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Arijit Mukherjee

Publications and source records attributed to Arijit Mukherjee.

16 recordsLinked to original sources
arXiv2026

Classification of products of Fano varieties with Picard number one

Given a partition $(n_1,\ldots,n_r)$ of a positive integer $n$, one has the associated $n$-dimensional multiprojective space $\mathbb{P}^{n_1}\times \cdots \times \mathbb{P}^{n_r}$. We show that distinct partitions of $n$ yield non-isomorphic multiprojective spaces, giving a new proof via the extremal contractions of their closed cone of curves. In contrast to the earlier approaches, the argument here is uniform across all partitions, and extends beyond multiprojective spaces. In fact, we further extend it to a more general setting, namely to products of Fano varieties of Picard number one: we prove that a fixed such factor in each dimension makes the products attached to distinct partitions pairwise non-isomorphic. As a consequence, a complete classification of products of smooth quadrics, of dimension $\geq 3$, has been obtained.

math.AG↗
arXiv2026

Commutators of finite multiplicative order

This article studies the equation $[A,B]^k = \Id_n$ for matrices over $\CC$,characterizing the pairs $(k,n)$ for which solutions exist via a classical result of Lam and Leung on sums of roots of unity. The problem is next generalized to matrix rings $M_n(S)$ over arbitrary unital rings $S$, where a sufficient condition on the unity of $S$ is established and explicit constructions of solutions are provided. Beyond matrix rings, the structural implications of the equation $[a,b]^n = 1$ in a general unital ring $R$ are investigated, yielding a collection of idempotents whose properties govern the ring's structure. We prove that under a suitable condition on these idempotents, $[a,b]^n = 1$ implies $R$ is isomorphic to $M_n(S)$ for some unital ring $S$. We also provide an alternative proof using a result on characterisation of matrix rings by Goyal and Khurana. These results together establish a framework connecting commutator equations and classical criteria for recognizing full matrix rings.

math.RA↗
arXiv2026

On the classification of products of Hilbert schemes of points over a surface

Let $S$ be a smooth projective surface over $\mathbb{C}$ and $S^{[n]}$ be the Hilbert scheme of $n$ points over $S$, for any positive integer $n$. Under suitable hypotheses, we show that the products of Hilbert schemes $S^{[{\bf a}]}$ and $S^{[{\bf b}]}$ associated to two distinct partitions ${\bf a}$ and ${\bf b}$ of $n$ are non-isomorphic, using Betti numbers, Hodge numbers and Euler characteristics of the individual factors. The classifications obtained using Betti numbers and Euler characteristics extend over $\overline{\F}_q$ as well, where $\F_q$ is the finite field with $q$ elements. We also provide a complete classification of such product spaces for K3 surfaces using their inherent symplectic structure.

math.AG↗
arXiv2026

Photometric Redshift Estimation Using Scaled Ensemble Learning

The development of the state-of-the-art telescopic systems capable of performing expansive sky surveys such as the Sloan Digital Sky Survey, Euclid, and the Rubin Observatory's Legacy Survey of Space and Time (LSST) has significantly advanced efforts to refine cosmological models. These advances offer deeper insight into persistent challenges in astrophysics and our understanding of the Universe's evolution. A critical component of this progress is the reliable estimation of photometric redshifts (Pz). To improve the precision and efficiency of such estimations, the application of machine learning (ML) techniques to large-scale astronomical datasets has become essential. This study presents a new ensemble-based ML framework aimed at predicting Pz for faint galaxies and higher redshift ranges, relying solely on optical (grizy) photometric data. The proposed architecture integrates several learning algorithms, including gradient boosting machine, extreme gradient boosting, k-nearest neighbors, and artificial neural networks, within a scaled ensemble structure. By using bagged input data, the ensemble approach delivers improved predictive performance compared to stand-alone models. The framework demonstrates consistent accuracy in estimating redshifts, maintaining strong performance up to z ~ 4. The model is validated using publicly available data from the Hyper Suprime-Cam Strategic Survey Program by the Subaru Telescope. Our results show marked improvements in the precision and reliability of Pz estimation. Furthermore, this approach closely adheres to-and in certain instances exceeds-the benchmarks specified in the LSST Science Requirements Document. Evaluation metrics include catastrophic outlier, bias, and rms.

astro-ph.GA↗
arXiv2025

Gram-like matrix preserving extensions and completions of noncommutative polynomials

Given a positive noncommutative polynomial $f$, equivalently a sum of Hermitian squares (SOHS), there exists a positive semidefinite Gram matrix that encrypts all the structural essence of $f$. There are no available methods for extending a noncommutative polynomial to a SOHS keeping the Gram matrices unperturbed. As a remedy, we introduce an equally significant notion of Gram-like matrices and provide linear algebraic techniques to get the desired extensions. We further use positive semidefinite completion problem to get SOHS and provide criteria in terms of chordal graphs and 2-regular projective algebraic sets.

