Work
Currently, my research focuses on the intersection of quantum algebras and category theory. I spend a lot of my time thinking about ways in which abstraction can help us understand physical structures that are otherwise poorly behaved.
Research
Published in arXiv, 2025
This article offers an intuitive introduction to monoidal categories through the lens of painting, presenting abstract mathematical concepts with visual and tactile analogies. Aimed at curious undergraduates and non-specialists, it seeks to demystify category theory by showing how ideas like the tensor product, associators, and braidings can be understood as compositional tools on a canvas. [Submitted for Publication]
Recommended citation: Komalan, K. (2025). Brushstrokes and Tensor Products: Painting with a Monoidal Category. arXiv preprint arXiv:2508.05482.
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Published in arXiv, 2025
We introduce “quantum barcodes,” a theoretical framework that applies persistent homology to classify topological phases in quantum many-body systems. By mapping quantum states to classical data points through strategic observable measurements, we create a “quantum state cloud” analyzable via persistent homology techniques. Our framework establishes that quantum systems in the same topological phase exhibit consistent barcode representations with shared persistent homology groups over characteristic intervals. We prove that quantum phase transitions manifest as significant changes in these persistent homology features, detectable through discontinuities in the persistent Dirac operator spectrum. Using the SSH model as a demonstrative example, we show how our approach successfully identifies the topological phase transition and distinguishes between trivial and topological phases. While primarily developed for symmetry-protected topological phases, our framework provides a mathematical connection between persistent homology and quantum topology, offering new methods for phase classification that complement traditional invariant-based approaches. [Submitted for Publication]
Recommended citation: Komalan, K. (2025). Quantum Barcodes: Persistent Homology for Quantum Phase Transitions. arXiv preprint arXiv:2504.10468.
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Published in arXiv, 2024
In this paper, we provide a construction of a Topological Quantum Field Theory from a Non-Hermitian Ribbon Fusion Category. This is a simple method that does not involve enriching over Fusion Categories, or using other complicated structures. To substantiate this construction, we also prove theorems on the Müger center, braiding, and spherical structure of such a fusion category. [Submitted for Publication]
Recommended citation: Komalan, K. (2024). Constructing TQFTs Using Non-Hermitian Ribbon Fusion Categories. arXiv preprint arXiv:2410.16993.
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