BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Prof. BV Rajarama Bhat (ISI Bangalore) DTSTART:20200819T103000Z DTEND:20200819T113000Z DTSTAMP:20260422T225753Z UID:WOTOA/1 DESCRIPTION:Title: A caricature of dilation theory\nby Prof. BV Rajarama Bhat (ISI Bangalor e) as part of Webinars on Operator Theory and Operator Algebras\n\n\nAbstr act\nWe present a set-theoretic version of some basic dilation results of operator theory. The results we have considered are Wold decomposition\, H almos dilation\, Sz. Nagy dilation\, inter-twining lifting\, commuting and non-commuting dilations\, BCL theorem etc. We point out some natural gene ralizations and variations. This is a joint work with Sandipan De and Na rayan Rakshit.\n LOCATION:https://researchseminars.org/talk/WOTOA/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Sameer Chavan (IIT Kanpur) DTSTART:20200909T113000Z DTEND:20200909T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/3 DESCRIPTION:Title: Di richlet-type spaces on the unit ball and joint 2-isometries\nby Sameer Chavan (IIT Kanpur) as part of Webinars on Operator Theory and Operator A lgebras\n\n\nAbstract\nWe discuss a formula that relates the spherical mom ents of the multiplication tuple on a Dirichlet-type space to a complex mo ment problem in several variables. This can be seen as the ball-analogue o f a formula originally invented by Richter. One may capitalize on this for mula to study Dirichlet-type spaces on the unit ball and joint 2-isometrie s. This talk is based on a joint work with Rajeev Gupta and Md Ramiz Reza. \n LOCATION:https://researchseminars.org/talk/WOTOA/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Sutanu Roy (NISER) DTSTART:20200916T113000Z DTEND:20200916T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/4 DESCRIPTION:Title: Qu antum group contraction and bosonisation\nby Sutanu Roy (NISER) as par t of Webinars on Operator Theory and Operator Algebras\n\n\nAbstract\nAbst ract: In 1953 İnönü and Wigner introduced group contraction: a systema tic (limiting) process to obtain from a given Lie group a non-isomorphic L ie group. For example\, the contraction of SU(2) group (with respect to it s closed subgroup T) is isomorphic to the double cover of E(2) group. The q-deformed C*-algebraic analogue of this example was introduced and invest igated by Woronowicz during the mid '80s to early '90s. More precisely\, t he C*-algebraic deformations of SU(2) and (the double cover of) E(2) with respect to real deformation parameters 0<|q|<1 become compact (denoted by SUq(2)) and non-compact locally compact (denoted by Eq(2)) quantum groups\ , respectively. Furthermore\, the contraction of SUq(2) groups becomes (is omorphic) to Eq(2) groups. However\, for complex deformation parameters 0< |q|<1\, the objects SUq(2) and Eq(2) are not ordinary but braided quantum groups. More generally\, the quantum analogue of the normal subgroup of a semidirect product group becomes a braided quantum group and the reconstr uction process of the semidirect product quantum group from a braided quan tum group is called bosonisation. In this talk\, we shall present a braide d version of the contraction procedure between SUq(2) and Eq(2) groups (fo r complex deformation parameters 0<|q|<1) and address its compatibility wi th bosonisation. This is based on a joint work with Atibur Rahaman.\n LOCATION:https://researchseminars.org/talk/WOTOA/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Jyotishman Bhowmick (ISI Kolkata) DTSTART:20200923T113000Z DTEND:20200923T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/5 DESCRIPTION:Title: Me tric-compatible connections in noncommutative geometry\nby Jyotishman Bhowmick (ISI Kolkata) as part of Webinars on Operator Theory and Operator Algebras\n\n\nAbstract\nLevi-Civita's theorem in Riemannian geometry stat es that if $(M\, g)$ is a Riemannian manifold\, then there exists a unique connection on $M$ which is torsionless and compatible with $g$. The curva ture of the manifold is then computed from this particular connection. \n\ nWe will try to explain the notions to state and prove Levi-Civita's theor em in the context of a noncommutative differential calculus. In particula r\, we will describe two notions of metric-compatibility of a connection. The talk will be based on joint works with D. Goswami\, S. Joardar\, G. La ndi and S. Mukhopadhyay.\n\nThe geometric notions appearing in the lecture will be defined and explained in the beginning.