TEW - A Panorama of Geometry (2023)

Speakers and Syllabus

 

Name of the Speaker with affiliation

No. of Lecture s

Detailed Syllabus

Sushmita Venugopalan (SV) Reader, IMSc, Chennai

9 hours

 Title :  An introduction to algebraic geometry
Syllabus :
* Affine varieties, Zariski topology, Hilbert's Nullstellensatz, Projective space, Projective varieties.
* Examples : Conics and Cubics, Twisted cubic, Veronese and Segre maps.
* Grassmanians.
* Dimension and degree.
* Blowing up points.

Vijay Ravikumar (VR)
Azim Premji University, Bengaluru

9 hours

 Title : Discovering geometry through our physical senses

Part 1 : Geometry of vision
* From perspective drawing to RP2 as the extended Euclidean plane.
* Projective geometry: Duality; Desargues' Theorem, Pappus's Theorem.
* Fundamental Theorem of Projective Geometry.
* Cross ratio : an invariant of perspective change.
* Three point perspective, RP3 as the extended euclidean space.

Part 2 : Topology of movement
* SO(3) as the set of spatial rotations.
* Topology of SO(3) via a series of movement challenges.
* Quaternion representation of spatial rotations.
* Proof that SO(3) = RP3.
* Time permitting : Hopf fibration.

Parameshwaran Sankaran (PS) Professor,

CMI, Chennai

4.5

hours

  Title: Geometry of the Poincaré upper half plane.
*  Surfaces in R^3.
* Arc length--geodesics.  Examples--spheres, torus. Riemannian metric. Poincaré upper half space H.
* Isometry groups of S^2,R^2, and H. Properties of (geodesic) triangles.  Area formula.
* Notion of curvature (for surfaces)--examples. Euler characteristics, Gauss-Bonnet formula.
* Generalization to higher dimensions, if time permits.

Amritanshu Prasad (AP) Professor, IMSc, Chennai

4.5

hours

 Title : Groebner bases
*Monomial orders, division algorithm for multivariate
polynomials.
* Dickson's lemma and Hilbert basis theorem
using Groebner bases.
* Properties of Groebner bases.

 

 

References:

 

  1.  Algebraic Geometry by Joe Harris.

  2. An invitation to algebraic geometry by Karen E. Smith , Lauri Kahanpää , Pekka Kekäläinen , William Traves. 

  3.  The Four Pillars of Geometry by John Stillwell.

  4. Perspective and Projective Geometry by Annalisa Crannell, Fumiko Futamura, and Marc Frantz.

  5.  Projective Geometry by HSM Coxeter

  6.  Naive Lie Theory by John Stillwell

  7. Lecture Notes on Elementary Topology and Geometry (Springer UTM) by  I. M. Singer and J A Thorpe.

  8. Hyperbolic Geometry by A F Beardon.

  9. Ideals, Varieties, and Algorithms (Springer UTM) by Cox, Little and O'Shea.

 

 

Name of the tutors

S. No.

Name

Affiliation

1

Abhirup Chatterjee (AC)

IMSc

2

S Velmurugan (VM)

IMSc

3

Manika Gupta (MG)

IMSc

Time Table

 

Day

Date

Lecture 1

9:30–11:00

Tea 11.00
to
11:30

Lecturer 2

11.30–1.00

Lunch
1.00
to
2.00

Lecture 4

2.00-3.30

Tea 3.35
to
3.55

Discussion 4.00-5.00

Snacks 5.05
to

5.30

 

 

(Speaker’s name)

 

(Speaker’s name)

 

(Speaker’s name)

 

(Tutor’s name)

 

Mon

20

SV

 

VR

 

PS

 

SV+AC+MG

 

Tue

21

SV

 

VR

 

PS

 

VR+AC+VM

 

Wed

22

SV

 

VR

 

PS

 

PS+VM+MG

 

Thu

23

SV

 

VR

 

AP

 

SV+AC+MG

 

Fri

24

SV

 

VR

 

AP

 

VR+AC+VM

 

Sat

25

SV

 

VR

 

AP

 

AP+VM+MG

 
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