# TEW Algebra, Discrete Mathematics, Complex Analysis (2017)

Venue: |
The Institute of Mathematical Sciences, Chennai, Tamil Nadu |

Date: |
22nd, May 2017 to 27th, May 2017 |

Name |
K. N. Raghavan | K. Srinivas |

Mailing Address |
Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113. |
Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113. |

** **

**Algebra**: by** Sankaran Viswanath** (10.5 hours)

**Topics covered**

- Group actions on sets: orbits and stabilizers
- Burnside’s Lemma
- The class equation and applications
- Applications of group actions to Group Theory: the Sylow Theorems
- Applications of group actions to Combinatorics: counting necklace configurations, weighted version of Burnside’s Lemma, simple instances of Polya’s enumeration theorem.
- Application of group actions to elementary number theory: orbits for the action of the automorphism group of a cyclic group and an identity involving d(n) (the number of divisors of n ) and φ(n) (Euler’s Totient function).

**Complex Analysis**: by** Parameswaran Sankaran** (8.5 hours)

**Topics covered**

- Holomorphic functions, Cauchy-Riemann equations, power series.
- Local properties of holomorphic functions, singularities, meromorphic functions, Laurent series.
- Complex integration, residue calculus, Rouché’s theorem.
- Infinite product, Weierstrass’ theorem; growth of an entire function, Hadarmard’s theorem.

The emphasis was on explaining the concepts with the help of examples rather than on proving theorems.

**Complex Analysis**: by** K. Srinivas** (2.5 hours)

The Riemann zeta function was introduced. It was shown that

- The Riemann zeta function ζ(s), s = σ + it , is analytic in the half-plane σ > 1
- ζ(s) admits analytic continuation in σ > 0
- by means of functional equation, ζ(s) extends to a meromorphic function in the entire complex
- place with a simple pole at s = 1 with residue 1
- ζ(s) has real zeros at s = −2, −4, · · ·
- The following topics were briefly discussed:
- The complex zeros of ζ(s) , Riemann Hypothesis, and Hardy’s theorem.
- prime number theorem.

**Discrete Mathematics**: by** K. N. Raghavan **(11.5 hours)

Topics covered

- Colourings and exact colourings of graphs; the chromatic polynomial; acyclic orientations and statement of Stanley’s theorem that the chromatic polynomial evaluated at −1 gives, up to sign,the number of acyclic orientations.
- Definition of heaps of pieces, enumeration of heaps.
- Generating functions of heaps, the inversion lemma.
- proof of Stanley’s theorem using the inversion lemma.
- calculations with generating functions: Fibonacci sequence; Fibonacci polynomials; Dyck paths.
- Matching polynomials of graphs; statement about all their roots being real.

The emphasis was more on examples and calculations rather than on theory.

** Time Table**

Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
Saturday |

09:30 - 11:00 Discrete Mathematics K N Raghavan |
09:30 - 11:00 Complex Analysis P Sankaran |
09:30 - 11:00 Discrete Mathematics K N Raghavan |
09:30 - 11:00 Discrete Mathematics K N Raghavan |
09:30 - 11:00 Algebra S Viswanath |
09:30 - 11:00 Discrete Mathematics K N Raghavan |

Coffee/Tea | Coffee/Tea | Coffee/Tea | Coffee/Tea | Coffee/Tea | Coffee/Tea |

11:30 - 13:00 Algebra S Viswanath |
11:30 - 13:00 Discrete Mathematics K N Raghavan |
11:30 - 13:00 Complex Analysis P Sankaran |
11:30 - 13:00 Complex Analysis P Sankaran |
11:30 - 13:00 Algebra S Viswanath |
11:30 - 13:00 Algebra S Viswanath |

Lunch | Lunch | Lunch | Lunch | Lunch | Lunch |

14:00 - 15:00 Algebra S Viswanath |
14:00 - 15:00 Algebra S Viswanath | 14:00 - 15:00 Algebra S Viswanath | 14:00 - 15:00 Discrete Mathematics K N Raghavan | 14:00 - 15:00 Complex Analysis P Sankaran | 14:00 - 15:00 Complex Analysis K Srinivas |

Coffee/Tea | Coffee/Tea | Coffee/Tea | Coffee/Tea | Coffee/Tea | Coffee/Tea |

15:15 - 16:45 Tutorial Session: Discrete Mathematics K N Raghavan |
15:15 - 16:45 Tutorial Session: Algebra S Viswanath |
15:15 - 16:45 Tutorial Session: Complex Analysis P Sankaran |
15:15 - 16:45 Tutorial Session: Complex Analysis P Sankaran |
15:15 - 16:45 Tutorial Session: Discrete Mathematics K N Raghavan |
15:15 - 16:45 Tutorial Session: Complex Analysis K Srinivas |

