Recent advances in complex differential geometry - Program

Summer School: June 13-17, 2016

Mini-courses by

In the first lecture we will present some background in convex geometry, the Brunn-Minkowski inequality, the Alexander Fenchel inequality and the Prekopa-Leindler inequality. In the second lecture we will discuss some theorems on positivity of holomorphic vector bundles and argue that they can be seen as complex analogs of (some of) the results on convexity from the first lecture. We will also sketch some applications of these results, related to the convexity of K-energy and the Ohsawa-Takegoshi theorem. After that we will discuss a more recent generalization of these methods to the study of variations of complex structures; this last part is (ongoing, i e. not completely finished) joint work with Xu Wang.

In this series of lectures, we will survey a number of recent results related to positivity : positive cones, intersection theory of chomology classes, duality results, multiplier ideal sheaves, vanishing and extension theorems. We will also try to discuss several a few open problems connected to these questions (existence of rational curves, abundance conjecture, transcendental Morse inequalities, deformations of Kähler manifolds ...)

  • Duong H. Phong“Non-linear Partial Differential Equations in Complex Geometry"

This series of lectures will be devoted to an informal and mostly self-contained survey of some non-linear partial differential equations arising in complex geometry. Starting from the classical theory of the Yang-Mills equation on holomorphic vector bundles, we shall discuss topics ranging from the Ricci flow on Riemann surfaces with conic singularities, to elements of the theory of fully non-linear equations, and to recent equations from non-K\”ahler geometry such as the Gauduchon conjecture, the Fu-Yau equations, and Strominger systems.


Conference: June 17-June 22, 2016

Mini-courses by


Speakers :

  • V. Apostolov (UQAM)
  • T. Collins (Harvard Univ.)
  • V. Datar (UC Berkeley)
  • R. Dervan (Univ. of Cambridge) 
  • T. Delcroix (Univ. de Grenoble)
  • A. Fino (Univ. of Torino)                                            Slides of A. Fino's talk: A-Fino-Toulouse-2016-main.pdf
  • L. Foscolo (Stony Brook Univ.)
  • C. Lu (Scuola Normale, Pisa) 
  • G. Marinescu (Univ. of Cologne)
  • T. Murphy (Cal State Fullerton)                                 Slides of T. Murphy's talk:  T-Murphy-Toulouse-talk.pdf
  • D. Panov (King's College, London)
  • C. Spotti (Univ. of Cambridge)
  • D. W. Nyström (Univ. of Gothenburg)
  • B. Taji (Univ. of Freiburg) 
  • X. Wang (Rutgers Univ.)
  • J. Xiao (Univ. de Grenoble)

Abstracts and titles of the 1h talks can be found here : file type icon titles-abstracts.pdf

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