Description

Mathematics Colloquium.

Speaker: Saikat Mazumdar.

Affiliation: IIT Bombay.

Date and Time: Wednesday 14 August, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Yamabe problem and beyond: an interplay of geometry and PDE.

Abstract: Motivated by the theory of compact surfaces, Yamabe wanted to

show that on a given compact Riemannian manifold of any dimension there

always exists a (conformal) metric with constant scalar curvature. It

turns out that solving the Yamabe problem amounts to solving a nonlinear

elliptic partial differential equation (PDE). The solution of the Yamabe

problem by Trudinger, Aubin and Schoen highlighted the local and global

nature of the problem and the unexpected role of the positive mass theorem

of general relativity. In the first part of my talk, I will survey the

Yamabe problem and the related issues of the compactness of solutions.

In the second part of the talk, I will discuss the higher-order or

polyharmonic version of the Yamabe problem: "Given a compact Riemannian

manifold (M, g), does there exists a metric conformal to g with constant

Q-curvature?" The behaviour of Q-curvature under conformal changes of the

metric is governed by certain conformally covariant powers of the

Laplacian. The problem of prescribing the Q-curvature in a conformal class

then amounts to solving a nonlinear elliptic PDE involving the powers of

Laplacian called the GJMS operator. In general the explicit form of this

GJMS operator is unknown. This together with a lack of maximum principle

makes the problem difficult to tackle. I will present some of my results

in this direction and mention some recent progress.

Speaker: Saikat Mazumdar.

Affiliation: IIT Bombay.

Date and Time: Wednesday 14 August, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Yamabe problem and beyond: an interplay of geometry and PDE.

Abstract: Motivated by the theory of compact surfaces, Yamabe wanted to

show that on a given compact Riemannian manifold of any dimension there

always exists a (conformal) metric with constant scalar curvature. It

turns out that solving the Yamabe problem amounts to solving a nonlinear

elliptic partial differential equation (PDE). The solution of the Yamabe

problem by Trudinger, Aubin and Schoen highlighted the local and global

nature of the problem and the unexpected role of the positive mass theorem

of general relativity. In the first part of my talk, I will survey the

Yamabe problem and the related issues of the compactness of solutions.

In the second part of the talk, I will discuss the higher-order or

polyharmonic version of the Yamabe problem: "Given a compact Riemannian

manifold (M, g), does there exists a metric conformal to g with constant

Q-curvature?" The behaviour of Q-curvature under conformal changes of the

metric is governed by certain conformally covariant powers of the

Laplacian. The problem of prescribing the Q-curvature in a conformal class

then amounts to solving a nonlinear elliptic PDE involving the powers of

Laplacian called the GJMS operator. In general the explicit form of this

GJMS operator is unknown. This together with a lack of maximum principle

makes the problem difficult to tackle. I will present some of my results

in this direction and mention some recent progress.

Description

Ramanujan Hall, Department of Mathematics

Date

Wed, August 14, 2019

Start Time

4:00pm-5:00pm IST

Duration

1 hour

Priority

5-Medium

Access

Public

Created by

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Updated

Mon, August 12, 2019 6:56pm IST