Student Differential Geometry seminar
Mathematics Department
Stony Brook University
About the seminar
Our goal is to bring together students in the Differential Geometry
group Stony Brook to discuss topics in the interface
between Geometric Analysis, Complex Geometry and affine areas.
The seminar is jointly organized by
Marlon de Oliveira Gomes and Lisandra Hernandez Vazquez.
Fall 2018
The topic for this Fall is Einstein metrics and special holonomy. Below is a
summary of our goals for the semester.
 Understand what are some of the obstructions to the existence of Einstein metrics in low dimensions.
 Understand what is holonomy, Berger's classification of
irreducible honolomy representations, and why metrics of
special holonomy are Einstein.
 Understand various constructions of Einstein metrics with
special holonomy
 Understand the convergence and moduli theory of Einstein
metrics.
 If time permits, discuss a few constructions of Einstein
with generic holonomy. In particular, how some of them arise
as limits (in an appropriate sense) of metrics of special
holonomy.
 If time permits, use the existence of Einstein metrics
to derive information about the topology of the
underlying manifolds.
Time and location: We meet weekly on Mondays, from
4:15 pm to 5:15 pm, in 5127.
The schedule for our seminar is below. Abstracts for talks will be linked in the titles as they become available.
Date 
Title 
Speaker 
References 
8/27 
Organizational meeting 


9/3 
Labour day. 


9/10 
Introduction to Einstein manifolds. 
Lisandra Hernandez 
[Be87] 
9/17 
Holonomy representations and Einstein metrics. 
Marlon Gomes 
[Be87] 
9/24 
KählerEinstein metrics: examples and obstructions. 
Michael Albanese 
[Jo00] 
10/1 
The Calabi conjecture (and KählerRicci flow). 
Jae Ho Cho 
[Jo00] 
10/8 
Fall break! 


10/15 
HyperKähler geometry. 
John Sheridan 
[Bv99], [Hi91], [Hu97], [Hu11] 
10/22 
QuaternionKähler geometry. 
Marlon Gomes 
[BG08], [HKLR87], 
10/29 
G_2 geometry: introduction. 
Jordan Rainone 
[Jo00], [Jo07] 
11/5 
Joyce's construction of compact G_2 manifolds: part I . 
Marlon Gomes 
[Jo00], [Jo07] 
11/12 
Joyce's construction of compact G_2 manifolds: part II. 
JeanFrançois Arbour 
[Jo00], [Jo07] 
11/19 
Spin(7) geometry 
Matthew Lam 
[Jo00], [Jo07] 
11/26 
Moduli and convergence theory. 
Zhongshan An 
[LW99] 
12/3 
Aspects of Ricciflat, ALE 4manifolds. 
Demetre Kazaras 
[LV16] 
List of possible extra topics : 3manifolds, SasakiEinstein metrics, applications to 4manifold topology.
References
 [Be87] : Arthur L. Besse, Einstein manifolds. Springer, Berlin, 1987.
 [Bv99] : Arnaud Beuville, Riemannian Holonomy and Algebraic Geometry , arXiv:math/9902110v1.
 [BG08] : Charles Boyer and Krzysztof Galicki, Sasakian Geometry. Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.
 [GL88] : Krzysztof Galicki and H. Blaine Lawson Jr., Quaternionic reduction and quaternionic orbifolds , Math. Ann. 282 (1988), no. 1, pp. 121.
 [HKLR87] : Nigel J. Hitchin, Anders Karlhede, Ulf Lindström, and Martin Roček, HyperKähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), no. 4, pp. 535589.
 [Hi91] : Nigel J. Hitchin, Hyperkähler manifolds, Séminaire Bourbaki (19911992) 34 (1991), pp. 137166.
 [Hu97] : Daniel Huybrechts, Compact Hyperkaehler Manifolds: Basic results, arXiv:alggeom/9705025v1.
 [Hu11] : Daniel Huybrecths, Hyperkähler manifolds and sheaves , Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), v. 2, (2011), pp. 450460.
 [Jo00] : Dominic Joyce, Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs, Oxford University Press,
Oxford, 2000.
 [Jo07] : Dominic Joyce, Riemannian holonomy groups and Calibrated Geometry. Oxford Graduate Texts in Mathematics, v.12, Oxford University Press,
Oxford, 2007.
 [LW99] : Claude LeBrun and McKenzie Wang (editors), Essays on Einstein Manifolds , Surveys in Differential Geometry, v.6, International Press, Boston, 2001.
 [LV16] : Michael T. Lock and Jeff A. Viaclovsky, Quotient singularities, eta invariants, and selfdual metrics., Geometry and Topology 20 (2016), pp. 17731806.
