List of Talks
Forthcoming talks:
Speaker: Prof. Susanna Spektor (Sheridan College Institute of Technology and Advanced Learning)
Title: On the restricted Kahane-Khintchine type inequalities.
Abstract: We prove a Kahane-Khintchine type inequality under the assumption that the sum of Rademacher random variables equals zero. We will show a generalization of this result in vector space and to non-commutative Banach spaces with q-Schatten norm. We also show a new tail-bound for a hypergeometric random variable, as well as application of this theoretical results in Statistics and Speech recognition.
Schedule: 7th October, 2021, Thursday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Sachi Srivastava (University of Delhi )
Title: Delay Semigroups and Polynomial Stability
Abstract: We study the notion of "polynomial stability" for the delay semigroups associated with abstract delay differential equations. In particular, we find conditions under which a delay semigroup associated with certain delay operators is polynomially stable. Some new perturbation results for stability and polynomial stability of strongly continuous semigroups are also obtained and provide another approach to the study.
Schedule: Postponed to a later date.
Speaker: Prof. Bata Krishna Das (IIT Bombay)
Title: Commutant lifting in the Schur-Agler class.
Abstract: D. Sarason (1967) first introduced commutant lifting theorem to provide an operator theoretic approach to the classical function theoretic problem known as interpolation. Subsequently, it has found many applications in operator theory and other areas. There are also quite a lot of efforts have been put to obtain its several variable analogues. In this talk we will consider the problem for different weighted Bergman spaces over the unit ball in \mathbb{C}^n and find a necessary and sufficient condition for the commutant lifting theorem to hold in the Schur-Agler class. Apart from being a several variable analogue, it unifies and provides new proofs, along with more qualitative insight, of several classical commutant lifting theorems. This is joint work with S. Barik and M. Bhattacharjee.
Schedule: Postponed to a later date.
Past talks:
Speaker: Prof. BV Rajarama Bhat (Indian Statistical Institute, Bangalore)
Title: A CARICATURE OF DILATION THEORY
Abstract: We present a set-theoretic version of some basic dilation results of operator theory. The results we have considered are Wold decomposition, Halmos dilation, Sz. Nagy dilation, inter-twining lifting, commuting and non-commuting dilations, BCL theorem etc. We point out some natural generalizations and variations. This is a joint work with Sandipan De and Narayan Rakshit.
Schedule: 19th August, Wednesday at 16:00hrs (Indian Standard Time)
Speaker: Prof. Gadadhar Misra (Indian Institute of Science, Bangalore)
Title: Which homogeneous operators are subnormal
Abstract: The bi-holomorphic automorphism group Möb of the unit disc $\mathbb{D}$ acts naturally on the unit ball in the algebra of bounded linear operators $\mathscr{L}(\mathscr{H})$ defined on a separable complex Hilbert space $\mathscr{H}$. A contraction is said to be homogeneous if the orbit of this action modulo unitary equivalence is a singleton. In this talk, a class of homogeneous operators will be described. Among these, the ones that are the restriction of a normal operator to an invariant subspace (these are said to be subnormal) would be identified. The relationship of this question with induced representations, in the sense of Mackey, of Möb will be discussed. This is a report on work in progress.
(Click here to view the pdf file of the Abstract)
Schedule: 26th August, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Sameer Chavan (Indian Institute of Technology, Kanpur)
Title: Dirichlet-type spaces on the unit ball and joint 2-isometries
Abstract: We discuss a formula that relates the spherical moments of the multiplication tuple on a Dirichlet-type space to a complex moment problem in several variables. This can be seen as the ball-analogue of a formula originally invented by Richter. One may capitalize on this formula to study Dirichlet-type spaces on the unit ball and joint 2-isometries. This talk is based on a joint work with Rajeev Gupta and Md Ramiz Reza.
Schedule: 9th September, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Sutanu Roy (NISER, Bhubaneswar)
Title: Quantum group contraction and bosonisation
Abstract: In 1953 İnönü and Wigner introduced group contraction: a systematic (limiting) process to obtain from a given Lie group a non-isomorphic Lie group. For example, the contraction of SU(2) group (with respect to its 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 investigated by Woronowicz during the mid '80s to early '90s. More precisely, the 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 (isomorphic) 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 reconstruction process of the semidirect product quantum group from a braided quantum group is called bosonisation. In this talk, we shall present a braided version of the contraction procedure between SUq(2) and Eq(2) groups (for complex deformation parameters 0<|q|<1) and address its compatibility with bosonisation. This is based on a joint work with Atibur Rahaman.
Schedule: 16th September, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Jyotishman Bhowmick (Indian Statistical Institute, Kolkata)
Title: Metric-compatible connections in noncommutative geometry
Abstract: Levi-Civita's theorem in Riemannian geometry states that if (M, g) is a Riemannian manifold, then there exists a unique connection on M which is torsionless and compatible with g. The curvature of the manifold is then computed from this particular connection.
