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J. Anderson, Hyperbolic Geometry, 1st ed., Springer Undergraduate Mathematics Series, Springer-Verlag, Berlin, New York, 1999.
S. Katok, Fuchsian Groups, Chicago Lecture Notes in Mathematics, Chicago University Press, 1992.
A. Beardon, The Geometry of Discrete Groups, Springer-Verlag, Berlin, New York, 1983.

MTH713A: Differential Topology

2020 - 2021, Semester II

Course Description: This is an introductory course in Differential Topology. The aim of this course is to introduce basic tools to study the topology and geometry of manifolds. We start with reviewing two key results from several variable calculus, namely the inverse function theorem and implicit function theorem which are essential to study differential manifolds. Throughout this course, we will discuss the theory of manifolds and a way to generalise differential, integral and vector calculus. By the end of the course, we should understand and able to work with manifolds, tangent and co-tangent bundles, transversality, Morse Lemma, Morse function, Whitney embedding theorem, Poincare – Hopf theorem, Sards theorem and its applications and many other things listed below in the course contents.

Course Contents:

Review of inverse function theorem and implicit function theorem.

Introduction to differential manifolds, sub-manifolds and manifolds with boundary.
Smooth maps between differential manifolds, tangent space, differential of smooth maps,
(local) diffeomorphism. Immersions, embeddings, regular value, level sets, submersions.

Tangent and cotangent bundles, vector bundles, vector fields, integral curves.
Transversality, Whitney embedding theorem, Sards theorem, Morse lemma, Morse
functions.

Oriented intersection Theory: degree, Lefchetz fixed point theory, the Poincare-Hopf
theorem, the Euler characteristic and triangulations (time permitting: Integration on manifolds).

Recommended books:

1. Differential Topology, Victor Guillemin and Alan Pollack (AMS Chelsea Book Series)
2. Topics in Differential Topology, Amiya Mukherjee (Hindustan Book Agency)
3. Topology from the Differentiable Viewpoint, John W. Milnor (Princeton University Press)
4. An Introduction to Differentiable Manifolds and Riemannian Geometry, William M.
  Boothby (Elsevier)
5. An introduction to manifolds, Loring W Tu (Springer).

MTH305A : Several variable calculus and differential geometry

2020 - 2021, Semester I

Syllabus:

Differentiation: Definition and examples, Mean value inequality, Tangent planes to level sets of functions; Implicit mapping theorem, Inverse mapping theorem and applications; Taylor's theorem and applications.
Curves: Definition and examples, Regular curves, Plane curves, Curvature of plane curves, Isoperimetric inequality for plane curves; Space curves, FrenetSerret formula for space curves; Local existence theorem curves.
Surfaces: Definition and examples; Tangent planes, Maps between surfaces; First fundamental and second fundamental forms; Curvature of surface; Hilbert's theorem for compact surfaces; Gauss theorem a Egregium.

References:

1. Spivak: Calculus on manifolds, Springer.

2. M P do Carmo: Differential geometry of curves and surfaces, Prentice Hall.

3. W. Rudin: Principles of Mathematical Analysis.

4. Tom M. Apostol: Mathematical Analysis, Narosa Publishing House, India.

5. A Pressley: Elementary differential geometry, Springer India.

Important Dates:

Course begins: September 1, 2020.

Ref. for first few lectures Chapter 1 and 2, Spivak: Calculus on manifolds

MTH 301A -Analysis I

2021-2022, Semester I

Summer Term 2021:

MTH305A: Several variables calculus \& differential geometry

Notes:

Introduction

MTH304A : Topology

2019 - 2020, Semester II

MTH713A : Differential Topology

2019 - 2020, Semester I

MTH102A Tutor

2018 - 2019, Semester II

Group

PhD students

current.

1) Bhola Nath Saha

2) Achintya Dey (Joint with Dr Abhijit Pal)

UG Project students.

Current.

......

Past.

1) Nupur Jain 2022-23(I):

Topic: Topological graph theory

Major results-

Kuratowski's theorem,
Cayley Graphs and Frucht’s Theorem
Existence of regular graphs of a given Girth

Report

MSc project students

Current.

Papiya Sur

Topic. Spherical Geometry

Past.

1) Arghys Sinha (2020-21): Report

Summer research programe:

1) Arya Vadnere (2019-20).

Topic. Aougab-Huang conjecture

Publication.