Snacks | Snacks | Snacks | Snacks | Snacks | Snacks |

Actual Participants |

Sr.n |
SID |
Full Name |
Affiliation |
City |
State |

1 | 12068 | Mr. Mani Kandan K | GURU NANAK COLLEGE | CHENNAI | Tamil Nadu |

2 | 12112 | Mr. Hari Krishnan Thanushkodi | GURU NANAK COLLEGE (AUTONOMOUS) | CHENNAI | Tamil Nadu |

3 | 12201 | Mrs. N.Murugammal Narayanan Prabu | GURUNANAK COLLEGE | CHENNAI | Tamil Nadu |

4 | 12299 | Mr. Rajaji Ramasamy | Patrician College of Arts and Science | Chennai | Tamil Nadu |

5 | 12854 | Mr. Jayasankar Sivaraman | RKM Vivekananda College, Chennai 600004 | Chennai | Tamil Nadu |

6 | 12874 | Dr Maria Joseph | St. Joseph's College(Autonomous) | Trichy | |

7 | 13004 | Dr. Manikandan Mariappan | Bishop Heber College | Tiruchirappalli | Tamil Nadu |

8 | 13043 | Dr. Devipriya Ganeshan | Stella Maris College | Chennai | Tamil Nadu |

9 | 13053 | Ms Dhanalakshmi Arumugasamy | Stella Maris College | Chennai | Tamil Nadu |

10 | 13084 | Mr. Mahadev Ota | B S Abdur Rahman University | Chennai | Tamil Nadu |

11 | 13119 | Dr. S. Sarah Surya | Stella Maris College | Chennai | Tamil Nadu |

12 | 13125 | Dr Saravanakumar Dhanabal | Kalasalingam University | Krishnankoil | Tamil Nadu |

13 | 13138 | Mr. Christopher A | Don Bosco College (Arts and Science) | Karaikal | Pondicherry |

14 | 13158 | Mrs Prabhavathi Viswanathan Krishnan | Apollo arts &science college | Chennai | Tamil Nadu |

15 | 13162 | Mrs. Prabha Kumar | Ethiraj college for women, chennai 600 008 | chennai | Tamil Nadu |

16 | 13163 | Dr V.Vijaya Bharathi | National Institute of Technology | Trichy | Tamil Nadu |

17 | 13169 | Mrs. Archana Rajaraman | kumarmangalam solutions | CHENNAI | Tamil Nadu |

18 | 13177 | Mr Carmel Pushpa Raj J | St.Joseph's College | Tiruchirappalli | Tamil Nadu |

19 | 13182 | Mr Thirumal P | D G Vaishnav College | chenai. | Tamil Nadu |

20 | 13187 | Mr Bala Samuvel J. | Alpha Arts and Science College | Chennai | Tamil Nadu |

21 | 13190 | Mrs. Sujatha M | K.S.RANGASAMY COLLEGE OF ARTS AND SCIENCE(AUTONOMOUS) | THIRUCHENGODE | Tamil Nadu |

22 | 13203 | Mrs Subhashini A | Anna university | chennai | Tamil Nadu |

23 | 13220 | Mrs. Esakkiammal Esakki Sundaram | S.I.V.E.T. College | CHENNAI | Tamil Nadu |

24 | 13230 | Mrs. Ramya Murugesan | Chevalier T. Thomas Elizabeth College for Women | Chennai | Tamil Nadu |

25 | 13268 | Mrs Shanthi As | Stella Maris College | Chennai | Tamil Nadu |

26 | 13269 | Dr. K. Bagee Rathi | Aditanar college of Arts and Science | Thoothukudi | Tamil Nadu |

27 | 13297 | Dr Sadiq Basha | ANNA UNIVERSITY | CHENNAI | Tamil Nadu |

28 | 13299 | Dr Bharathi S | Bharathiar University PG Extension Centre | Erode | Tamil Nadu |

29 | 13313 | Prof Priya Dharshini | Dr.MGR Janaki college of arts and science for women | chennai | Tamil Nadu |

30 | 13321 | Ms Tamilselvi Selvaraj | DG VAISHNAVA COLLEGE | chennai | Tamil Nadu |

31 | 13322 | Mrs. Enitha Dorothy | Women's Christian College, Chennai. | Chennai | Tamil Nadu |

