Summer 2018 will feature two mini-courses:
Math 8910 - Topics in Analysis
Instructor: Neil Lyall
Ten Lectures on June 18-22 (MTWRF) and 25-29 (MTWRF), 10:00-11:00 AM, Boyd 410
Course Description: I would plan to cover some large subset (depending on the interest of students) of the following self-contained topics. I have a slight preference for the first three topics as I believe they will be of broader appeal.
1. Prime Number Theorem (PNT) [2-3 lectures]
Quick review of complex analysis
Newman's short proof of the PNT
What is a Tauberian Theorem? Overview of other proofs of the PNT?
2. What is Ergodic Theory? [2-3 lectures]
Quick review of measure theory
Mean and pointwise theorems
Normal numbers, Continued fractions and Khinchine’s constant
3. The Isoperimetric Inequality [2-3 lectures]
Basics about Fourier Series and a first proof
The Brunn-Minkowski Theorem and a second proof
Other proofs? More about Fourier Series?
4. The Gauss Circle Problem [2-3 lectures]
Fourier transform of arc-length measure
Poisson Summations and the Gauss Circle problem
More on oscillatory integrals. Other lattice point problems?
5. Differentiation [2-3 lectures]
Cricket averages, Covering Lemmas and Maximal functions (beautiful topic no longer covered in Math 8100)
Thin sets and Besicovitch sets (sets of measure zero that contain line segments that point in all possible directions)
Math 8920 - A crash course on homotopy theory
Instructor: Weiwei Wu
Seven Lectures on June 11-14 (MTWR) and 20-22 (WRF), 1:30-3:00 PM, Boyd 410
Course Description: The course is to cover basic notions of homotopy groups, homotopy fibrations and cofibrations, Puppe sequences, Hurewicz and Whitehead theorems. Time allowing I will explain how this fits into the framework of model categories, or in a different direction, I could go towards generalized homology and spectra, depending on student interests. The main reference will be Chapters 6 and 8 of "Lecture Notes in Algebraic Topology" by Davis and Kirk.