## Departmental PhD Thesis Exam – Peter Crooks

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, May 4, 2016
11:10 a.m.
BA1200

PhD Candidate:  Peter Crooks
Supervisor:  Lisa Jeffrey
Thesis title:  The Equivariant Geometry of Nilpotent Orbits and Associated Varieties

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Abstract:

Nilpotent orbits are highly structured algebraic varieties lying at the interface of algebraic geometry, Lie theory, symplectic geometry, and geometric representation theory. The interest in these objects has been long-standing, ranging from Kostant’s foundational work in the 1950s and 1960s to Kronheimer’s realization of nilpotent orbits as instanton moduli spaces. At the same time, nilpotent orbits are studied for the sake of understanding closely associated varieties, such as nilpotent Hessenberg varieties.

In this thesis, we study the equivariant algebraic geometry and topology of nilpotent orbits and related varieties. Our first group of results is principally concerned with presentations of $T$-equivariant cohomology rings. More specifically, we give concrete presentations of the $T$-equivariant cohomology rings of the regular and minimal nilpotent orbits, with the latter presentation providing an equivariant counterpart to existing work on the ordinary cohomology of the minimal nilpotent orbit. We also examine the family of Hessenberg varieties arising from the minimal nilpotent orbit, showing them to be GKM and obtaining presentations of their $T$-equivariant cohomology rings. In Lie type $A$, we explain how one would compute the Poincar\'{e} polynomials and irreducible components of these Hessenberg varieties.

Our second group of results includes a characterization of those semisimple real Lie algebras for which every complex nilpotent orbit contains a real one, building on Rothschild’s criterion for such a Lie algebra to be quasi-split. We also consider the role of nilpotent orbits in quaternionic K\”ahler geometry by giving a new, self-contained proof of the LeBrun-Salamon Conjecture for equivariant contact structures on partial flag varieties. This approach allows us to give an intrinsic description of the standard contact structure on the isotropic Grassmannian of $2$-planes in $\mathbb{C}^{2n}$.

We conclude on a somewhat different note by computing the generalized $T$-equivariant cohomology of a direct limit of smooth projective varieties. As a brief application, we obtain the generalized $T$-equivariant cohomology of the affine Grassmannian.

A copy of the thesis can be found here: Thesis3