# Annual Foundation School - Part I (2012)

Venue: |
CMI, Chennai |

Dates: |
3rd Dec - 29th Dec, 2012 |

Convener(s) |
Speakers, Syllabus and Time table |
Applicants/Participants |

Name |
C. D 'Cruz | S. Shrihari |

Mailing Address |
Chennai Mathematical Institute |
Chennai Mathematical Institute |

**1.1. Group Theory.**

**Speaker: S. Senthamarai Kannan**- Group actions, Sylow Theory, direct and semi-direct products, simplicity of the alternating groups, solvable groups, p-groups, nilpotent groups, Jordan-Holder theorem.
**Speaker: Krishna Hanumanthu**- free groups, generators and relations, finite subgroups of SO(3), SU(2), simplicity of PSL(V).
**Speaker: Purusottam Rath**- Representations and characters of finite groups: Maschke’s Theorem, Schur’s lemma, characters, orthogonality relations, character tables of some groups, Burnside’s theorem.
**Speaker: Sanoli Gun**- Modules over PIDs: Modules, direct sums, free modules, finitely generated modules over a PID, structure of finitely generated abelian groups, rational and Jordan canonical form.

**1.2. Analysis.**

**Speaker: Amritanshu Prasad**- Abstract measures, outer measure, completion of a measure, construction of the Lebesgue measure, nonmeasurable sets.
**Speaker: B. V. Rao**- Measurable functions, approximation by simple functions, Cantor function, almost uniform convergence, Ego-roff and Lusin’s theorems, convergence in measure.
**Speaker: E. K. Narayanan**- Integration, monotone and dominated convergence theorems, comparison with the Riemann integral, signed measures and Radon-Nikodym theorem.
**Speaker: R. Srinivasan**- Fubini’s theorem, Lp - spaces.

**1.3. Differential Topology.**

**Speaker: Shiva Shankar**- Review of differential calculus of several variables: Inverse and Implicit function theorems. , Richness of smooth functions; Smooth partition of unity, Submanifolds of Euclidean spaces (without and with boundary), Tangent space, embeddings, immersions and submersions, Regular values, pre-image theorem , Transversality and Stability
**Speaker: TBA**- Topological and smooth manifolds, partition of unity , Fundamental gluing lemma and classication of 1-manifolds, Vector bundle; Tangent bundle, Morse - Sard theorem, Easy Whitney embedding theorems.
**Speaker: TBA**- Orientation on manifolds. , Transverse Homotopy theorem and oriented intersection number , Degree of maps both oriented and non oriented case, Winding number, Jordan Brouwer Separation theorem, Borsuk - Ulam Theorem, Vector eld and isotopies (statement of theorems only) with application to Hopf Degree theorem.
**Speaker: TBA**- Morse functions, Morse Lemma, Connected sum, attaching handles , Handle decompostion theorem, Application to smooth classication of compact smooth surfaces.

Number of lectures allotted for each topic |
|||

Subject |
Topic |
Speaker |
Duration (in hours) |

Algebra |
Module I
Module II Module III Module IV |
S. S. Kannan
K. Hanmanthu P. Rath S. Gun |
6
6 6 6 |

Total |
24 | ||

Analysis |
Module I
Module II Module III Module IV |
A. Prasad
B. V. Rao E. K. Narayanan R. Srinivasan |
6
6 6 6 |

Total |
24 | ||

Topology & Geometry |
Module I
Module II Module III Module IV |
S. Shankar | 6
6 6 6 |

Total |
24 | ||

Grand Total |
72 | ||

Tutorials Total |
40 | ||

UM Lectures Total |
5 |

**Tutors:** Research Scholars and Post-Docs from CMI and IMSc. The list will be conﬁrmed by the instructors.

Date |
UM Speakers |
Date |
UM Speakers |

Fri 11/12 | C. S. Seshadri | Fri 18/12 | A. Sitram |

Thu 21/12 | R. Balasubramanian | Tue 28/12 | S. Kesavan |

Thu 24/12 | R. Karandikar | -- | -- |

**References:**

[G-P] V. Gullemin and A. Pollack, Differential Topology, Englewood Cliff,

N.J. Prentice Hall (1974).

[H] W. Hirsch, Differential Topology, Springer-Verlag.

[M] J. W. Milnor, Topology from the Differential Viewpoint, Univ. Press,

Verginia.

