PDF-RS Annual Symposium 2025

Abstract. Given a finite twisted ideal polygon in \(\mathbb{H}^3\), there exists a harmonic map heat flow \(u_t : \mathbb{C} \to \mathbb{H}^3\) such that the image of \(u_t\) is asymptotic to that polygon for all \(t \in [0, \infty)\). We extract a sequence \(\hat{u}_n\) such that the image of leaves corresponding to certain quadratic differential under \(\hat{u}_n\) has bounded geodesic curvature and the bound is independent of \(n\). Moreover, if the bound is less than 1, then there exists a sequence \(\hat{u}_n\) which subconverges to a harmonic map \(\hat{u}_\infty : \mathbb{C} \to \mathbb{H}^3\) asymptotic to a twisted ideal polygon.

Abstract. In their foundational work, Murray and von Neumann studied a specific class of unbounded operators that are, in a precise sense, associated with a given von Neumann algebra. These operators are now referred to as Murray-von Neumann affiliated operators. Their study led to significant structural results concerning the underlying von Neumann algebra. It was later proven that for finite von Neumann algebras, the set of affiliated operators, as defined by Murray and von Neumann, forms a well-structured algebraic object, specifically a topological \(*\)-algebra. However, the algebraic structure of affiliated operators in the general case remains mysterious. In this talk, we introduce a new class of unbounded operators associated with a given von Neumann algebra, which we term affiliated operators. We show that for a von Neumann algebra \(\mathscr{M}\), the set of affiliated operators, denoted \(\mathscr{M}_{\textrm{aff}}\), is closed under the operator sum, product, Kaufman inverse, and adjoint operations. Furthermore, we demonstrate that \(\mathscr{M}_{\textrm{aff}}\) has the structure of a (right) near-semiring. Additionally, we prove that the near-semiring \(\mathscr{M}_{\textrm{aff}}\) transforms functorially under normal \(*\)-homomorphisms, establishing it as an intrinsic construction of \(\mathscr{M}\). As a key result, we show that this new notion of affiliation encompasses the classical Murray-von Neumann affiliation. By leveraging the functoriality of \(\mathscr{M}_{\textrm{aff}}\), we also demonstrate that the set of Murray-von Neumann affiliated operators is intrinsic to \(\mathscr{M}\), thus independent of any specific representation.

Abstract. The idea of paired operators can be traced back at least to Widom's 1960s work in the context of singular integral equations. In this talk, we will discuss about paired operators on the Hilbert space \(L^2(\mathbb{T})\) of all Lebesgue square-integrable functions on the unit circle \(\mathbb{T}\). Toeplitz and Hankel operators have different facades and flavors, but the classification problem of paired operators on \(L^2(\mathbb{T})\) prompted us to investigate Toeplitz + Hankel operators on the Hardy space, and they have also been classified. We will also introduce the notion of inner- paired operators defined on the classical model spaces and discuss about its classification. This talk is based on a joint work with Nilanjan Das and Jaydeb Sarkar.

Abstract. I will introduce the Lefschetz algebra \(L^*(X)\) on a smooth complex projective variety \(X\) and construct an example which does not satisfy the analogues of Hard Lefschetz and Poincaré Duality. To do this I will introduce Grassmanians and flag varieties. 8 The contents of this talk are mostly from the paper by June Huh and Botong Wang - Lefschetz Classes on Projective Varieties.

Abstract. We prove a Hankel-variant commutant lifting theorem. This also uncovers the complete structure of the Beurling-type reducing and invariant subspaces of Hankel operators. Kernel spaces of Hankel operators play a key role in the analysis.

Abstract. In 2000, VanderKam proved that for sufficiently large primes \(p\), the Hecke operators are linearly independent in their actions on the cycle \(\mathbf{e}\) from \(0\) to \(\infty\) in the space of weight \(2\) cuspidal modular symbols for \(\Gamma_0(p)\). I will talk about the generalization of VanderKam's result to modular symbols of higher weights. This is a joint work with D. Banerjee.

Abstract. The problem of completing partially specified matrices to positive matrices has been extensively studied due to its theoretical significance and practical applications. In this talk, we address an analogous problem: completing a linear map on \(C^*\)-algebras to a completely positive map. It is shown that whenever such a completion is feasible there exists a unique minimal completion. This theorem is used to show that under some very general conditions a completely positive map almost everywhere equivalent to a quasi-pure map is actually equal to that map. This talk is based on a joint work with my advisor Prof. B. V. R Bhat.

Abstract. In this talk, we will consider de Branges-Rovnyak spaces of a considerably large class of reproducing kernel Hilbert spaces and find a characterization for them to be complete Nevanlinna-Pick spaces. We will show that a non-trivial de Branges-Rovnyak space, associated to a contractive multiplier, of the Hardy space over the bidisc or the Bergman space over the unit disc is a complete Nevanlinna-Pick space if and only if it is isometrically isomorphic to the Hardy space over the unit disc as reproducing kernel Hilbert spaces. On the contrary, it will be shown that non-trivial de Branges-Rovnyak spaces of the Hardy space over the \(n\)-disc with \(n\geq 3\) are never complete Nevanlinna-Pick spaces. The talk will be based on joint work with H. Ahmed, B.K. Das.

Abstract. The notions of joint and outer spectral radii are extended to the setting of Hilbert \(C^\ast\)-bimodules. A Rota-Strang type characterisation is proved for the joint spectral radius. In this general setting, an approximation result for the joint spectral radius in terms of the outer spectral radius has been established. 8 This work leads to a new proof of the Wielandt-Friedland's formula for the spectral radius of positive maps. It is observed that algebras generated by tuples of matrices can be determined and their dimensions can be computed by realizing them as linear span of Choi-Kraus coefficients of some easily computable completely positive maps. This talk is based on a joint work with B V Rajarama Bhat and Prajakta Sahasrabuddhe.

