Mon-Fri, Jan 7-11, 14-18, 22-25, 28-1, 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, 2-285, 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: Alan Edelman, 2-343, x3-7770, firstname.lastname@example.org
Random Walks, Discrete Harmonic Functions, and Electrical Circuits
Mon Jan 7, 01-02:30pm, 2-190
Polya's Random Walk Theorem
ABSTRACT: The probability that the simple random walk returns to its initial position is one in dimensions one and two, but strictly less than one in dimensions three and higher. This surprising law of nature was discovered
by George Polya in 1921. We'll give a proof of Polya's theorem by cobbling together elementary results from combinatorics, analysis, and special functions.
Wed Jan 9, 01-02:30pm, 2-190
When unknowns outnumber equations
When there are more unknowns than equations, linear algebra tells us there are infinitely many solutions. How can we choose which one is the best? One approach is to seek the sparsest solution ??? the one with the most zeros. We will discuss a few areas where there are more unknowns than measurements, such as image processing and genomics. Then we will analyze a method of finding sparse solutions to matrix equations.
Fri Jan 11, 01-02:30pm, 2-190
Numerical integration: Trapezoids to Chebyshev
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).
Mon Jan 14, 01-02:30pm, 2-190
A Sobolev inequality and a Strichartz estimate.
In this lecture I will present an almost self contained version of two very useful tools in the study of certain Partial Differential Equations (PDE). These are a Sobolev inequality and a Strichartz estimate. One essential mathematical object that I will be using is the Fourier transform and some powerful identities and isometries that involve it.
Wed Jan 16, 01-02:30pm, 2-190
Discrepancy theory Or: How much balance is possible?
Fri Jan 18, 01-02:30pm, 2-190
Determinants That Count
How many ways can you cover a chessboard with dominoes? How many ways can 3 salesmen visit 17 cities with no overlap? How many ways can a grid of tiny magnets align or misalign
with each other? Astonishingly, all of these questions can be answered by computing the determinant of an integer-valued matrix. We'll learn these powerful counting techniques and discover what they have to do with the physics of ferromagnets.
Wed Jan 23, 01-02:30pm, 2-190
"Spreadsheets, Big Tables, and the Algebra of Associative Arrays"
Spreadsheets (e.g., Microsoft Excel) and triple stores (e.g., Google Big Table) are widely used every day. Associative arrays capture this data mathematically and allow it be manipulated using linear algebra. This class introduces associative arrays, adjacency matrices, and incidence matrices. D4M (http://www.mit.edu/~kepner/D4M ) provides an associative array interface in Matlab and will be used in class homework.
Fri Jan 25, 01-02:30pm, 2-190
Mon Jan 28, 01-02:30pm, 2-190
Extremal Combinatorics studies the maximum or minimum size of discrete structures such as graphs, set systems, and simplicial complexes. The field is experiencing a rapid growth, partly due to its connection to other fields of Mathematics such as Number Theory, Geometry, Probability Theory, and so on. In this lecture, I will give a brief overview of this field by discussing some fundamental results.
Wed Jan 30, 01-02:30pm, 2-190