AIS - Advanced Linear Algebra (2020)

Speakers and Syllabus


 

Name of the Speaker with affiliation

No. of Lectures

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

3

Title: Complex solutions to 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

 Pre-requisites: 

Basic algebra and linear algebra at the level of Artin's Algebra

[Note: Detailed lecture notes will be provided]

 

Dilip Patil
IISc

3

Title: Trace form and Applications

Lecture 1: Bilinear forms and Sylvester's inertia Theorem for real symmetric matrices.
Lecture 2&3: Hermite's Theorem for counting real solutions of polynomial equations using trace forms.

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

3

Title: Efficient computation of top singular vectors/principal components

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

 

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

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

(3.35-3.55)

Tutorial

(4.00-5.00)

Snacks

5.05-5.30

 

 

(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

 

 

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