Research

All preprints are available on my arxiv page.

Reviews and citations are available on MathSciNet (author ID 852611) and Google Scholar.

My work has been supported by NSF Grants DMS-1404670/1622541 (2014 – 2017), DMS-1708493/1904354 (2017 – 2020), and DMS-2005280 (2020 – 2023).

Collaborators:

Manuel Amann (Augsburg), Jason DeVito (U. Tennessee, Martin), Judy Holdener (Kenyon College), Elahe Khalili Samani (Syracuse), Catherine Searle (Wichita State), Zhixu Su (Washington), Yusheng Wang (Beijing Normal University), Michael Wiemeler (Muenster), Burkhard Wilking (Muenster), William Wylie (Syracuse), Dmytro Yeroshkin (Free University of Brussels), Matthew Zaremsky (UAlbany).

Work in progress:

  • (with E. Khalili Samani and C. Searle) Positive curvature and finite abelian symmetry.
    • We are generalizing results in the Grove symmetry program from the case of torus actions to the case of actions by (finite) elementary abelian groups. In the case of 2-groups, Fang and Grove proved the analogue of the Grove-Searle maximal symmetry rank theorem, and we prove here analogues of Wilking’s homotopy classification for half-maximal symmetry rank and Rong and Su’s result for quarter-maximal symmetry rank. All three of these results are optimal due to examples on the sphere, complex projective space, and quaternionic projective space, respectively.
    • This project is supported by the Summer Research for Women in Mathematics (SWiM) Program at MSRI.
    • Slides of a talk Elahe gave in the Differential Geometry Seminar, UC Riverside, April 2021.
  • (with Y. Wang)
    • We consider the maximal symmetry rank (MSR) problem for positively curved manifolds with non-cyclic fundamental groups. The computation is done in dimensions of the form 4k + 3 by work of Rong and Wilking and in dimensions 12k + 5 by work of Frank, Rong, and Wang. The computation is open in general. In the project, we compute the MSR for positively curved manifolds with non-cyclic fundamental groups in a new infinite family of dimensions, 49 + 60k.
    • Slides and video of a talk I gave in the Topology-Geometry Zoom Seminar, University of Oregon, April 2021.
  • (with M. Wiemeler and B. Wilking)
    • Building on work of Wilking’s 2003 paper and my 2013 paper on positively curved manifolds with torus symmetry, we prove a 1930s conjecture of Hopf on the positivity of the Euler characteristic, under the additional assumption that the manifold admits an isometric action by a 5-dimensional torus. The best known previous results involved torus ranks that grew logarithmically (if n = 4k) and linearly (if n = 4k + 2) in the manifold dimension n.
    • We also prove results that involve additional assumptions (including topological assumptions on the manifold, geometric assumptions on the torus action, or stronger symmetry assumptions) and we correspondingly prove stronger topological conclusions (in the best case, recovering the rational cohomology of the ambient manifold).
    • Slides and video of a talk I gave in the Virtual seminar on geometry with symmetries in May 2020.

Publications:

  1. (with J. DeVito) Cohomogeneity one manifolds with singly generated rational cohomology.
    Doc. Math., 25: 1835–1863, 2020.
  2. (with M. Amann) Positive curvature and symmetry in small dimensions.
    Commun. Contemp. Math.
    , 22(6): 57 pp., 2020.
  3. (with W. Wylie and D. Yeroshkin) The weighted connection and sectional curvature for manifolds with density.
    J. Geom. Anal., 29(1): 957–1001, 2019.
  4. (with Z. Su) On dimensions supporting a rational projective plane.
    J. Topol. Anal., 11(3): 535–555, 2019.

    Video of a talk I gave in the Topology Seminar at Indiana University. Thanks to Carmen Rovi (now at Heidelberg) for recording the talk.

  5. Fundamental groups of manifolds with positive sectional curvature and torus symmetry.
    J. Geom. Anal., 27: 2894–2925, 2017.
  6. (with M. Amann) On a generalized conjecture of Hopf with symmetry.
    Compos. Math., 153: 313–322, 2017.
  7. (with W. Wylie) Positive weighted sectional curvature.
    Indiana Univ. Math. J., 66(2): 419–462, 2017.
  8. (with M. Amann) Positive curvature and rational ellipticity.
    Algebr. Geom. Topol., 15(4): 2269–2301, 2015.
  9. (with M. Amann) Topological properties of positively curved manifolds with symmetry.
    Geom. Funct. Anal., 24(5): 1377–1405, 2014.
  10. Positively curved Riemannian manifolds with logarithmic symmetry rank bounds.
    Comm. Math. Helv., 89(4): 937–962, 2014.
  11. On the Hopf conjecture with symmetry.
    Geom. Topol., 17(1): 563–593, 2013.

Publications by students:

  1. L. Kennard, Y. Wu. Halperin’s conjecture in formal dimensions up to 20.
    Submitted.

    Slides and video of a talk Yantao gave in the SU Algebra Seminar, April 2021.

  2. E. Khalili Samani. Obstructions to free actions on Bazaikin spaces.
    Transform. Groups (to appear).

    Slides of a talk Elahe gave at the Midwest Geometry Conference, September 2019.

  3. L. Kennard, J. Rainone. Characterizations of the round two-dimensional sphere in terms of closed geodesics.
    Involve, 10(2): 243–255, 2017.

Additional publications:

  1. L. Kennard. On the Hopf conjecture with symmetry, in Geometry of manifolds with non-negative sectional curvature.
    Lecture Notes in Math., 2110: 111-116, 2014
  2. J. Holdener, L. Kennard, M. Zaremsky. Generalized Thue-Morse sequences and the von Koch curve.
    Int. J. Pure Appl. Math., 47(3): 397–403, 2008. (Undergraduate publication)