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 (Notre Dame), Lawrence Mouillé (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), Yusheng Wang (Beijing Normal), Matthew Zaremsky (UAlbany).

Publications:

  1. (with L. Mouillé) Positive intermediate Ricci curvature with maximal symmetry rank (preprint).
    • Abstract: Generalizing the foundational work of Grove and Searle, the second author proved upper bounds on the ranks of isometry groups of closed Riemannian manifolds with positive intermediate Ricci curvature and established some topological rigidity results in the case of maximal symmetry rank and positive second intermediate Ricci curvature. Here, we recover even stronger topological rigidity, including results for higher intermediate Ricci curvatures and for manifolds with nontrivial fundamental groups.
  2. (with M. Wiemeler and B. Wilking) Positive curvature, torus symmetry, and matroids (submitted).
    • Slides and video of a talk I gave at the Workshop on Curvature and Global Shape at the University of Muenster in August 2021.
    • Abstract: We identify a link between regular matroids and torus representations all of whose isotropy groups have an odd number of components. Applying Seymour’s 1980 classification of the former objects, we obtain a classification of the latter. In addition, we prove optimal upper bounds for the cogirth of regular matroids up to rank nine, and we apply this to prove the existence of fixed-point sets of circles with large dimension in a torus representation with this property up to rank nine. Finally, we apply these results to prove new obstructions to the existence of Riemannian metrics with positive sectional curvature and torus symmetry.
  3. (with E. Khalili Samani and C. Searle) Positive curvature and finite abelian symmetry (submitted).
    • Slides of a talk Elahe gave in the Differential Geometry Seminar, UC Riverside, April 2021. 
    • This project is supported by the Summer Research in Mathematics (SRiM) Program at MSRI.
    • Abstract: 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, we extend results of Fuquan 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.
  4. (with M. Wiemeler and B. Wilking) Splitting of torus representations and applications in the Grove symmetry program (submitted).
    • Slides and video of a talk I gave in the Virtual seminar on geometry with symmetries in May 2020.
    • Abstract: A 1930s conjecture of Hopf states that an even-dimensional compact Riemannian manifold with positive sectional curvature has positive Euler characteristic. We prove this conjecture under the additional assumption that the isometry group has rank at least five. The fundamental new tool used to achieve this is a reduction to, and structural results concerning, a representation theoretic problem involving torus representations all of whose isotropy groups are connected.
  5. (with J. DeVito) Cohomogeneity one manifolds with singly generated rational cohomology.
    Doc. Math., 25: 1835–1863, 2020.
  6. (with M. Amann) Positive curvature and symmetry in small dimensions.
    Commun. Contemp. Math.    , 22(6): 57 pp., 2020.
    • Part of this paper confirms Halperin’s conjecture in dimensions up to 16. In the publication below with Yantao Wu, this is simplified and extended up to dimension 20.
    • Abstract: Extending existing work in small dimensions, Dessai computed the Euler characteristic, signature, and elliptic genus for 8-manifolds of positive sectional curvature in the pres- ence of torus symmetry. He also computes the diffeomorphism type by restricting his results to classes of manifolds known to admit non-negative curvature, such as biquotients. The first part of this paper extends Dessai’s calculations to even dimensions up to 16. In particular, we obtain a first characterization of the Cayley plane in such a setting. The second part studies a closely related family of manifolds called positively elliptic manifolds, and we prove a conjecture of Halperin in this context for dimensions up to 16 or Euler characteristics up to 16.
  7. (with W. Wylie and D. Yeroshkin) The weighted connection and sectional curvature for manifolds with density.
    J. Geom. Anal., 29(1): 957–1001, 2019.
  8. (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.
  9. Fundamental groups of manifolds with positive sectional curvature and torus symmetry.
    J. Geom. Anal., 27: 2894–2925, 2017.
  10. (with M. Amann) On a generalized conjecture of Hopf with symmetry.
    Compos. Math., 153: 313–322, 2017.
  11. (with W. Wylie) Positive weighted sectional curvature.
    Indiana Univ. Math. J., 66(2): 419–462, 2017.
  12. (with M. Amann) Positive curvature and rational ellipticity.
    Algebr. Geom. Topol., 15(4): 2269–2301, 2015.
  13. (with M. Amann) Topological properties of positively curved manifolds with symmetry.
    Geom. Funct. Anal., 24(5): 1377–1405, 2014.
  14. Positively curved Riemannian manifolds with logarithmic symmetry rank bounds.
    Comm. Math. Helv., 89(4): 937–962, 2014.
  15. On the Hopf conjecture with symmetry.
    Geom. Topol., 17(1): 563–593, 2013.

Publications by / with students:

  1. L. Kennard, Y. Wu (B.S. 2021, SU). Halperin’s conjecture in formal dimensions up to 20. (PDF)
    Comm. Algebra, 51(8): 3606–3622, 2023.
    • Yantao was awarded the University Scholar Award in 2021 (one of 12, which is about 0.5% of the total graduating class). He started a Ph.D. program in mathematics at Johns Hopkins University in 2021.
  2. E. Khalili Samani (Ph.D. 2021, SU). Obstructions to free actions on Bazaikin spaces.
    Transform. Groups, 27:1515–1532, 2022.
  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: