (Jadavpur University)
DTSTART:20210413T160000Z
DTEND:20210413T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/46
DESCRIPTION:Title: On certain "differently" characterized subgroups of the
circle group\nby Pratulananda Das (Jadavpur University) as part of Top
ological Groups\n\nLecture held in Elysium.\n\nAbstract\nThe talk is based
on recent joint works with Prof. Dikran Dikranjan\, Prof. Wei He\, my Ph.
D. students Kumardipta Bose and Ayan Ghosh. In this talk\, we will discuss
certain new versions of characterized subgroups of the circle group $\\ma
thbb{T}$ which we name as statistically characterized subgroups and $\\alp
ha$-statistically characterized subgroups. Our investigations show that th
ese subgroups (for arithmetic sequences of integers) are essentially diff
erent and strictly larger in size than the much investigated class of char
acterized subgroups\, having cardinality $\\mathfrak{c}$ but remaining non
trivial (i.e. different from $\\mathbb{T}$) though remaining topologically
nice. As a natural consequence\, we consider an extended version of Armac
ost's problem of ``description of topologically torsion elements" of the c
ircle group and describe topologically $s$-torsion and topologically $\\al
pha$-torsion elements (which form the statistically characterized subgroup
s and $\\alpha$-statistically characterized subgroups respectively) in ter
ms of the support and provide a complete solution.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linus Kramer (Universität Münster)
DTSTART:20210427T160000Z
DTEND:20210427T180000Z
DTSTAMP:20260404T094531Z
UID:TopologicalGroups/47
DESCRIPTION:Title: Automatic continuity for topological groups\nby Linu
s Kramer (Universität Münster) as part of Topological Groups\n\nLecture
held in Elysium.\n\nAbstract\nAn abstract homomorphism $f\\colon G\\righta
rrow H$ between topological groups \nis a group homomorphism which is not
assumed to be continuous. I will\ndiscuss old and new results which say th
at under certain assumptions\non $G$ or $H$\, such an abstract homomorphis
m is automatically continuous.\n\nThis is based on joint work with Karl H.
Hofmann and Olga Varghese.\n
LOCATION:https://stable.researchseminars.org/talk/TopologicalGroups/47/
END:VEVENT
END:VCALENDAR