ATML School in Diff. Geometry & Measure Theory (2009)

Academic Programme:

The programme will have two main streams: one on Measure Theory and the other on Differential Geometry.
Daily programme and Course contents:
The daily programme will consist of 3 lecture hours in the morning session and two problem sessions in the afternoon sessions.

Measure and Integration

Riemann Integral and its deficiencies, Extension theory and construction of Lebesgue Measure, Lebesgue integral and its properties. Measure and Integration on Product spaces.

Differential Geometry

Evolution of the notion of curvature, Parameterized Curves in E² Frenet-Serret Equations, moving frames, First and Second Fundamental Forms for a surface in E³, Euler's theorem on normal curvature, Mean and Gaussian curvatures, Theorema Egregium, classification of compact surfaces (discussion), idea of a manifold.

Two applications to complex analysis:
i) Hyperbolic geometry and Schwarz's lemma,
ii) Weierstrass representation for a minimal surface in E³.

Resource persons:

• Prof. Ravi S. Kulkarni, Department of Mathematics, IIT Bombay
• Prof. Sudeb Mitra, Department of Mathematics, City University of New York
• Prof. Inder K. Rana, Department of Mathematics, IIT Bombay

Tutors:

(i) Mr. Vikram Aithal IIT Bombay.
(ii) Ms. Rupali Khedkar, IIT Bombay.
(iii) Ms. Sridevi Masuti, IIT Bombay.
(iv) Dr. Vivek Sinha, NIT Suratkal.
(v) Mr. Surinderpal Singh, IIT Bombay.