AIS  Advanced Linear Algebra (2020)
Speakers and Syllabus
Name of the Speaker with affiliation 
No. of Lectures 
Detailed Syllabus 
R. B. Bapat 
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, PerronFrobenius theorem for irreducible matrices, primitive, cyclic,reducible matrices (statements and examples), Inequalities for Perron root. Matrices over the max algebra. 
Apoorva Khare 
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 blackbox result on the continuity of the roots of complex polynomials as functions of the coefficients. 
Jugal Verma 
3 
Title: Complex solutions to polynomial equations via eigenvalues. Syllabus/Outline: Lecture 1+2: Hilbert's Nullstellensatz and its consequences using linear algebra Prerequisites: Basic algebra and linear algebra at the level of Artin's Algebra [Note: Detailed lecture notes will be provided]

Dilip Patil 
3 
Title: Trace form and Applications Lecture 1: Bilinear forms and Sylvester's inertia Theorem for real symmetric matrices. 
Gautam Bharali 
4 
Title: The role of linear algebra in complex analysis Syllabus/Outline: Day 1: The meaning of the CauchyRiemann condition 
Praneeth Netrapalli 
3 
Title: Efficient computation of top singular vectors/principal components Lecture 1: Power method and Lanczos method for computing top eigenvectors/singular vectors.

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 
4 
Title: Representation theory

Rajesh Sharma Himachal Pradesh University 
2 
Title: Numerical range Syllabus/Outline: Properties of numerical range of matrices, Convexity of the numerical range, ToeplitzHausdorff theorem, Consequences of convexity of numerical range, Configuration of numerical range of twobytwo matrices, Circulatory of numerical range of threebythree 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 
Manjunath Krishnapur IISc 
3 
Title: On graphs and matrices 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. 
B. Sury ISI Bangalore 
3 
Title: Linear groups Abstract: We introduce and study various groups of matrices. Orthogonal, Unitary and Symplectic groups are discussed and the exponential mapping on matrix groups will be studied. Various types of decompositions of these matrix groups will be described. Finally, symmetry groups of solids are analyzed. 
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 (3.353.55) 
Tutorial (4.005.00) 
Snacks 5.055.30 


(name of the speaker) 

(name of the speaker) 

(name of the speaker + tutors) 

(name of the speaker + tutors) 

Mon 
11 May 
Inamdar 

Manish 

PM 

AM 

Tues 
12 May 
Inamdar 

Manish 

PM 

AM 

Wed 
13 May 
Inamdar 

Manish 

PM 

AM 

Thu 
14 May 
Inamdar 

Manish 

PM 

AM 

Fri 
15 May 
Bapat 

Bharali 

GS 

GD 

Sat 
16 May 
Bapat 

Bharali 

GS 

GD 

SUNDAY : OFF 

Mon 
18 May 
Bapat 

Bharali 

GS 

GD 

Tues 
19 May 
Bapat 

Bharali 

GS 

GD 

Wed 
20 May 
Manjunath 

Khare 

BG 

PV 

Thu 
21 May 
Manjunath 

Khare 

BG 

PV 

Fri 
22 May 
Manjunath 

Khare 

BG 

PV 

Sat 
23 May 
Rajesh 

Rajesh 

SB 

SB 

SUNDAY : OFF 

Mon 
25 May 
Verma 

Patil 

SM 

SM 

Tues 
26 May 
Verma 

Patil 

SM 

SM 

Wed 
27 May 
Verma 

Patil 

SM 

SM 

Thu 
28 May 
Misra 

Praneeth 

SK 

Sury 

Fri 
29 May 
Misra 

Praneeth 

SK 

Sury 

Sat 
30 May 
Misra 

Praneeth 

SK 

Sury 
Speakers: Apoorva Khare, Dilip Patil, Manish Kumar, Gadadhar Misra, Rajesh Sharma, Praneeth Netrapalli, Gautam Bharali, Manjunath Krishnapur, Sreedhar Inamdar, Jugal Verma, R B Bapat. B. Sury.
Tutorial Assistants:
S. No. 
Name 
Affiliation 
1 
Gopinath Sahoo (GS) 
ISI Delhi 
2 
Prateek Kumar Viswakarma (PV) 
IISc 
3 
Satyendra Kumar Mishra (SM) 
ISI Bangalore 
4 
Surjit Kumar (SK) 
IISc 
5 
Arunava Mandal (AM) 
ISI Bangalore 
6 
Pratik Mehta (PM) 
ISI Bangalore 
7 
Gopal Datt (GD) 
IISc 
8 
B S Jnaneshwar (BG) 
IISc 
9 
Snehasish Bose (SB) 
ISI Bangalore 