32 | 13325 | Mrs Anandha Selvam | THE AMERICAN COLLEGE | MADURAI | Tamil Nadu |

33 | 13329 | Mrs. Shanthi Muniasamy | Guru Nanak College | Chennai | Tamil Nadu |

34 | 13337 | Mr Gnanam Mahesh Kalakaoi | AGURCHAND MANMULL JAIN COLLEGE | Chennai | Tamil Nadu |

35 | 13343 | Ms. Logu S | M. G. R Janaki college of arts and science | Chennai | Tamil Nadu |

36 | 12203 | Mr. Thilakraj.E Elumalai.V | GURUNANAK COLLEGE(AUTONOMOUS) | CHENNAI | Tamil Nadu |

37 | 12983 | Ms Krishnaveni Cheruvu | MARIS STELLA COLLEGE | VIJAYAWADA | Andhra Pradesh |

38 | 13266 | Dr. Uma S | Thiruvalluvar University Constituent Arts and Science College, Kallakurichi - 606213 Villupuram District |
Kallakurichi, Villupuram | Tamil Nadu |

39 | 13301 | Dr Kalamani D | Bharathiar University PG extension Centre, Erode | erode | Tamil Nadu |

40 | K.Karunagaran | Bharat university | Chennai | ||

41 | NANDAKUMAR VELU | Bharat university | Chennai | ||

42 | P.BHATHMANABAN | Bharat university | Chennai | ||

43 | Usha Sangale | SRTM University | Nanded, Maharashtra | ||

44 | Sharanya C | Bhaktavatsalam memorial college for women | Chennai | ||

45 | A Murugan | Anna University | Chennai | ||

46 | G. Jagadeesh | Anna University | Chennai | ||

47 | Divya S. Raj | DGVC | Chennai | ||

48 | V. Pusparaj | MOHAMED SATHAK COLLEGE OF ARTS AND SCIENCE | Chennai | ||

49 | A. Marimuthu | MOHAMED SATHAK COLLEGE OF ARTS AND SCIENCE | Chennai | ||

50 | P. Gangeshwari | Kumararani Meena Muthiah College of Arts and Sciences | Chennai | ||

51 | S. Elumalai | RMKV College | Chennai | ||

52 | R. Siluvai Dasan | Tagore Group of Institutions | Chennai | ||

53 | Hanna Bright | Ethiraj College for Women | Chennai | ||

54 | 13301 | Dr Kalamani D | Bharathiar University PG extension Centre, Erode | erode | Tamil Nadu |

55 | 13143 | Mr G Jagadeesh | ANNA UNIVERSITY-CEG CAMPUS | GUINDY | Tamil Nadu |

56 | 13179 | Mr. Marimuthu A | MOHAMED SATHAK COLLEGE OF ARTS AND SCIENCE | CHENNAI | Tamil Nadu |

57 | 13275 | Mr. Pusparaj V | MOHAMED SATHAK COLLEGE OF ARTS AND SCIENCE | CHENNAI | Tamil Nadu |

58 | 13333 | Ms. Saranya.C Saranya.C R.Chandra Mohan | Bhaktavatsalam memorial college for women | chennai | Tamil Nadu |

59 | 13346 | Ms Divya S Raj | DG VAISHNAV COLLEGE | CHENNAI | Tamil Nadu |

**How to reach**

The Chennai neé Madras, and to IMSc, the Institute of Mathematical Sciences. Here is some information on getting to the Institute. Address and phone numbers are

THE INSTITUTE OF MATHEMATICAL SCIENCES

C I T CAMPUS, TARAMANI, CHENNAI 600 113

Telephone Numbers

Here are some maps to give you a rough idea of the layout of Madras and the area around the Institute.

- A simplified schematic map of Madras, showing the major roads and Hotels and marking the area in which the CIT Campus is located.
- A postscript version of this figure if you want to download it to print at your end.
- A detailed Madras map.
- A gzipped postscript version of this figure
- A schenatic map of the location of IMSc within CIT Campus and its environs
- Detailed map of the location of IMSc within CIT Campus and other landmarks in the Greater Adyar area.

The Institute of Mathematical Sciences is located in a campus-like area variously known as the ``CIT Campus'' or the ``CPT Campus''. Many people in Madras may not have heard of the Institute, but almost everyone would have heard of the M.G.R. Film City and Tidel Park which are further down in the same campus area. IMSc has two complexes. One houses the main Institute building, the library building and the new building. The other has the students' hostel and Guest House. These two complexes are across the road from each other.