AIS - Advanced Topics in Finite Fields (2023)
Speakers and Syllabus
|
Name of the Speaker with affiliation |
No. of Lectures |
Detailed Syllabus |
|
Sartaj Ul Hasan |
6x1.5hrs |
The Structure of Finite Fields: Field extensions, Characterizations of finite fields, Roots of irreducible polynomials, Traces, Norms, Bases, Cyclotomic polynomials, Representation of elements of finite fields. |
|
Amritanshu Prasad
|
6x1.5hrs |
Counting subspaces and matrices by rank, similarity classes, JoRKan canonical form over finite field, similarity class type, centralizer caRKinality, counting invariant subspaces. |
|
Mrinmoy Datta |
6x1.5hrs |
Order of polynomials, primitive polynomials, irreducible polynomials and their construction, linearized polynomials, permutation polynomials, Hermite and AGW criteria for permutation polynomials, special types of permutation polynomials. |
|
Ayineedi Venkateswarlu |
6x1.5hrs |
Linear recurring sequences, Linear Feedback Shift Registers (LFSRs), Periodicity, Characteristic polynomial, Minimal polynomial, Generating functions, Maximal period sequences, Trace Representation, Golomb's randomness postulates, Linear complexity and related measures, Berlekamp-Massey algorithm, LFSR based cipher designs, Coding-theoretic connections, Modes of implementation of LFSRs, WoRK-oriented LFSRs and multi-sequences |
|
Sudhir R. Ghorpade |
6x1.5hrs |
Equations and Varieties over Finite Fields: Elementary bounds, Theorems of Konig-Rados, Warning and Chevalley, Elementary estimates on the number of zeros of multivariate polynomial over finite fields, The case of system of multivariate polynomials of bounded degree, Variants of Nullstellensatz for algebraic varieties over finite fields, Outline of Weil bounds for the number of points of algebraic varieties over finite fields, and related results. |
|
K Srinivas |
6x1.5hrs |
Characters, Gauss sums, Jacobi sums, and exponential sums over finite fields. |
References
1. Lidl, R. and Niederreiter, H., Finite Fields, Cambridge University Press, 1997.
2. Mullen, G. L. and Panario, D., Handbook of Finite Fields, CRC Press, 2013.
(Lecture notes from the speakers, if available)
Time Table
|
Day |
Date |
Lecture 1 (9:30–11:00) |
Tea (11:05 –11:25) |
Lecture 2 (11:30–1:00) |
Lunch (1:05–2:25) |
Tutorial (2:30–3:30) |
Tea (3:35-3:55) |
Tutorial (4:00-5:00) |
Snacks (5:05-5:30) |
|
|
|
speaker |
|
speaker |
|
speaker+tutors |
|
speaker + tutors |
|
|
Mon |
10 |
SH |
|
AP |
|
SH+KG+RK |
|
AP+SR+TB |
|
|
Tues |
11 |
SH |
|
AP |
|
SH+KG+RK |
|
AP+SR+TB |
|
|
Wed |
12 |
SH |
|
AP |
|
SH+KG+RK |
|
AP+SR+TB |
|
|
Thu |
13 |
SH |
|
AP |
|
SH+KG+RK |
|
AP+SR+TB |
|
|
Fri |
14 |
SH |
|
AP |
|
SH+KG+RK |
|
AP+SR+TB |
|
|
Sat |
15 |
SH |
|
AP |
|
SH+KG+RK |
|
AP+SR+TB |
|
|
SUNDAY OFF |
|||||||||
|
Mon |
17 |
MD |
|
AV |
|
MD+SM+HK |
|
AV+DA+AA |
|
|
Tues |
18 |
MD |
|
AV |
|
MD+SM+HK |
|
AV+DA+AA |
|
|
Wed |
19 |
MD |
|
AV |
|
MD+SM+HK |
|
AV+DA+AA |
|
|
Thu |
20 |
MD |
|
AV |
|
MD+SM+HK |
|
AV+DA+AA |
|
|
Fri |
21 |
MD |
|
AV |
|
MD+SM+HK |
|
AV+DA+AA |
|
|
Sat |
22 |
MD |
|
AV |
|
MD+SM+HK |
|
AV+DA+AA |
|
|
SUNDAY OFF |
|||||||||
|
Mon |
24 |
SG |
|
KS |
|
SG+RL+SD |
|
KS+MS+SV |
|
|
Tues |
25 |
SG |
|
KS |
|
SG+RL+SD |
|
KS+MS+SV |
|
|
Wed |
26 |
SG |
|
KS |
|
SG+RL+SD |
|
KS+MS+SV |
|
|
Thu |
27 |
SG |
|
KS |
|
SG+RL+SD |
|
KS+MS+SV |
|
|
Fri |
28 |
SG |
|
KS |
|
SG+RL+SD |
|
KS+MS+SV |
|
|
Sat |
29 |
SG |
|
KS |
|
SG+RL+SD |
|
KS+MS+SV |
|
Tutorial Assistants:
|
S. No. |
Name |
Affiliation |
|
1 |
Divya Aggarwal |
IIIT Delhi |
|
2 |
Akansha Arora |
IIIT Delhi |
|
3 |
Suhas Rao |
IMSc Chennai |
|
4 |
Subrata Manna |
IIT Hyderabad |
|
5 |
M. Subramani |
IIITDM Kancheepuram |
|
6 |
Rati Ludhani |
IIT Bombay |
|
7 |
Sakshi Dang |
IIT Bombay |
|
8 |
Kirpa Garg |
IIT Jammu |
|
9 |
Hridesh Kumar |
IIT Jammu |
|
10 |
Ramandeep Kaur |
IIT Jammu |
|
11 |
S Velmurugan |
IMSc Chennai |
|
12 |
Tirtharaj Basu |
IMSc Chennai |
Full forms for the abbreviations of speakers and tutors:
| AP: Amritanshu Prasad | SR: Suhas Rao | TB: Tirtharaj Basu |
| SH: Sartaj Ul Hasan | KG: Kirpa Garg | RK: Ramandeep Kaur |
| MD: Mrinmoy Datta | SM: Subrata Manna | HK: Hridesh Kumar |
| AV: Ayineedi Venkateswarlu | DA: Divya Aggarwal | AA: Akansha Arora |
| SG: Sudhir R. Ghorpade | RL: Rati Ludhani | SD: Sakshi Dang |
| KS: K. Srinvas | MS: M. Subramani | SV: S Velmurugan |