[S] Anant R. Shastri, Elements of Differential Topology, CRC Press, 2011.

Selected Applicants |

Sr. |
SID |
Full Name |
Gender |
Affiliation |
Ph.D. Deg. Date |

1 | 1636 | Prasanta Kumar Barik | M | Inst. of Math, Bhubaneshwar | PhD |

2 | 1645 | Aunrag Singh | M | IIT Kanpur | PhD |

3 | 1648 | Mustaq Ahmed Bhat | M | Kashmir University | PhD |

4 | 1670 | Goutam Anand | M | IISER Thiruvanthanpuram | Integrated Phd |

5 | 1696 | Murugan Krishnan | M | Bhararhiar University | M.Phil |

6 | 1705 | Praveen Kumar Lavudi | M | Andhra University | Phd |

7 | 1715 | Azeef Muhammad PA | M | University of Kerala | PhD |

8 | 1739 | Mitu Gupta | F | University of Jammu | PhD |

9 | 1756 | Tarun Kumaru | F | Tezpur Univesity | PhD |

10 | 1791 | Neha Prabhu | F | IISER Pune | PhD |

11 | 1803 | Rukumani Srinivasan | F | Anna University | PhD |

12 | 1804 | Prema Jayachandran | F | Anna University | PhD |

13 | 1805 | Alagamanni Renganathan | F | Anna University | PhD |

14 | 1813 | Kavita Pradeep | F | Anna University | PhD |

15 | 1827 | Sudipta Paul | M | Gauhati University | PhD |

16 | 1837 | Amey Sriphad Deshpande | M | Universit of Pune | PhD |

17 | 1918 | Gunja Sachdeva | F | IISER Pune | Phd |

18 | 1919 | Sanjeev Singh | M | IIT Madras | PhD |

19 | 1906 | Paltur Sarkar | M | Jadavpur University | PhD |

20 | 1999 | Nisha Godani | F | Daylbagh Educational Institute, Agra | PhD |

21 | 2001 | Jatin Trikambal Majithia | M | IISER Pune | PhD |

22 | 2007 | Souvrav Das | M | IIT Roorkee | PhD |

23 | 2023 | Sandeep KT | M | IIT Bombay | PhD |

24 | 2036 | Papri Dey | F | IIT Bombay | PhD |

25 | 1738 | Kurian Soman | M | University of Kerala | Phd |

26 | 1694 | Nagarajan Murugesan | M | Bharathiar University | PhD |

27 | 1687 | Tawheeda Rasool | M | NIT Srinagar | PhD |

28 | 1744 | Manoj Kumar Verma | M | BHU | PhD |

29 | 1650 | Suhail Gulzar Mattoo | M | University of Kashmir | PhD |

30 | 1905 | Sudipta Purakit | M | Jadavpur University | PhD |

How to reach |

**Location**

The Chennai Mathematical Institute is located in the SIPCOT campus in Siruseri on the Rajiv Gandhi Salai ( formerly Old Mahabalipuram Road OMR) which is on the southern outskirs of the city. It is approximately 40 kms from the Chennai Central station and about 30 kms from the airport. Siruseri which is between Thriuvanmiyur and Kelambakkam. Chennai is 29 km north and Kelambakkam is 5 km south. The nearest railway station is Thiruvanmiyur which takes 35 minutes to reach. Nearest airport is Chennai International Airport.

**For those arriving at Chennai Central:**

You can take a pre-paid auto or taxi from the railway station either to the Bus terminus at Parrys or to Fort Railway station. The rates for the pre-paid autos and pre-paid taxi's are fixed.

The bus terminus at Parrys is 1.4 kms from Chennai Central railway station. From Parrys take any to either take a bus to either Adayar or Siruseri.

You can also take a train from Chennai Fort station towards Velachery and alight at Kasturba nagar. A link to the suburban trains is available on the Southern railway website. Kasturba nagar (or Madhya Kailash) is an important stop for MTC buses. There are several buses from Madhya Kailash to Siruseri IT park main gate. CMI is 1.2 kms from themain gate.

**For those arriving at Egmore:**

From Egmore you can either take a train to Saidapet station or bus to Saidapet (main bus stop). From Saidapet station you will have walk to the Saidapet bus stop.

**For those arriving at the Aiport:**

From the airport you can avail the pre-paid taxi service to Siruseri Both the private and the TTDC pre-paid taxi service are very reliable and efficient.