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Semi-Free Actions with Manifold Orbit Spaces (2020)
Journal Article
Harvey, J., Kerin, M., & Shankar, K. (2020). Semi-Free Actions with Manifold Orbit Spaces. Documenta Mathematica, 25, 2085-2114. https://doi.org/10.25537/dm.2020v25.2085-2114

In this paper, we study smooth, semi-free actions on closed, smooth, simply connected manifolds, such that the orbit space is a smoothable manifold. We show that the only simply connected 5-manifolds admitting a smooth, semi-free circle action with f... Read More about Semi-Free Actions with Manifold Orbit Spaces.

Fake Lens Spaces and Non-Negative Sectional Curvature (2020)
Book Chapter
Goette, S., Kerin, M., & Shankar, K. (2020). Fake Lens Spaces and Non-Negative Sectional Curvature. In O. Dearricott, W. Tuschmann, Y. Nikolayevsky, T. Leistner, & D. Crowley (Eds.), Differential Geometry in the Large (285-290). Cambridge University Press. https://doi.org/10.1017/9781108884136.016

In this short note we observe the existence of free, isometric actions of finite cyclic groups on a family of 2-connected 7-manifolds with non-negative sectional curvature. This yields many new examples including fake, and possible exotic, lens space... Read More about Fake Lens Spaces and Non-Negative Sectional Curvature.

Highly connected 7-manifolds and non-negative sectional curvature (2020)
Journal Article
Goette, S., Kerin, M., & Shankar, K. (2020). Highly connected 7-manifolds and non-negative sectional curvature. Annals of Mathematics, 191(3), 829-892. https://doi.org/10.4007/annals.2020.191.3.3

In this article, a six-parameter family of highly connected 7-manifolds which admit an S O ( 3 ) -invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that a... Read More about Highly connected 7-manifolds and non-negative sectional curvature.

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.

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.

Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension (2013)
Journal Article
Galaz-Garcia, F., & Kerin, M. (2014). Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension. Mathematische Zeitschrift, 276(1-2), 133-152. https://doi.org/10.1007/s00209-013-1190-5

Let Mn, n ∈ {4, 5, 6}, be a compact, simply connected n-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on Mn by a torus Tn−2 is equivariantly... Read More about Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension.

A note on totally geodesic embeddings of Eschenburg spaces into Bazaikin spaces (2012)
Journal Article
Kerin, M. (2013). A note on totally geodesic embeddings of Eschenburg spaces into Bazaikin spaces. Annals of Global Analysis and Geometry, 43(1), 63-73. https://doi.org/10.1007/s10455-012-9333-1

In this note it is shown that every 7-dimensional Eschenburg space can be totally geodesically embedded into infinitely many topologically distinct 13-dimensional Bazaikin spaces. Furthermore, examples are given which show that, under the known const... Read More about A note on totally geodesic embeddings of Eschenburg spaces into Bazaikin spaces.