Local Organizing Committee : S. A. Katre, V. V. Acharya, V. M. Sholapurkar.

As a part of ATM schools held under NCM, we propose to hold Annual Foundation School – I at Bhaskaracharya Pratishthana, Pune, during 30th November to 26th December 2015. The target audience for this AFS-I  are the research scholars who are normally in their first year of Ph.D. in any of the Universities or research Institutions. There will be three hours of lectures and 2 hours of tutorials every day (Monday to Saturday). The resource persons have confirmed their participation and topics.

PROPOSAL FOR AFS-I

Teachers :

• Algebra (Group theory)
  1) Rabeya Basu (IISER, Pune)                
  2) Chetan Balwe (TIFR, Mumbai)
  3) Rahul Kitture (S. P. College, Pune)          
  4) Chandrasheel Bhagwat (IISER, Pune)
• Analysis (Real Analysis-Measure Theory)
  1) Diganta Borah (IISER, Pune)              
  2) Sameer Chavan (IIT, Kanpur)
  3) H. Bhate (S. P. Pune University)          
  4) V. M. Sholapurkar (S. P. College, Pune)
• Topology
  1) V. V. Acharya (Fergusson College, Pune)
  2) Mangala Narlikar (BP, Pune)
  3) K. D. Joshi (COEP, Pune)                
  4) A. J. Parameswaran  (TIFR, Mumbai)

Associate teachers :

1) Rakesh Pawar (TIFR, Mumbai)           
2) Siddharth Bhalerao (S. P. College, Pune)
3) Deepa Krishnamurty (MIT College, Pune)
4) Varun Thakare (HRI, Allahabad)

 Syllabus and Speakers

1. Group Theory.

Rabeya Basu  (6 hours)

    (1) Basic examples of groups such as cyclic groups, dihedral and quaternion groups, matrix groups and permutation groups. review of normal subgroups and isomorphism theorems, internal and direct products.

Chetan Balwe (6 hours)

    (2) Isometries of Rn and plane. group actions, finite subgroups of SO(2) and SO(3).

Rahul Kitture (6 hours)

    (3) Sylow Theory, classification of finite groups of order 12, simplicity of the alternating groups and PSL(V) solvable groups, p-groups, Jordan-H ̈older theorem.

Chandrasheel Bhagwat (6 hours)

    (4) Linear groups Classical groups, SU (2), latitudes and longitudes on the 3-sphere, simplicity of SO(3), normal subgroups of SL(2, F ).

References:

    (1) M. Artin, Algebra, Second Edition, Prentice Hall of India, 2011.

    (2) N. Jacobson, Basic Algebra Volume 1, Second Edition, Dover Pub lications, 2009.

    (3) S. Lang, Algebra, Third Edition, Springer India, 2001.

2. Real Analysis.

Diganta Borah (6 hours)

    (1) Basics Abstract measure spaces and the concept of measurability, simple functions, basic properties of measures, Lebesgue integration of positive functions and complex values functions, measure zero sets, completion of a measure and outer measure.

Sameer Chavan (6 hours)

    (2) Positive Borel Measures: Topological preliminaries on locally compact Hausdorff spaces, Riesz representation theorem (outline of the proof), Borel measures, Lebesgue measure on Rk , comparison     with Riemann integration. Approximation by continuous functions,Generalized Riesz representation theorem.

H. Bhate  (6 hours)

(3) Differentiation Maximal functions, Lebesgue points,Fundamental Theorem of integral calculus, Absolutely continuous functions, Change of variable formula.

V. M. Sholapurkar  (6 hours)

(4) Integration on products Monotone classes, algebra on products, product measure, Fubini, completion, convolution.

References:

    (1) I. K. Rana, Introduction to Measure and Integration, II edition, Narosa Publishing House, New Delhi.

    (2) Royden H. L., Real Analysis, III edition, Macmillan, New York, 1963.

    (3) W. Rudin, Real And Complex Analysis, III edition, Tata McGraw- Hill Higher Education, New Delhi 1987.

3. Topology.

V. V. Acharya   (6 hours)

(1) Review of Multivariable Differential Calculus Differentiability of functions on open subsets of Rn , relation with par tial/directional derivative, Taylor’s theorem etc. Inverse and implicit function theorems, rank theorem,Differentiability of functions on arbitrary subsets of subsets of Rn , diffeomorphisms, smooth version of invariance of domain.Richness of smooth functions, smooth partition of unity and consequenceson subspaces of Rn such as approximation of continuous functions by smooth functions.

Mangala Narlikar  (6 hours)
 Sard’s theorem for smooth functions Rn → Rm and some applications.

(2) Basic point-set topology part (a) Open sets and closed sets, limit points, closure and boundary  points, subspace. Bases and subbases.Continuous functions, open functions, closed functions, homeomorphisms.Separation axioms: Hausdorffness regularity and normality. Urysohn’s  lemma and Tietze extension theorem.

K.D. Joshi    (6 hours)
Compactness and Lindeloff property, local compactness. countability, separability. Path connectedness, connectedness, local connectedness. Product topology.

(3) Basic point-set topology part (b)  Induced and coinduced topologies. Quotient topology, separation axioms under quotient topology, criterion for a restriction of a quotient map to be a quotient map examples such as cones, cylinders, Mobius strips etc.

A. J.  Parameswaran (6 hours)
Paracompactness and partition of unity, Stone’s theorem (paracompactness of metric topology). Topological groups and orbit spaces. Examples from matrix groups. Function spaces, compact-open-topology and exponential correspondence.

References:

   (1) M.A. Armstrong, Basic Topology, Springer.

   (2) K. D. Joshi, Introduction to General Topology, New Age International (P) Limited.

   (3) J. R. Munkres, Topology, A First Course, Prentice-Hall of India, New Delhi 1987.

   (4) W. Rudin, Principles of Mathematical Analysis 3rd edition, McGraw Hill, 1976.

   (5) A. R. Shastri, Elements of Differential Topology, CRC Press, Taylor  and Francis Group, Boca Raton, 2011.

   (6) A. R. Shastri, Basic Algebraic Topology, CRC Press, Taylor and Francis Group, Boca Raton. 2013.

   (7) M. Spivak, Calculus on Manifolds, Benjamin Ink., New York, 1965.

Time Table:
It is planned to follow the standard time table of AFS.

 

 

	Weekly Schedule for AFS-I (30th Nov.-26th Dec. 2015)
	09.30	11.00	11.30	1.00	2.30	3.30	4.00	5.00
Mon	Algebra-L1	T E A	Analysis-L1	L U N C H	Algebra-T1	T E A	Algebra-T2	S N A C K S
Tues	Topology-L1		Algebra-L2		Analysis-T1		Analysis-T2
Wed	Analysis-L2		Topology-L2		Topology-T1		Topology-T2
Thur	Algebra-L3		Analysis-L3		Algebra-T3		Algebra-T4
Fri	Topology-L3		Algebra-L4		Analysis-T3		Analysis-T4
Sat	Analysis-L4		Topology-L4		Topology-T4		Topology-T4