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: Roberto Svaldi. 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 Undergraduate Math Office,
E18-366.
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 E18-401W (e-mail Roberto 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 5,
Andrew Sutherland: p-adic numbers.
Abstract:
The real numbers are typically constructed as a "completion" of the
rational numbers. But there is more than one way to complete the
rational numbers (infinitely many, in fact), and each leads to a new
number system that is as useful in its own way as the real numbers.
Introduced by Hensel at the beginning of the 20th century, p-adic
numbers are now widely used in number theory and arithmetic geometry,
and have applications in many other areas of mathematics, including
analysis, topology, and dynamics.
Lecture notes.
Homework 1, Part 1.
Homework 1 (with parts 1-2) due Friday 1/9 by 4pm in E18-366
(held in 4-270) Lecture 2. W, Jan 7,
Omer Tamuz: Convergence of
majority dynamics.
Abstract:
What happens when everyone on Facebook tries to have the same opinions as
their friends?
Does everyone eventually settle on one opinion?
It turns out that this depends on the friendship graph.
Lecture notes.
Homework 1, Part 2.
There is a mistake in the homework. In exercise 1b the correct bound for the cardinality of the dynamic monopoly should be 50 and not 80.
If you can't come up with a proof for the new bound, you may try to give a counterexample for the previous bound of 80. Hint: try to construct a graph with 10 vertices, degree 3 and a dynamic monopoly of 3 vertices. If you can do this, take 20 copies of this graph and this will give a counterxample.
Homework 1 (with parts 1-2) due Friday 1/9 by 4pm in E18-366
(held in 4-270) Lecture 3. F, Jan 9,
Pierre-Thomas Brun: A journey into the world of Euler's elastica: when rods, droplets and pendula behave the same.
Abstract:
Every high school student has encountered the pendulum equation in the course of his or her studies. Surprisingly, the very same equation applies to a broad range of physical systems ranging from pendant drops to the deformations of thin elastic bodies and was thoroughly investigated by Euler in 1744.
In this lecture we revisit Euler's elastica equation to introduce the concepts of boundary value problem, bifurcation and singularity. Our study is conducted combining analytical solutions and symbolic computational tools such as shooting methods and contour plots. Our findings are examined to bridge theory and applications be it in the natural or the man-made worlds.
Lecture notes.
Homework 2, Part 1.
Homework 2 (with parts 1-3) due Friday 1/16 by 4pm in E18-366
(held in 4-270) Lecture 4. M, Jan 12,
Steven Johnson : Numerical integration: Trapezoids to Chebyshev.
Abstract:
Because most functions cannot be integrated in closed-form, numerical techniques for evaluating integrals on a computer are essential.
In this lecture, we start with one of the simplest techniques, the trapezoidal rule, and show that it has deep connections to Fourier theory and, by a simple transformation, turns into one of the most sophisticated integration techniques (involving something called Chebyshev polynomials).
Lecture notes, part 1. Contains the homework in the last page.
Lecture notes, Part 2.
Homework 2 (with parts 1-3) due Friday 1/16 by 4pm in E18-366
(held in 4-270) Lecture 5. W, Jan 14,
Joern 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 2, Part 3.
Homework 2 (with parts 1-3) due Friday 1/16 by 4pm in E18-366
(held in 4-270) Lecture 6. F, Jan 16,
Henry Cohn:
Sphere packing
Abstract:
What is the densest packing of congruent spheres in high dimensions? In this talk, we'll look at what is known about this problem and why it matters.
Lecture notes.
Homework 3, Part 1.
Homework 3 (with parts 1-2) due Friday 1/23 by 4pm in E18-366
(held in 4-270) Lecture 7. W, Jan 21,
Tom Mrowka: Knots and how to detect knotting.
Abstract:
The problem of understanding whether a closed curve in 3-space is knotted or more
generally whether two curves are equivalent has a long history.
In the late 1800’s Tait and Lord Kelvin hoped to link knot types to the then new understood periodic table.
Since their initial investigations
knot theory has becoming intertwined with geometry, analysis, high energy physics and representation
theory and even biology (DNA can be knotted!).
Despite all the tools available the question of whether
there is polynomial time algorithm that can recognize if a curve is knotted remain open. This talk will
introduce to you some of the interesting mathematics behind this.
Lecture notes.(The file is about 15MB, so please be careful with the data usage if you are on some kind of mobile device.)
Homework 3, Part 2.
Homework 3 (with parts 1-2) due Friday 1/23 by 4pm in E18-366
(held in 4-270) Lecture 8. F, Jan 23,
Spencer Hughes
: On The Convergence of Series.
Abstract:
Deciding whether or not a given infinite sum of numbers converges (and, when it does, proving that this is so) is perhaps the most fundamental thing an analyst does. And so gaining intuition about questions of convergence is a very important skill. This lecture will be all about this type of problem.
We will discuss some standard methods that can be applied to various special cases and then we will go on to look at more exotic and interesting series. We will perhaps end with some mention of open such problems, where the question of convergence of a certain series encodes a problem of a different flavour.
Lecture notes.
Homework 4, Part 1.
Homework 4 (with parts 1-3) due Friday 1/30 by 4pm in E18-366
(held in 4-270) Lecture 9. M, Jan 26,
Semyon Dyatlov: Chaos in dynamical systems.
Abstract:
We will study several mathematical ways to describe when a given dynamical system exhibits chaotic, or unpredictable, behavior, such as the notions of ergodicity and mixing.
These concepts will be illustrated on several examples,
both basic (where I will attempt to give a rigorous proof of ergodicity) and more interesting ones, such as chaotic billiards (which will be demonstrated by numerical simulations).
Lecture notes and homework 4, Part 2.(Exercises are at the end of the notes.)
Homework 4 (with parts 1-3) due Friday 1/30 by 4pm in E18-366
(class will be held in E25-111) Lecture 10., Th. 1/29, 1pm.
Sigurdur Helgason:Lie-theoretic Approach to Differential Equations.
Abstract:
Around 1870 Sophus Lie conceived the idea of a theory for differential equations in analogy to Galois theory for algebraic equations.
In this lecture we shall explain his methods, prove his first main theorem (which oddly enough seems to have disappeared from customary texts on differential equations) and use it on several examples. Several basic exercises will be stated.
Prerequisites: Familiarity with the notion of a group and the definition of a differential equation.
I will mostly limit myself to relationship to differential equations in the spirit of Lie's work. However, following the great work of Elie Cartan and others, Lie group theory took off on its own, reaching considerable depth, but also exerting ever increasing influence on other branches of mathematics, differential geometry, number theory, finite group theory and mathematical physics.
Lecture notes.
Sophus Lie and the role of Lie Groups in Mathematics
Homework 4 (with parts 1-2) due Friday 1/30 by 4pm in E18-366.