Syllabus:
Name of the Speaker 
Affiliation 
No. of Lecturers 
Detailed Syllabus 
Parvati Shastri 
Mumbai University (Retried) 
4 
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. 
J. K Verma 
I.I.T. Bombay 
4 
Isometries of R^{n }and plane. Group actions, finite subgroups of SO(2) and SO(3). 
Ritumoni Sarma 
I. I.T. Delhi 
4 
Sylow Theory, classification of finite groups of order 12, simplicity of the alternating groups and PSL(V) solvable groups, pgroups, JordanHÖlder theorem. 
Prof. K. Ramesh 
ISI Kolkata 
4 
Linear groups Classical groups, SU (2), latitudes and longitudes on the 3sphere, simplicity of SO(3), normal subgroups of SL(2, F). 
Prachi Mahajan 
I.I.T. Bombay 
4 
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. Positive Borel Measures. 
Anand P. Singh 
Central University, Ajmer 
4 
Topological preliminaries on locally compact Hausdorff spaces, Riesz representation theorem (outline of the proof), Borel measures, Lebesgue measure on R^{n}, comparison with Riemann integration. Approximation by continuous functions, Generalized Riesz representation theorem. 
Parasar Mohanty 
I.I.T. Kanpur 
4 
Differentiation Maximal functions, Lebesgue points, First Fundamental Theorem of integral calculus, Absolutely continuous functions, Second Fundamental theorem of integral calculus. 
Sameer Chavan 
I.I.T Kanpur 
4 
Change of variable formula. Integration on products Monotone classes, algebra on products, product measure, Fibini, completion, convolution. 
PASS Krishna

I. I. T. Guwahati

5 
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. Compactness and LindelÖff property, local compactness. I and II countability separability. Path connectedness, connectedness, local connectedness. Product topology. 
A. R. Shastri

I I T Bombay

6 
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. Paracompactness and partition of unity, Stone’s theorem (paracompactness of metric topology). Topological groups and orbit spaces. Examples from matrix groups. Function spaces, compactopentopology and exponential correspondence. 
B. Subhash 
NIICER, Bhubaneswar 
5 
Differentiability of functions on open subsets of R^{n}, relation with partial/directional derivative, Taylor’s theorem etc. Inverse and implicit function theorems, rank theorem, Differentiability of functions on arbitrary subsets of subsets of R^{n}, diffeomorphisms, smooth version of invariance of domain. Richness of smooth functions, smooth partition of unity and consequences on subspaces of R^{n} such as approximation of continuous functions by smooth functions. Sard’s theorem for smooth functions R^{n} →R^{m} and some applications. 
References:
 M. Artin, Algebra, Second Edition, Prentice Hall of India, 2011.
 N. Jacobson, Basic Algebra Volume 1, Second Edition, Dover Publications, 2009.
 Joseph A. Gallian, Contemporary Abstract Algebra, Narosa Publishing House, New Delhi.
 I. K. Rana, Introduction to Measure and Integration, II edition, Narosa Publishing House, New Delhi.
 Royden H. L. Real Analysis, III edition, Macmillan, New York, 1963.
 S.C. Malik and Savita Arora, Mathematical Analysis, New Age International (P) Limited.
 M.A. Armstrong, Basic Topology, Springer.
 K. D. Joshi, Introduction to General Topology, New Age International (P) Limited.
 A.R. Shastri, Elements of Differential Topology, CRC Press, Taylor and Francis Group, Boca Raton, 2011.
TimeTable
Day 
Date 
Lecture 1 (09.3011.00) 
Tea 11.00 
Lecture 2 (11.3001.00) 
Lunch 01.0002.30) 
Tutorial (02305.00) 
Tea
(5.00) 


Lecture Subject(speaker) 

Lecture Subject(Speaker) 

Tutorial subject(speaker/tutors) 

Mon 
30.11.2015 
Algebra (PAS) 

Analysis(PrM) 

Algebra(PAS)(DK) 

Tues 
1.12.2015 
Topology(BS) 

