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Basic metric geometry of the bottleneck distance (2024)
Journal Article
Che, M., Galaz-García, F., Guijarro, L., Membrillo Solis, I., & Valiunas, M. (2024). Basic metric geometry of the bottleneck distance. Proceedings of the American Mathematical Society, https://doi.org/10.1090/proc/16776

Given a metric pair (X, A), i.e. a metric space X and a distinguished closed set A ⊂ X, one may construct in a functorial way a pointed pseudometric space D∞(X, A) of persistence diagrams equipped with the bottleneck distance. We investigate the basi... Read More about Basic metric geometry of the bottleneck distance.

Kurdyka–Łojasiewicz functions and mapping cylinder neighborhoods (2022)
Journal Article
Cibotaru, D., & Galaz-García, F. (in press). Kurdyka–Łojasiewicz functions and mapping cylinder neighborhoods. Annales de l'Institut Fourier,

Kurdyka–Łojasiewicz (KŁ) functions are real-valued functions characterized by a differential inequality involving the norm of their gradient. This class of functions is quite rich, containing objects as diverse as subanalytic, transnormal or Morse fu... Read More about Kurdyka–Łojasiewicz functions and mapping cylinder neighborhoods.

Three-dimensional Alexandrov spaces: A survey (2022)
Book Chapter
Galaz-García, F., & Núñez-Zimbrón, J. (2022). Three-dimensional Alexandrov spaces: A survey. In G. Arizmendi Echegaray, L. Hernández-Lamoneda, & R. Herrera Guzmán (Eds.), Recent Advances in Alexandrov Geometry (49-88). Springer Verlag. https://doi.org/10.1007/978-3-030-99298-9_2

We survey several results concerning the geometry and topology of threedimensional Alexandrov spaces with the aim of providing a panoramic and up-to-date view of the subject. In particular we present the classification of positively and nonnegatively... Read More about Three-dimensional Alexandrov spaces: A survey.

Stability of the cut locus and a Central Limit Theorem for Fréchet means of Riemannian manifolds (2021)
Journal Article
Eltzner, B., Galaz-García, F., Huckemann, S. F., & Tuschmann, W. (2021). Stability of the cut locus and a Central Limit Theorem for Fréchet means of Riemannian manifolds. Proceedings of the American Mathematical Society, 149(9), 3947-3963. https://doi.org/10.1090/proc/15429

We obtain a central limit theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin’s Omnibus central limit theorem for Fréchet means. We obtain our CLT assuming certain... Read More about Stability of the cut locus and a Central Limit Theorem for Fréchet means of Riemannian manifolds.

Collapsed 3-Dimensional Alexandrov Spaces: A Brief Survey (2020)
Book Chapter
Galaz-García, F., Guijarro, L., & Núñez-Zimbrón, J. (2020). Collapsed 3-Dimensional Alexandrov Spaces: A Brief Survey. In O. Dearricott, W. Tuschmann, Y. Nikolayevsky, T. Leistner, & D. Crowley (Eds.), Differential geometry in the large (291-310). Cambridge University Press. https://doi.org/10.1017/9781108884136.017

We survey two recent developments in the topic of three-dimensional Alexandrov spaces: the topological classification of closed collapsed threedimensional Alexandrov spaces and the geometrization of sufficiently collapsed closed three-dimensional Ale... Read More about Collapsed 3-Dimensional Alexandrov Spaces: A Brief Survey.

Cohomogeneity one Alexandrov spaces in low dimensions (2020)
Journal Article
Galaz-García, F., & Zarei, M. (2020). Cohomogeneity one Alexandrov spaces in low dimensions. Annals of Global Analysis and Geometry, 58(2), 109-146. https://doi.org/10.1007/s10455-020-09716-7

Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space, they are said to be of coho... Read More about Cohomogeneity one Alexandrov spaces in low dimensions.

Positive Ricci curvature on simply-connected manifolds with cohomogeneity-two torus actions (2020)
Journal Article
Corro, D., & Galaz-García, F. (2020). Positive Ricci curvature on simply-connected manifolds with cohomogeneity-two torus actions. Proceedings of the American Mathematical Society, 148(7), 3087-3097. https://doi.org/10.1090/proc/14961

We show that for each n 1, there exist infinitely many spin and non-spin diffeomorphism types of closed, smooth, simply-connected (n + 4)- manifolds with a smooth, effective action of a torus T n+2 and a metric of positive Ricci curvature invariant u... Read More about Positive Ricci curvature on simply-connected manifolds with cohomogeneity-two torus actions.

Torus actions on rationally elliptic manifolds (2020)
Journal Article
Galaz-García, F., Kerin, M., & Radeschi, M. (2021). Torus actions on rationally elliptic manifolds. Mathematische Zeitschrift, 297, 197-221. https://doi.org/10.1007/s00209-020-02508-6

An upper bound is obtained on the rank of a torus which can act smoothly and effectively on a smooth, closed (simply connected) rationally elliptic manifold. In the maximal-rank case, the manifolds admitting such actions are classified up to equivari... Read More about Torus actions on rationally elliptic manifolds.

Cohomogeneity one topological manifolds revisited (2017)
Journal Article
Galaz-García, F., & Zarei, M. (2018). Cohomogeneity one topological manifolds revisited. Mathematische Zeitschrift, 288(3-4), 829-853. https://doi.org/10.1007/s00209-017-1915-y

We prove a structure theorem for closed topological manifolds of cohomogeneity one; this result corrects an oversight in the literature. We complete the equivariant classification of closed, simply-connected cohomogeneity one topological manifolds in... Read More about Cohomogeneity one topological manifolds revisited.

Torus Orbifolds, Slice-Maximal Torus Actions, and Rational Ellipticity (2017)
Journal Article
Galaz-García, F., Kerin, M., Radeschi, M., & Wiemeler, M. (2018). Torus Orbifolds, Slice-Maximal Torus Actions, and Rational Ellipticity. International Mathematics Research Notices, 2018(18), 5786-5822. https://doi.org/10.1093/imrn/rnx064

In this work, it is shown that a simply connected, rationally elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to t... Read More about Torus Orbifolds, Slice-Maximal Torus Actions, and Rational Ellipticity.