arXiv subjects

Ayodeji Lindblad

Publications and source records attributed to Ayodeji Lindblad.

4 recordsLinked to original sources

arXiv2026

Asymptotically short generalizations of $t$-design curves

Ehler and Gröchenig defined spherical $t$-design curves to be curves whose associated line integrals exactly average all degree at most $t$ polynomials. These authors posed the question of finding spherical $t$-design curves $γ_t$ on $S^d$ of asymptotically optimal arc length $\ell(γ_t)\asymp t^{d-1}$ as $t\to\infty$. This work investigates analogues of this question for $\textit{$\varepsilon_t$-approximate}$ and $\textit{weighted $t$-design curves}$, proving existence of such curves on $S^d$ achieving this asymptotic arc length for odd $d\in\Bbb N_+$ in the approximate setting (where $\varepsilon_t\asymp1/t$ as $t\to\infty$) and all $d\in\Bbb N_+$ in the weighted setting (where these curves have weight functions which are strictly positive at all but finitely many points). Formulas for such weighted $t$-design curves for $d\in\{2,3\}$ are presented.

math.MG↗

arXiv2026

Designs related through projective and Hopf maps

We verify a construction which, for $\Bbb K$ the reals, complex numbers, quaternions, or octonions, builds a spherical $t$-design by placing a spherical $t$-design on each $\Bbb K$-projective or $\Bbb K$-Hopf fiber associated to the points of a $\lfloor t/2\rfloor$-design on a quotient projective space $\Bbb{KP}^n\neq\Bbb{OP}^2$ or sphere. This generalizes work of König and Kuperberg, who verified the $\Bbb K=\Bbb C$ case of the projective settings, and of Okuda, who (inspired by independent observation of this construction by Cohn, Conway, Elkies, and Kumar) verified the $\Bbb K=\Bbb C$ case of the generalized Hopf settings.

math.MG↗

arXiv2026

Boundary Dehn twists are often commutators

For $X$ any complete intersection of even complex dimension or any connected sum thereof (or, more generally, any space among certain broad classes of smooth manifolds), we concretely construct diffeomorphisms $a,c$ of punctured $X$ rel boundary whose commutator $[a,c]$ represents the smooth mapping class (rel boundary) of the boundary Dehn twist. This shows that boundary Dehn twists on 4-manifolds known to be nontrivial in the smooth mapping class group rel boundary by work of Baraglia-Konno, Kronheimer-Mrowka, J. Lin, and Tilton become trivial after abelianization, generalizing work of Y. Lin, who applied an argument based on the global Torelli theorem and an obstruction of Baraglia-Konno to prove that the abelianized boundary Dehn twist on the punctured $K3$ surface is trivial.

math.GT↗

arXiv2025

Asymptotically optimal $t$-design curves on $S^3$

A $\textit{spherical $t$-design curve}$ was defined by Ehler and Gröchenig to be a continuous, piecewise smooth, closed curve on the sphere with finitely many self-intersections whose associated line integral applied to any polynomial of degree at most $t$ evaluates to the average of this polynomial on the sphere. These authors posed the problem of proving that there exist sequences $(γ_t)_{t=0}^\infty$ of $t$-design curves on $S^d$ of asymptotically optimal length $\ell(γ_t)\asymp t^{d-1}$ as $t\to\infty$ and solved this problem for $d=2$. This work solves the problem for $d=3$ by proving that there exists a constant $\mathscr C>0$ such that for any $C\geq\mathscr C$ and $t\in\Bbb N_+$, there exists a simple $t$-design curve on $S^3$ of length $Ct^2$.

math.MG↗