Student Projects offered by Xu Wang
Bachelor and master program 1. Classical Brunn-Minkowski theory
- Minkowski sum of convex bodies, Brunn-Minkowski inequality, isoperimetric inequality;
- Legendre transform, mixed volume, Alexandrov-Fenchel inequality;
- Prekopa theorem, the Brascamp-Lieb proof.
Reference:
- Berndtsson's notes: Convex and complex geometry
- Section 2 in A remark on the Alexandrov-Fenchel inequality
Bachelor and master program 2. From linear algebra to Hodge theory
- Jordan matrix, hard Lefschetz condition, Lefschetz star operator;
- symplectic structure, almost complex structure;
- variation of Lefschetz star operator, generalized Kähler identity;
- (need some calculus) Hodge decomposition, Alexandrov-Fenchel inequality;
- (advanced level) curvature of direct images.
Reference:
Early PhD program. Complex Brunn-Minkowski theory
- subharmonicity of Bergman kernel and Green operator and its applications;
- (key) positivity of direct image: Ohsawa-Takegoshi extension, effectivity of relative canonical bundle, Demailly-Kollar openness conjecture, convexity of Mabuchi functional, find more applications;
- curvature of direct image: variation of Hodge structure, Higgs bundle, Viehweg-Zuo sheaf, etc.
Reference:
- Berndtsson's notes: Convex and complex geometry
- A video of Berndtsson's lecture: complex Brunn-Minkowski theory
- Section 2 in A remark on the Alexandrov-Fenchel inequality
PhD and postdoc program. Extremal Kahler metrics
- Moment map, toric variety, Kempf-Ness theorem;
- Donaldson-Fujiki picture of Kahler geometry, Tian-Yau-Donaldson conjecture;
- geometry of GIT fibration.
Reference:
- Székelyhidi's notes: Introduction to Extremal metrics
- Book by Guillemin: Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces.
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