We will try to explain the notions to state and prove Levi-Civita's theorem in the context of a noncommutative differential calculus. In particular, 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. Landi and S. Mukhopadhyay.
The geometric notions appearing in the lecture will be defined and explained in the beginning.
Schedule: 23rd September, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Tirthankar Bhattacharyya (Indian Institute of Science, Bangalore)
Title: On the geometry of the symmetrized bidisc.
Abstract: We study the action of the automorphism group of the $2$ complex dimensional manifold symmetrized bidisc $\mathbb G$ on itself. The automorphism group is $3$ real dimensional. It foliates $\mathbb G$ into leaves all of which are $3$ real dimensional hypersurfaces except one, viz., the royal variety. This leads us to investigate Isaev's classification of all Kobayashi-hyperbolic $2$ complex dimensional manifolds for which the group of holomorphic automorphisms has real dimension $3$ studied by Isaev. Indeed, we produce a biholomorphism between the symmetrized bidisc and the domain
\[\{(z_1,z_2)\in \mathbb{C} ^2 : 1+|z_1|^2-|z_2|^2>|1+ z_1 ^2 -z_2 ^2|, Im(z_1 (1+\overline{z_2}))>0\}\]
in Isaev's list. Isaev calls it $\mathcal D_1$. The road to the biholomorphism is paved with various geometric insights about $\mathbb G$.
Several consequences 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 the fact that $\mathcal D_1$ is a ``symmetrization''. When we symmetrize (appropriately 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 mentioned that these are biholomorphic to $\mathbb D \times \mathbb D$. We produce explicit biholomorphisms between these domains and $\D \times \D$.
(Click here to view the pdf file of the Abstract)
Schedule: 30th September, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Mizanur Rahaman (BITS Pilani KK Birla Goa Campus)
Title: Bisynchronous Games
Abstract: For some games played by two cooperating but non-communicating players, the players can use entanglement as a resource to improve their outcomes beyond what is possible classically. Graph colouring game, graph homomorphism game 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 its connection with many challenging problems in operator algebras.
In this 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 connection between bisynchronous games and the theory of quantum groups. Moreover, when the number of inputs is equal to the number of outputs, each bisynchronous correlation gives rise to a completely positive map which will be shown to be factorizable in the sense of Haagerup and Musat. This is a joint work with Vern Paulsen. No background in quantum theory is needed for this talk.
Schedule: 7th October, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Soumyashant Nayak (Indian Statistical Institute, Bangalore)
Title: What is a Murray-von Neumann algebra?
Abstract: It was observed by Murray and von Neumann in their seminal paper on rings of operators (1936) that the set of closed, densely-defined operators affiliated with a finite von Neumann algebra has the structure of a *-algebra. The algebra of affiliated operators naturally appears in many contexts; 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 various notions associated with such affiliated operators. In fact, we will view Murray-von Neumann algebras as ordered complex topological *-algebras arising from a functorial construction over finite von Neumann algebras.
Schedule: 14th October, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. S Sundar (IMSc, Chennai)
Title: An asymmetric multiparameter CCR flow
Abstract: The theory of E_0-semigroups initiated by R.T.Powers and developed extensively by Arveson has been an active area of research for well over thirty years. An E_0-semigroup is a 1-parameter semigroup of unital normal *-endomorphisms of B(H) where H is a Hilbert space.
However, nothing prevents us from considering semigroups of endomorphisms indexed by more general semigroups. This was analysed in collaboration with Anbu Arjunan, S.P. Murugan and R. Srinivasan.
I will explain a few similarities between the one parameter theory and the multiparameter theory. Also, there are significant differences. I will attempt to illustrate one difference by explaining that a multiparameter CCR flow need not be symmetric.
Schedule: 21st October, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Soumalya Joardar (IISER Kolkata)
Title: Quantum symmetry of graph C* -algebras
Abstract: Graph C*-algebras are examples of C*-algebras generated by partial isometries. The notion of quantum symmetry of graph C*-algebras will be discussed. Emphasis will be given on the invariance of KMS states of graph C*-algebras at critical inverse temperature under such quantum symmetry. The richness of quantum symmetry will be exhibited by a particular consideration. Also a unitary easy quantum group will be shown to appear as the quantum symmetry of a particular graph C*-algebra. The talk is based on a joint project with Arnab Mandal.