Algebra (PAS) 

Analysis(PrM)(MJ) 

Wed 
2.12.2015 
Analysis(PrM) 

Topology(BS) 

Topology(BS)(KR) 

Thu 
3.12.2015 
Algebra (PAS) 

Analysis(PrM) 

Algebra(PAS)(DK) 

Fri 
4.12.2015 
Topology(BS) 

Algebra (PAS) 

Analysis(PrM)(MJ) 

Sat 
5.12.2015 
Analysis(PrM) 

Topology(BS) 

Topology(BS)(KR) 

Mon 
7.12.2015 
Algebra (JKV) 

Analysis(APS) 

Algebra(JKV)(DK) 

Tues 
8.12.2015 
Topology(BS) 

Algebra (JKV) 

Analysis(APS)(MJ) 

Wed 
9.12.2015 
Analysis(APS) 

Topology(PK) 

Topology(PK)(KR) 

Thu 
10.12.2015 
Algebra (JKV) 

Analysis(APS) 

Algebra(JKV)(DK) 

Fri 
11.12.2015 
Topology(PK) 

Algebra (JKV) 

Analysis(APS)(MJ) 

Sat 
12.12.2015 
Analysis(APS) 

Topology(PK) 

Topology(PK)(KR) 

Mon 
14.12.2015 
Algebra (RS) 

Analysis(PaM) 

Algebra (RS) (DK/SK) 

Tues 
15.12.2015 
Topology(PK) 

Algebra (RS) 

Topology(PK)(SK/BS) 

Wed 
16.12.2015 
Analysis(PaM) 

Topology(PK) 

Analysis(PaM)(SK/BS) 

Thu 
17.12.2015 
Algebra (RS) 

Analysis(PaM) 

Algebra (RS) (DK/SK) 

Fri 
18.12.2015 
Topology(ARS) 

Algebra (RS) 

Analysis(PaM)(SK/BS) 

Sat 
19.12.2015 
Analysis(PaM) 

Topology(ARS) 

Topology(PK)(SK/BS) 

Mon 
21.12.2015 
Algebra (KR) 

Analysis(SC) 

Algebra (IS) (DK/SK) 

Tues 
22.12.2015 
Topology(ARS) 

Algebra (KR) 

Topology(ARS)(SK/BS) 

Wed 
23.12.2015 
Analysis(SC) 

Topology(ARS) 

Analysis(SC)(SK/BS) 

Thu 
24.12.2015 
Algebra (KR) 

Analysis(SC) 

Algebra (IS) (DK/SK) 

Fri 
25.12.2015 
Topology(ARS) 

Algebra (KR) 

Analysis(SC)(SK/BS) 

Sat 
26.12.2015 
Analysis(SC) 

Topology(ARS) 

Topology(ARS)(SK/BS) 

 (PAS) Parvati Shastri
 (PrM) Prachi Mahajan
 (BS) B. Subhash
 (JKV) Jugal Kishore Verma
 (APS) Anand Prakash Singh
 (PK) P. Krishna
 (RS) Ritumoni Sarma
 (PaM) Parasar Mohonty
 (ARS) A. R. Shastri
 (DK) Devendra Kumar
 (KR) Prof. K. Ramesh
 (SC) Sameer Chavan
 (MJ) Monika Jain
 (SK) Surjit Kumar
Tutorial Assistants:
S. No. 
Name 
Affiliation 
Subject 
No. of weeks 
1 
Dr. Devendra Kumar (DK) 
JECRC University, Jaipur 
Algebra 
I to IV 
2 
Dr. Monika Jain(MJ) 
JECRC University, Jaipur 
Real Analysis 
I to IV 
3 
Dr. Surjit Kumar (SK) 
I.I.Sc. Bangalore 
Analysis + Topology 
III and IV 
4 
 
IIT Bombay 


5 
 
IIT Delhi 


Please Note: We have approached a few more course associates, in order to have a full quota of course associates. Confirmation from them is pending and hence the names are not shown in the above table.