AIS - Advanced Linear Algebra (2020)

Speakers and Syllabus


Name of the Speaker with affiliation No. of Lect. Detailed Syllabus
R. B. Bapat
ISI Delhi
4 Title: Topics in Nonnegative Matrices

Syllabus/Outline:
Review of eigenvalues and eigenvectors, Nonnegative Matrices, Graph associated with a matrix, Brouwer's Fixed Point Theorem, Perron's theorem for positive matrices, strongly connected graphs,irreducibility, Perron-Frobenius theorem for irreducible matrices, primitive, cyclic,reducible matrices (statements and examples), Inequalities for Perron root. Matrices over the max algebra.

Apoorva Khare
IISc
3 Title: Totally nonnegative (TN) and totally positive (TP) matrices

Syllabus/Outline:
(a) Definitions, examples. (b) TP is dense in TN. (c) Eigenvalues of square TP/TN matrices are positive/nonnegative.

Parts (b) and (c) should remind one of the exact same statements for positive (semi)definite matrices -- the analogue of (c) is Sylvester's criterion. The proofs for TN/TP matrices will including my covering Perron's theorem for matrices with positive entries, Kronecker's theorem for compound matrices, and a black-box result on the continuity of the roots of complex polynomials as functions of the coefficients.

Jugal Verma
IIT Bombay
6 Title: Solutions of real and complex polynomial equations via eigenvalues

Syllabus/Outline:
Lecture 1+2: Hilbert's Nullstellensatz and its consequences using linear algebra
Lecture 3: Construction of the complex solutions of polynomial equations using eigenvalues
Lecture 4: Bilinear forms and Sylvester's inertia Theorem for real symmetric matrices.
Lecture 5 + 6: Hermite's Theorem for counting real solutions of polynomial equations using trace forms.

Pre-requisites:
Basic algebra and linear algebra at the level of Artin's Algebra
[Note: Detailed lecture notes will be provided]

Gautam Bharali
IISc
4 Title: The role of linear algebra in complex analysis

Syllabus/Outline:
Day 1: The meaning of the Cauchy-Riemann condition
Days 2 and 3: Almost complex structures
Day 4: Integrable complex structures OR the Pick interpolation theorem, depending on the audience's mathematical inclination

Praneeth Netrapalli
Microsoft Research, Bangalore
4 Title: Efficient computation of top singular vectors/principal components

Syllabus/Outline:
Lecture 1: Power method and Lanczos method for computing top eigenvectors/singular vectors.
Lecture 2: Alternating minimization for low rank matrix completion
Lecture 3: Streaming PCA

Sreedhar Inamdar 4 Title: Advanced linear algebra

Syllabus/Outline:
Cholesky decomposition, Singular value decomposition, Spectral theorem, Jordan canonical form, positive matrices, Positive definite functions, geometry of positive matrices.
Manish Kumar
ISI Bangalore
4 Title: Representation theory

Syllabus/Outline:

TBA
Rajesh Sharma
Himachal Pradesh University
2 Title: Numerical Range:

Syllabus/Outline:

Properties of numerical range of matrices, Convexity of the numerical range, Toeplitz-Hausdorff theorem, Consequences of convexity of numerical range, Configuration of numerical range of two-by-two matrices, Circulatory of numerical range of three-by-three matrices, Geometry of the numerical range of matrices, boundary points, sharp points and related results, Positive unital linear maps and bounds on the diameter of numerical range.
Gadadhar Misra
IISc
3 Title: Curvature inequalities

Syllabus/Outline: TBA

Manjunath Krishnapur
IISc
3 Title: Curvature inequalities

Syllabus/Outline:
Graph Laplacian and the relationships between spectral properties of the Laplacian and the properties of the graph. This will include counting spanning trees of a graph and Cayley's theorem. Cheeger's inequality on graphs. Nodal domain theorem. Resistance metric on a graph.

 

Tutorial Assistants:

Sr.n Name Affiliation
1 TBA TBA
2 TBA TBA
3 TBA TBA

Time Table

Names of speakers and course associates/tutors

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)
5.05-5.30
    speaker   speaker L
U
N
C
H
speaker + tutors   speaker + tutors S
N
A
C
K
S
Mon 11 May SI   MK SI +   MK +
Tues 12 May SI   MK SI +   MK +
Wed 13 May SI   MK SI +   MK +
Thu 14 May SI   MK SI +   MK +
Fri 15 May RB   GB RB+   GB+
Sat 16 May RB   GB RB+   GB+
 SUNDAY : OFF
Mon 18 May RB   GB L
U
N
C
H

RB+   GB+ S
N
A
C
K
S
Tues 19 May RB   GB RB+   GB+
Wed 20 May KR   RS KR+   RS+
Thu 21 May KR   AK KR+   AK+
Fri 22 May KR   AK KR+   AK+
Sat 23 May RS   AK RS+   AK+
SUNDAY : OFF
Mon 25 May PN   JV L
U
N
C
H

PN+   JV+ S
N
A
C
K
S
Tues 26 May PN   JV PN+   JV+
Wed 27 May PN   JV PN+   JV+
Thu 28 May GM   JV GM+   JV+
Fri 29 May GM   JV GM+   JV+
Sat 30 May GM   JV GM+   JV+
  • AK: Apoorva Khare
  • MK: Manish Kumar
  • GM: Gadadhar Misra
  • RS: Rajesh Sharma
  • PN: Praneeth Netrapalli
  • GB: Goutam Bharali
  • KR: Manjunath Krishnapur
  • SI: Sreedhar Inamdar
  • JV: Jugal Verma
  • RB: R B Bapat

 

 

 

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