ATMW Analysis and Geometry (2014)

A brief  description of the Workshop: 

The subject Geometric Analysis has been  a very active area of research for the past several decades  and the  studies  in the area have  been instrumental   in solving several important problems in  Mathematics. To mention a few, Ricci flow and proof  of Poincare  Conjecture,  Study of harmonic maps, minimal surfaces,  Yamabe problem, prescribing the curvatures,   Complex  manifolds and CR Yamabe problem, General relativity, Positive mass theorem, etc.  Geometric analysis lies in the interface of several subjects like   Geometry, PDE, Harmonic analysis,  Topology and Nonlinear functional analysis.
The aim of this workshop is to expose  young mathematicians to the  state of the art in the subject  leading them to possible research career in the field and as such this workshop is a sequel to the workshop on the same topic being held from  8th to 27th  July 2013.

Apart from others the following are expected  to lecture in the workshop.

• Speaker: Professor Sun-Yung Alice Chang, Department  of Mathematics, Princeton University, USA
  Title of lecture:- Prescribing Gaussian curvature, Q-curvature equations and related PDE
  Abstract and outline:-In this mini lecture series, I will cover some non-linear partial differential equations which arise in the study of prescribing curvature problems in conformal geometry.

Lecture I:We will discuss the classical problem of prescribing Gaussian curvature on compact surfaces; basic background and an analytic tool of sharp Moser-Trudinger Sobolev inequality.
(Reference [1])

Lecture II:Uniformization Theorem on compact surface, generalized Moser-Trudinger inequality on high dimensional manifolds, Polyakov formula, Log determinant functional and definition of Q curvature on 4-manifolds. (Reference [1])

Lecture III:Conformal covariant operators, Paneitz and GJMS operators, basic existence result of ”prescribing Q-curvature” problem.
(Reference [2], [3])

Lecture IV:Positivity of Paneitz operators, some geometric application of the Q-curvature equation.

(Reference [4], [5] and more recent literature). If time permits, I plan also to cover some relationship between Q-curvature and study of a class of symmetric functions of the Riccitensor called the σk of Schouten tensors.

References
( pdf files of [1], [2], [3], [4]) are available at www.math.princeton.edu/ chang)

[1] Sun-Yung A. Chang ”Nonlinear elliptic equations in conformal geometry”, Nachdiplom lectures in Mathematics, ETH, Zurich, Springer-Birkh¨auser, 2004.

[2] Sun-Yung A. Chang , Michael Eastwood, Bent Orsted, and Paul Yang), ”What is Q-curvature?” Acta Appl. Math. 102 (2008), no.2-3 119-125.

[3] Sun-Yung A. Chang and Paul Yang, “Extremal metrics of zeta function determinants on 4-manifolds,” Annals of Math. 142 (1995) pp. 171-212.

[4] [4] Sun-Yung A. Chang, J. Qing and P.C. Yang, “On the Chern-Gauss-Bonnet integral for conformal metrics on R4 ,” Duke Math. J., vol 102, no.3 (2000) pp. 523-544.
[5] Matthew Gursky,” The principal eigenvalue of a conformally invariant differential operator, with an application to semi-linear elliptic PDE” Comm. Math. Phys. 207
(1999), 131-143.

• Speaker:- Professor Paul Yang, Department  of Mathematics, Princeton University, USA
  Title of lecture:- CR Yamabe problem and CR conformal Geometry.
  Abstract and Outline:-In this lecture series, I will discuss the CR Yamabe problem. This includes back ground material on CR structures, the CR embedding problem and the d-bar problem, and the CR Paneitz operator.
  Lecture 1: The CR structure. The Tanaka Webster connection, curvature and torsion. Holomorphic functions and the embedding problem.
  Lecture 2: The CR Paneitz operator and the solution of the embedding problem.
  Lecture 3: The CR positive mass theorem.
  Lecture 4: The solution of the CR Yambe problem.
  Material for Lectures 1 and 2 can be downloaded here
  http://www.math.rutgers.edu/~chanillo/crduke.pdf
  and
  http://www.math.rutgers.edu/~chanillo/contemp_math_final.pdf
  References for Lectures 3 and 4:
  [1] R. Schoen and S.-T. Yau, Proof of the Positive Mass Theorem II, Communications in Math. Physics, 79 (1981), 231-260
  [2] E. Witten, A New Proof of the Positive Mass Theorem, Communications in Math. Physics, 80, (1981), 381-402.
  [3] J.-H. Cheng, A. Malchiodi and Paul Yang, slides on the proof of the CR Positive Mass Theorem and CR Yamabe Problem
  http://www.math.rutgers.edu/~chanillo/malchiodi.pdf
• Speaker: Professor Xiaojun Huang, Department of Mathematics, Rutgers University, USA
  Title of lecture:- Chern-Moser invariants and Normal  forms.
  Abstract and Outline: The equivalence problem in Several Complex Variables studies when two domains in a complex space  are holomorphically equivalent. When the domains are strongly pseudoconvex and have smooth boundaries, the result of Bochner and C. Fefferman can be applied to reduce this equivalence problem to that for real hypersurfaces. In this short course,we will introduce the Cartan-Chern-Moser method for solving the equivalence problem for real hypersurfaces. We will introduce the normal form method and the moving frame method. If time permits, we will also study the equivalence problem for real submanifolds with CR ( Cauchy-Riemann)singularities.
  Background for the lectures of Prof. Huang can be found in:

  [1] Xiaojun Huang, Local Equivalence Problems for Real Submanifolds in Complex Spaces, Springer Lecture Notes in Mathematics, 1848, Real methods in Complex and CR Geometry, 109-163, eds. D. Zaitsev, G. Zampieri, Lectures given at the CIME Summer school, Martina Franca, Italy, 2002.

• Speaker:-Professor Rupert L. Frank, Department  of Mathematics, California Institute of Technology,USA
  Title of lecture:- Sharp inequalities

Each of the speakers above will give 5 hours of lectures (4+1).

Schedule of Lectures

Speakers and title of lectures:

1. Professor Sun-Yung Alice Chang, Department of Mathematics, Princeton University, USA
   Title of lecture:- Prescribing Gaussian curvature, Q-curvature equations and related PD
2. Professor Paul Yang, Department of Mathematics, Princeton University, USA
   Title of lecture: - CR Yamabe problem and CR conformal Geometry.
3. Professor Xiaojun Huang, Department of Mathematics, Rutgers University, USA
   Title of lecture:-Chern-Moser invariants and Normal forms.
4.  Professor Cyril Tintarev, Department of Mathematics, Uppsala University, Sweden
   Title of lectures:-Cocompactness and cocompactness of Sobolev-type imbeddings on manifolds & Compactness properties of Moser-Trudinger functional
5.   Professor Sandeep K, Tata Institute of Fundamental Research-Centre for Applicable Mathematics, Bangalore
   Title of lecture:- Symmetry properties and uniqueness of extremals for the Sobolev and Hardy-Sobolev Maz’ya inequalities.
6. The titles of lecture by Prof. Sagun Chanillo will be announced in due course.

VENUE:  Lecture Hall - 111, 1st floor, TIFR CAM, Bangalore

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