This workshop aims to cover some of the topics in Linear Algebra which are beyond the scope of a standard M.Sc. curriculum. The first week will be devoted to Normal operators, Bilinear forms, together with some elementary Linear Algebraic techniques in Combinatorics. The second week will be devoted to Matrix groups andthe Structure of finitely generated modules over a Principal Ideal Domain leading to Canonical Forms of Linear operators. More advanced Linear Algebraic methods in Combinatoricswill be discussed towards the end. Young faculty members and young researchers are encouraged to apply.
SPEAKERS AND SYLLABUS
|
Name and Affiliation |
Week & Hours |
Detailed Syllabus |
|
Speaker 1. K.N. Raghavan (KNR) IMsc, Chennai |
Week 1: Lecture 8 hours Tutorial 6 hours |
Module I (Bilinear Forms; the Spectral theorem for Normal operators). Real and complex inner product spaces. Gram-Schmidt orthogonalization. The Riesz representation theorem. The Adjoint of a linear operator. Orthogonal projections, orthogonal resolutions of the identity. Quadratic maps. Symmetric forms, orthogonal bases. Symmetric forms over ordered fields. Hermitian forms. The spectral theorems in Hermitian and Symmetric cases. Alternating forms. The Pfaffian. Witt’s theorem. The Witt group. |
|
Speaker 2. Indranath Sengupta (ISG) IIT Gandhinagar |
Week 1: Lecture 3 hours Tutorial 2 hours
|
Module I (Eigenvalues and eigenvectors). Linear operators, eigenvalues and eigenvectors, geometric and algebraic multiplicities. Triangularizability and Schur’s lemma, diagonalizable operators. |
|
Speaker 3. J.K. Verma (JKV) IIT Bombay |
Week 1: Lecture 6 hours Tutorial 4 hours |
Module II (Applications of the Spectral theorem). Normal operators. Spectral theorem for normal operators (statement only). Self-adjoint operators, unitary operators and Isometries. The polar decomposition of an operator. Singular Value Decomposition. Differential Equations. |
|
Speaker 4. Ananthnarayan H. (AH) IIT Bombay |
Week 2: Lecture 8 hours Tutorial 5 hours |
Module III (Modules). Basic properties. Free and Noetherian modules. The Hilbert basis theorem. Modules over a PID: annihilators and orders, cyclic modules, the decomposition of cyclic modules, free modules over a PID, the primary cyclic decomposition. The structure of a linear operator: the characteristic polynomial, the Cayley-Hamilton theorem, Jordan Canonical Form, the Rational Canonical form. |
|
Speaker 5. Ritumoni Sharma (RS) IIT Delhi |
Week 2: Lecture 6 hours Tutorial 4 hours |
Module IV (Matrix Groups). Matrix groups: definition and examples, examples of compact linear groups, examples of connected linear groups, the polar decomposition for the special linear groups, one-parameter subgroups. |
|
Speaker 6. Neeldhara Mishra (NM) IIT Gandhinagar |
Week 1: Lecture 1 hour Week 2: Lecture 4 hours Tutorial 3 hours |
Module V (Combinatorics). Arguments based on Linear Independence: Elementary examples including Oddtown (the linear bound), Eventown (the bound; the Graham Pollak Theorem, Nonuniform Fisher Inequality, Introduction to VC-dimension and a proof of the Sauer-Shelah lemma. An overview of families those are extremal with respect to VC-dimension. Polynomial Space Arguments: Applications in Extremal Set Theory. The theorems of Bollobás (uniform, non-uniform and skew versions). Applications in Geometry. Unit Distance Graphs and a counterexample for Borsuk's Conjecture. Tensor Product Methods: Setting up the machinery. An introduction to wedge products. The theorems of Bollobás revisited. Bollobás' theorem for subspaces (due to Furedi). |
Texts and References
1. Artin, M. Algebra. 2nd Edition. Prentice Hall India. 2011.
2. Babai, László and Frankl, Péter. Linear Algebra Methods in Combinatorics with Applications in Geometry and Computer Science. Unpublished. 1995.
3. Hall, Brian C. Lie Groups, Lie Algebras, and Representations. Springer. 2004.
4. Jacobson, N. Basic Algebra. Volume I. Edition 2. Dover Publications Inc. NY. 2009.
5. Lang, S. Algebra. 3rd Edition. Addison-Wesley. 1993.
6. Matoušek, Jiří. Thirty-three miniatures: Mathematical and Algorithmic Applications of Linear Algebra. Student Mathematical Library. American Mathematical Society. 2010.
