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SUMMARY:Juan Manuel Lorenzo Naveiro (University of Santiago de Compostela)
DTSTART;VALUE=DATE-TIME:20211201T130000Z
DTEND;VALUE=DATE-TIME:20211201T140000Z
DTSTAMP;VALUE=DATE-TIME:20240329T115803Z
UID:SymmetricSpacesSeminar/2
DESCRIPTION:Title: Introduction to Polar Actions\nby Juan Manuel Lorenzo N
aveiro (University of Santiago de Compostela) as part of Symmetric Spaces
Seminar\n\n\nAbstract\nAn isometric action of a Lie group on a complete Ri
emannian manifold is said to be polar if there exists a submanifold that i
ntersects every orbit orthogonally. Such a submanifold is known as a secti
on and is totally geodesic. These actions give a generalization of several
well-known concepts in geometry\, such as the polar coordinate system in
the Euclidean plane or the Spectral Theorem for self-adjoint operators.\n\
nThe aim of this talk is to explore several examples and properties of pol
ar actions\, with an application to the theory of real semisimple Lie alge
bras. From these properties\, one can obtain an explicit description of th
eir sections. Afterwards\, we will derive an algebraic criterion to determ
ine if a given action is polar when the ambient manifold is a symmetric sp
ace of compact (or noncompact) type.\n
LOCATION:https://researchseminars.org/talk/SymmetricSpacesSeminar/2/
END:VEVENT
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SUMMARY:Luis Pedro Castellanos Moscoso (Osaka City University)
DTSTART;VALUE=DATE-TIME:20211215T130000Z
DTEND;VALUE=DATE-TIME:20211215T140000Z
DTSTAMP;VALUE=DATE-TIME:20240329T115803Z
UID:SymmetricSpacesSeminar/4
DESCRIPTION:Title: Moduli spaces and left-invariant symplectic structures on L
ie groups\nby Luis Pedro Castellanos Moscoso (Osaka City University) a
s part of Symmetric Spaces Seminar\n\n\nAbstract\nIn the setting of Lie gr
oups\, it is natural to ask about the existence of left-invariant structur
es. A symplectic Lie group is a Lie group endowed with a left-invariant sy
mplectic form. There are interesting results on the structure of symplecti
c Lie groups and some classifications in low dimensions\, but the general
picture is far from complete.\n\nIn this talk we present an approach to cl
assify left-invariant symplectic structures on Lie groups. The procedure i
s based on the moduli space of left-invariant nondegenerate 2-forms\, whic
h is a certain orbit space in the set of all nondegenerate 2-forms on a Li
e algebra. We present some of the results obtained so far with this approa
ch\, including a classification of left-invariant symplectic structures on
some almost abelian Lie algebras.\n
LOCATION:https://researchseminars.org/talk/SymmetricSpacesSeminar/4/
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BEGIN:VEVENT
SUMMARY:Ivan Solonenko (King's College London - LSGNT)
DTSTART;VALUE=DATE-TIME:20220126T130000Z
DTEND;VALUE=DATE-TIME:20220126T140000Z
DTSTAMP;VALUE=DATE-TIME:20240329T115803Z
UID:SymmetricSpacesSeminar/5
DESCRIPTION:Title: The index of symmetry of a Riemannian manifold\nby Ivan
Solonenko (King's College London - LSGNT) as part of Symmetric Spaces Sem
inar\n\n\nAbstract\nRiemannian symmetric spaces constitute arguably the mo
st well-understood class of Riemannian manifolds\, mainly because they can
be extensively studied and ultimately classified by means of Lie theory.
