This course consists of independent
lectures given by faculty members of the mathematics department at MIT.

The lecturers assign homework problems related to the material presented.

Organizer and contact: Prof. Alan Edelman (for registration issues etc). Email

Teaching assistant: Yun William Yu (for all other issues, including questions about homework, attendance, etc.). Email

Lectures
are held MWF 1-2:30 in room
4-270.

Homeworks
are due each Friday by 4pm and need to be turned in
at the pset boxes outside
4-174.. P-sets will be posted on this site shortly after each lecture.

Recitations
are held every Thursday at 10.30am and 1pm in room
36-156
(both sessions cover the same material).

We suggest you come to the morning session if you have no time
constraints, as we expect that to be less
popular.

Office hours are every Friday 3 - 4 PM, after class, in 32-G550 (e-mail Yun William Yu if you have trouble finding this area!)

This course is offered with the P/D/F grading option.
To receive a passing grade, we ask that you attend lectures and put forth an
effort on the problem sets.

Homeworks will be collected every Friday,
graded and returned to you the following week.

You can check your grades and registration status at the
Stellar website for the class.

(held in 4-270) Lecture 1. M, Jan 4,
Andrew Sutherland: Perfect Forward Secrecy

*Abstract:*
Over the past five years virtually every major website (Google, Facebook, Dropbox, Twitter, Amazon, Wikipedia, ...) has switched over to the ECDHE-RSA (Elliptic Curve Diffie-Hellman Ephemeral Rivest-Shamir_Adleman) protocol for secure key exchange. They have done this in order to achieve what is known as "perfect forward secrecy". I will explain how this protocol works, the mathematics that lies behind it, and why it is so important.

Lecture notes.

Homework 1, Part 1.

Homework 1 (with parts 1-2) due Friday 1/8 by 4pm in 4-174

(held in 4-270) Lecture 2. W, Jan 6,
Gilbert Strang: The Singular Value Decomposition (SVD) of a Matrix.

*Abstract:*
The SVD completes the ' big picture ' of linear algebra. It produces orthonormal bases for all 4 fundamental subspaces(the column space and nullspace of A and A transpose). And those bases of v's and u's diagonalize the matrix. In the end A = U SIGMA V' = (orthogonal) (diagonal) (orthogonal). This turns out to be a good way to understand a matrix of data.

Lecture notes. (excerpt from Prof. Strang's *Linear Algebra and Its Applications*)

Homework 1, Part 2.

Homework 1 (with parts 1-2) due Friday 1/8 by 4pm in 4-174

(held in 4-270) Lecture 3. F, Jan 8,
David Spivak: A category-theoretic approach to understanding the steady states of coupled dynamical systems.

*Abstract:*
The same series, parallel, and feedback composition diagrams that describe coupling of dynamical systems also describe matrix arithmetic: multiplication, Kronecker product, and trace. Each dynamical system has a corresponding steady state matrix,sometimes called a bifurcation diagram. If a certain system is presented as the composite of coupled sub-systems, its steady states can be computed using matrix arithmetic.

Lecture notes. (paper excerpt from David Spivak)

Homework 2, Part 1. (for lec 3; updated 2016-Jan-11 to correct typos in problems 2c and 3c. Also, Problem 5 has been made optional as the lecture didn't really get to bifurcation diagrams.)

Homework 2 (with parts 1-3) due Friday 1/15 by 4pm in 4-174

(held in 4-270) Lecture 4. M, Jan 11,
Steven Johnson : Delta functions and distributions: When functions have no value(s).

*Abstract:*
Changing the definition of a function from the freshman-calculus definition, to something called a "distribution," circumvents a lot of annoyances in analysis. It allows you to define delta functions (e.g. the density of a "point mass"), differentiate discontinuous functions, interchange limits and derivatives, and more. This is essentially what scientists and engineers are "really" doing, though they never tell you!

Lecture notes.

Homework 2, Part 2.

Homework 2 (with parts 1-3) due Friday 1/15 by 4pm in 4-174

(held in 4-270) Lecture 5. W, Jan 13,
Philippe Rigollet: TBA.

*Abstract:*
Is the average always the best estimator? Often it is but sometimes it’s not. The James-Stein estimator illustrates this phenomenon by
showing that when several problems are done simultaneously, even if they are independent, it might be a good idea to couple them to get a better performance.

Lecture notes. & Pset 2, Part 3.

Homework 2 (with parts 1-3) due Friday 1/15 by 4pm in 4-174

(held in 4-270) Lecture 6. F, Jan 15,
Haynes Miller: Knots and Numbers.

*Abstract:*
Algebraic topology attempts to extract discrete invariants from flexible geometric configurations. One of the most easily visualized type of topological object is a knot -- a closed loop in Euclidean space. Appearance of simplicity notwithstanding, knot theory has been an active area of study for the past 150 years, and especially in the past three decades. In this lecture we'll study fragments of knots known as tangles, and discover a rational number that serves as a complete invariant for a subclass of tangles called rational tangles. The verification will be by square-dance.

Lecture notes.

Knot Table.

Homework 3, Part 1.

Homework 3 (with parts 1-2) due Friday 1/22 by 4pm in 4-174

(held in 4-270) Lecture 7. W, Jan 20,
Tom Mrowka: How to detect knottiness

*Abstract:*
How to detect if a given closed loop in three space is knotted and how to understand
if one closed loop is deformable to another are long studied questions in mathematics.
We’ll review a tiny bit of the history of this problem and give some examples of the tools
and some of the methods used.

Lecture notes.

Homework 3, Part 2.

Homework 3 (with parts 1-2) due Friday 1/22 by 4pm in 4-174

(held in 4-270) Lecture 8. F, Jan 22,
Homer Reid
: Determinants that Count.

*Abstract:*
How many ways can you cover a chessboard with dominoes? How many ways can 4 salesmen visit 17 cities without overlapping? How many ways can a grid of iron atoms (mis)align with each other? Amazingly, all of these questions can be answered by writing down a matrix of integers and computing its determinant. We will introduce these powerful counting tools and explain their connection to the physics of ferromagnets.

Lecture notes.

Homework 4, Part 1.

Homework 4 (with parts 1-3) due Friday 1/29 by 4pm in 4-174

(held in 4-270) Lecture 9. M, Jan 25,
Michael Brenner: Mathematical models of baseball games.

*Abstract:*
I will discuss a simple mathematical model of a baseball game, developed by Frederick Mostellar in the 1950s, that asks what is the probability that the best team wins the world series. We will discuss the strengths and weaknesses of this model and use it to discuss what it means for a mathematical model to say something meaningful about the world.

Pre-lecture reading

Lecture notes.

Homework 4, Part 2.
(There's a bug in the Matlab code on page 2. Please find here corrected code for the worldseries.m function and simulating many world series.)

Homework 4 (with parts 1-3) due Friday 1/29 by 4pm in 4-174

(held in 4-270) Lecture 10., W, Jan 27,
Jörn Dunkel: Overdamped dynamics of small objects in fluids.

*Abstract:*
The dynamics of small particles in fluids affects a wide spectrum of physical and biological phenomena, ranging from sedimentation processes in the oceans to transport of chemical messenger substances between and within microorganisms. After discussing these and other relevant examples, we will introduce the mathematical equations that describe such particle motions and study their solutions for basic test cases.

Lecture notes

Homework 4, Part 3

Homework 4 (with parts 1-3) due Friday 1/29 by 4pm in 4-174.