arXiv subjects

Montek Singh Gill

Publications and source records attributed to Montek Singh Gill.

3 recordsLinked to original sources

arXiv2026

2-Linearizability of Geometric 3-Manifold Groups Over Commutative Rings

The fundamental groups of compact 3-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by only allowing finite groups of the form $\mathrm{PGL}(2,R)$, where $R$ is a finite commutative ring. In earlier work, this conjecture was disproven in full generality. The conjecture arose in the context of orientable connected compact 3-manifolds which are geometrizable. By constructing explicit faithful linear representations using rings with nilpotent elements, we demonstrate that the conjecture holds for six of the eight Thurston model geometries, namely all but $\mathbb{S}^3$ and $\widetilde{\mathrm{SL}_2}$. In the case of $\mathbb{S}^3$, the conjecture holds if we replace $\mathrm{PGL}(2,R)$ with $\mathrm{GL}(2,R)$. A spherical counterexample for the projective variant is the Poincaré homology sphere $Σ(2,3,5)$. In the case of $\widetilde{\mathrm{SL}_2}$, the conjecture fails to hold for both the projective and non-projective variants; a counterxample is provided by the Brieskorn sphere $Σ(2,3,7)$.

math.GT↗

arXiv2025

Stabilizations of $\mathbb{E}_\infty$ Operads and $p$-Adic Stable Homotopy Theory

We study differential graded operads and $p$-adic stable homotopy theory. We first construct a new class of differential graded operads, which we call the stable operads. These operads are, in a particular sense, stabilizations of $\mathbb{E}_\infty$ operads. For example, we construct a stable Barratt-Eccles operad. We develop a homotopy theory of algebras over these stable operads and a theory of (co)homology operations for algebras over these stable operads. We note interesting properties of these operads, such as that, non-equivariantly, in each arity, they have (almost) trivial homology, whereas, equivariantly, these homologies sum to a certain completion of the generalized Steenrod algebra and so are highly non-trivial. We also justify the adjective "stable" by showing that, among other things, the monads associated to these operads are additive in the homotopy coherent, or $\infty$-, sense. We then provide an application of our stable operads to $p$-adic stable homotopy theory. It is well-known that cochains on spaces yield examples of algebras over $\mathbb{E}_\infty$ operads. We show that in the stable case, cochains on spectra yield examples of algebras over our stable operads. Moreover, a result of Mandell says that, endowed with the $\mathbb{E}_\infty$ algebraic structure, cochains on spaces provide algebraic models of $p$-adic homotopy types. We show that, endowed with the algebraic structure encoded by our stable operads, spectral cochains provide algebraic models for $p$-adic stable homotopy types.

math.AT↗

arXiv2023

Interpreting the Euler-Lagrange Equations as the Gradient of the Action Functional

We study the smooth path spaces of Euclidean spaces $\mathbb{R}^N$, as diffeological spaces. We show that the tangent spaces of the free path space $\mathscr{P}$ are isomorphic to $\mathscr{P}$ itself, and that the tangent spaces of the space $\mathscr{P}_{\mathbf{p}, \mathbf{q}}$ of paths with fixed endpoints $\mathbf{p}$ and $\mathbf{q}$ are isomorphic to the smooth loop space of $\mathbb{R}^N$ based at the origin. We also define cotangents and gradients of smooth maps from these path spaces, and then show that, in the case of the action functional which arises in the calculus of variations, the gradient is precisely the path formed out of the terms of the Euler-Lagrange equations. We show that solutions of the Euler-Lagrange equations correspond precisely to the zeros of the gradient, and also provide analogous interpretations for the constrained Euler-Lagrange equations. This gives an illuminating geometric perspective on these equations. Finally, we illustrate the theory with several concrete examples from geometry, mechanics and machine learning.

math.GM↗