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

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:

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

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.

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

2017-12-18, Xu Wang