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

B Sury
ISI 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
Complete reducibility, Schurs' lemma, character theory. Time permitting: Induced representation and Frobenius reciprocity.

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.

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

09 May

Mandeep

Manish

Mandeep

Manish
&
Arunava

Tues

10 May

Mandeep

Manish

Mandeep

Manish
&
Arunava

Wed

11 May

Bharali

Manish

Bharali & Gopal

Manish
&
Arunava

Thu

12 May

Rajesh

Manish

Rajesh

Manish
&
Arunava

Fri

13 May

Bapat

Bharali

Bapat & Gopinath

Bharali
&
Gopal

Sat

14 May

Bapat

Mandeep

Bapat & Gopinath

Mandeep

SUNDAY : OFF

Mon

16 May

Bapat

Bharali

Bapat & Gopinath

Bharali
&
Gopal

Tues

17 May

Bapat

Bharali

Bapat & Gopinath

Bharali
&
Gopal

Wed

18 May

Manjunath

Khare

Manjunath & Jnaneshwar

Khare
&
Projesh

Thu

19 May

Manjunath

Khare

Manjunath & Jnaneshwar

Khare
&
Projesh

Fri

20 May

Manjunath

Rajesh

Manjunath & Jnaneshwar

Rajesh

Sat

21 May

Rajesh

Khare

Rajesh

Khare
&
Projesh

SUNDAY : OFF

Mon

23 May

Sury

Patil

Sury

Patil
&
Parnashree

Tues

24 May

Sury

Patil

Sury

Patil
&
Parnashree

Wed

25 May

Sury

        Patil

Sury

Patil
&
Parnashree

Thu

26 May

Misra

Verma

Misra
&
Pradhan

Verma
&
Parnashree

Fri

27 May

Misra

Verma

Misra
&
Pradhan

Verma
&
Parnashree

Sat

28 May

Misra

Verma

Misra
&
Pradhan

Verma
&
Parnashree

S. No.

Name

Affiliation

1

Gopinath Sahoo

Bennett University

2

Projesh Nath Choudhury

IISc

3

Parnashree Ghosh

ISI Kolkata

4

Deepak Pradhan

ISI Bangalore

5

Arunava Mandal

ISI Bangalore

6

Gopal Datt

BBAU, Lucknow

7

B S Jnaneshwar

IISc