arXiv subjects

Punya Plaban Satpathy

Publications and source records attributed to Punya Plaban Satpathy.

5 recordsLinked to original sources

arXiv2026

Near Isospectrality and Spectral Rigidity for Compact Locally Symmetric Manifolds

The inverse spectral problem asks to what extent the Laplace--Beltrami spectrum determines the geometry of a Riemannian manifold. We study a natural weakening, called \emph{near isospectrality}, in which the spectra of two compact manifolds agree outside a finite set, counted with multiplicity. We prove that for compact quotients of a fixed simply connected symmetric space of nonpositive sectional curvature, near isospectrality already forces full isospectrality. We then extend this rigidity to a broad collection of compact quotients of irreducible symmetric spaces of noncompact type. In this larger setting, near isospectrality determines enough heat invariants to identify the universal cover within the class under consideration, and the fixed-cover rigidity result then implies full isospectrality. Thus, within the class studied here, eventual agreement of the Laplace spectrum already forces complete spectral agreement.

math.SP↗

arXiv2025

Borel--Serre type bordifications and Canonical Pairs for Loop groups

To the symmetric space of the (positive half) of a real loop group, we attach a Borel--Serre type bordification and equip it with a Hausdorff topology. The attached boundary, indexed by certain rational parabolics of the loop group, is shown to be homotopic to an affine, rational Tits building. A loop analogue of an arithmetic group is also shown to act continuously on the bordification and its quotient by this action is studied using the reduction theory of H. Garland. While the quotient is no longer compact (as in the Borel--Serre construction from finite-dimensions) we relate the non-compactness to the center of the loop group. We also introduce a notion of semi-stability for loop groups, following works of Harder--Narasimhan, Behrend, and most recently Chaudouard, and use this to describe a partition of our loop symmetric space. This partition is then related to the rational bordification and its quotient.

math.RT↗

arXiv2025

Erdős-Kac type theorem for the number of scattering geodesics on modular surface

In 1917, Hardy and Ramanujan showed that if $ω(n)$ is the number of distinct prime factors of a randomly chosen positive integer $n,$ then the normal order of $ω(n)$ is $\log \log \, n.$ This led Erdős and Kac to prove their celebrated result showing a Gaussian behaviour for $ω(n).$ In this article we prove an Erdős-Kac kind result for the number of scattering geodesics on the modular surface with a common sojourn time.

math.NT↗

arXiv2025

Prime scattering geodesic theorem

The modular surface, given by the quotient $\mathcal{M} = \Ha/\text{PSL}(2,\Z)$, can be partitioned into a compact subset $\Mm$ and an open neighborhood of the unique cusp in $\mathcal{M}$. We consider scattering geodesics in $\mathcal{M}$, first introduced by Victor Guillemin in \cite{Guillemin1976-xr} for hyperbolic surfaces with cusps. These are geodesics in $\mathcal{M}$ that lie in $\mathcal{M} \setminus \Mm$ for both large positive and negative times. Associated with such a scattering geodesic in $\mathcal{M}$, a finite \textit{sojourn time} is defined in \cite{Guillemin1976-xr}. In this article, we study the distribution of these scattering geodesics in $\mathcal{M}$ and their associated \textit{sojourn times}. In this process, we establish a connection between the counting of scattering geodesics on the modular surface and the study of positive integers whose prime divisors lie in arithmetic progression. This article is the first such result for scattering geodesics.

math.NT↗

arXiv2020

Scattering theory on higher $\mathbb{Q}$-rank locally symmetric spaces

In 1977, Victor Guillemin published a paper discussing geometric scattering theory, in which he related the Lax-Phillips Scattering matrices (associated to a noncompact hyperbolic surface with cusps) and the sojourn times associated to a set of geodesics which run to infinity in either direction. This work was later extended to $\mathbb{Q}$-rank one Locally symmetric spaces coming from Semisimple Lie groups by Lizhen Ji and Maciej Zworski. Here, we will extend some of the above mentioned results to higher rank locally symmetric spaces, in particular we will introduce higher dimensional analogues of scattering geodesics called $\textbf{Scattering Flat}$ and study these flats in the case of the locally symmetric space given by the quotient $SL(3,\mathbb{Z}) \backslash SL(3,\mathbb{R}) / SO(3)$. A parametrization space is discussed for such scattering flats as well as an associated vector valued parameter (bearing similarities to sojourn times) called $\textbf{sojourn vector}$ and these are related to the frequency of oscillations of the associated scattering matrices coming from the minimal parabolic subgroups of $\text{SL}(3,\mathbb{R})$. The key technique is the factorization of higher rank scattering matrices.

math.SP↗