# AIS - Advanced Linear Algebra (2020)

## Speakers and Syllabus

 Name of the Speaker with affiliation No. of Lectures Detailed Syllabus R. B. BapatISI 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 KhareIISc 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 VermaIIT Bombay 3 Title: Complex solutions to polynomial equations via eigenvalues.  Syllabus/Outline: Lecture 1+2: Hilbert's Nullstellensatz and its consequences using linear algebraLecture 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 PatilIISc 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 BharaliIISc 4 Title: The role of linear algebra in complex analysis Syllabus/Outline: Day 1: The meaning of the Cauchy-Riemann conditionDays 2 and 3: Almost complex structuresDay 4: Integrable complex structures OR the Pick interpolation theorem, depending on the audience's mathematical inclination Praneeth NetrapalliMicrosoft 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 completionLecture 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 KumarISI 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

File Attachments: