ATMW Geometry & Topology (2015) -Speakers and Syllabus

Speakers 

Speaker’s Name

Affiliation

Prof. Satya Deo

HRI, Allahabad

Prof. Hemangi Shah

HRI, Allahabad

Prof. KingshookBiswas

Vivekananada University, Belur

Prof. SomnathBasu

Vivekananada University, Belur

 

By Prof. Satya Deo:-

Syllabus (Topology) six lectures of one and half hour duration and a fewtutorials:

Recall of the basic topological and homotopical concepts. A brief introduction of the Alexander- SpanierCohomology (Cechcohomology) and their properties. Covering dimension and the Cohomological dimension of a space, Basic Results and Examples,The relation between the two dimensions, Hopf's extension theorem, Cohomological dimension over various coefficients, Interesting examples, Alexandroff's problem. Dranishnikov's theorem showing that the two dimensions can differ even for compacta.

References:

1. Dimension theory, by Allen Pears, Cambridge University Press

2. Cohomological Dimensions from K Nagami's book (Appendix), Academic Press

3. The basic paper of N. Dranishnikov solving Alexandroff's Problem.

4. Foundations of Algebraic Topology by Eilenberg and Steenrod (CechCohomolgy).

By Prof. SomnathBasu:-

Counting geodesics on manifolds:

(1) Riemannian manifolds, tangent bundle, connections and curvature, geodesics and geodesic flow.
(2) Morse theory (brief survey), existence of closed geodesics, geodesics as critical points of energy functional.
(3) Counting geodesics, free loop space, Morse theory revisited (via Morse-Smale functions).
(4) Relating the geodesic counting problem to Betti numbers, some examples (surfaces, spaces with fundamental group).

 

By Prof. KingshookBiswas:-

Geodesic flow in negatively curved manifolds:

Exponential map of Riemannian manifolds, Jacobi fields, Hopf-Rinow and Cartan-Hadamard Theorems. Negatively curved manifolds and CAT(-1) spaces, Busemann functions, horospheres, boundary at infinity, visual metrics, cross-ratio. Ergodicity of geodesic ow of closed negatively curved manifolds.Equivalence between marked length spectrum, topological conjugacy of geodesic ow, and Moebius structure of boundary at infinity for closed negatively curved manifolds.

By Prof. Hemangi M. Shah:-

Harmonic Manifolds:

Introduction of harmonic manifolds, Equivalent definitions, Examples, Classification.Busemann functions in a harmonic manifold. Mean curvature of horospheres, Analyticity of Busemann functions, Applications of analyticity of Busemann function, Continuity of total Busemann function. Convexity of spheres in a manifold without conjugate points. Topological Lemma.Proof of convexity.Applications to harmonic manifolds.

Prerequisites needed from Riemannian Geometry:

Definition of conjugate points, Jacobi field, Exponential map, Index lemma, geodesic submanifold defined by a vector.

 


Date

9:30 - 11:00

Tea Break

11:30 -13:00

Lunch Break

14:00 -15:30

Tea Break

16:00-17:00

02-Nov

KB

SD

KB

Tutorial

03-Nov

KB

SD

KB

Tutorial

04-Nov

SB

SD

HS

Tutorial

05-Nov

SB

SD

HS

Tutorial

06-Nov

SB

SD

HS

Tutorial

07-Nov

SB

SD

HS

Guest Lecture By Prof. M.G. Nadkarni


Lecturer’s Name

Lecturer’s Code

Topic Titles

Prof. Satya Deo

SD

Advanced Topics in Algebraic Topology

Prof. Hemangi M. Shah

HS

Harmonic Manifolds

Prof. SomnathBasu

SB

Counting geodesics on manifolds

Prof. KingshookBiswas

KB

Geodesic flows on manifolds



Tutor’s Name

Affiliation

Pradeep Das

HRI, Allahabad

S. Manikandan

HRI, Allahabad