To an arbitrary Riemannian manifold\, one can naturally assign a number\,
called the index of symmetry\, which measures how badly the manifold fails
to be a symmetric space. If the manifold is complete\, it admits a natura
l (in general\, singular) foliation\, called the foliation of symmetry\, w
hich is invariant under isometries and such that the index of symmetry of
the manifold is precisely the dimension of the smallest leaf. What is more
\, each leaf turns out to be a symmetric space in the induced metric. In c
ase the manifold is homogeneous\, its foliation of symmetry has all its le
aves of the same dimension and thus becomes a genuine foliation. Although
its construction is very natural and intrinsic\, the foliation (and thus t
he index) of symmetry is fairly hard to compute even for homogeneous space
s. \n\nIn this talk\, I will define the foliation of symmetry and introduc
e its main properties and then tell about some situations when it can be e
xplicitly computed\, namely for normal compact homogeneous spaces and a ce
rtain class of compact naturally reductive spaces. I will also present the
classification of compact homogeneous spaces with a sufficiently high ind
ex of symmetry. I will follow the papers by Olmos\, Reggiani\, Tamaru\, an
d Berndt [2014\, 2017].\n
LOCATION:https://researchseminars.org/talk/SymmetricSpacesSeminar/5/
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BEGIN:VEVENT
SUMMARY:Juan Manuel Lorenzo Naveiro (University of Santiago de Compostela)
DTSTART;VALUE=DATE-TIME:20220202T130000Z
DTEND;VALUE=DATE-TIME:20220202T140000Z
DTSTAMP;VALUE=DATE-TIME:20240329T115803Z
UID:SymmetricSpacesSeminar/6
DESCRIPTION:Title: Symmetric spaces of noncompact type I: The Cartan and Iwasa
wa decompositions\nby Juan Manuel Lorenzo Naveiro (University of Santi
ago de Compostela) as part of Symmetric Spaces Seminar\n\n\nAbstract\nThe
main property of symmetric spaces is the fact that they can be described i
n terms of two Lie groups together with an involution representing the geo
desic symmetry at a given point. Because of this\, one can extensively stu
dy the geometry of these manifolds by means of purely algebraic methods. D
uring the course of this and the next two talks\, Tomás Otero\, Ivan Solo
nenko\, and I will introduce some of those methods in the context of symme
tric spaces of noncompact type.\n\nThroughout this first talk\, we will ex
hibit two decompositions of the isometry group and its Lie algebra for suc
h manifolds. The first one is the Cartan decomposition (valid for any symm
etric space)\, which gives a way to express some invariants of a symmetric
space (connection\, curvature\, geodesics\, etc.) in Lie-theoretic terms.
On the other hand\, one has the Iwasawa decomposition (only valid in the
noncompact setting)\, which serves as a generalization of the Gram-Schmidt
process for the isometries of a noncompact symmetric space and allows to
realize the space as a simply connected solvable Lie group with a left-inv
ariant metric.\n
LOCATION:https://researchseminars.org/talk/SymmetricSpacesSeminar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivan Solonenko (King's College London - LSGNT)
DTSTART;VALUE=DATE-TIME:20220209T130000Z
DTEND;VALUE=DATE-TIME:20220209T140000Z
DTSTAMP;VALUE=DATE-TIME:20240329T115803Z
UID:SymmetricSpacesSeminar/7
DESCRIPTION:Title: Symmetric spaces of noncompact type II: Roots and Dynkin di
agrams\nby Ivan Solonenko (King's College London - LSGNT) as part of S
ymmetric Spaces Seminar\n\n\nAbstract\nLast week we discussed some basic g
eometric properties of symmetric spaces of noncompact type in the context
of the Cartan and Iwasawa decompositions. This time\, we will look more cl
osely at the algebraic side of the picture. We will dive in greater detail
into the restricted root space decomposition of the isometry Lie algebra
of a symmetric space of noncompact type and study its main properties. The
re are two stark distinctions between this decomposition and the – proba
bly more well-known – root space decomposition of a complex semisimple L
ie algebra: restricted root systems may be nonreduced\, while restricted r
oot subspaces may have dimension greater than 1. Somewhat surprisingly\, b
y allowing a root system to be nonreduced\, one only gets one additional s
