Annual Foundation School - I (2015) - JECRC University, Jaipur - Speakers and Syllabus

Syllabus:

Name of the Speaker

Affiliation

No. of Lecturers
(1.5 hrs duration)

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 Rn 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, p-groups, Jordan-HÖlder theorem.

Prof. K. Ramesh

ISI Kolkata

4

Linear groups Classical groups, SU (2), latitudes and longitudes on the 3-sphere, 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 Rn, 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, compact-open-topology and exponential correspondence.

B. Subhash

NIICER, Bhubaneswar

5

Differentiability of functions on open subsets of Rn, 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 Rn, diffeomorphisms, smooth version of invariance of domain. Richness of smooth functions, smooth partition of unity and consequences on subspaces of Rn such as approximation of continuous functions by smooth functions. Sard’s theorem for smooth functions Rn →Rm 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.

 Time-Table

Day

Date

Lecture 1

(09.30-11.00)

Tea

11.00

Lecture 2

(11.30-01.00)

Lunch

01.00-02.30)

Tutorial

(0230-5.00)

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)

 

  1. (PAS) Parvati Shastri
  2. (PrM) Prachi Mahajan
  3. (BS) B. Subhash
  4. (JKV) Jugal Kishore Verma
  5. (APS) Anand Prakash Singh
  6. (PK) P. Krishna
  7. (RS) Ritumoni Sarma
  8. (PaM) Parasar Mohonty
  9. (ARS) A. R. Shastri
  10. (DK) Devendra Kumar
  11. (KR) Prof. K. Ramesh
  12. (SC) Sameer Chavan
  13. (MJ) Monika Jain
  14. (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.