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Harsh Patil

Publications and source records attributed to Harsh Patil.

6 recordsLinked to original sources
arXiv2026

A Finitary Approach to Coarse Separation of Euclidean spaces

We give a novel proof of the fact that every coarsely separating family of subsets of the Euclidean space $\mathbb{R}^{d}$ must have asymptotic dimension at least $d-1$. The proof only uses singular homology/cohomology and standard facts from algebraic topology, such as Alexander duality. We do this by first reducing the problem to a finitary version of it. Using our approach, it follows immediately that every coarsely separating family of subsets of a $d$-dimensional Euclidean building or a product of $d$ geodesic, geodesically complete metric spaces has asymptotic dimension at least $d-1$. As a corollary, we obtain obstructions to coarse embeddings of Euclidean spaces into certain fundamental groups of graphs of groups.

math.MG↗
arXiv2026

The Second Challenge on Cross-Domain Few-Shot Object Detection at NTIRE 2026: Methods and Results

Cross-domain few-shot object detection (CD-FSOD) remains a challenging problem for existing object detectors and few-shot learning approaches, particularly when generalizing across distinct domains. As part of NTIRE 2026, we hosted the second CD-FSOD Challenge to systematically evaluate and promote progress in detecting objects in unseen target domains under limited annotation conditions. The challenge received strong community interest, with 128 registered participants and a total of 696 submissions. Among them, 31 teams actively participated, and 19 teams submitted valid final results. Participants explored a wide range of strategies, introducing innovative methods that push the performance frontier under both open-source and closed-source tracks. This report presents a detailed overview of the NTIRE 2026 CD-FSOD Challenge, including a summary of the submitted approaches and an analysis of the final results across all participating teams. Challenge Codes: https://github.com/ohMargin/NTIRE2026_CDFSOD.

cs.CV↗
arXiv2025

Relative cohomological dimension of a relatively hyperbolic pair

We show that the relative cohomological dimension $\cd(G,H)$ of a relatively hyperbolic pair $(G,H)$ is always finite when $G$ is torsion-free. We also show that this dimension is preserved under quasi-isometries, provided that $G$ is torsion-free and the peripheral subgroup $H$ is unconstricted and of type $F_{\infty}$. As a corollary of our methods, we compute $\cd(G,H)$ in a range of cases.

math.GR↗
arXiv2023

Finiteness properties and Relatively Hyperbolic Groups

We show that properties $F_n$ and $FP_n$ hold for a relatively hyperbolic group if and only if they hold for all the peripheral subgroups. As an application we show that there are at least countably many distinct quasi-isometry classes of one-ended groups that are type $F_n$ but not $F_{n+1}$ and similarly of type $FP_n$ and not $FP_{n+1}$ for all positive integers $n$.

math.GR↗
arXiv2021

Generating the liftable mapping class groups of regular cyclic covers

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 1$, and let $\mathrm{LMod}_{p}(X)$ be the liftable mapping class group associated with a finite-sheeted branched cover $p:S \to X$, where $X$ is a hyperbolic surface. For $k \geq 2$, let $p_k: S_{k(g-1)+1} \to S_g$ be the standard $k$-sheeted regular cyclic cover. In this paper, we show that $\{\mathrm{LMod}_{p_k}(S_g)\}_{k \geq 2}$ forms an infinite family of self-normalizing subgroups in $\mathrm{Mod}(S_g)$, which are also maximal when $k$ is prime. Furthermore, we derive explicit finite generating sets for $\mathrm{LMod}_{p_k}(S_g)$ for $g \geq 3$ and $k \geq 2$, and $\mathrm{LMod}_{p_2}(S_2)$. For $g \geq 2$, as an application of our main result, we also derive a generating set for $\mathrm{LMod}_{p_2}(S_g) \cap C_{\mathrm{Mod}(S_g)}(\iota)$, where $C_{\mathrm{Mod}(S_g)}(\iota)$ is the centralizer of the hyperelliptic involution $\iota \in \mathrm{Mod}(S_g)$. Let $\mathcal{L}$ be the infinite ladder surface, and let $q_g : \mathcal{L} \to S_g$ be the standard infinite-sheeted cover induced by $\langle h^{g-1} \rangle$ where $h$ is the standard handle shift on $\mathcal{L}$. As a final application, we derive a finite generating set for $\mathrm{LMod}_{q_g}(S_g)$ for $g \geq 3$.

math.GT↗