eries in the classification of irreducible root systems\, namely the nonre
duced system (BC)_r. We will examine the classification of noncompact symm
etric spaces through the lens of restricted root systems and their Dynkin
diagrams. Since irreducible noncompact symmetric spaces are essentially in
one-to-one correspondence with simple noncompact real Lie algebras\, this
will give us a nice perspective on the classification of the latter. Fina
lly\, we will investigate the Weyl and automorphism groups of the restrict
ed root system and observe how they can be thought of in terms of isometri
es of the underlying space.\n
LOCATION:https://researchseminars.org/talk/SymmetricSpacesSeminar/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomás Otero Casal (University of Santiago de Compostela)
DTSTART;VALUE=DATE-TIME:20220216T130000Z
DTEND;VALUE=DATE-TIME:20220216T140000Z
DTSTAMP;VALUE=DATE-TIME:20240329T115803Z
UID:SymmetricSpacesSeminar/8
DESCRIPTION:Title: Symmetric spaces of noncompact type III: Parabolic subgroup
s and subalgebras\nby Tomás Otero Casal (University of Santiago de Co
mpostela) as part of Symmetric Spaces Seminar\n\n\nAbstract\nParabolic sub
groups are of special interest when studying isometric actions on symmetri
c spaces of noncompact type. Geometrically speaking\, given a symmetric sp
ace of noncompact type M=G/K\, proper parabolic subgroups of G are the sta
bilizers of points at infinity of M. Throughout this third talk\, we will
introduce parabolic subalgebras from both the geometric and algebraic view
points and explain how one can construct "standard" parabolic subalgebras
from a choice of a subset of simple roots.\n\nWe will also talk about some
decomposition results for parabolic Lie subalgebras and the decomposition
s they induce at the group/manifold level. In particular\, this will allow
us to introduce a special type of totally geodesic submanifolds of M call
ed boundary components\, which are symmetric spaces behaving in an especia
lly nice way with respect to the root space decomposition.\n
LOCATION:https://researchseminars.org/talk/SymmetricSpacesSeminar/8/
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BEGIN:VEVENT
SUMMARY:Alberto Rodríguez-Vázquez (University of Santiago de Compostela)
DTSTART;VALUE=DATE-TIME:20220302T130000Z
DTEND;VALUE=DATE-TIME:20220302T140000Z
DTSTAMP;VALUE=DATE-TIME:20240329T115803Z
UID:SymmetricSpacesSeminar/9
DESCRIPTION:Title: Totally geodesic submanifolds and Hermitian symmetric space
s\nby Alberto Rodríguez-Vázquez (University of Santiago de Compostel
a) as part of Symmetric Spaces Seminar\n\n\nAbstract\nTotally geodesic sub
manifolds in symmetric spaces are those submanifolds with the simplest geo
metry and they admit a nice algebraic characterization in terms of Lie tri
ple systems. A special class of symmetric spaces where totally geodesic su
bmanifolds can be studied is that of Hermitian symmetric spaces. In these
spaces we can use the notion of Kähler angle to measure how a submanifold
fails to be complex.\n\nIn this talk\, I will report on an ongoing work w
here a method to construct totally geodesic submanifolds with non-trivial
constant Kähler angle in non-flat Hermitian symmetric spaces is given.\n
LOCATION:https://researchseminars.org/talk/SymmetricSpacesSeminar/9/
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BEGIN:VEVENT
SUMMARY:Yuji Kondo (Hiroshima University)
DTSTART;VALUE=DATE-TIME:20220309T130000Z
DTEND;VALUE=DATE-TIME:20220309T140000Z
DTSTAMP;VALUE=DATE-TIME:20240329T115803Z
UID:SymmetricSpacesSeminar/10
DESCRIPTION:Title: A classification of left-invariant pseudo-Riemannian metri
cs on some nilpotent Lie groups\nby Yuji Kondo (Hiroshima University)
as part of Symmetric Spaces Seminar\n\n\nAbstract\nIt is known that a conn
ected and simply-connected Lie group admits only one left-invariant Rieman
nian metric up to scaling and isometry if and only if it is isomorphic to
the Euclidean space\, the Lie group of the real hyperbolic space\, or the
direct product of the three dimensional Heisenberg group and the Euclidean
space of dimension n-3.\n\nIn this talk\, I am going to talk about a clas
sification of left-invariant pseudo-Riemannian metrics of an arbitrary sig
nature for the third Lie groups with n>3 up to scaling and automorphisms.
This completes the classifications of left-invariant pseudo-Riemannian met
rics for the above three Lie groups up to scaling and automorphisms.\n\nOu
r classification can be obtained by a certain isometric action of cohomoge
neity zero on a pseudo-Riemannian symmetric space. This action is not prop
er\, in fact there exists a non-closed orbit\, which allows us to consider
degenerations of orbits. Finally\, I explain that degenerations of orbits
can have an implication with respect to curvature properties.\n\nThis stu
dy is based on the joint work with Hiroshi Tamaru from Osaka City Universi
ty.\n
LOCATION:https://researchseminars.org/talk/SymmetricSpacesSeminar/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuri Nikolayevsky (La Trobe University\, Melbourne)
DTSTART;VALUE=DATE-TIME:20220406T090000Z
DTEND;VALUE=DATE-TIME:20220406T100000Z
DTSTAMP;VALUE=DATE-TIME:20240329T115803Z
UID:SymmetricSpacesSeminar/11
DESCRIPTION:Title: Einstein hypersurfaces in irreducible symmetric spaces
\nby Yuri Nikolayevsky (La Trobe University\, Melbourne) as part of Symmet
ric Spaces Seminar\n\n\nAbstract\nIn this talk\, I will present the result
s of the joint paper of Jeong Hyeong Park and myself in which we give a cl
assification of Einstein hypersurfaces in irreducible symmetric spaces. Th
e main theorem states that there are three classes of such hypersurfaces\,
belonging to three very different "geometries": homogeneous geometry\, Le
gendrian geometry and affine geometry. I will give a brief introduction in
to these three geometries and explain how they fit together in our classif
ication.\n
LOCATION:https://researchseminars.org/talk/SymmetricSpacesSeminar/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Miguel Domínguez-Vázquez (University of Santiago de Compostela)
DTSTART;VALUE=DATE-TIME:20220420T120000Z
DTEND;VALUE=DATE-TIME:20220420T130000Z
DTSTAMP;VALUE=DATE-TIME:20240329T115803Z
UID:SymmetricSpacesSeminar/12
DESCRIPTION:Title: Polar foliations on symmetric spaces\nby Miguel Domín
guez-Vázquez (University of Santiago de Compostela) as part of Symmetric
Spaces Seminar\n\n\nAbstract\nA polar foliation is a decomposition of a Ri
emannian manifold into equidistant submanifolds (called leaves) of possibl
y different dimensions\, and such that through any point there exists a su
bmanifold (called section) intersecting all leaves perpendicularly. These
objects arise as generalizations of important concepts\, such as isoparame
tric hypersurfaces and polar actions\, whose study has produced many beaut
iful and profound results over the last decades. In this expository talk I
will present an introduction to polar foliations and isoparametric hypers
urfaces in symmetric spaces\, and report on some results concerning their
classification problem.\n
LOCATION:https://researchseminars.org/talk/SymmetricSpacesSeminar/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:José Carlos Díaz-Ramos (University of Santiago de Compostela)
DTSTART;VALUE=DATE-TIME:20220504T120000Z
DTEND;VALUE=DATE-TIME:20220504T130000Z
DTSTAMP;VALUE=DATE-TIME:20240329T115803Z
UID:SymmetricSpacesSeminar/13
DESCRIPTION:Title: Cohomogeneity one actions on quaternionic hyperbolic space
s\nby José Carlos Díaz-Ramos (University of Santiago de Compostela)
as part of Symmetric Spaces Seminar\n\n\nAbstract\nIn this talk I will pre
sent the classification procedure to obtain the classification of cohomoge
neity one actions on quaternionic hyperbolic spaces\, as well as some bypr
oducts of this study\, such as the existence of inhomogeneous isoparametri
c hypersurfaces with constant principal curvatures.\n
LOCATION:https://researchseminars.org/talk/SymmetricSpacesSeminar/13/
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