**Overview**: Basic focus in this workshop is a discussion of the topology of curves, surfaces, and to give an introduction to Algebraic Topology through a discussion of homotopy, fundamental group, covering spaces and simplicial homology. Main emphasis will be to explain concepts through examples.Below is a more descriptive information about the syllabus.

**Syllabus****Theme 1**: Definition of manifolds as submanifolds of Rn , and tie this up with the standard abstract definition by discussing a proof of ‘medium form’ of the Whitney embedding theorem for compact manifolds.**Theme 2**: Proof of triangulation of closed surfaces, classification theorem for surfaces. Discuss orientability, and show that closed surfaces embedded in R3 are necessarily orientable.**Theme 3**: A discussion of quotient topology, group actions, quotient spaces, projective plane and Klein bottle. Embeddings of projective plane and Klein bottle in R4 .**Theme 4**: A discussion of homotopy, path homotopy, homotopy equivalence, fundamental groups, covering spaces. Discuss fundamental groups of sphere, projective plane, torus, Klein bottle and higher genus surfaces.**Theme 5**: Simplicial homology, singular homology, homotopy invariance, equivalence of simplicial and singular homology, homology and fundamental group.

** Schedule**

Date |
9:00- 10:00 |
10:30-11:30 |
11:45- 12:45 |
15:00-16:00 |
16:30- 17:30 |

07/12/15 | L1(CSA*) | L2(BT) | L3(CSA) | T1 | T2 |

08/12/15 | L4(BT) | L5(AC) | L6(BT) | T3 | T4 |

09/12/15 | L7(CSA) | L8(AC) | L9(BT) | T5 | T6 |

10/12/15 | L10(CSA) | L11(AC) | L12(BT) | T7 | T8 |

11/12/15 | L13(CSA) | L14(AC) | L15(BT) | T9 | T10 |

12/12/15 | L16(CSA) | L17(AC) | L18(KV) | – | – |

S | U | N | D | A | Y |

14/12/15 | L19(HAG) | L20(SP) | L21(KV) | T11 | T12 |

15/12/15 | L22(HAG) | L23(SP) | L24(KV) | T13 | T14 |

16/12/15 | L25(HAG) | L26(SP) | L27(KV) | T15 | T16 |

17/12/15 | L28(HAG) | L29(SP) | L30(KV) | T17 | T18 |

18/12/15 | L31(HAG) | L32(SP) | L33(KV) | T19 | T20 |

19/12/15 | L34(HAG) | L35(SP) | L36(AC) | – | – |

Ln indicates Lecture no. n, and Tn indicates Tutorial no. n

Tea Breaks: 10:00 - 10:30 and 16:00 - 16:30

Lunch Break: 12:45 - 15:00

**Tutors**

1) Arijit Ganguly, School of Mathematics, TIFR, Mumbai.

2) Amith Shasthri, School of Mathematics, TIFR, Mumbai.

3) Sanjit Das, Vellore Institute of Technology, Vellore.

4) Binoy Raveendran, TIFR CAM, Bangalore.

Names of speaker |
Affiliation |
Topics |

C S Aravinda (CSA*) | TIFR CAM, Bangalore |
Brief review of multivariable calculus. |

Bankteshwar Tiwari (BT) | BHU-DST Centre, Varanasi | Bump functions, partition of unity, smooth Urysohn lemma; quotient topology, Group actions, quotient spaces, Projective plane, Klein bottle. |

Anisa Chorwadwala (AC) | IISER, Pune | Definition of smooth manifolds as submani- folds of Rn, abstract definition by smoothness of transition functions, ‘medium form’ of the Whitney embedding theorem for compact manifolds, embeddings of Projective plane and Klein bottle in R4 . |

C S Aravinda (CSA) | TIFR CAM, Bangalore | Proof of triangulation of closed surfaces, and Classification theorem for surfaces. |

Suhas Pandit (SP) | IIT, Chennai | Homotopy, path homotopy, homotopy equivalence, Fundamental groups, covering spaces. |

H A Gururaja (HAG) | TIFR CAM, Bangalore | Transversality, proof of orientability of closed surfaces in R3 . |

Keerti Vardhan Madahar (KV) | Panjab University, Chandigarh | Simplicial homology, singular homology, homotopy invariance, equivalence of simplicial and singular homology, homology and fundamental group. |

**Reference Books:** The following books are the suggested references:

(1) A first course in Topology By Munkres.

(2) Elementary Differential Geometry By Christian B ̈ar.

(3) Differential Topology By V. Guillemin and G. Pollack.

(4) Algebraic Topology By A. Hatcher.