MIT: Independent Activities Period: IAP 2016

IAP 2016 Subjects: Mathematics

Boris Hanin
Mon-Fri, Jan 4-8, 11-15, 19-22, 25-29, 12-01:00pm, 54-100, Recitation: TR 10am or 2pm

Pre-register on WebSIS and attend first class.
No listeners
Prereq: GIR:CAL1
Level: U 12 units Standard A - F Grading    CALC II

First half is taught during the last six weeks of the Fall term; covers material in the first half of 18.02 (through double integrals). Second half of 18.02A can be taken either during IAP (daily lectures) or during the second half of the Spring term; it covers the remaining material in 18.02. Contact: Theresa Cummings, E18-366, x3-4977,

System Functions and the Laplace Transform
Haynes Miller, Jeremy Orloff
Mon-Fri, Jan 19-22, 25-29, 01-03:00pm, 4-261

Pre-register on WebSIS and attend first class.
Prereq: 18.03
Level: U 3 units Graded P/D/F   

Studies basic continuous control theory as well as representation of functions in the complex frequency domain. Covers generalized functions, unit impulse response, and convolution; and Laplace transform, system (or transfer) function, and the pole diagram. Includes examples from mechanical and electrical engineering. Contact: Haynes Miller, E17-446, x3-7569,

Mathematics Lecture Series
Alan Edelman
Pre-register on WebSIS and attend first class.
Listeners welcome at individual sessions <b>(series)</b>
Prereq: GIR:CAL1
Level: U 6 units Graded P/D/F Can be repeated for credit   

Ten lectures by mathematics faculty members on interesting topics from both classical and modern mathematics. All lectures accessible to students with calculus background and an interest in mathematics. At each lecture, reading and exercises are assigned. Students prepare these for discussion in a weekly problem session.
Students taking 18.095 for credit are expected to attend regularly and to do problem sets. Recitation Thursday at 10:30 or 1:00.
Contact: Alan Edelman, E17-418, x3-7770,

Perfect Forward Secrecy
Andrew Sutherland
Over the past five years virtually every major website (Google, Facebook, Dropbox, Twitter, Amazon, Wikipedia, ...) has switched 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 behind it, and why it is important.
Mon Jan 4, 01-02:30pm, 4-270

The Singular Value Decomposition (SVD) of a Matrix
Gilbert Strang
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.
Wed Jan 6, 01-02:30pm, 4-270

A category-theoretic approach to understanding the steady states of coupled dynamical systems
David Spivak
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.
Fri Jan 8, 01-02:30pm, 4-270

Delta functions and distributions: When functions have no value(s)
Steven Johnson
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!
Mon Jan 11, 01-02:30pm, 4-270

Philippe Rigollet
Wed Jan 13, 01-02:30pm, 4-270

Haynes Miller
Fri Jan 15, 01-02:30pm, 4-270

Tomasz Mrowka
Wed Jan 20, 01-02:30pm, 4-270

Determinants that Count
Homer Reid
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.
Fri Jan 22, 01-02:30pm, 4-270

Mathematical models of baseball games.
Michael Brenner
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.
Mon Jan 25, 01-02:30pm, 4-270

Overdamped dynamics of small objects in fluids
Joern Dunkel
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.
Wed Jan 27, 01-02:30pm, 4-270

Special Subject in Mathematics
Introduction to Proofs
Vladislav Voroninski
Mon-Fri, Jan 4-8, 11-15, 01-03:00pm, 4-149

Pre-register on WebSIS and attend first class.
Limited to 50 participants.
Listeners allowed, space permitting
Prereq: Permission of instructor Calculus I (GIR)
Level: U 3 units Graded P/D/F Can be repeated for credit   

Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. 18.S097 is graded P/D/F.
An introduction to writing mathematical proofs, including discussion of mathematical notation, methods of proof, and strategies for formulating and communicating mathematical arguments. Topics include: introduction to logic and sets, rational numbers and proofs of irrationality, quantifiers,
mathematical induction, limits and working with real numbers, countability and uncountability, introduction to the notions of open and closed sets.
Additional topics may be discussed according to student interest. There will be some assigned homework problems --there is no textbook. Space may be limited; please email ( to reserve a spot.
Contact: Vladislav Voroninski, E18-304,