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) edelman [ ] math.mit.edu
Teaching assistant: Sheel Ganatra
, ganatra
[ ] math.mit.edu, 2-490,
office hours: 4 - 5 PM in 2-490 (e-mail if you have trouble finding this area!)
Lectures are held MWF 1-2:30
in room 2-190. The first lecture, however, will be in 35-225 from 1-2:30.
Homeworks are due each
Friday by 4pm and need to be turned in at the Undergraduate Math
Office, 2-108.
Recitations are held every
Thursday at 10.30am in room 2-142 and 1pm in room 2-146
(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.
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.
(held in 35-225) Lecture 1. M, Jan 9, Tara Holm: Dance
of the Astonished Topologist (...or how I left squares and hexes for math)
Lecture notes are available on Stellar, here. Homework 1 part 1.
I will give a friendly introduction to some key ideas and tools in topology, including covering spaces and monodromy. The main example will come from square dancing, a hobby I picked up whilst a gradate student at MIT. No prior experience with topology or square dancing will be assumed.
HW 1 (with parts 1-2) due Friday 1/13/2012 at 4pm in
2-108.
Lecture 2. W, Jan. 11, Steven Johnson: Invisibility Cloaks
On the mathematics by which objects can be rendered invisible in some wave equations, and the prospects of achieving this in reality.
Lecture notes are available here. Some additional figures from the lecture. Homework 1 part 2.
HW 1 (with parts 1-2) due Friday 1/13/2012 at 4pm in
2-108.
Lecture 3. F, Jan. 13, Jonathan Novak:
Hilbert's third problem; or, why freshmen have to learn calculus
In 1900 Hilbert asked: given two polyhedra of equal volume, is it always possible to cut one of them into finitely many polyhedral pieces which can be reassembled into the other? If the answer to this question is yes, then we don't need limiting processes to compute volumes of polyhedra. If it's no, we're stuck with calculus. Come and find out which way the cookie crumbles.
Official lecture notes are available here. Homework 2 part 1.
HW 2 (with parts 1-2) due Friday 1/20/2012 at 4pm in
2-108.
No lecture on M, Jan 16. Martin Luther King, Jr Day - holiday.
Lecture 4. W, Jan. 18, Alan Edelman:
Random Matrix Theory: Cutting edge research and applications in science, engineering, and finance
Random matrix theory is the natural third member of the sequence: scalar probability, vector probability, matrix probability. It came last because it was harder, but it is also richer. Pure mathematics loves that there is still so much to discover. New applications are found every day. Learn a bit today and even more in 18.338 this upcoming semester.
Blackboard lecture notes and Powerpoint Slides are available on Stellar, here. Homework 2 part 2.
HW 2 (with parts 1-2) due Friday 1/20/2012 at 4pm in
2-108.
Lecture 5. F, Jan. 20, Paul Hand: A Game That Everyone Thinks is Fair
Suppose a hat contains two distinct real numbers. You pick one number at random and have to guess whether the other number is bigger or smaller than it. Can you be correct more than 50% of the time? Almost everyone says no, but, surprisingly, the answer is yes. Come learn some probability in order to see why.
Lecture notes are available here. Homework 3 part 1.
HW 3 (with parts 1-3) due Friday 1/27/2012 at 4pm in
2-108.
Lecture 6. M, Jan. 23, Haynes Miller:
Knots and Numbers
How are knots enumerated and told apart? We'll see that knots and numbers are inseparably entangled, and conduct a proof by square dance.
Lecture notes are available here. Homework 3 part 2.
HW 3 (with parts 1-3) due Friday 1/27/2012 at 4pm in
2-108.
The theme of this lecture is to show how geometry can be used to understand the rational number solutions to a polynomial equation.
Lecture notes are available here. Homework 3 part 3.
HW 3 (with parts 1-3) due Friday 1/27/2012 at 4pm in
2-108.
Lecture 8. F, Jan. 27, Tanya Khovanova: Geometry: applications, research and art
Why are manhole covers round? Bring your answer to this famous interview question. We will use manhole covers as a starting point to discuss some modern research in convex geometry. In the second part of the lecture I will explain the topology behind the drawings of Anatoly Fomenko.
Homework 4 part 1.
HW 4 (with parts 1-3) due Friday 1/27/2012 at 1pm in
2-108.
Lecture 9. M, Jan. 30, Alexander Postnikov: Grassmannian: From Geometry to Combinatorics
The classical geometric objects, called Grassmannians, lead to beautiful combinatorial structures. The Grassmannians show up everywhere, in Hilbert's fifteenth problem, in almost all areas of modern algebraic combinatorics, and they may play a crucial role in the future theoretical physics of 21-st century.
Lecture notes are available here. Homework 4 part 2.
HW 4 (with parts 1-3) due Friday 1/27/2012 at 1pm in
2-108.
Lecture 10. W, Feb. 1, Jacob Fox: Ramsey Theory
The philosophy of Ramsey theory is that "Every large system contains a large well-organized subsystem." It is currently one of the most active areas of research within combinatorics, overlapping substantially with number theory, geometry, analysis, logic and computer science. This lecture will introduce some of the fundamental results and problems in the area.
Lecture notes are available here. Homework 4 part 3.
HW 4 (with parts 1-3) due Friday 1/27/2012 at 1pm in
2-108.