Hello there..!! This is Soumyadip. Welcome to the Blog.
Feel free to contact me via E-mail.
For all published papers, notes, Blog posts, updates on my research.
- E-mail:- elatednandi@gmail.com,
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https://orcid.org/0009-0000-2053-5254
Posts
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Open Problems solved by Frontier AI
Over the last year, and a better part of this year, a number of open problems have been solved by Frontier AI models. In this post, I have handpicked a few problems :- Erdos’s Unit distance problem, Claude Cycles (an open problem by Donald Knuth), Cosmic String Spectra Integral and the recently disproven Jacobian conjecture (counter-example).
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The Jacobian Conjecture - (CounterExample)
Note: The following concerns a counterexample to the Jacobian Conjecture announced by Levent Alpöge on X on July 19–20, 2026. As of July 23, 2026, no formal paper , apart from this has appeared, The arithmetic has been checked by multiple independent sources. I have provided one hand-checked verification below. The $n=2$ case remains open. This note presents the announced example and its structure; it will be revised as formal publications appear. This post is also part of another detailed post, I made about how Frontier models have contributed to solving Open Problems across disciplines.
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$\mathcal{N}=2$ Supersymmetry Algebra and BPS Bounds
For $\mathcal{N}=2$ supersymmetry, the derivation of supercharges is naturally understood as an extension of the Poincare algebra. The Poincare algebra $SO(1,3)$ is generated by translations $P_\mu$ ($\mu=0,1,2,3$) and the Lorentz generators $M_{\mu\nu}=-M_{\nu\mu}$. The main focus of this post is the derivation of BPS bounds from the central charges, BPS states and the the Bogomol’nyi Equations. A topological defect of $1d$ (cosmic strings) is taken as the foundation for the Bogomol’nyi decompositions.
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The Witten Index
Supersymmetric quantum mechanics possesses a remarkably rigid structure: the energy spectrum is non-negative, and non-zero energy states come in boson–fermion pairs. This pairing suggests a natural topological invariant — the Witten index — which counts the difference between the number of bosonic and fermionic ground states and, crucially, does not change under continuous deformations of the theory. In these notes, we build up to the definition of the Witten index starting from the supersymmetry algebra, explain why it is a robust diagnostic for spontaneous supersymmetry breaking, illustrate it with a concrete example, and finally sketch its path-integral representation.
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Supersymmetry Formalism from Commutation Relations
This article develops the mathematical construction of supersymmetry from the algebra of bosonic and fermionic operators. The aim is pedagogical rather than historical, emphasizing the logical progression from operator algebras to the superfield formalism.
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Paper - Towards a Human-Centered AGI
Published :- January, 2026
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Paper:- Topological Operators in Quantum Gravity:- A Treatment within Effective Field Theory Consistency
Download the pdf here
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Paper:- Global Symmetry Breaking:- A Perspective on Symmetry Inconsistencies .
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QFT Formalism
An introduction to Quantum field Theory formalism from relativistic fields. Symmetries and Noether’s theorem and interacting fields are the focal point of discussion.
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