BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Juan Manuel Lorenzo Naveiro (University of Santiago de Compostela) DTSTART:20211201T130000Z DTEND:20211201T140000Z DTSTAMP:20260404T111009Z UID:SymmetricSpacesSeminar/2 DESCRIPTION:Title: Introduction to Polar Actions\nby Juan Manuel Lo renzo Naveiro (University of Santiago de Compostela) as part of Symmetric Spaces Seminar\n\n\nAbstract\nAn isometric action of a Lie group on a comp lete Riemannian manifold is said to be polar if there exists a submanifold that intersects every orbit orthogonally. Such a submanifold is known as a section and is totally geodesic. These actions give a generalization of several well-known concepts in geometry\, such as the polar coordinate sys tem in the Euclidean plane or the Spectral Theorem for self-adjoint operat ors.\n\nThe aim of this talk is to explore several examples and properties of polar actions\, with an application to the theory of real semisimple L ie algebras. From these properties\, one can obtain an explicit descriptio n of their sections. Afterwards\, we will derive an algebraic criterion to determine if a given action is polar when the ambient manifold is a symme tric space of compact (or noncompact) type.\n LOCATION:https://stable.researchseminars.org/talk/SymmetricSpacesSeminar/2 / END:VEVENT BEGIN:VEVENT SUMMARY:Luis Pedro Castellanos Moscoso (Osaka City University) DTSTART:20211215T130000Z DTEND:20211215T140000Z DTSTAMP:20260404T111009Z UID:SymmetricSpacesSeminar/4 DESCRIPTION:Title: Moduli spaces and left-invariant symplectic structur es on Lie groups\nby Luis Pedro Castellanos Moscoso (Osaka City Univer sity) as part of Symmetric Spaces Seminar\n\n\nAbstract\nIn the setting of Lie groups\, it is natural to ask about the existence of left-invariant s tructures. A symplectic Lie group is a Lie group endowed with a left-invar iant symplectic form. There are interesting results on the structure of sy mplectic Lie groups and some classifications in low dimensions\, but the g eneral picture is far from complete.\n\nIn this talk we present an approac h to classify left-invariant symplectic structures on Lie groups. The proc edure is based on the moduli space of left-invariant nondegenerate 2-forms \, which is a certain orbit space in the set of all nondegenerate 2-forms on a Lie algebra. We present some of the results obtained so far with this approach\, including a classification of left-invariant symplectic struct ures on some almost abelian Lie algebras.\n LOCATION:https://stable.researchseminars.org/talk/SymmetricSpacesSeminar/4 / END:VEVENT BEGIN:VEVENT SUMMARY:Ivan Solonenko (King's College London - LSGNT) DTSTART:20220126T130000Z DTEND:20220126T140000Z DTSTAMP:20260404T111009Z UID:SymmetricSpacesSeminar/5 DESCRIPTION:Title: The index of symmetry of a Riemannian manifold\n by Ivan Solonenko (King's College London - LSGNT) as part of Symmetric Spa ces Seminar\n\n\nAbstract\nRiemannian symmetric spaces constitute arguably the most well-understood class of Riemannian manifolds\, mainly because t hey can be extensively studied and ultimately classified by means of Lie t heory. To an arbitrary Riemannian manifold\, one can naturally assign a nu mber\, called the index of symmetry\, which measures how badly the manifol d fails to be a symmetric space. If the manifold is complete\, it admits a natural (in general\, singular) foliation\, called the foliation of symme try\, which is invariant under isometries and such that the index of symme try 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 metri c. In case the manifold is homogeneous\, its foliation of symmetry has all its leaves of the same dimension and thus becomes a genuine foliation. Al though its construction is very natural and intrinsic\, the foliation (and thus the index) of symmetry is fairly hard to compute even for homogeneou s spaces. \n\nIn this talk\, I will define the foliation of symmetry and i ntroduce its main properties and then tell about some situations when it c an be explicitly computed\, namely for normal compact homogeneous spaces a nd a certain class of compact naturally reductive spaces. I will also pres ent the classification of compact homogeneous spaces with a sufficiently h igh index of symmetry. I will follow the papers by Olmos\, Reggiani\, Tama ru\, and Berndt [2014\, 2017].\n LOCATION:https://stable.researchseminars.org/talk/SymmetricSpacesSeminar/5 / END:VEVENT BEGIN:VEVENT SUMMARY:Juan Manuel Lorenzo Naveiro (University of Santiago de Compostela) DTSTART:20220202T130000Z DTEND:20220202T140000Z DTSTAMP:20260404T111009Z UID:SymmetricSpacesSeminar/6 DESCRIPTION:Title: Symmetric spaces of noncompact type I: The Cartan an d Iwasawa decompositions\nby Juan Manuel Lorenzo Naveiro (University o f Santiago de Compostela) as part of Symmetric Spaces Seminar\n\n\nAbstrac t\nThe main property of symmetric spaces is the fact that they can be desc ribed in terms of two Lie groups together with an involution representing the geodesic symmetry at a given point. Because of this\, one can extensiv ely study the geometry of these manifolds by means of purely algebraic met hods. During the course of this and the next two talks\, Tomás Otero\, Iv an Solonenko\, and I will introduce some of those methods in the context o f symmetric spaces of noncompact type.