Name of the Speakers with their affiliation.

No. of
Lectures

Detailed Syllabus

Professor Rajendra Bhatia

Department of Mathematics Ashoka University, Haryana

6 hrs

Use of Fourier Analysis to solve some problems of matrix theory: Computation of norms of some infinite matrices like the Hilbert matrix, Cesaro matrix, etc. as well as of the Toeplitz and Hankel matrices. Some generalizations of these that are useful in number theory. Solutions of the Sylvester Equation using Fourier analysis. Positive definite sequences and functions, and their use in deriving matrix inequalities.

Professor Peter Semrl

Institute of Mathematics, Physics and Mechanics, Slovenia,

3 hrs

From Loewner's characterization of operator monotone functions to fundamental theorem of chronogeometry:
 
Local order isomorphisms of matrix and operator domains will be discussed. A connection with Loewner's theorem and the fundamental theorem of chronogeometry will be explained. The first one characterizes operator monotone functions while the second one describes the general form of bijective preservers of light-likeness on the classical Minkowski space. We will present an improvement of the fundamental theorem of chronogeometry.

Professor Shreemayee Bora

Department of Mathematics IIT, Guwahati

9 hrs

LU factorization:
Gaussian Elimination as an LU factorization of a matrix; Existence and uniqueness of LU factorization; computing LU factors; variants of LU factorization; Cholesky factorization of positive definite matrices: Positive definite matrices and their properties; existence and uniqueness of Cholesky factorization of positive definite matrices and its relationship with LU factorization; computing Cholesky factors;

QR factorizations of matrices: Existence and uniqueness of QR factorizations of matrices; computing QR factors; applications of QR factorizations; relationship of QR factorizations with other factorizations;

Polar decompositions of matrices: Existence of polar decompositions; relationship with other decompositions; their applications;

Singularvaluedecompositionsofmatrices:Existenceanduniqueness, properties and applications. Relationship with other decompositions. Decompositions related to the matrix eigenvalue problem:Diagonalizability; Jordan canonical form (in brief);Schur's

Theorem (real and complex versions); spectral theorems; a brief introduction to computing eigenvalues and eigenvectors for dense matrices.

Dr. Apoorva Khare

Associate Professor
Department of Mathematics
IISC, Bangalore

4.5 hrs

Entrywise positivity preservers: Theory of functions preserving matrix
positivity. Positivity and total positivity of matrices, classical results of
Schoenberg and Karlin; inertia of matrices; classical results on
entrywise powers as well as some recent developments on positivity

Dr. Tanvi Jain

Associate Professor
Theoretical Statistics and
Mathematics Unit,
Indian Statistical Institute,
New Delhi.

4.5 hrs

Day

Date

Lecture 1

(9.30–11.00)

Tea

(11.05–11.25)

Lecture 2

(11.30–1.00)

Lunch

(1.05–1.55)

Lecture 3

(2.00–3.30)

Tea

(3.35-3.55)

Discussion

(4.00-5.00)

Mon

18/12

SB

RB

TJ

JS/YK

Tues

19/12

SB

RB

TJ

JS/YK

Wed

20/12

SB

RB

TJ

JS/YK

Thu

21/12

SB

PS

AK

JS/YK

Fri

22/12

PS

SB

AK

JS/YK

Sat

23/12

AK

SB

RB

JS/YK