TEW - Algebra & its applications (2019)

Speakers and Syllabus


The Teacher’s Enrichment Workshop (TEW) is an outreach activity of the National Centre for Mathematics (NCM), which sponsors instructional schools and workshops at several levels in various places across the country. It is meant for college teachers to revisit, update and enrich their mathematical knowledge. An important component of the workshop is the discussion hour during which the teachers will have opportunities to get their doubts cleared and work out exercises, both routine and advanced.

This workshop on Algebra and its applications is primarily for Mathematics Teachers from colleges / universities in the region of Delhi NCR. The sessions will be aimed to discuss the fundamental concepts in Group theory, Ring theory, Linear Algebra, Algebraic Coding theory and their interesting applications. It may be noted that accommodation will NOT be provided to participants.

Speakers and Tutors

S. No.

Name

Affiliation

1

Prof. Geetha Venkataraman

Ambedkar University Delhi

2

Prof. Shanta Laishram

ISI, Delhi

3

Prof. Amit Kulshreshtha

IISER, Mohali

4.

Prof. Anuradha Sharma

IIIT, Delhi

 Syllabus

Name of the Speaker with affiliation

No. of Lectures

Title

Detailed Syllabus

Prof. Geetha Venkataraman (GV), Ambedkar University Delhi

6

Groups and Symmetry

Intuitive introduction and mathematical definition of Symmetry, Finite planar figures, Strip or Frieze Patterns, Classifying Frieze patterns, Wallpaper Patterns, Symmetry in art, architecture and around us, Symmetry Groups of finite Planar Objects, Group Actions, Orbit-Stabilizer Theorem, Platonic Solids (duality, verifying Euler’s formula), Symmetry Groups of Platonic Solids, Colouring Problems, Cauchy’s Theorem, Sylow’s Theorems, Class Equation, Cayley’s Theorem, Index Theorem etc.

Prof. Shanta Laishram (SL),

ISI Delhi

7

Ring Theory with applications to Number Theory

Rings and Fundamentals, Homomorphisms and Ideals, Quotient Rings, UFD, PID, Applications to Number Theory

Prof. Amit Kulshreshtha (AK),

IISER Mohali

4

Linear algebra and its applications

Sums of squares over fields, Fibonacci series and other similar iterative processes, Google’s ranking algorithm (PageRank), Multivariable calculus, Linear algebraic groups.

Prof. Anuradha Sharma (AS),

IIIT Delhi

7

Applications of Algebra to Coding Theory

Block codes, Hamming distance of a block code,  Sphere Packing bound, Singleton bound, Linear codes over finite fields, Dual codes, Hamming weight enumerator,  MacWilliams identity, Equivalence of Linear codes, Encoding and Decoding of linear codes, Cyclic codes, Generator polynomials, Generating idempotents, Decoding of cyclic codes, Burst-error-correcting codes

 References:

  1. Groups & Symmetry
    (a) M. A. Armstrong, Groups and Symmetry, Springer-Verlag, 1988.
    (b) Michael Artin, Algebra, (2nd Edition), Pearson, 2015.
    (c) Norman L. Biggs, Discrete Mathematics, (2nd Edition), OUP, 2003.
    (d) David S. Dummit and Richard M. Foote, Abstract Algebra, (3rd Edition), John Wiley and Sons, 2004.
    (e) Joseph A. Gallian, Contemporary Abstract Algebra, (8th Edition), Cengage Learning, 2013.

  2. Ring Theory with applications to Number Theory
    (a) Robert Ash, Basic Abstract Algebra: For Graduate Students and Advanced Undergraduates, Dover Publications, 2007. (Chapter 2).
    (b) V. V. Prasolov, Polynomials, Springer-Verlag, 2004. (Chapter 2, Section 2.1).

  3. Linear Algebra and its applications
    (a) Gilbert Strang, Linear Algebra and Its Applications, (4th Edition), Thompson Learning, 2006.
    (b) Sheldon Axler, Linear Algebra Done Right, (3rd Edition), Springer-Verlag, 2015.

  4. Applications of Algebra to Coding Theory
    (a) S. Ling and C. Xing, Coding Theory, Cambridge University Press, 2004
    (b) MacWilliams and Sloane, The Theory of error-correcting codes, North-Holland publishing company, 1983.
    (c) W. Cary Huffman and Vera Pless, Fundamental of error-correcting codes, Cambridge University Press, 2003.
    (d) Raymond Hill, A first course in Coding Theory, Oxford University Press, 1990.


Time Table

 Time-table (with names of speakers and tutors):

Day

Date

Lecture
1
(9.30
to
10.30)

Tea

Lecture
2
(11.00
to
12.00)

Lecture
3
(12.00
to
1.00)

Lunch

(1.00
to
2.20)

Lecture 4
(2.30
to
3.30)

Tea

Discussion

(4.00
to
5.00)

Snacks

(5.05
to
5:20)

 

 

 

 

 

 

 

 

 

 

 

Mon

06.05.2019

GV

 

GV

AK

 

AK

 

AK

 

Tue

07.05.2019

GV

 

GV

AK

 

AK

 

AK

 

Wed

08.05.2019

AS

 

AS

SL

 

SL

 

SL

 

Thu

09.05.2019

AS

 

SL

AS

 

SL

 

SL+AS

 

Fri

10.05.2019

SL

 

SL

AS

 

AS

 

AS

 

Sat

11.05.2019

SL

 

AS

GV

 

GV

 

GV

Valedictory

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