"Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection." --- Hermann Weyl, Symmetry (1952)
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Research Overview: Operator Algebras, Subfactors & Quantum Symmetries
My research is in operator algebras, with a primary focus on Vaughan Jones' subfactor theory and finite-index inclusions of von Neumann and C*-algebras. A central goal of my program is to understand how a finite-index inclusion N ⊆ M encodes algebraic, geometric, combinatorial, and quantum structures.
1. Subfactor Commuting Squares, Planar Subalgebras & Biprojection Lattices
We study Popa's smooth commuting squares, planar subalgebra embeddings, Bisch's biprojections, and the lattice L(N ⊆ M) of intermediate subalgebras. Key works include our planar subalgebra characterization of smooth commuting squares (with V. Kodiyalam, J. Operator Theory 2021), providing a bound for the cardinality of the finite lattice of intermediate C*-subalgebras (with V. P. Gupta, J. London Math. Soc. 2021), and structural/dynamical properties of subfactor commuting squares (with C. Silambarasan, arXiv:2608.23125).
2. Regular Inclusions, Pimsner–Popa Bases & Unitary Normalizers
We investigate two-sided Pimsner–Popa bases, unitary orthonormal bases, normalizers, and Weyl group crossed product decompositions for II₁ subfactors, simple unital C*-inclusions, and finite-dimensional C*-inclusions (with B. V. Rajarama Bhat, IMRN 2025; with V. P. Gupta, Münster J. Math. 2025; with C. Silambarasan and B. Pal, arXiv:2605.12221).
3. Two Subfactors: Hadamard / Biunitary / Spin-Vertex Models, Angles & Relative Entropy
We investigate the relative position of pairs of subfactors (P, Q) inside a finite-index inclusion--combining concrete constructions of subfactor pairs from complex Hadamard matrices, noncommutative biunitary matrices, spin models, and vertex models (with S. Guin and Guruprasad, J. Operator Theory 2025; J. Aust. Math. Soc. 2025, 2026; IJM 2025; PIAS 2025) with geometric and information-theoretic invariants including operator angles and quadrilateral rigidity (with S. Das, Z. Liu, and Y. Ren, Trans. Amer. Math. Soc. 2019), relative entropy of intermediate subfactors (Proc. Amer. Math. Soc. 2022), and von Neumann entropy of the angle operator (with S. Guin and B. Pal, J. Noncommut. Geom. 2024).
4. Weak Quantum Hypergroups, Fourier-Theoretic Inequalities & Higher Reflections in C*-Inclusions
We study quantum symmetries and harmonic analysis on finite-index C*-inclusions B ⊆ A—constructing weak quantum hypergroups associated with finite-index C*-inclusions (with D. Goswami and B. Pal, IMRN 2026), proving Fourier-theoretic inequalities for inclusions of simple C*-algebras (with S. Guin and Sruthymurali, NYJM 2023), and analyzing higher reflections and canonical shift entropy for C*-inclusions with finite Watatani index (with S. Guin, B. Pal, and Sruthymurali, IJM 2025).
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