Schedule: 4th November, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Dr. Devarshi Mukherjee (University of Göttingen)
Title: Isoradial embeddings and non-commutative geometry
Abstract: In this talk, we describe a framework to study non-commutative geometry as a relative version of differential geometry. More precisely, given a C*-algebra A, we would like to make sense of a "smooth" subalgebra A^\infty \subseteq A, and deduce properties about A using such a subalgebra. Such a smooth subalgebra should be analogous to the Frechet algebra C^\infty(M) \subseteq C(M) for a smooth manifold M, in the world of commutative C*-algebras. We shall review the fundamental properties and applications of such embeddings, called \textit{isoradial embeddings}, due to Ralf Meyer. If time permits, I will mention an ongoing research program with Meyer, Corti\~nas and Cuntz, that uses such embeddings to develop non-commutative geometry over finite fields.
I will not assume that the audience has any background beyond familiar examples of C*-algebras. A lot of the motivation would however be clearer to those familiar with cyclic homology or operator algebraic K-theory.
Schedule: 11th November, Wednesday at 17:00hrs (Indian Standard Time)
Click here for the pdf file of the abstract.
Speaker: Dr. Samya Kumar Ray (Wuhan University, China)
Title: Maximal ergodic inequalities on non-commutative L_p-spaces .
Abstract: In an influential paper, Junge and Xu established a non-commutative analogue of Dunford-Schwartz maximal ergodic inequality, solving a longstanding open problem in ergodic theory. However, there are very few non-commutative ergodic theorems beyond L_1-L_\infty contractions of Junge-Xu. In this talk, we consider the problem of finding more general non-commutative ergodic theorems than L_1-L_\infty contractions. En route we discuss how our work is related to various results of Haagerup, Ruan and Pisier on non-commutative L_p spaces. This is a joint work with Guixiang Hong and Simeng Wang.
Schedule: 18th November, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Baruch Solel (Technion - Israel Institute of Technology )
Title: Invariant subspaces for certain tuples of operators.
Abstract: In this talk we will generalize results of Sarkar and of Bhattacharjee-Eschmeier-Keshari-Sarkar concerning invariant subspaces for commuting tuples of operators. These authors prove Beurling-Lax-Halmos type results for commuting tuples $T=(T_1,\ldots,T_d)$ operators that are contractive and pure; that is $\Phi_T(I)\leq I$ and $\Phi_T^n(I)\searrow 0$ where $$\Phi_T(a)=\Sigma_i T_iaT_i^*.$$
Here we generalize some of their results to commuting tuples $T$ satisfying similar conditions but for $$\Phi_T(a)=\Sigma_{\alpha \in \mathbb{F}^+_d} x_{|\alpha|}T_{\alpha}aT_{\alpha}^*$$ where $\{x_k\}$ is a sequence of non negative numbers satisfying some natural conditions (where $T_{\alpha}=T_{\alpha(1)}\cdots T_{\alpha(k)}$ for $k=|\alpha|$). In fact, we deal with a more general situation where each $x_k$ is replaced by a $d^k\times d^k$ matrix.
We also apply these results to subspaces of certain reproducing kernel correspondences $E_K$ (associated with maps-valued kernels $K$) that are invariant under the multipliers given by the coordinate functions.
Click here for the pdf file of the abstract.
Schedule: 2nd December, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Prahlad Vaidyanathan (IISER Bhopal)
Title: Rokhlin Dimension for Group Actions on C*-algebras
Abstract: Rokhlin Dimension was introduced by Hirshberg, Winterand 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.
The talk is meant to be expository, and accessible to a large audience.
Schedule: 9th December, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Ved Prakash Gupta (JNU, Delhi)
Title: Lattice of intermediate subalgebras of a pair of simple C*-algebras
Abstract: The study of the lattice of intermediate objects of a pair $B \subset A$ in any category is quite a natural and fundamental 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 direction has been done in the category of finite groups, both of qualitative and quantitave flavour. Its natural analogue in the theory of operator algebras has had some success, though mainly quantitative in nature and based on some existing tools. Continuing the trend, in a recent work with Keshab Chandra Bakshi, we developed certain tools in the category of simple C*-algebras (motivated by and analogous to the ones existing in the category of simple von Neumann algebras) to answer a quantitative question of Roberto Longo regarding the lattice of intermediate von Neumann subalgebras of an inclusion of type III factors. We shall present some essence of this development with an attempt to make the talk accessible to a larger audience.
Schedule: 16th December, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Xiang Tang (Washington University in St. Louis, USA)
Title: Analytic Grothendieck Riemann Roch Theorem
Abstract: In this talk, we will introduce an interesting index problem naturally associated to the Arveson-Douglas conjecture in functional analysis. This index problem is a generalization of the classical Toeplitz index theorem and connects to many different branches of Mathematics. 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.
Schedule: 13th January 2021, Wednesday at 09:30hrs (Indian Standard Time).
Speaker: Mr. Sugato Mukhopadhyay (ISI Kolkata)
Title: Levi-Civita connections on bicovariant differential calculus
Abstract: In this talk, we will propose a definition of Levi-Civita connections on bicovariant differential calculi of Hopf algebras, which satisfy a technical property. Given a bi-invariant metric on such a calculus, we will present a sufficient condition for the existence 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.
