# Annual Foundation School - Part I (2011)

Venue: |
(IISER), Pune |

Dates: |
5 Dec - 31 Dec, 2011 |

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

Name |
Dr. Rama Mishra | S. A. Katre |

Mailing Address |
Indian Institute of Science Education and Research (IISER), First floor, Central Tower, Sai Trinity Building Garware Circle, Sutarwadi, Pashan Pune, Maharashtra 411021, India |
Professor and Coordinator, Centre for Advanced Study in Mathematics, Dept. of Mathematics, Univ. of Pune Pune-411007 |

**Speakers**

Algebra |
Real Analysis |
Topology |

Rabeya Basu
S. R. Ghorpade
Gurmeet Kaur Bakshi S.A. Katre |
Sameer Chavan
V. M. Sholapurkar
Diganta Borah
Debraj Chakraborty |
H. Bhate
Rama Mishra
Krishna Kaipa
Sreekar Shastry |

No. of lectures for each teacher: 4 lectures of one and half hours duration.

Associate Teachers: Shilpa Gondhali, Deepa Krishnamurty, Rahul Kitture, Vikas Jadhav

Unity of Mathematics Lectures:

1. Prof. Ravi Kulkarni: Some subtle points in Differential Topology (2 lectures)

2. Prof. S. S. Sane: Finite simple groups (2 lectures)

**Syllabus**

**Group Theory.**- Module 1: (6 lectures) Group actions, Sylow Theory, direct and semi-direct products, simplicity of the alternating groups, solvable groups, p-groups, nilpotent groups, Jordan-Holder theorem.
- Module 2: (6 lectures) free groups, generators and relations, ﬁnite subgroups of SO(3), SU(2), simplicity of PSL(V).
- Module 3: (6 lectures) Representations and characters of ﬁnite groups: Maschke’s Theorem, Schur’s lemma, characters, orthogonality relations, character tables of some groups, Burnside’s theorem.
- Module 4: (6 lectures) 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.
**Real Analysis.**- Module 1: 6 lectures: Abstract measures, outer measure, completion of a measure, construction of the Lebesgue measure, non-measurable sets.
- Module 2: 6 lectures: Measurable functions, approximation bysimple functions, Cantor function, almost uniform convergence, Ego-roﬀ and Lusin’s theorems, convergence in measure.
- Module 3: 6 lectures: Integration, monotone and dominated convergence theorems, comparison with the Riemann integral, signed measures and Radon-Nikodym theorem.
- Module 4: 6 lectures: Fubini’s theorem, L
^{p}- spaces. **Diﬀerential Topology.**Numbers in the bracket refer to sections from [S]- (1) Module 1: 6 lectures: Review of diﬀerential calculus of several variables: inverse and implicit function theorems. (1.4), Richness of smooth functions; smooth partition of unity.(1.6, 1.7), Submanifolds of Euclidean spaces (without and with boundary) (3.1, 3.2), Tangent space, embeddings, immersions and submersions (3.3,3.4), Regular

values, pre-image theorem (3.4), Transversality and Stability (3.5, 3.6) [The above material should be supported by exercises at the end of each section and examples matrix groups (9.1).] - (2) Module 2: 6 lectures: Abstract topological and smooth mani-folds, partition of unity (5.1,5.2), Fundamental Gluing lemma and classiﬁcation of 1-manifolds (5.3,5.4), Deﬁnition of a vector bundle;tangent bundle. (5.5), Morse-Sard theorem. Easy Whitney embedding theorems.
- (3) Module 3: 6 lectures: Orientation on manifolds. (4.1), Transverse Homotopy theorem and oriented intersection number (7.1,7.2), Degree of maps both oriented and non oriented case(7.3,7.4), Winding number, Jordan Brouwer Separation theorem. (7.5),Borsuk-UlamTheorem (7.6), Vector ﬁelds and isotopies (statement of theorems only) with appliaction to Hopf-Degree theorem. (6.3 and 7.7).
- (4) Module 4: 6 lectures: Morse functions (8.1), Morse Lemma (8.2),Connected sum, attaching handles (8.3), Handle decompostion the-orem.(8.4), Application to smooth classiﬁcation of compact smooth surfaces (8.5) (2 lectures)

**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.