math.OC↗
arXiv2024

A representation theoretic classification of multiprojective spaces

For a partition $(n_1,\ldots,n_r)$ of a positive integer $n$, consider the associated multiprojective space $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_r}$. That multiprojective spaces attached to distinct partitions of $n$ are pairwise non-isomorphic is known, having been established by algebro-geometric methods. In this paper we give a new, representation-theoretic proof. We prove that the complex cohomology ring of a multiprojective space carries a natural $\mathfrak{sl}(2,\mathbb{C})$-module structure compatible with the K\"unneth decomposition. The classification then follows from C. S. Rajan's theorem on the unique decomposition of tensor products of irreducible representations of a simple Lie algebra. We further show, by a dimension count argument, that this approach is intrinsic to the genus zero curve, distinguishing $\mathbb{P}^1$ from curves of higher genus.

math.AG↗
arXiv2024

ERATTA: Extreme RAG for Table To Answers with Large Language Models

Large language models (LLMs) with retrieval augmented-generation (RAG) have been the optimal choice for scalable generative AI solutions in the recent past. Although RAG implemented with AI agents (agentic-RAG) has been recently popularized, its suffers from unstable cost and unreliable performances for Enterprise-level data-practices. Most existing use-cases that incorporate RAG with LLMs have been either generic or extremely domain specific, thereby questioning the scalability and generalizability of RAG-LLM approaches. In this work, we propose a unique LLM-based system where multiple LLMs can be invoked to enable data authentication, user-query routing, data-retrieval and custom prompting for question-answering capabilities from Enterprise-data tables. The source tables here are highly fluctuating and large in size and the proposed framework enables structured responses in under 10 seconds per query. Additionally, we propose a five metric scoring module that detects and reports hallucinations in the LLM responses. Our proposed system and scoring metrics achieve >90% confidence scores across hundreds of user queries in the sustainability, financial health and social media domains. Extensions to the proposed extreme RAG architectures can enable heterogeneous source querying using LLMs.

cs.AI↗
arXiv2024

Dynamics of two by two symmetric matrices of trace zero

In this paper, we describe the entire structure of the vector space $Sym_2^0$ of all symmetric matrices of size $2$ having trace zero. This is motivated by the geometrical interpretation of any arbitrary element of $Sym_2^0$. We further study the orbits and stable sets of these elements. As an application of the obtained structure of $Sym_2^0$, we obtain the symmetric matrices of size $2$, trace of whose product with any trace zero symmetric matrix is zero. Finally some well known trigonometric formulas are interpreted geometrically incorporating the anatomy of $Sym_2^0$.

math.RA↗
arXiv2023

Diagonal property and weak point property of higher rank divisors and certain Hilbert schemes

In this paper, we introduce the notion of the diagonal property and the weak point property for an ind-variety. We prove that the ind-varieties of higher rank divisors of integral slopes on a smooth projective curve have the weak point property. Moreover, we show that the ind-variety of $(1,n)$-divisors has the diagonal property and is a locally complete linear ind-variety and calculate its Picard group. Furthermore, we obtain that the Hilbert schemes of a curve associated to the good partitions of a constant polynomial satisfy the diagonal property. In the process of obtaining this, we provide the exact number of such Hilbert schemes up to isomorphism by proving that the multi symmetric products associated to two distinct partitions of a positive integer $n$ are not isomorphic.

math.AG↗
arXiv2023

Automated Heterogeneous Low-Bit Quantization of Multi-Model Deep Learning Inference Pipeline

Multiple Deep Neural Networks (DNNs) integrated into single Deep Learning (DL) inference pipelines e.g. Multi-Task Learning (MTL) or Ensemble Learning (EL), etc., albeit very accurate, pose challenges for edge deployment. In these systems, models vary in their quantization tolerance and resource demands, requiring meticulous tuning for accuracy-latency balance. This paper introduces an automated heterogeneous quantization approach for DL inference pipelines with multiple DNNs.