\n LOCATION:https://researchseminars.org/talk/WOTOA/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Tirthankar Bhattacharyya (IISc Bangalore) DTSTART:20200930T113000Z DTEND:20200930T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/6 DESCRIPTION:Title: On the geometry of the symmetrized bidisc\nby Tirthankar Bhattacharyya ( IISc Bangalore) as part of Webinars on Operator Theory and Operator Algebr as\n\n\nAbstract\nWe study the action of the automorphism group of the $2$ complex dimensional manifold symmetrized bidisc $\\mathbb G$ on itself. T he automorphism group is $3$ real dimensional. It foliates $\\mathbb G$ in to leaves all of which are $3$ real dimensional hypersurfaces except one\, viz.\, the royal variety. This leads us to investigate Isaev's classifica tion of all Kobayashi-hyperbolic $2$ complex dimensional manifolds for wh ich the group of holomorphic automorphisms has real dimension $3$ studied by Isaev. Indeed\, we produce a biholomorphism between the symmetrized bid isc and the domain\n\n \\[\\{(z_1\,z_2)\\in \\mathbb{C} ^2 : 1+|z_1|^2-|z_ 2|^2>|1+ z_1 ^2 -z_2 ^2|\, Im(z_1 (1+\\overline{z_2}))>0\\}\\]\n\nin Isaev 's list. Isaev calls it $\\mathcal D_1$. The road to the biholomorphism is paved with various geometric insights about $\\mathbb G$. \n\nSeveral con sequences of the biholomorphism follow including two new characterizations of the symmetrized bidisc and several new characterizations of $\\mathcal D_1$. Among the results on $\\mathcal D_1$\, of particular interest is th e fact that $\\mathcal D_1$ is a ``symmetrization''. When we symmetrize (a ppropriately defined in the context) either $\\Omega_1$ or $\\mathcal{D}^{ (2)} _1$ (Isaev's notation)\, we get $\\mathcal D_1$. These two domains $ \\Omega_1$ and $\\mathcal{D}^{(2)} _1$ are in Isaev's list and he mentione d that these are biholomorphic to $\\mathbb D \\times \\mathbb D$. We prod uce explicit biholomorphisms between these domains and $\\D \\times \\D$.\ n LOCATION:https://researchseminars.org/talk/WOTOA/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Mizanur Rahaman (BITS Pilani Goa campus) DTSTART:20201007T113000Z DTEND:20201007T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/7 DESCRIPTION:Title: Bi synchronous Games\nby Mizanur Rahaman (BITS Pilani Goa campus) as part of Webinars on Operator Theory and Operator Algebras\n\n\nAbstract\nFor s ome games played by two cooperating but non-communicating players\, the pl ayers can use entanglement as a resource to improve their outcomes beyond what is possible classically. Graph colouring game\, graph homomorphism ga me and graph isomorphism game are a few examples of these games. Over the last few years\, a remarkable progress has been taken place in the theory of these non-local games. One significant aspect of this development is it s connection with many challenging problems in operator algebras.\n\nIn th is talk\, I will review the theory of these games and explain the relevant connection with operator algebras. In particular\, I will introduce a new class of games which is called bisynchronous and will show a close connec tion between bisynchronous games and the theory of quantum groups. Moreove r\, when the number of inputs is equal to the number of outputs\, each bis ynchronous correlation gives rise to a completely positive map which will be shown to be factorable in the sense of Haagerup and Musat. This is a jo int work with Vern Paulsen. No background in quantum theory is needed for this talk.\n LOCATION:https://researchseminars.org/talk/WOTOA/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Soumyashant Nayak (ISI Bangalore) DTSTART:20201014T113000Z DTEND:20201014T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/8 DESCRIPTION:Title: Wh at is a Murray-von Neumann algebra?\nby Soumyashant Nayak (ISI Bangalo re) as part of Webinars on Operator Theory and Operator Algebras\n\n\nAbst ract\nIt was observed by Murray and von Neumann in their seminal paper on rings of operators (1936) that the set of closed\, densely-defined operato rs affiliated with a finite von Neumann algebra has the structure of a *-a lgebra. The algebra of affiliated operators naturally appears in many cont exts\; for instance\, in the setting of group von Neumann algebras in the study of non-compact spaces and infinite group actions. In this talk\, we will give an intrinsic description of Murray-von Neumann algebras avoiding reference to a Hilbert space\, thus\, revealing the intrinsic nature of v arious notions associated with such affiliated operators. In fact\, we wil l view Murray-von Neumann algebras as ordered complex topological *-algebr as arising from a functorial construction over finite von Neumann algebras .