Mon-Fri, Jan 3-7, 10-14, 18-21, 24-28, 12-01:00pm, 54-100, Recitation: TR 10am or 2pm
Pre-register on WebSIS and attend first class.
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 first half of the Spring term; it covers the remaining material in 18.02.
Contact: Galina Lastovkina, 2-108, x3-4977, email@example.com
Mathematics Lecture Series
Pre-register on WebSIS and attend first class.
Listeners welcome at individual sessions (series)
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: Michael Eichmair, 2-171, x3-4385, firstname.lastname@example.org
The talk will contain an elementary introduction to elliptic functions and a discussion of their connection with combinatorics and elementary number theory. In particular, I will explain how to use elliptic functions to prove Jacobi's formula for the number of decompositions of an integer into a sum of four squares.
Wed Jan 5, 01-02:30pm, 2-190
Numerical integration and the redemption of the trapezoidal rule
Unfortunately, most functions cannot be integrated by hand, 18.01-style. On a computer, we can approximate an area as a sum of trapezoids, and this simple method has a beautiful connection to the theory of Fourier cosine series. Using Fourier analysis to understand the approximation, we can transform the trapezoidal rule from one of the worst integration methods into one of the best, Clenshaw-Curtis integration.
Fri Jan 7, 01-02:30pm, 2-190
How much information is contained in a high-resolution digital picture? For a long time, Fourier series were the dominant mathematical tool to answer this question. The discovery of wavelets in the mid 1980s, however, changed the way engineers nowadays think about image compression. Wavelets are a series expansion scheme for functions -- much like Fourier -- but with very different convergence
Mon Jan 10, 01-02:30pm, 2-190
"C,H,O and all that: the curious properties of Quaternions and Octonions"
The complex numbers, quaternions and octonions are some of the most famous objects in classical mathmematics. In this talk I will introduce these number systems and describe some of their history, geometry and arithmetic.
Fri Jan 14, 01-02:30pm, 2-190
Quantum gravity via the LIFO hamburger-cheeseburger model
An old and famous physics problem is to rigorously describe the large n limit of the surface formed by stitching together n squares in a random fashion. The result should be a kind of random two-dimensional surface. I will explain how to reduce one version of this question to a tractable puzzle about inventory accumulation at a LIFO retailer with two products.
Tue Jan 18, 01-02:30pm, 2-190
A "doodle" is a smooth closed curve in the plane, like a figure eight. How can you tell dissimilar doodles apart?
Moving up a dimension you have surfaces "immersed" in 3-space. Steve Smale won a Fields medal for proving that you can turn the standard 2-sphere inside out by deforming through immersions. I will describe a classification of immersed surfaces by means of the "Kervaire invariant."
Wed Jan 19, 01-02:30pm, 2-190
We will discuss two-player perfect information games in which whoever moves last wins, and such that the game must end after finitely many moves. There is no element of uncertainty or randomness. Two classes of games have a particularly elegant theory, which we illustrate with the games Left-Right Hackenbush and Neutral Hackenbush (which includes the famous game Nim).
Fri Jan 21, 01-02:30pm, 2-190
This lecture presents the tools of dimensional analysis frequently used for pilot studies. Which physical variables should we select to model a phenomenon? Is there a simple law that relates them? If so, can we guess it before any further work? How many experiments should we do to check this law? How to build a relevant scale-down? Applications in physics, biology and engineering.
Mon Jan 24, 01-02:30pm, 2-190
The Schrodinger equation on lines, planes, circles and tori.
The Schr\"odinger equation is one of those second order PDE that usually are not presented in traditional PDE courses. In this class I will introduce the Fourier transform and I will use it to find explicit solutions to the linear Schr\"odinger equations on lines, circles, planes and tori. I will also present some cool properties that these solutions have and how one has to use analytic number theory to get them.
Wed Jan 26, 01-02:30pm, 2-190
Tricky Problems in Probability and Statistics
We will discuss several famous problems in probability including the Monty Hall problem and the Two Children problem (with variations). It took some time for mathematicians to agree on their solutions. In the second part of the lecture we will examine certain statistical
studies to expose the fallacies in their conclusions. Once you practice analyzing several examples, you will never look at statistics the same way again.
Fri Jan 28, 01-02:30pm, 2-190