\n\nThroughout this first talk\, we will exhibit two decompositions of the isometry group and its Lie algebra for such manifolds. The first one is the Cartan decomposition (valid for a ny symmetric space)\, which gives a way to express some invariants of a sy mmetric 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 all ows to realize the space as a simply connected solvable Lie group with a l eft-invariant metric.\n LOCATION:https://stable.researchseminars.org/talk/SymmetricSpacesSeminar/6 / END:VEVENT BEGIN:VEVENT SUMMARY:Ivan Solonenko (King's College London - LSGNT) DTSTART:20220209T130000Z DTEND:20220209T140000Z DTSTAMP:20260404T111009Z UID:SymmetricSpacesSeminar/7 DESCRIPTION:Title: Symmetric spaces of noncompact type II: Roots and Dy nkin diagrams\nby Ivan Solonenko (King's College London - LSGNT) as pa rt of Symmetric Spaces Seminar\n\n\nAbstract\nLast week we discussed some basic geometric properties of symmetric spaces of noncompact type in the c ontext of the Cartan and Iwasawa decompositions. This time\, we will look more closely at the algebraic side of the picture. We will dive in greater detail into the restricted root space decomposition of the isometry Lie a lgebra of a symmetric space of noncompact type and study its main properti es. There are two stark distinctions between this decomposition and the – probably more well-known – root space decomposition of a complex sem isimple Lie algebra: restricted root systems may be nonreduced\, while res tricted root subspaces may have dimension greater than 1. Somewhat surpris ingly\, by allowing a root system to be nonreduced\, one only gets one add itional series in the classification of irreducible root systems\, namely the nonreduced system (BC)_r. We will examine the classification of noncom pact symmetric spaces through the lens of restricted root systems and thei r Dynkin diagrams. Since irreducible noncompact symmetric spaces are essen tially in one-to-one correspondence with simple noncompact real Lie algebr as\, this will give us a nice perspective on the classification of the lat ter. Finally\, we will investigate the Weyl and automorphism groups of the restricted root system and observe how they can be thought of in terms of isometries of the underlying space.\n LOCATION:https://stable.researchseminars.org/talk/SymmetricSpacesSeminar/7 / END:VEVENT BEGIN:VEVENT SUMMARY:Tomás Otero Casal (University of Santiago de Compostela) DTSTART:20220216T130000Z DTEND:20220216T140000Z DTSTAMP:20260404T111009Z UID:SymmetricSpacesSeminar/8 DESCRIPTION:Title: Symmetric spaces of noncompact type III: Parabolic s ubgroups and subalgebras\nby Tomás Otero Casal (University of Santiag o de Compostela) as part of Symmetric Spaces Seminar\n\n\nAbstract\nParabo lic subgroups are of special interest when studying isometric actions on s ymmetric spaces of noncompact type. Geometrically speaking\, given a symme tric space of noncompact type M=G/K\, proper parabolic subgroups of G are the stabilizers of points at infinity of M. Throughout this third talk\, w e will introduce parabolic subalgebras from both the geometric and algebra ic viewpoints and explain how one can construct "standard" parabolic subal gebras from a choice of a subset of simple roots.\n\nWe will also talk abo ut some decomposition results for parabolic Lie subalgebras and the decomp ositions they induce at the group/manifold level. In particular\, this wil l allow us to introduce a special type of totally geodesic submanifolds of M called boundary components\, which are symmetric spaces behaving in an especially nice way with respect to the root space decomposition.\n LOCATION:https://stable.researchseminars.org/talk/SymmetricSpacesSeminar/8 / END:VEVENT BEGIN:VEVENT SUMMARY:Alberto Rodríguez-Vázquez (University of Santiago de Compostela) DTSTART:20220302T130000Z DTEND:20220302T140000Z DTSTAMP:20260404T111009Z UID:SymmetricSpacesSeminar/9 DESCRIPTION:Title: Totally geodesic submanifolds and Hermitian symmetri c spaces\nby Alberto Rodríguez-Vázquez (University of Santiago de Co mpostela) as part of Symmetric Spaces Seminar\n\n\nAbstract\nTotally geode sic submanifolds in symmetric spaces are those submanifolds with the simpl est geometry and they admit a nice algebraic characterization in terms of Lie triple systems. A special class of symmetric spaces where totally geod esic submanifolds can be studied is that of Hermitian symmetric spaces. In these spaces we can use the notion of Kähler angle to measure how a subm anifold fails to be complex.