\n LOCATION:https://researchseminars.org/talk/WOTOA/8/ END:VEVENT BEGIN:VEVENT SUMMARY:S Sundar (IMSc Chennai) DTSTART:20201021T113000Z DTEND:20201021T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/9 DESCRIPTION:Title: An asymmetric multiparameter CCR flow\nby S Sundar (IMSc Chennai) as par t of Webinars on Operator Theory and Operator Algebras\n\n\nAbstract\nThe theory of E_0-semigroups initiated by R.T. Powers and developed extensivel y by Arveson has been an active area of research for well over thirty year s. An E_0-semigroup is a 1-parameter semigroup of unital normal *-endomorp hisms of B(H) where H is a Hilbert space.\n\nHowever\, nothing prevents us from considering semigroups of endomorphisms indexed by more general sem igroups. This was analysed in collaboration with Anbu Arjunan\, S.P. Muru gan and R. Srinivasan. \n\nI will explain a few similarities between the one parameter theory and the multiparameter theory. Also\, there are sign ificant differences. I will attempt to illustrate one difference by explai ning that a multiparameter CCR flow need not be symmetric.\n LOCATION:https://researchseminars.org/talk/WOTOA/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Soumalya Joardar (IISER Kolkata) DTSTART:20201104T113000Z DTEND:20201104T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/11 DESCRIPTION:Title: Q uantum symmetry of graph C* -algebras\nby Soumalya Joardar (IISER Kolk ata) as part of Webinars on Operator Theory and Operator Algebras\n\n\nAbs tract\nGraph C*-algebras are examples of C*-algebras generated by partial isometries. The notion of quantum symmetry of graph C*-algebras will be di scussed. Emphasis will be given on the invariance of KMS states of graph C *-algebras at critical inverse temperature under such quantum symmetry. Th e richness of quantum symmetry will be exhibited by a particular considera tion. Also a unitary easy quantum group will be shown to appear as the qua ntum symmetry of a particular graph C*-algebra. The talk is based on a joi nt project with Arnab Mandal.\n LOCATION:https://researchseminars.org/talk/WOTOA/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Devarshi Mukherjee (University of Goettingen) DTSTART:20201111T113000Z DTEND:20201111T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/12 DESCRIPTION:Title: I soradial embeddings and non-commutative geometry\nby Devarshi Mukherje e (University of Goettingen) as part of Webinars on Operator Theory and Op erator Algebras\n\n\nAbstract\nIn this talk\, we describe a framework to s tudy non-commutative geometry as a relative version of differential geomet ry. More precisely\, given a C*-algebra A\, we would like to make sense of a "smooth" subalgebra $A^\\infty \\subseteq A$\, and deduce properties ab out A using such a subalgebra. Such a smooth subalgebra should be analogo us to the Frechet algebra $C^\\infty(M) \\subseteq C(M)$ for a smooth mani fold M\, in the world of commutative C*-algebras. We shall review the fun damental properties and applications of such embeddings\, called $\\textit {isoradial embeddings}$\, due to Ralf Meyer. If time permits\, I will ment ion an ongoing research program with Meyer\, Corti\\~nas and Cuntz\, that uses such embeddings to develop noncommutative geometry over finite fields . \n\nI will not assume that the audience has any background beyond famil iar examples of C*-algebras. A lot of the motivation would however be clea rer to those familiar with cyclic homology or operator algebraic K-theory. \n LOCATION:https://researchseminars.org/talk/WOTOA/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Samya Kumar Ray (Wuhan University) DTSTART:20201118T113000Z DTEND:20201118T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/13 DESCRIPTION:Title: M aximal ergodic inequalities on non-commutative L_p-spaces\nby Samya Ku mar Ray (Wuhan University) as part of Webinars on Operator Theory and Oper ator Algebras\n\n\nAbstract\nIn an influential paper\, Junge and Xu establ ished a non-commutative analogue of Dunford-Schwartz maximal ergodic inequ ality\, solving a longstanding open problem in ergodic theory. However\, t here are very few non-commutative ergodic theorems beyond L_1-L_\\infty co ntractions of Junge-Xu. In this talk\, we consider the problem of finding more general non-commutative ergodic theorems than L_1-L_\\infty contracti ons. En route we discuss how our work is related to various results of Haa gerup\, Ruan and Pisier on non-commutative L_p spaces. This is a joint wor k with Guixiang Hong and Simeng Wang.