Mathematical Toolkit - Autumn 2018


TTIC 31150/CMSC 31150

T Th 9:30 - 10:50, TTIC 526

Discussion: W 4-5 pm, TTIC 526

Office Hours: T 11-12, TTIC 534 (Madhur's office)

Instructor: Madhur Tulsiani

TA: Omar Montasser




 

The course is aimed at first-year graduate students and advanced undergraduates. The goal of the course is to collect and present important mathematical tools used in different areas of computer science. The course will mostly focus on linear algebra and probability. We intend to cover the following topics and examples:

  • Abstract linear algebra: vector spaces, linear transformations, Hilbert spaces, inner product, Gram-Schmidt orthogonalization, eigenvalues and eigenvectors, SVD.
  • Least squares, iterative solvers for systems of linear equations, spectral graph theory, perturbation and stability of eigenvalues
  • Discrete probability: random variables, Markov, Chebyshev and Chernoff-Hoeffding bounds
  • Gaussian variables, concentration inequalities, dimension reduction.
  • Spectral partitioning and clustering.
  • Additional topics (to be chosen from based on time and interest): Martingales, Markov Chains, Random Matrices, Tensors, Chaining Methods

The course will have 4 homeworks (40 percent), 2 quizzes (10 percent), a midterm (20 percent) and a final (30 percent).


There is no textbook for this course. Please see the "Resources" and "Recommended Books" sections below for some useful references.



Homeworks and Announcements


  • No discussion on October 3. Discussion sections will start from October 10.
  • The first quiz will be in (the beginning of) class on October 18.
  • Homework 1 (Due on 10/26/18).


Lecture Plan and Notes


  • 10/2: Fields and vector spaces.
    [Notes]
  • 10/4: Bases of vector spaces, Lagrange interpolation.
    [Notes]
  • 10/9: Bases of infinite-dimensional spaces, Linear Transformations.
    [Notes]
  • 10/11: Eigenvalues and eigenvectors, Inner Product Spaces
    [Notes]
  • 10/11: Orthogonality and orthonormality, Adjoints
    [Notes]


Resources




Recommended Books


There is no textbook for this course. The following books may be useful as references: