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:

1. Berndtsson's notes: Convex and complex geometry
2. 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:

1. Based on Notes on variation of Lefschetz star operator and T-Hodge theory

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:

1. Berndtsson's notes: Convex and complex geometry
2. A video of Berndtsson's lecture: complex Brunn-Minkowski theory
3. 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:

1. Székelyhidi's notes: Introduction to Extremal metrics
2. Book by Guillemin: Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces.

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