\n LOCATION:https://researchseminars.org/talk/WOTOA/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Baruch Solel (Technion - Israel Institute of Technology) DTSTART:20201202T113000Z DTEND:20201202T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/14 DESCRIPTION:Title: I nvariant subspaces for certain tuples of operators\nby Baruch Solel (T echnion - Israel Institute of Technology) as part of Webinars on Operator Theory and Operator Algebras\n\n\nAbstract\nIn this talk we will generaliz e results of Sarkar and of Bhattacharjee-Eschmeier-Keshari-Sarkar concern ing invariant subspaces for commuting tuples of operators. These authors p rove Beurling-Lax-Halmos type results for commuting tuples $T=(T_1\,\\ldot s\,T_d)$ operators that are contractive and pure\; that is $\\Phi_T(I)\\le q I$ and $\\Phi_T^n(I)\\searrow 0$ where $$\\Phi_T(a)=\\Sigma_i T_iaT_i^*. $$\n\nHere we generalize some of their results to commuting tuples $T$ sat isfying similar conditions but for $$\\Phi_T(a)=\\Sigma_{\\alpha \\in \\m athbb{F}^+_d} x_{|\\alpha|}T_{\\alpha}aT_{\\alpha}^*$$ where $\\{x_k\\}$ i s a sequence of non negative numbers satisfying some natural conditions (w here $T_{\\alpha}=T_{\\alpha(1)}\\cdots T_{\\alpha(k)}$ for $k=|\\alpha|$) . In fact\, we deal with a more general situation where each $x_k$ is repl aced by a $d^k\\times d^k$ matrix.\n\nWe also apply these results to subsp aces of certain reproducing kernel correspondences $E_K$ (associated with maps-valued kernels $K$) that are invariant under the multipliers given by the coordinate functions.\n LOCATION:https://researchseminars.org/talk/WOTOA/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Prahlad Vaidyanathan (IISER Bhopal) DTSTART:20201209T113000Z DTEND:20201209T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/15 DESCRIPTION:Title: R okhlin Dimension for Group Actions on C*-algebras\nby Prahlad Vaidyana than (IISER Bhopal) as part of Webinars on Operator Theory and Operator Al gebras\n\n\nAbstract\nRokhlin Dimension was introduced by Hirshberg\, Wint erand Zacharias as a higher rank version of the Rokhlin property. It maybe thought of as a noncommutative analogue of a ‘free’ action of a group on a topological space. We discuss this idea\, and what it means for the corresponding crossed product C*-algebra.\n\nThe talk is meant to be expos itory\, and accessible to a large audience.\n LOCATION:https://researchseminars.org/talk/WOTOA/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Ved Prakash Gupta (JNU) DTSTART:20201216T113000Z DTEND:20201216T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/16 DESCRIPTION:Title: L attice of intermediate subalgebras of a pair of simple C*-algebras\nby Ved Prakash Gupta (JNU) as part of Webinars on Operator Theory and Operat or Algebras\n\n\nAbstract\nThe study of the lattice of intermediate object s of a pair $B \\subset A$ in any category is quite a natural and fundamen tal question and has a significant say in obtaining a better understanding of the structures of the objects A and B. A good deal of work in this dir ection has been done in the category of finite groups\, both of qualitativ e and quantitave flavour. Its natural analogue in the theory of operator a lgebras has had some success\, though mainly quantitative in nature and ba sed on some existing tools. Continuing the trend\, in a recent work with K eshab Chandra Bakshi\, we developed certain tools in the category of simpl e C*-algebras (motivated by and analogous to the ones existing in the cate gory of simple von Neumann algebras) to answer a quantitative question of Roberto Longo regarding the lattice of intermediate von Neumann subalgebra s of an inclusion of type III factors. We shall present some essence of th is development with an attempt to make the talk accessible to a larger aud ience.\n LOCATION:https://researchseminars.org/talk/WOTOA/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Xiang Tang (Washington University in St. Louis\, USA) DTSTART:20210113T040000Z DTEND:20210113T053000Z DTSTAMP:20260422T225753Z UID:WOTOA/17 DESCRIPTION:Title: A nalytic Grothendieck Riemann Roch Theorem\nby Xiang Tang (Washington U niversity in St. Louis\, USA) as part of Webinars on Operator Theory and O perator Algebras\n\n\nAbstract\nIn this talk\, we will introduce an intere sting index problem naturally associated to the Arveson-Douglas conjecture in functional analysis. This index problem is a generalization of the cla ssical Toeplitz index theorem and connects to many different branches of M athematics. In particular\, it can be viewed as an analytic version of the Grothendieck Riemann Roch theorem. This is joint work with R. Douglas\, M . Jabbari\, and G. Yu.\n LOCATION:https://researchseminars.org/talk/WOTOA/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Sugato Mukhopadhyay (ISI Kolkata) DTSTART:20210120T113000Z DTEND:20210120T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/18 DESCRIPTION:Title: L evi-Civita connections on bicovariant differential calculus\nby Sugato Mukhopadhyay (ISI Kolkata) as part of Webinars on Operator Theory and Ope rator Algebras\n\n\nAbstract\nIn this talk\, we will propose a definition of Levi-Civita connections on bicovariant differential calculi of Hopf alg ebras\, which satisfy a technical property. Given a bi-invariant metric on such a calculus\, we will present a sufficient condition for the existenc e of a unique bicovariant Levi-Civita connection on the calculus. We will discuss examples of Hopf algebras that fit into this framework. This talk is based on a joint work with Jyotishman Bhowmick.