7. Roman, Steven. Advanced Linear Algebra. 3rd Edition. Springer. 2011.
8. Stasys Jukna. Extremal Combinatorics - With Applications in Computer Science. Texts in Theoretical Computer Science. An {EATCS} Series. Springer. 2011.
TUTORS:
|
S. No. |
Name |
Affiliation |
|
1 |
BIBEKANANDA MAJI |
RESEARCH ASSOCIATE, IIT GANDHINAGAR, PALAJ, GANDHINAGAR 382355, GUJARAT. |
|
2 |
JOYDIP SAHA |
RESEARCH ASSOCIATE, IIT GANDHINAGAR, PALAJ, GANDHINAGAR 382355, GUJARAT. |
TIME TABLE
Venue: Block 6; Room 202
Week 1:
|
July 10 Monday |
July 11 Tuesday |
July 12 Wednesday |
July 13 Thursday |
July 14 Friday |
July 15 Saturday |
|
Inauguration (9:15-9:45) |
KNR (9:30 – 11:00) |
KNR (9:30 – 11:00) |
ISG (9:30 – 10:30) |
ISG (9:30 – 10:30) |
NM (9:30 – 10:30) |
|
Tea Break (9:45-10:00) |
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|
KNR (10:00 – 11:30) |
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|
Tea Break (11:30 – 12:00) |
Tea Break (11:00 – 11:30) |
Tea Break (11:00 – 11:30) |
Tea Break (10:30 – 11:00) |
Tea Break (10:30 – 11:00) |
Tea Break (10:30 – 11:00) |
|
ISG (12:00 – 13:30) |
JKV (11:30 – 13:00) |
JKV (11:30 – 13:00) |
JKV (11:00 – 13:00) |
KNR (11:00 – 13:00) |
KNR (11:00 – 13:00) |
|
Lunch Break (13:30 – 15:00) |
Lunch Break (13:00 – 14:30) |
Lunch Break (13:00 – 14:30) |
Lunch Break (13:00 – 14:30) |
Lunch Break (13:00 – 14:30) |
Lunch Break (13:00 – 14:30) |
|
JKV (15:00 – 17:00) |
Tutorial (JKV+Tutors) (14:30 – 16:30) |
Tutorial (KNR +Tutors) (14:30 – 16:30) |
Tutorial (JKV+Tutors) (14:30 – 16:30) |
Tutorial (KNR +Tutors) (14:30 – 16:30) |
Tutorial (KNR+Tutors) (14:30 – 16:30)
|
|
Tea & Snacks |
Tea & Snacks |
Tea & Snacks |
Tea & Snacks |
Tea & Snacks |
Tea & Snacks |
Week 2:
|
July 17 Monday |
July 18 Tuesday |
July 19 Wednesday |
July 20 Thursday |
July 21 Friday |
July 22 Saturday |
|
NM (9:30 – 11:00) |
NM (9:30 – 11:00) |
AH (9:30 – 11:00) |
AH (9:30 – 11:00) |
NM (9:30 – 10:30) |
AH (9:30 – 10:30) |
|
Tea Break (11:00 – 11:30) |
Tea Break (11:00 – 11:30) |
Tea Break (11:00 – 11:30) |
Tea Break (11:00 – 11:30) |
Tea Break (10:30 – 11:00) |
Tea Break (10:30 – 11:00) |
|
RS (11:30 – 13:00) |
RS (11:30 – 13:00) |
RS (11:30 – 13:00) |
RS (11:30 – 13:00) |
AH (11:00 – 13:00) |
AH (11:00 – 13:00) |
|
Lunch (13:00 – 14:30) |
Lunch (13:00 – 14:30) |
Lunch (13:00 – 14:30) |
Lunch (13:00 – 14:30) |
Lunch (13:00 – 14:30) |
Lunch (13:00 – 14:30) |
|
Tutorial (RS +Tutors) (14:30 – 16:30) |
Tutorial (NM+Tutors) (14:30 – 16:30) |
Tutorial (RS +Tutors) (14:30 – 16:30) |
Tutorial (AH+Tutors) (14:30 – 16:30) |
Tutorial (NM+Tutors) (14:30 – 15:30) |
Tutorial (AH+Tutors) (14:30 – 16:30) |
|
Tutorial (AH+Tutors) (15:30 – 16:30) |
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|
Tea & Snacks |
Tea & Snacks |
Tea & Snacks |
Tea & Snacks |
Tea & Snacks |
Tea & Snacks |