Special Day on Complex Geometry and Analysis on real analytic Riemannian manifolds
Mathematics Department
Northwestern University
Saturday February 21, 2015
Swift Hall, 107
Local Map
All are invited to attend, there is no registration.
Schedule
9.30am  10.30am

László Lempert (Purdue)
Adapted complex structures I  Abstract
Adapted complex structures, also known as Grauert tubes, first arose in
the early 1990s in connection with the homogeneous complex MongeAmpère
equation, as complex structures on the tangent or cotangent bundle of a
Riemannian manifold M (or on certain subsets of these bundles). They
also arise in Lie theory, geometric quantization and in the study of
semiclassical pseudodifferential operators.
Adapted complex structures can be viewed as complex structures on the
phase space of M that are compatible with the natural symmetries of
phase space, and this is the approach I will take in my lectures. I will
discuss various definitions of these structures and their fundamental
properties, such as existence, uniqueness, regularity; also the
connection with the MongeAmpère equation and with curvature estimates for
the metric of M.

11.00am  12.00pm

Junehyuk Jung (IAS)
Quantum ergodicity and the number of nodal domains

2.00pm  3.00pm

László Lempert (Purdue)
Adapted complex structures II  Abstract
Adapted complex structures, also known as Grauert tubes, first arose in
the early 1990s in connection with the homogeneous complex MongeAmpère
equation, as complex structures on the tangent or cotangent bundle of a
Riemannian manifold M (or on certain subsets of these bundles). They
also arise in Lie theory, geometric quantization and in the study of
semiclassical pseudodifferential operators.
Adapted complex structures can be viewed as complex structures on the
phase space of M that are compatible with the natural symmetries of
phase space, and this is the approach I will take in my lectures. I will
discuss various definitions of these structures and their fundamental
properties, such as existence, uniqueness, regularity; also the
connection with the MongeAmpère equation and with curvature estimates for
the metric of M.

3.15pm  4.15pm

Giorgio Patrizio (University of Florence)
MongeAmpère Manifolds: rigidity vs. deformability
 Abstract
I will consider complex manifolds equipped with a plurisubharmonic exhaustion
satisfying the top dimensional complex degenerate MongeAmpère equation outside its minimal set.
The minimal set is always small, and in the usual examples it is either a complex submanifold
(Stoll's parabolic manifolds) or a maximal dimensional totally real submanifold (Grauert Tubes).
The different nature of the minimal set largely determines properties of the ambient manifold, as rigidity properties and deformability features.
This, in turn, has consequences in classification and/or characterization results.
After a discussion of these aspects, I will outline a class of examples arising in
the theory of almost homogeneous manifolds which naturally includes classical examples along with new ones with minimal set of "mixed nature".

4.30pm  5.30pm

Brian Hall (University of Notre Dame)
Magnetic complex structures on cotangent bundles  Abstract
I will describe joint work with Will Kirwin on using "imaginary time
flows" to construct complex structures on cotangent bundles. First, I will discuss
the geodesic flow. If one starts with the vertical polarization on a cotangent
bundle and pushes forward by the "imaginary time geodesic flow", the result is a
complex polarization, which turns out to be the adapted complex structure of
LempertSzőke and GuilleminStenzel. If one instead uses a magnetic flow—describing
the motion of a charged particle in the presence of a magnetic field—the resulting
complex structure is new. I will describe several different ways of interpreting the
imaginarytime flow and will discuss two concrete examples.

Organizers:
Questions? email to: emphasisGA@math.northwestern.edu