ISI Bangalore PDF-RS Symposium 2026

About

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The PDF-RS Symposium is an annual academic event organised by the Research Scholars at the Theoretical Statistics and Mathematics Unit (SMU) of the Indian Statistical Institute, Bangalore that serves as a platform for postdoctoral fellows and research scholars to present their research contributions.

In 2026, the event was organised over two days (02-03 March 2026) and featured 10 lectures. The organisers were Arghya Chongdar (SRF, ISIBC) and Ritaman Ghosh (SRF, ISIBC).

Photos from the symposium are available here.

Lecture Schedule

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Day 1 (Monday, 02 March 2026)

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Day 2 (Tuesday, 03 March, 2026)

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Lecture Abstracts

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Day 1 (March 02 2026, Monday)

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Lecture 1

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A transference principle for involution-invariant functional Hilbert spaces

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Speaker: Shubham Jain

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Affiliation: Postdoctoral Fellow, SMU, ISI Bangalore.

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Abstract: For any domain \(\Omega \subset \mathbb{C}^n\), the Bergman space is canonically defined. Moreover, if domains \(\Omega_1\) and \(\Omega_2\) are related by a biholomorphic or a proper map, then the corresponding Bergman spaces are naturally connected, and their Bergman kernels satisfy a transformation identity determined by the map.

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In contrast, for a general domain \(\Omega\), there is no canonical definition of the Hardy space. In this talk, we describe a construction of a Hardy space for a certain class of domains and investigate the relationship between the corresponding kernels when \(\Omega_1\) and \(\Omega_2\) are related by a \(2\)-proper map. This talk is based on a joint work with Prof. Sameer Chavan and Santu Bera.

Lecture 2

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Solvability of Automorphism group of Commutative algebras and Isolated Hypersurface Singularities

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Speaker: Dibyendu Das

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Affiliation: Senior Research Fellow, SMU, ISI Bangalore.

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Abstract: Let \(A\) be a finite-dimensional commutative associative algebra with unity over an algebraically closed field \(K\), and \(\text{Aut}_K(A)\) denotes the group of all \(K\)-algebra automorphisms of \(A\). The aim of the talk is to discuss about the solvability of \(\text{Aut}_K(A)\) and its connection with isolated hypersurface singularities.

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Lecture 3

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Visibility domains in complex manifolds

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Speaker: Rumpa Masanta

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Affiliation: Postdoctoral Fellow, SMU, ISI Bangalore.

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Abstract: In this talk, we extend the notion of visibility with respect to the Kobayashi distance to domains in arbitrary complex manifolds. Visibility here is a weak notion of negative curvature and refers to a property resembling visibility in the sense of Eberlein–O’Neill for Riemannian manifolds. However, we do not assume Cauchy-completeness, with respect to the Kobayashi distance, of the domains in question. The visibility property of a domain \(D\) can be used to deduce many properties of certain holomorphic mappings into \(D\), ranging from their continuous extendibility to the iterative dynamics of such self-maps of \(D\). Here, we present a few sufficient conditions for visibility in the above setting, and with these conditions, we see that the class of domains with the visibility property is very large. Finally, we will discuss an application of visibility and establish a far-reaching generalisation of the classical Wolff–Denjoy theorem.

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Lecture 4

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Hyponormal contractions and analytic shifts

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Speaker: Sneha B

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Affiliation: Senior Research Fellow, SMU, ISI Bangalore.

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Abstract: Hyponormal operators are known to be among the most difficult operators to analyze. In this work, we focus on two finite types of hyponormal operators. The first type becomes analytic shifts, while the second type admits analytic models. A basic model for hyponormal operators plays a key role in our analysis. This talk is based on joint work with Neeru Bala and Jaydeb Sarkar.

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Lecture 5

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Invariant subspaces of Brownian shifts

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Speaker: Soma Das

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Affiliation: NBHM Postdoctoral Fellow.

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Abstract: Brownian shifts were introduced by Agler and Stankus in the study of \(m\)-isometries, and they resolved the invariant subspace problem for these operators in the classical setting. In this talk, we will present classification results concerning the unitary equivalence of restrictions of Brownian shifts to their invariant subspaces in the classical case. Furthermore, we will extend the discussion to Brownian shifts acting on vector-valued Hardy spaces. In this broader framework, we obtain a characterization of invariant subspaces and provide a solution to the corresponding unitary equivalence problem for the invariant subspaces of these shifts.