cs.CV↗
arXiv2023

Hallucination-minimized Data-to-answer Framework for Financial Decision-makers

Large Language Models (LLMs) have been applied to build several automation and personalized question-answering prototypes so far. However, scaling such prototypes to robust products with minimized hallucinations or fake responses still remains an open challenge, especially in niche data-table heavy domains such as financial decision making. In this work, we present a novel Langchain-based framework that transforms data tables into hierarchical textual data chunks to enable a wide variety of actionable question answering. First, the user-queries are classified by intention followed by automated retrieval of the most relevant data chunks to generate customized LLM prompts per query. Next, the custom prompts and their responses undergo multi-metric scoring to assess for hallucinations and response confidence. The proposed system is optimized with user-query intention classification, advanced prompting, data scaling capabilities and it achieves over 90% confidence scores for a variety of user-queries responses ranging from {What, Where, Why, How, predict, trend, anomalies, exceptions} that are crucial for financial decision making applications. The proposed data to answers framework can be extended to other analytical domains such as sales and payroll to ensure optimal hallucination control guardrails.

cs.CL↗
arXiv2023

Estimates of K\"ahler metrics on noncompact finite volume hyperbolic Riemann surfaces, and their symmetric products

Let $X$ denote a noncompact finite volume hyperbolic Riemann surface of genus $g\geq 2$, with only one puncture at $i\infty$ (identifying $X$ with its universal cover $\mathbb{H}$). Let $\overline{X}:=X\cup\lbrace i\infty\rbrace$ denote the Satake compactification of $X$. Let $\Omega_{\overline{X}}$ denote the cotangent bundle on $\overline{X}$. For $k\gg1$, we derive an estimate for $\mu_{\overline{X}}^{\mathrm{Ber},k}$, the Bergman metric associated to the line bundle $\mathcal{L}^{k}:=\Omega_{\overline{X}}\otimes \mathcal{O}_{\overline{X}}\big((k-1)\infty\big)$. For a given $d\geq 1$, the pull-back of the Fubini-Study metric on the Grassmannian, which we denote by $\mu_{\mathrm{Sym}^d(\overline{X})}^{\mathrm{FS},k}$, defines a K\"ahler metric on $\mathrm{Sym}^d(\overline{X})$, the $d$-fold symmetric product of $\overline{X}$. Using our estimates of $\mu_{\overline{X}}^{\mathrm{Ber},k}$, as an application, we derive an estimate for $\mu_{\mathrm{Sym}^d(\overline{X}),\mathrm{vol}}^{\mathrm{FS},k}$, the volume form associated to the (1,1)-form $\mu_{\mathrm{Sym}^d(\overline{X})}^{\mathrm{FS},k}$.

math.CV↗
arXiv2022

Diagonal property and weak point property of higher rank divisors and certain Hilbert schemes

In this paper, we introduce the notion of the diagonal property and the weak point property for an ind-variety. We prove that the ind-varieties of higher rank divisors of integral slopes on a smooth projective curve have the weak point property. Moreover, we show that the ind-variety of $(1,n)$-divisors has the diagonal property. Furthermore, we obtain that the Hilbert schemes associated to the good partitions of a constant polynomial satisfy the diagonal property. On the process of obtaining this, we provide an upper bound on the number of such Hilbert schemes up to isomorphism. Furthermore, we prove that the obtained upper bound is attained in case of genus zero curves and hence conclude that the bound is sharp.

math.AG↗
arXiv2020

Tautological algebra of the moduli stack of semistable bundles of rank 2 on a general curve

Our aim is to determine the tautological algebra generated by the cohomology classes of the Brill-Noether loci in the rational cohomology of the moduli stack $\mathcal{U}_C(n,d)$ of semistable bundles of rank $n$ and degree $d$. We show that for a general smooth projective curve $C$ of genus $g\geq 2$, $d=2g-2$, the tautological algebra of $ \mathcal{U}_C(2,2g-2)$ (resp. the moduli stack $\mathcal{SU}_C(2,\mathcal{L})$ of semistable bundles of rank $2$ and determinant $\mathcal{L}$ with $\deg(\mathcal{L})=2g-2$) is generated by the divisor classes (resp. the class of the Theta divisor $\Theta$). This is previously known in rank one situation, called the (classical) Porteous formula.

math.AG↗
arXiv2019

Cross-Lingual Training for Automatic Question Generation

Automatic question generation (QG) is a challenging problem in natural language understanding. QG systems are typically built assuming access to a large number of training instances where each instance is a question and its corresponding answer. For a new language, such training instances are hard to obtain making the QG problem even more challenging. Using this as our motivation, we study the reuse of an available large QG dataset in a secondary language (e.g. English) to learn a QG model for a primary language (e.g. Hindi) of interest. For the primary language, we assume access to a large amount of monolingual text but only a small QG dataset. We propose a cross-lingual QG model which uses the following training regime: (i) Unsupervised pretraining of language models in both primary and secondary languages and (ii) joint supervised training for QG in both languages. We demonstrate the efficacy of our proposed approach using two different primary languages, Hindi and Chinese. We also create and release a new question answering dataset for Hindi consisting of 6555 sentences.

cs.CL↗