# 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.

Want to know more, just come to his office. **Contact info**