Special Day on Complex Geometry and PDE
Columbia UniversityFriday April 20, 2012
Department of Mathematics
All are invited to attend, there is no registration.
Schedule
10.00am  11.00am
Room Math 520 
YumTong Siu (Harvard University)
Recent results and open problems in the theory of multiplier ideal sheaves

11.15am  12.15pm
Room Math 520 
Jian Song (Rutgers University)
Geometric surgery by partial differential equations
 Abstract
We will discuss recent developments on how geometric PDEs
can perform canonical geometric surgery. We propose the analytic
minimal model program with Ricci flow to classify algebraic varieties
via geometric surgeries in GromovHausdorff topology equivalent to
birational surgeries such as contractions and flips. This approach can
also be applied to the study of degeneration of CalabiYau metrics. As
an application, we prove a conjecture of Candelas and de la Ossa for
conifold flops and transitions.

12.15pm  2.00pm 
Lunch Break

2.00pm  3.00pm
Room Math 520 
Ahmed Zeriahi (Université Paul Sabatier Toulouse)
Stability of solutions to complex MongeAmpère equations in big cohomology classes
 Abstract
We establish various stability results for solutions of complex
MongeAmpère equations in big cohomology classes, generalizing
results that were known in the context of Kähler classes.
(This is a joint work with Vincent Guedj).

3.00pm  3.30pm 
Coffee Break

3.30pm  4.30pm
Room Math 520 
Sławomir Dinew (Rutgers University Newark)
Liouville and CalabiYau type theorems for complex
Hessian equations  Abstract
I will discuss the existence of classical and weak solutions to the
complex Hessian equation on compact Kähler manifolds. In particular I
will present a gradient a priori estimate for the solutions. By a standard
blowup argument it is linked to a certain Liouville type theorem for
maximal msubharmonic functions.

4.45pm  5.45pm
Room Math 520 
Valentino Tosatti (Columbia University)
Collapsing of solutions of complex MongeAmpère equations
 Abstract
One of the key problems in geometric analysis is to understand limits
of sequences of Riemannian manifolds which collapse (in the sense that
their volume goes to zero). We will discuss this problem in the case when
the metrics are solutions of complex MongeAmpère equations on
compact Kähler manifolds. This applies for example to the case of
limits of Ricciflat CalabiYau metrics or to certain solutions of the
KählerRicci flow. This is partly joint work with M.Gross and
Y.Zhang.

6.00pm

Conference Dinner

Organizers: D.H. Phong, V. Tosatti
http://www.math.columbia.edu/~tosatti/specialday.html