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 resolving the Alekseevskii Conjecture (in its various, modern forms).

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, applications of Geometric Invariant Theory to the geometry of Lie groups, the structure of isometry groups for homogeneous spaces, and questions about embeddings of homogeneous spaces.

My recent work has been supported by the National Science Foundation, grant DMS-1906351, the OU Research Council, and the Simons Foundation (#360562, michael jablonski).

Publications

1. Infinitesimal maximal symmetry and Ricci soliton solvmanifolds (with Carolyn Gordon),
   Transactions of the AMS, 377 (2024), No. 8, 5673-5704.
   Abstract. Article
2. Survey: Homogeneous Einstein Manifolds,
   Revista de la Uníon Matemática Argentina, 64 (2023), No. 2, 461-485.
   Abstract. Article.
3. Maximal symmetry and unimodular solvmanifolds,
   Pacific Journal of Math. 298 (2019), No. 2, pp. 417-427.
   Abstract. Article.
4. Einstein solvmanifolds have maximal symmetry (with Carolyn Gordon),
   Journal of Differential Geometry, Vol. 111, No. 1 (2019), pp. 1-38.
   Abstract. Article.
5. A step towards the Alekseevskii Conjecture (with Peter Petersen),
   Mathematische Annalen, 368 (2017), 197--212.
   Abstract. Article.
6. 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.
7. Homogeneous Ricci solitons are algebraic,
   Geometry & Topology 18-4 (2014), 2477--2486.
   Abstract. Article.
8. Strongly Solvable Spaces,
   Duke Math. Journal 164 (2015), no. 2, 361--402.
   Abstract. Article.
9. Homogeneous Ricci solitons,
   Journal für die reine und angewandte Mathematik (Crelle's journal) 699 (2015) 159--182.
   Abstract. Article.
10. Ricci Yang-Mills solitons on nilpotent Lie groups (with Andrea Young),
    Journal of Lie Theory 23 (2013) 177--202.
    Abstract. Article.
11. Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups,
    Geometry & Topology 15 (2011) 735--764.
    Abstract. Article.
12. Moduli of Einstein and non-Einstein nilradicals,
    Geometriae Dedicata 152 (2011), 63--84.
    Abstract. Article.
13. Detecting orbits along subvarieties via the moment map,
    Münster J. Math. 3 (2010), 67--88.
    Abstract. Article.
14. Distinguished Orbits of Reductive Groups,
    Rocky Mtn. Journal of Math. 42 (2012), 1521--1549.
    Abstract. Article.
15. 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.
16. 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.
2. Counterexample to a conjecture of Taketomi-Tamaru, arXiv:1810.08173 [math.DG].
3. Einstein solvmanifolds as submanifolds of symmetric spaces, arXiv:1810.11077 [math.DG].
4. Semisimple Lie algebras are determined by their Iwsawa subalgebras (with Jon Epstein), arXiv:2310.00799

Mentoring

PhD students:
   Jordan Danser (2018 -- 2022)
   Dustin Gaskins (2018 -- 2023)

Masters students:
   Michael Tilles (2017)
   Thomas Lane (2017)

Postdocs:
   Jon Epstein (2016 -- 2019)

Notes

1. Survey: Non-compact Homogeneous Spaces.  Oberwolfach Reports 2014.
2. Survey: Riemannian Geometry meets Geometric Invariant Theory.  Oberwolfach 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)