Schedule: 20th January 2021, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Dr. P Muthukumar (ISI Bangalore)
Title: Shift invariant subspaces of composition operators
Abstract: In this talk, we discuss about joint invariant subspaces of multiplication (shift) operator and composition operators on the Hardy space H2. In particular, we discuss a characterization of shift invariant subspaces to be invariant under composition operators. Also, I will point out some open questions along this direction.
Schedule: 17th February 2021, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. Apoorva Khare (IISc Bangalore)
Title: Total positivity: history, basics, and modern connections
Abstract: I will give a gentle introduction to totally positive matrices and Polya frequency functions. This includes basic examples, history, and fundamental results on total positivity, variation diminution, 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 Polya frequency functions connect to the Laguerre–Polya class and hence Polya–Schur multipliers, and mention 21st century incarnations of the latter.
Schedule: 3rd March 2021, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Prof. CR Jayanarayanan (IIT Palakkad)
Title: Ball proximinality of M-ideals of compact operators
Abstract: In this lecture, we will discuss the proximinality of closed unit ball of $M$-ideals of compact operators on Banach spaces. We will show that every positive (self-adjoint) operator on a Hilbert space has a positive (self-adjoint) compact approximant from the closed unit ball of space of compact operators. This is a joint work with Sreejith Siju.
Schedule: 10th March 2021, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Ms. Anshu Nirbhay (IISER Bhopal)
Title: Some Dimension Theories of C*-algebras and Rokhlin-type Properties
Abstract: There are many ranks associated with a $C^*$-algebra. Rieffel defined the notion of stable ranks in the 1980s. We will mainly focus on two of these ranks namely connected stable rank and general stable rank. If we are given a group $G$, which acts on a $C^*$-algebra $A$ 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.
(Click here for the pdf file of the abstract.)
Schedule: 17th March 2021, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Dr. Keshab Chandra Bakshi (Chennai Mathematical Institute)
Title: On a question of Vaughan Jones.
Abstract: Given a subgroup H of a finite group G, as an application of famous Hall's Marriage Theorem, we can obtain a set of coset representatives which acts simultaneously as representatives of both left and right cosets of H in G. Given a subfactor $N\subset M$ with finite Jones 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, right) N-module. About a decade back, Vaughan Jones asked whether one can find a common set which acts simultaneously as a left and a right generating set. As a naive attempt in this direction, we answer this question in the affirmative for a large class of integer index subfactors. We also discuss some applications of our results.
Based on a recent joint work with Ved Gupta.
Schedule: 24th March 2021, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Ms. Apurva Seth (IISER Bhopal)
Title: AF- algebras and Rational Homotopy Theory
Abstract: In this talk, we will give a procedure to compute the rational homotopy group of the group of quasi-unitaries of an AF-algebra. As an application, we show that an AF-algebra is K-stable if and only if it is rationally K-stable.
Schedule: 31st March 2021, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Dr. Haonan Zhang (IST Austria)
Title: Gradient estimates of symmetric quantum Markov semigroups
Abstract: The lower bound of Ricci curvature has many geometric consequences and applications in proving functional inequalities. It can be characterized using \Gamma-calculus in Bakry-Émery theory, or via the geodesic semi-convexity of entropy with respect to 2-Wasserstein distance which has been pioneered by Lott-Sturm-Villani. In this talk I will present some attempts in recent years to generalize Ricci curvature lower bounds to noncommutative setting. These different notions around noncommutative Ricci curvature lower bounds, including the gradient estimate that we are concerned with, have many interesting applications. In particular, one can deduce a number of functional inequalities from a strictly positive Ricci curvature lower bound, which are very useful in noncommutative analysis and quantum information theory. Some recent work will also be discussed if time permits. This is based on joint work (arXiv:2007.13506) with Melchior Wirth (IST Austria).
Schedule: 7th April 2021, Wednesday at 17:00hrs (Indian Standard Time)
Speaker: Dr. Tiju Cherian John (University of South Carolina )
Abstract: Click here for the pdf file of the abstract.
Schedule: 14th April 2021, Wednesday at 17:00hrs (Indian Standard Time)
Mini-course on Planar Algebras.
Speaker: Dr. Keshab Chandra Bakshi (DST INSPIRE Faculty, CMI)
Abstract: Jones initiated the study of modern theory of subfactors in the early 1980's. Subsequently, he invented planar algebras as an axiomatization of the `standard invariant of a subfactor'. The mini-course is aimed to give an exposition of some aspects of this theory, including a discussion of the famous GJS-construction which reconstructs a subfactor from a `good planar algebra'.
Prerequisites: Functional analysis and a basic course on operators on Hilbert space.
Schedule: 19th May 2021 -- 9th June 2021, Wednesday at 17:00hrs (Indian Standard Time)