Mathematics

Research Interests

My main interests are in Riemannian geometry, with a particular emphasis on homogeneous spaces G/H and the G-invariant structures they can admit. Much of my work has been dedicated to the classification of non-compact homogeneous Einstein and Ricci soliton metrics, and working towards the following conjecture.

Strong Generalized Alekseevsky Conjecture: Let M be a non-compact homogeneous space with Ricci soliton metric. Then M is isometric to a simply-connected solvable Lie group with algebraic Ricci soliton metric.

While this conjecture is still open even in the special case of Einstein metrics, there has been substantial progress in the past five years which has reignited the belief that the original Alekseevsky conjecture (and its modern version) is true.

Further, I am interested in understanding the existence/non-existence of left-invariant Einstein and Ricci soliton metrics on homogeneous spaces (especially solvable Lie groups), geometric evolutions on homogeneous spaces, and applications of Geometric Invariant Theory to the geometry of Lie groups.

My recent work has been supported by the National Science Foundation, grant DMS-1612357, the OU Research Council, and the Simons Foundation (#360562, michael jablonski).

Publications

  1. Einstein solvmanifolds have maximal symmetry (with Carolyn Gordon), Journal of Differential Geometry - in press - arXiv:1507.08321 [math.DG].
    Abstract.
  2. A step towards the Alekseevskii Conjecture (with Peter Petersen),
    Mathematische Annalen, 368 (2017), 197--212. 
    AbstractArticle.
  3. Linear stability of algebraic Ricci solitons (with Peter Petersen and Michael B. Williams),
    Journal für die reine und angewandte Mathematik 713 (2016), 181--224.
    Abstract
    Article.
  4. Homogeneous Ricci solitons are algebraic,
    Geometry & Topology 18-4 (2014), 2477--2486.
    Abstract. Article.
  5. Strongly Solvable Spaces,
    Duke Math. Journal 164 (2015), no. 2, 361--402.
    Abstract. Article.
  6. Homogeneous Ricci solitons,
    Journal für die reine und angewandte Mathematik (Crelle's journal) 699 (2015) 159--182.
    Abstract. Article.
  7. Ricci Yang-Mills solitons on nilpotent Lie groups (with Andrea Young),
    Journal of Lie Theory 23 (2013) 177--202.
    Abstract. Article.
  8. Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups,
    Geometry & Topology 15 (2011) 735--764.
    Abstract. Article.
  9. Moduli of Einstein and non-Einstein nilradicals,
    Geometriae Dedicata 152 (2011), 63--84.
    Abstract. Article.
  10. Detecting orbits along subvarieties via the moment map,
    Münster J. Math. 3 (2010), 67--88.
    Abstract. Article.
  11. Distinguished Orbits of Reductive Groups,
    Rocky Mtn. Journal of Math. 42 (2012), 1521--1549.
    Abstract. Article.
  12. Closed orbits of semisimple group actions and the real Hilbert-Mumford function (with Pat Eberlein),
    New Developments in Lie Theory and Geometry, 283--321, Contemp. Math., 491 (2009).
    Abstract. Article.
  13. An Improved upper bound for the pebbling threshold of the n-path (with Adam Wier, Julia Salzman, and Anant Godbole),
    Discrete Mathematics (2004) 275, 367--373.
    Abstract. Article.

Preprints

  1. On the linear stability of expanding Ricci solitons (with Peter Petersen and Michael B. Williams), arXiv:1409.3251 [math.DG].
    Abstract.

Notes

  1. Survey: Non-compact Homogeneous SpacesOberwolfach Reports 2014.
  2. Survey: Riemannian Geometry meets Geometric Invariant TheoryOberwolfach Reports 2014.
  3. Good representations and homogeneous spaces, April 2008, arXiv:0804.3343 [math.AG]
    -- After the first posting of this note on the arXiv, I was informed that the results of this work have previously appeared in the literature.
  4. Real Geometric Invariant Theory and Ricci Soliton Metrics on Two-step Nilmanifolds, April 2008 (THESIS)

Upcoming Conferences and Workshops
(see below for funding information)

Lie Group Actions in Riemannian Geometry
 
Dartmouth College

June 26-30, 2017

The primary theme of the conference will be curvature properties in the presence of symmetry, with emphasis on homogeneous Einstein and Ricci soliton metrics, metrics of positive curvature, and Ricci flow.

Lectures will be held in the morning and late afternoon to allow time for informal discussions each afternoon. We thus hope to encourage progress on open problems and the formation of new collaborations.

For further information, see  https://math.dartmouth.edu/~lga/

Through a grant from Dartmouth College, we can provide support for a limited number of researchers.  Priority will be given to graduate students, recent Ph.D.'s, and others who do not have other sources of federal support.  To apply for funding, please contact Michael Jablonski at mjablonski@math.ou.edu.  Graduate students and recent Ph.D.'s should include a brief statement of interests and a CV.   Graduate students should also have their advisors send a supporting letter. For full consideration, please apply by April 15.  We especially encourage women and members of underrepresented minorities to apply.