\n\nIn this talk\, I will report on an ongoing work where a method to construct totally geodesic submanifolds with non-t rivial constant Kähler angle in non-flat Hermitian symmetric spaces is gi ven.\n LOCATION:https://stable.researchseminars.org/talk/SymmetricSpacesSeminar/9 / END:VEVENT BEGIN:VEVENT SUMMARY:Yuji Kondo (Hiroshima University) DTSTART:20220309T130000Z DTEND:20220309T140000Z DTSTAMP:20260404T111009Z UID:SymmetricSpacesSeminar/10 DESCRIPTION:Title: A classification of left-invariant pseudo-Riemannia n metrics on some nilpotent Lie groups\nby Yuji Kondo (Hiroshima Unive rsity) as part of Symmetric Spaces Seminar\n\n\nAbstract\nIt is known that a connected and simply-connected Lie group admits only one left-invariant Riemannian metric up to scaling and isometry if and only if it is isomorp hic to the Euclidean space\, the Lie group of the real hyperbolic space\, or the direct product of the three dimensional Heisenberg group and the Eu clidean space of dimension n-3.\n\nIn this talk\, I am going to talk about a classification of left-invariant pseudo-Riemannian metrics of an arbitr ary signature for the third Lie groups with n>3 up to scaling and automorp hisms. This completes the classifications of left-invariant pseudo-Riemann ian metrics for the above three Lie groups up to scaling and automorphisms .\n\nOur classification can be obtained by a certain isometric action of c ohomogeneity zero on a pseudo-Riemannian symmetric space. This action is n ot proper\, in fact there exists a non-closed orbit\, which allows us to c onsider degenerations of orbits. Finally\, I explain that degenerations of orbits can have an implication with respect to curvature properties.\n\nT his study is based on the joint work with Hiroshi Tamaru from Osaka City U niversity.\n LOCATION:https://stable.researchseminars.org/talk/SymmetricSpacesSeminar/1 0/ END:VEVENT BEGIN:VEVENT SUMMARY:Yuri Nikolayevsky (La Trobe University\, Melbourne) DTSTART:20220406T090000Z DTEND:20220406T100000Z DTSTAMP:20260404T111009Z UID:SymmetricSpacesSeminar/11 DESCRIPTION:Title: Einstein hypersurfaces in irreducible symmetric spa ces\nby Yuri Nikolayevsky (La Trobe University\, Melbourne) as part of Symmetric Spaces Seminar\n\n\nAbstract\nIn this talk\, I will present the results of the joint paper of Jeong Hyeong Park and myself in which we gi ve a classification of Einstein hypersurfaces in irreducible symmetric spa ces. The main theorem states that there are three classes of such hypersur faces\, belonging to three very different "geometries": homogeneous geomet ry\, Legendrian geometry and affine geometry. I will give a brief introduc tion into these three geometries and explain how they fit together in our classification.\n LOCATION:https://stable.researchseminars.org/talk/SymmetricSpacesSeminar/1 1/ END:VEVENT BEGIN:VEVENT SUMMARY:Miguel Domínguez-Vázquez (University of Santiago de Compostela) DTSTART:20220420T120000Z DTEND:20220420T130000Z DTSTAMP:20260404T111009Z UID:SymmetricSpacesSeminar/12 DESCRIPTION:Title: Polar foliations on symmetric spaces\nby Miguel Domínguez-Vázquez (University of Santiago de Compostela) as part of Sym metric Spaces Seminar\n\n\nAbstract\nA polar foliation is a decomposition of a Riemannian manifold into equidistant submanifolds (called leaves) of possibly different dimensions\, and such that through any point there exis ts a submanifold (called section) intersecting all leaves perpendicularly. These objects arise as generalizations of important concepts\, such as is oparametric hypersurfaces and polar actions\, whose study has produced man y beautiful and profound results over the last decades. In this expository talk I will present an introduction to polar foliations and isoparametric hypersurfaces in symmetric spaces\, and report on some results concerning their classification problem.\n LOCATION:https://stable.researchseminars.org/talk/SymmetricSpacesSeminar/1 2/ END:VEVENT BEGIN:VEVENT SUMMARY:José Carlos Díaz-Ramos (University of Santiago de Compostela) DTSTART:20220504T120000Z DTEND:20220504T130000Z DTSTAMP:20260404T111009Z UID:SymmetricSpacesSeminar/13 DESCRIPTION:Title: Cohomogeneity one actions on quaternionic hyperboli c spaces\nby José Carlos Díaz-Ramos (University of Santiago de Compo stela) as part of Symmetric Spaces Seminar\n\n\nAbstract\nIn this talk I w ill present the classification procedure to obtain the classification of c ohomogeneity one actions on quaternionic hyperbolic spaces\, as well as so me byproducts of this study\, such as the existence of inhomogeneous isopa rametric hypersurfaces with constant principal curvatures.\n LOCATION:https://stable.researchseminars.org/talk/SymmetricSpacesSeminar/1 3/ END:VEVENT END:VCALENDAR