18.995 Applied Mathematics for Pure Mathematicians

Time/place: Tuesday/Thursday 2:30-4:00 in 2-131 (located in Building 2)

Course description: This course will cover selected topics in applied mathematics, from a pure mathematical perspective. In other words, it will focus on material with important real-world applications (ranging from signal processing to statistics to numerical analysis to cryptography), but it will emphasize connections with pure mathematics and the naturality and beauty of the results. Many of the topics will have a computational angle to them, but we'll cover diverse areas. No special background will be required, just a general familiarity with basic analysis, algebra, and geometry.

There will be no class on the following dates:

• Tuesday, September 18
• Tuesday, October 9 (Columbus Day)
• Thursday, November 1
• Thursday, November 22 (Thanksgiving)

Problem sets:

• Problem Set 1 (due September 20)
• Problem Set 2 (due October 4)
• Problem Set 3 (due October 18)
• Problem Set 4 (due November 6, since we won't have class on November 1)
• Problem Set 5 (due November 15)

Students taking the class for credit will also give 45-minute presentations during the period of November 20 to December 11.

Handwritten lecture notes (not written for anyone but me, so they probably aren't useful if you weren't in class, but it can't hurt to provide them):

• September 6 (Numerical computation, Richardson extrapolation)
• September 11 (Asymptotics, Euler-Maclaurin summation)
• September 13 (Fourier series)
• September 20 (Fast Fourier transform and applications)
• September 25 (Orthogonal polynomials, quadrature)
• September 27 (Interpolation, Chebyshev nodes, Runge's phenomenon)
• October 2 (Fourier transform, band-limited functions)
• October 4 (Hardy spaces, uncertainty principles)
• October 11 (Compressive sensing; didn't cover last proof but included in notes for completeness)
• October 16 (Optimization)
• October 18 (Probability)
• October 23 (Central limit theorem)
• October 25 (Classical statistical tests)
• October 30 (Exponential families, Bayesian information criterion)
• November 6 (Principal component analysis, singular value decomposition)
• November 8 (Error-correcting codes)
• November 13 (Multiplicative weight algorithms, boosting)
• November 15 (Cryptography)