arXiv subjects

Jaime Calles Loperena

Publications and source records attributed to Jaime Calles Loperena.

3 recordsLinked to original sources

arXiv2025

A Center Transversal Theorem for mass assignments

In this paper, based on the ideas of Blagojević, Karasev & Magazinov, we consider an extension of the center transversal theorem to mass assignments with an improved Rado depth. In particular we substitute the marginal of a measure by a more general concept called a mass assignment over a flag manifold. Our results also allow us to solve the main problem proposed by Blagojević, Karasev & Magazinov in a linear subspace of lower dimension, as long as it is contained in a high-dimensional enough ambient space.

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arXiv2024

The ideal-valued index of fibrations with total space a $G_{2}$ flag manifold

Using the cohomology of the $G_2$-flag manifolds $G_2/U(2)_{\pm}$, and their structure as a fiber bundle over the homogeneous space $G_2/SO(4)$, we compute the $\mathbb{Z}_2$ Fadell-Husseini index of such fiber bundles, for the $\mathbb{Z}_2$ action given by complex conjugation. Also, considering the tautological bundle $γ$ over $\widetilde{G}_{4}(\mathbb{R}^{7})$, we compute the $\mathbb{Z}_2$ Fadell-Husseini index of the pullback bundle of $sγ$ along the composition of the fiber bundle $ G_2/U(2)_{\pm} \to G_2/SO(4)$, the embedding between $G_2/SO(4)$ and $\widetilde{G}_{3}(\mathbb{R}^{7})$, and the map that takes the orthogonal complement of a subspace. Here $sγ$ means the associated sphere bundle of $γ$. Furthermore, we derive a general formula for the $n$-fold product bundle $sγ^n$ for which we make the same computations. We finish our work with an application of our computations in a problem concerning discrete geometry.

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arXiv2022

Topology of the Grünbaum--Hadwiger--Ramos problem for mass assignments

In this paper, motivated by recent work of Schnider and Axelrod-Freed \& Soberón, we study an extension of the classical Grünbaum--Hadwiger--Ramos mass partition problem to mass assignments. Using the Fadell--Husseini index theory we prove that for a given family of $j$ mass assignments $μ_1,\dots,μ_j$ on the Grassmann manifold $G_{\ell}(\R^d)$ and a given integer $k\geq 1$ there exist a linear subspace $L\in G_{\ell}(\R^d)$ and $k$ affine hyperplanes in $L$ that equipart the masses $μ_1^L,\dots,μ_j^L$ assigned to the subspace $L$, provided that $d\geq j + (2^{k-1}-1)2^{\lfloor\log_2j\rfloor}$.

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