arXiv subjects

Gunja Sachdeva

Publications and source records attributed to Gunja Sachdeva.

6 recordsLinked to original sources

arXiv2026

$p$-adic Bi-Filtrations for Topological Machine Learning on Genomic Sequences

We introduce pVR, a topological machine learning framework for alignment-free genomic sequence classification that combines $p$-adic numbers with topological data analysis. Each DNA sequence is encoded along two complementary axes: a $p$-adic distance on $k$-mer prefixes, which captures hierarchical positional structure, and a compositional $L_1$ distance on $k$-mer frequencies, which captures local sequence content. The two distances jointly parameterise a bi-filtered Vietoris--Rips complex, and per-sequence topological summaries from this bi-filtration serve as features for standard machine learning classifiers. We establish theoretical guarantees for the construction: stability under metric perturbations and invariance to the choice of prime, alongside a result that explains why a single $p$-adic axis is topologically uninformative and why the bi-filtration recovers nontrivial homology. On twelve genomic benchmarks ($28$ to $500$ sequences, $3$ to $7$ classes), pVR outperforms four established alignment-free baselines on three of six low-sample datasets, with gains of up to $21$ percentage points; it underperforms only on a SARS-CoV-2 variant benchmark whose point-mutation divergence violates the hierarchical assumption, and all methods saturate in the large-sample regime. pVR also outperforms zero-shot frozen embeddings from the 500M-parameter Nucleotide Transformer v2 by $6.7$ to $11.4$ percentage points on three low-sample benchmarks. The pVR codebase is publicly available at https://github.com/MAHI-Group/pVR.

q-bio.QM↗

arXiv2025

A finite Linear Dependence of Discrete Series Multiplicities

Let $G$ be a connected semisimple simply connected Lie group with a compact Cartan subgroup and let $\Gamma$ be a uniform lattice in $G$. Let $\widehat{G}_d$ denote the set of equivalence classes of unitary discrete series representations of $G$. We prove that for any finite subset of $\widehat{G}_d$ satisfying a certain condition, the associated finite set of discrete series multiplicities in $L^2(\Gamma \backslash G)$ determines all discrete series multiplicities in $L^2(\Gamma \backslash G)$. This allows us to obtain a refinement of the strong multiplicity one result for discrete series representations. As an application, we deduce that for two given levels, the equality of the dimensions of the spaces of cusp forms over a suitable finite set of weights implies the equality of the dimensions of the spaces of cusp forms for all weights.

math.RT↗

arXiv2024

On Infinitesimal $\tau$-Isospectrality of Locally Symmetric Spaces

Let $(\tau, V_{\tau})$ be a finite dimensional representation of a maximal compact subgroup $K$ of a connected non-compact semisimple Lie group $G$, and let $\Gamma$ be a uniform torsion-free lattice in $G$. We obtain an infinitesimal version of the celebrated Matsushima-Murakami formula, which relates the dimension of the space of automorphic forms associated to $\tau$ and multiplicities of irreducible $\tau^\vee$-spherical spectra in $L^2(\Gamma \backslash G)$. This result gives a promising tool to study the joint spectra of all central operators on the homogenous bundle associated to the locally symmetric space and hence its infinitesimal $\tau$-isospectrality. Along with this we prove that the almost equality of $\tau$-spherical spectra of two lattices assures the equality of their $\tau$-spherical spectra.

math.RT↗

arXiv2024

Algebraicity of ratios of special $L$-values for $\mathrm{GL}(n)$

We prove, under certain assumptions, algebraicity of the ratio $L(m, \Pi \times \chi)/L(m, \Pi \times \chi')$, where $\Pi$ is a cuspidal automorphic cohomological unitary representation of $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$, and $\chi$, $\chi'$ are finite order Hecke characters such that $\chi_{\infty} = \chi'_{\infty} = \mathrm{sgn}^{r}$, and $m, r$ are specific positive integers which depends only on $\Pi_{\infty}$. The methods in this article are a generalization of those in the work of Mahnkopf [Cohomology of arithmetic groups, parabolic subgroups and the special values of $L$-functions of GL(n), J. Inst. Math. Jussieu, 4 (2005)].

math.NT↗

arXiv2023

On the Linear Algebraic Monoids Associated to Congruence of Matrices

This paper discusses the generalized congruence equation $X^tAX=B$, for $X \in M_n(k)$ over any field $k$, through the action of monoid $Sol_A \times Sol_B := \{X \ | \ X^tAX = A\} \times \{X \ | \ X^tBX = B\}$. We have completely characterized for what matrices $A$, the monoid $Sol_A$ is a Lie group. We have given the structure of the Lie group $Sol_A$ and $Sol_{A^2}$, and their Lie algebras when $A$ is $n \times n$ nilpotent matrix of nilpotency $n$. In this case, we have also proved that the invariants of $Sol_A$ for any $n$, and $Sol_{A^2}$ for $n$ even, are finitely generated.

math.RA↗

arXiv2022

Results on a Strong Multiplicity One Theorem

We prove an analogue of the strong multiplicity one theorem in the context of $\tau_n$-spherical representations of the group $G = SO(2,1)^\circ$ appearing in $L^2(\Gamma_i \backslash G)$ for uniform torsion-free lattices $\Gamma_i, i = 1, 2$ in $G$. This is a generalisation of a previous result by the first author and C. S. Rajan in \cite{B-R-2011} for the case of $G = SO(2,1)^\circ$.

math.RT↗