CIMPA - INDIA  Research School On Geometric Flows

 

                                                                   Jadavpur University, West Bengal, Kolkata – 700032, India.  

 

                                                                                                December  01 - 12, 2016

 

Geometric flows are burning topic now a days which involve the evolution of Riemannian metric along with some geometric concepts.

 

In this research school we shall discuss about evolution of Riemannian metric with respect to time. Ricci flow, mean curvature flow and some type of other Geometric flow  will be studied thoroughly. The main objective of this course will be to study the singularity formation in the case of Mean Curvature Flow and Ricci Flow. The concept of Ricci soliton was introduced by R. Hamilton in mid 80's and they are self-similar solutions to Hamilton's Ricci flows. The Ricci solitons and gradient Ricci solitons will also be studied. The discussions about Ricci Solitons as Contact Riemannian Metrics also be done.

 

We want to motivate researcher of our country and our neighbouring countries in this field, so that they can apply it in the various field of Physics and Mathematics. The aim of the Research School is to provide students with the basic as well as more advanced notions of both theories and applications.

 

Administrative and scientific coordinators:

 

Arindam Bhattacharyya (Jadavpur University, Kolkata, India), bhattachar1968@yahoo.co.in
Thomas Richard (Université Paris-Est Créteil, France), thomas.richard@u-pec.fr

 

Scientific Committee:

 

• Manjusha Majumdar (Tarafdar), University of Calcutta, India.

• Reto Muller, Queen Mary University of London, U.K.

• Ramesh Sharma, University Of New Haven, West Haven, CT 06516, U.S.A.

• Sayan Kar, IIT, Kharagpur, India.

• Thomas Richard, Université Paris-Est Créteil, France.

 

Local Organizing Committee:

 

• Arindam Bhattacharyya, Jadavpur University, India.

•  Alaka Das, Jadavpur University, India.

• Gopal Chandra Shit, Jadavpur University, India.

• Farook Rahaman, Jadavpur University, India.

 

Scientific Program:

 

All the courses and talks of the Research School will be in English.

 

Mini Courses:

 

I). Manjusha Majumdar (Tarafdar), India (Three Lectures of 50 min.)

Title: Basics of Riemannian manifold.

Abstract: In this talk we shall discuss about Riemannian metric and Riemannian connection on a Riemannian manifold (M,g). Properties of Riemannian curvature tensor, sectional curvature, Ricci tensor, Torsion tensor shall be studied. Covariant differentiation of tensors, Bianchi’s identities, concept of Lie derivative, pull back 1-form, isometry, Ricci identities will be studied rigorously. Concept of differential forms and volume form, wedge product, Laplace operator, Hessian of a function shall be introduced. Expression of Weyl conformal curvature tensor and results of conformally flat Riemannian manifold will also be discussed. 

 

II).  Alaka Das, India (Three lectures of 50 min.)

Title: Basics of PDE

Abstract: I shall  deliver the following courses on Partial Differential  Equation in CIMPA School on Geometric Flows.

1) Classification of second order partial differential equations to Hyperbolic,. Elliptic and Parabolic types. Reduction of linear and quasi-linear equations in two independent variables to their canonical forms, characteristic curves, Well-posed and ill-posed problems.

2) Solution of the problem of vibration of a string (Hyperbolic equation), Fundamental solutions of Laplace’s equation in two and three independent variables (elliptic equation), solution of Heat equation.

 

III). Sylvain Maillot, France (Five Lectures of 50 min.)

Title: Ricci flow and applications.

Abstract: In this lecture series the topics to be covered

1. Overview of Ricci flow and basic results, 2. Hamilton’s maximum principle and its consequences, 3. Overview of Perelman’s work on Geometrisation, 4. Hamilton’s compactness theorem, 5. Ancient solutions, 6. Perelman’s Canonical Neighbourhood Theorem.

 

IV). Thomas Richard, France (Five Lectures of 50 min.)

Title: Ricci flow on surfaces.

Abstract: In this lecture, we will introduce the Ricci flow on surfaces with a Riemannian metric. We will see how Ricci flow can be used to give a proof of the Uniformisation Theorem for compact smooth surfaces. We will then move to the more delicate subject of non compact surfaces and non smooth surfaces.

 

V). Reto Muller, U.K. (Five Lectures of 50 min.)

Title: The singularity formation in Mean Curvature Flow and Ricci Flow.

Abstract: Nonlinear Heat Flows have become an important tool in modern mathematics and in particular in Geometry and Topology. Their motivation is strikingly simple: they are designed to evolve rough initial data towards nice objects such as harmonic maps, geodesics, minimal surfaces, or manifolds with constant curvature. In particular, the Ricci Flow has proved to be extremely successful with Perelman's resolution of the Poincaré and Geometrisation conjectures, yielding a complete classification of three dimensional manifolds. Due to their nonlinearity, geometric flows typically develop singularities in finite time, and it is a challenging problem to understand what happens to a geometric flow when a singularity occurs. The main objective of this lecture course will be to study this singularity formation in the case of Mean Curvature Flow and Ricci Flow. We will answer some of the following key questions: Can the flow be continued past a singularity in some (weak) sense? What kind of singularities or singularity models can occur in genera and which singularities appear in a "generic" flows? The main tool to answer such questions will be monotonicity formulas, in particular Huisken's monotonicity formula for Mean Curvature Flow and Perelman's energy and entropy monotonicity for the Ricci Flow.

 

VI). Chenxu HE, U.S.A. (Five Lectures of 50 min.)

Title: Geometry of Gradient Ricci Solitons

Abstract: The concept of Ricci solitons was introduced by R. Hamilton in mid 80's. They are self-similar solutions to Hamilton's Ricci fows and often arise as limits of dilations of singularities in the Ricci  flow. They are also critical points of Perelman's  lamda and niu entropy. They can be viewed as natural generalization of Einstein manifolds and are of interests to physicists. They are called quasi Einstein metrics in physics literature. We would like to cover the following topics:

(1) Classifications in low dimensions.

(2) Known examples and constructions.

(3) Variational structures with respect to Perelman's entropy formulas.

 

VII). Sayan Kar, India (Five Lectures of 50 min.)

Title: Geometric flows with higher order and higher derivative terms.

Abstract: The inclusion of higher order and higher derivative terms (eg. Riemann curvature squared, Bach tensor etc.) in the equation of a geometric flow changes the evolution characteristics in various specific ways. The main aim of this talk would be to illustrate such changes through several examples, in diverse dimensions.

 

VIII). Ramesh Sharma, U.S.A. (Five Lectures of 50 min.)

Title: Ricci Solitons As Contact Riemannian Metrics.

Abstract: A Ricci soliton is a self-similar solution of the Hamilton's Ricci flow, and is a generalization of an Einstein metric. The study of a gradient Ricci soliton as a K-contact metric was initiated by Sharma (2008). For the non-gradient case, Sharma-Ghosh (2011) proved that a 3-dimensional Sasakian metric as a non-trivial Ricci soliton is homothetic to the standard Sasakian metric on Nil3. Recently, Ghosh - Sharma (2014) have generalized this result for all dimensions, and also have studied non-trivial contact Ricci solitons that are locally a 3-dimensional Gaussian soliton, or gradient shrinking rigid Ricci soliton, or non-gradient expanding (for example, Sol3).

 

Deadline for registration:

 

August 28, 2016

 

Applicants from India must contact local organizer: Arindam Bhattacharyya, Jadavpur University, Kolkata, India.

 

E-mail ID:    cimpa.ju2016@gmail.com                                         or                                     bhattachar1968@yahoo.co.in 

 

Contact Number: +91 9433949472