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Shah Faisal

Publications and source records attributed to Shah Faisal.

6 recordsLinked to original sources
arXiv2026

Lagrangian capacity and chain level string topology

We derive upper bounds for the Lagrangian capacities of Liouville domains with finite Gutt--Hutchings capacities and show that the Lagrangian capacity of a convex or concave toric domain of arbitrary dimension equals its diagonal. In particular, this completely settles the conjecture of Cieliebak-Mohnke on the Lagrangian capacity of ellipsoids. Our proof is based on an $S^1$-equivariant variant of the techniques of Fukaya and Irie, and does not use holomorphic curves with local tangency constraints, which would inevitably cause transversality issues. Moreover, we show that any extremal Lagrangian torus in an $2n$-dimensional ellipsoid must lie on the boundary. Applications of our results and techniques include new upper bounds on the Lagrangian width for aspherical Lagrangians in Liouville manifolds and the first computations of the Lagrangian capacities for many non-subcritical Weinstein domains in dimensions 4 and 6.

math.SG↗
arXiv2025

Stabilized symplectic embeddings of higher-dimensional ellipsoids

We provide a lower bound for the embedding capacity of higher-dimensional symplectic ellipsoids, formulated in terms of the Lagrangian capacity of ellipsoids. Our approach relies on examining the Borman--Sheridan class of a Weinstein neighborhood of a suitable monotone Lagrangian torus, using Tonkonog's string topology-based computation of the gravitational descendants of the torus.

math.SG↗
arXiv2025

Extremal Lagrangian tori in toric domains

Let $L$ be a closed Lagrangian submanifold of a symplectic manifold $(X,\omega)$. Cieliebak and Mohnke define the symplectic area of $L$ as the minimal positive symplectic area of a smooth $2$-disk in $X$ with boundary on $L$. An extremal Lagrangian torus in $(X,\omega)$ is a Lagrangian torus that maximizes the symplectic area among the Lagrangian tori in $(X,\omega)$. We prove that every extremal Lagrangian torus in the symplectic unit ball $(\bar{B}^{2n}(1),\omega_{\mathrm{std}})$ is contained entirely in the boundary $\partial B^{2n}(1)$. This answers a question attributed to Lazzarini and completely settles a conjecture of Cieliebak and Mohnke in the affirmative. In addition, we prove the conjecture for a class of toric domains in $(\mathbb{C}^n, \omega_{\mathrm{std}})$, which includes all compact strictly convex four-dimensional toric domains. We explain with counterexamples that the general conjecture does not hold for non-convex domains.

math.SG↗
arXiv2024

A proof of Gromov's non-squeezing theorem

The original proof of the Gromov's non-squeezing theorem [Gro85] is based on pseudo-holomorphic curves. The central ingredient is the compactness of the moduli space of pseudo-holomorphic spheres in the symplectic manifold $(\mathbb{CP}^1\times T^{2n-2}, \omega_{\mathrm{FS}}\oplus \omega_{\mathrm{std}})$ representing the homology class $[\mathbb{CP}^1\times\{\operatorname{pt}\}]$. In this article, we give two proofs of this compactness. The fact that the moduli space carries the minimal positive symplectic area is essential to our proofs. The main idea is to reparametrize the curves to distribute the symplectic area evenly and then apply either the mean value inequality for pseudo-holomorphic curves or the Gromov-Schwarz lemma to obtain a uniform bound on the gradient. Our arguments avoid bubbling analysis and Gromov's removable singularity theorem, which makes our proof of Gromov's non-squeezing theorem more elementary.

math.SG↗
arXiv2019

Ergodic Decomposition

Ergodic systems, being indecomposable are important part of the study of dynamical systems but if a system is not ergodic, it is natural to ask the following question: Is it possible to split it into ergodic systems in such a way that the study of the former reduces to the study of latter ones? Also, it will be interesting to see if the latter ones inherit some properties of the former one. This document answers this question for measurable maps defined on complete separable metric spaces with Borel probability measure, using the Rokhlin Disintegration Theorem.

math.DS↗