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This talk is mainly based on joint work with Nilanjan Das and Jaydeb Sarkar.

Day 2 (March 03 2026, Tuesday)

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Lecture 1

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Foguel-type operators similar to contractions

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Speaker: Nilanjan Das

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Affiliation: Postdoctoral Fellow, SMU, ISI Bangalore

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Abstract: Pisier’s celebrated counterexample to Halmos’s similarity problem was based on \(2 \times 2\) upper triangular block operator matrices involving three classical operators: forward and backward shifts on the diagonal and Hankel operators in the off-diagonal entry. Together with another classical object, namely Toeplitz operators, one can formulate another \(2^3 -1 = 7\) types of \(2 \times 2\) upper triangular block operator matrices, which we refer to as Foguel-type operators. Our plan is to review some existing literature on the similarity problem and to describe some of our own results, characterizing when the aforementioned Foguel-type operators are similar to contractions. For the most part, this talk will be based on a joint work with Dr. Soma Das and Prof. Jaydeb Sarkar.

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Lecture 2

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Instantaneous Everywhere Explosion of Interacting Diffusions

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Speaker: Shubham Ovhal

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Affiliation: Senior Research Fellow, SMU, ISI Bangalore.

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Abstract: We consider a system of interacting SDEs on the integer lattice with multiplicative noise and a drift satisfying the finite Osgood’s condition. We show instantaneous everywhere blowup for initial profiles decaying slower than \(\exp ( -\sqrt{\big|\log |x|\big|})\). We employ the splitting-up method to compare the interacting system to a one-dimensional SDE which blows up.

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Lecture 3

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Words Play with Groups and Algebras

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Speaker: Saikat Panja

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Affiliation: Postdoctoral Fellow, SMU, ISI Bangalore

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Abstract: A word here refers to an element of a finitely generated free group (respectively a free \(k\)-algebra; \(k\) is a field) on \(r\) letters.

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Given a group \(G\) (resp. an algebra \(A\)) and a word, one gets a natural map \(G^r\longrightarrow G\) (resp. \(A^r\longrightarrow A\)) by substitution. These are known as word maps. This survey talk will present a selection of results on word maps for groups as well as algebras, obtained over the past several decades. I will also touch upon some collaborative contributions to the subject.

Lecture 4

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Closed Extensions of a Closed Operator

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Speaker: Gahin Maiti

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Affiliation: Senior Research Fellow, SMU, ISI Bangalore.

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Abstract: The talk follows the paper by Christopher Fischbacher titled The Closed Extensions of a Closed Operator.

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Given a densely defined and closed operator \(A\) acting on a complex Hilbert space \(\mathcal{H}\), we establish a one-to-one correspondence between its closed extensions and subspaces \(\mathfrak{M}\subset\mathcal{D}(A^*)\), that are closed with respect to the graph norm of \(A^*\) and satisfy certain conditions. In particular, this will allow us to characterize all densely defined and closed restrictions of \(A^*\). After this, we will express our results using the language of Gel’fand triples generalizing the well-known results for the selfadjoint case.

As applications we construct:

• a sequence of densely defined operators that converge in the generalized sense to a non-densely defined operator,
• a non-closable extension of a symmetric operator and
• selfadjoint extensions of Laplacians with a generalized boundary condition.

Lecture 5

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Matricial ranges, dilations, and unital contractive maps

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Speaker: Pankaj Dey

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Affiliation: Postdoctoral Fellow, SMU, ISI Bangalore

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Abstract: Arveson introduced the notion of the matricial range of an operator in his remarkable article called Subalgebras of \(C^*\)-algebras II (Acta Math, 128, 1972) and established that every unital positive map on the operator system generated by the Jordan block of size \(2\) is completely positive. This describes the matricial range of the Jordan block of size \(2\) as the set of all matrices with numerical radius at most \(\frac{1}{2}\). This proof is far from trivial and depends on subtle results of Ando. After fifty years of the above result of Arveson, the matricial range of the Jordan block of size \(n\) has not been characterized for \(n\geq 3\). In this talk, we will discuss recent progress on this long-standing open problem for \(n=3\), using dilation-theoretic techniques. This talk is based on joint work with Mithun Mukherjee and Atanu Dhang.

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