arXiv subjects

Nancy Mae Eagles

Publications and source records attributed to Nancy Mae Eagles.

3 recordsLinked to original sources

arXiv2026

Invariants of Legendrian knots in thickened convex surfaces

We define a differential graded algebra associated to Legendrian knots in thickened convex surfaces $Σ\times \mathbb{R}$. The algebra is defined in the same spirit as the Chekanov-Eliashberg DGA for Legendrians in $\mathbb{R}^3$, but makes use of the data of the dividing set $Γ$ of $Σ$. The algebra is generated by countably many Reeb chords of the Legendrian $Λ$, and its differential counts certain immersed polygons in the projection $π:Σ\times \mathbb{R}\to Σ\times \{0\}$ with boundary on $π(Λ)\cup Γ$. We show that the differential squares to zero and that the stable tame isomorphism type of the DGA is invariant under Legendrian isotopy. Finally, we compute several examples and use the invariant to distinguish Legendrian knots in thickened convex surfaces that cannot be distinguished by the classical invariants.

math.SG↗

arXiv2022

A counterexample to symmetry of $L^p$ norms of eigenfunctions

We answer a question of Jakobson and Nadirashvili on the asymptotic behavior of the $L^p$ norms of positive and negative parts of eigenfunctions of the Laplacian. More precisely, we show that there exists a sequence of eigenfunctions $ψ_n$ on the flat $d$-torus for $d\geq 3$, with eigenvalues $λ_n\to\infty$ as $n\to\infty$, such that the ratio $\|ψ_nχ_{\{ψ_n>0\}}\|_p / \|ψ_nχ_{\{ψ_n<0\}}\|_p $ does not tend to $1$ as $n\to\infty$ for $1<p\leq \infty$. Our argument is elementary and computer-assisted.

math.SP↗

arXiv2021

H-chromatic symmetric functions

We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent if $X_{G_1}^{H} = X_{G_2}^{H}$, and use this idea to study uniqueness results for $H$-chromatic symmetric functions, with a particular emphasis on the case $H$ is a complete bipartite graph. We also show that several of the classical bases of the space of symmetric functions, i.e. the monomial symmetric functions, power sum symmetric functions, and elementary symmetric functions, can be realized as $H$-chromatic symmetric functions. We end with some conjectures and open problems.

math.CO↗