Abstract. I will discuss a broader category of risk measures referred to as spectral risk measures, along with their estimation in the context of left truncated and right censored (LTRC) data. LTRC are encountered frequently in insurance loss data due to deductibles and policy limits. Risk estimation is an important task in insurance as it is a necessary step for determining premiums under various policy terms. Spectral risk measures are inherently coherent and have the benefit of connecting the risk measure to the user's risk aversion. We propose a nonparametric estimator of spectral risk measure using the product limit estimator and establish the asymptotic normality for our proposed estimator. We also develop an Edgeworth expansion of our proposed estimator. The bootstrap is employed to approximate the distribution of our proposed estimator and is shown to be second-order accurate. Monte Carlo studies are conducted to compare the proposed spectral risk measure estimator with the existing parametric and nonparametric estimators for left-truncated and right-censored data. Based on our simulation study we estimate the exponential spectral risk measure for two data sets viz; Norwegian fire claims and French marine losses.

Abstract. A lemniscate of a monic complex polynomial \( Q_n \) is the sublevel set \( \{z \in \mathbb{C} \mid |Q_n(z)| < t\} \) for some \( t > 0 \). Erdős, Herzog, and Piranian initiated the study of lemniscates, focusing on geometric and topological properties such as area, length, and the number of connected components of unit lemniscates (\( t = 1 \)) in their 1958 paper [1]. While the maximal length of a lemniscate for a degree-\( n \) polynomial is known to be of the order \( n \), this talk explores the typical length of random polynomial lemniscates. Using an integral geometric formula, we show that for polynomials with i.i.d. roots uniformly distributed in \( \mathbb{D} \), the expected length of their lemniscates is bounded above by \( O(\log n) \), significantly less than the optimal order.

Abstract. Continuous piecewise monotone maps on compact intervals provide simple examples of dynamical systems, however their behavior can be very complex. In fact, even the simplest of such maps, which is piecewise linear and has only one turning point, can be chaotic in the sense of both Devaney and Li-Yorke, such as the so-called tent map. In this talk we will look at certain invariants for such maps that can help us in classifying them up to topological conjugacy.

Abstract. Let \(\mathbb K\) be a field, \(A\) be a matrix in \(M_n(\mathbb K)\), and \(\mathbb L\) be the splitting field of the characteristic polynomial of \(A\). In this talk, we will prove the existence and uniqueness of the Jordan-Chevalley decomposition of \(A\), viewed as a matrix in \(M_n(\mathbb L)\), in an elementary computational way. The main strategy is the transformation of \(A\) to an appropriate block-diagonal form via similarity transformations obtained from Roth's removal rule. Moreover, if \(\mathbb F\) is the fixed field of \(\mathrm{Aut}(\mathbb L/ \mathbb K)\) then the potentially- diagonalizable and nilpotent parts of \(A\) are both in \(M_n(\mathbb F)\); in particular, when \(\mathbb K\) is a perfect field, they are both in \(M_n(\mathbb K)\). With the block-diagonal form of \(A\) in hand, by appropriately using Roth's removal rule in the context of strictly upper-triangular matrices, we will arrive at the Jordan canonical form of \(A\).

Abstract. In this talk, we shall discuss the classification of bounded operators that are products of two orthogonal projections on a Hilbert space. Our results complement some of the classical results of Crimmins and von Neumann. We also characterize the products of two inner projections defined on the Hardy space over the unit polydisc. This talk is based on a joint work with Jaydeb Sarkar.

Abstract. Let \(\mathcal{H}\) be a separable Hilbert space. A one-parameter \(E_0\)-semigroup \(\{\alpha_t\}_{t \geq 0}\) is a family of unital, normal, \(*\)-endomorphisms of \(B(\mathcal{H})\) that obey the semigroup property, along with some continuity conditions. \(E_0\)-semigroups can be classified into three types depending upon the "abundance" of units and whether the said units generate the associated product system. One-parameter \(E_0\)-semigroups have been studied extensively by Powers, Arveson and other authors, and Arveson showed in his works that a one-parameter \(E_0\)-semigroup is a CCR flow if and only if it has units and is type I. He also introduced a numerical invariant, called the index, which provides an estimate about the number of units of an \(E_0\)-semigroup. 8 9 In this talk, I shall give a brief introduction to multi-parameter \(E_0\)-semigroups, focusing on CCR flows over closed convex cones in \(\mathbb{R}^d\), briefly touching on the differences between the one- parameter and multi-parameter theory. I shall also define Induced Isometric Representations, where we "induce" isometric representations from discrete semigroups onto continuous ones, and explain how these help with achieving uncountably many examples of prime type I CCR flows with index \(k\), for any \(k \in \mathbb{N} \cup \{\infty\}\).

Abstract. In this talk, we shall describe all the invariant subspaces of a completely nonunitary contraction \(T_c\) with nonzero scalar constant characteristic function \(c\), where \(|c|\lt 1 \). Moreover, we show that \(T_c\) and the restriction to its invariant subspaces are essentially normal. Using this description of \(T_c\)-invariant subspaces, we then identify the hyperinvariant subspaces.

Abstract. A function \(f\) on the real line is called an operator Lipschitz function if \[ \|f(A)-f(B)\|\le \text{const} \, \|A-B\| \] for arbitrary self-adjoint operators \(A\) and \(B\). Not all Lipschitz functions are operator Lipschitz. For example, the function \(|x|\) is not operator Lipschitz. The class of operator Lipschitz functions plays an important role in perturbation theory. I am going to give necessary conditions and sufficient conditions for a function to be operator Lipschitz. Similarly, one can define operator Lipschitz functions on the circle and on subsets of the complex plane.