\n LOCATION:https://researchseminars.org/talk/WOTOA/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Apoorva Khare (IISc Bangalore) DTSTART:20210303T113000Z DTEND:20210303T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/19 DESCRIPTION:Title: T otal positivity: history\, basics\, and modern connections\nby Apoorva Khare (IISc Bangalore) as part of Webinars on Operator Theory and Operato r Algebras\n\n\nAbstract\nI will give a gentle introduction to totally pos itive matrices and Polya frequency functions. This includes basic examples \, history\, and fundamental results on total positivity\, variation dimin ution\, and sign non-reversal – as well as a few proofs to show how the main ingredients fit together. Many classical results (and one Hypothesis) from before 1955 feature in this journey. I will end by describing how Po lya frequency functions connect to the Laguerre–Polya class and hence Po lya–Schur multipliers\, and mention 21st century incarnations of the lat ter.\n LOCATION:https://researchseminars.org/talk/WOTOA/19/ END:VEVENT BEGIN:VEVENT SUMMARY:CR Jayanarayanan (IIT Palakkad) DTSTART:20210310T113000Z DTEND:20210310T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/20 DESCRIPTION:by CR Jayanarayanan (IIT Palakkad) as part of Webinars on Oper ator Theory and Operator Algebras\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/WOTOA/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Anshu Nirbhay (IISER Bhopal) DTSTART:20210317T113000Z DTEND:20210317T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/21 DESCRIPTION:Title: S ome Dimension Theories of C*-algebras and Rokhlin-type Properties\nby Anshu Nirbhay (IISER Bhopal) as part of Webinars on Operator Theory and Op erator Algebras\n\n\nAbstract\nThere are many ranks associated with a $C^* $-algebra. Rieffel defined the notion of stable ranks in the 1980s. We wil l mainly focus on two of these ranks namely connected stable rank and gene ral stable rank. If we are given a group $G$\, which acts on a $C^*$-algeb ra $A$ via a map $\\alpha$\, the triple $(A\, G\, \\alpha)$ is said to be a $C^*$-dynamical system\, then we can associate a $C^*$-algebra called a crossed product $C^*$-algebra denoted by $A \\rtimes_{\\alpha}G$. We will discuss the homotopical stable ranks of a crossed product $C^*$-algebra by a finite group where the action involved has Rokhlin-type property.\n LOCATION:https://researchseminars.org/talk/WOTOA/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Keshab Chandra Bakshi (Chennai Mathematical Institute) DTSTART:20210324T113000Z DTEND:20210324T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/22 DESCRIPTION:Title: O n a question of Vaughan Jones\nby Keshab Chandra Bakshi (Chennai Mathe matical Institute) as part of Webinars on Operator Theory and Operator Alg ebras\n\n\nAbstract\nGiven a subgroup H of a finite group G\, as an applic ation of famous Hall's Marriage Theorem\, we can obtain a set of coset rep resentatives which acts simultaneously as representatives of both left an d right cosets of H in G. Given a subfactor $N\\subset M$ with finite Jone s index\, M can be regarded as a left as well as a right N-module. Pimsner and Popa proved that M is finitely generated as a left (equivalently\, ri ght) N-module. About a decade back\, Vaughan Jones asked whether one can f ind a common set which acts simultaneously as a left and a right generatin g set. As a naive attempt in this direction\, we answer this question in t he affirmative for a large class of integer index subfactors. We also disc uss some applications of our results.\n LOCATION:https://researchseminars.org/talk/WOTOA/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Apurva Seth (IISER Bhopal) DTSTART:20210331T113000Z DTEND:20210331T130000Z DTSTAMP:20260422T225753Z UID:WOTOA/23 DESCRIPTION:Title: A F- algebras and Rational Homotopy Theory\nby Apurva Seth (IISER Bhopal ) as part of Webinars on Operator Theory and Operator Algebras\n\n\nAbstra ct\nIn this talk\, we will give a procedure to compute the rational homoto py group of the group of quasi-unitaries of an AF-algebra. As an applicati on\, we show that an AF-algebra is K-stable if and only if it is rationall y K-stable.\n LOCATION:https://researchseminars.org/talk/WOTOA/23/ END:VEVENT END:VCALENDAR