JUNIOR SEMINAR ORGANIZATIONAL MEETING

JUNIOR SEMINAR ORGANIZATIONAL MEETINGTuesday, February 10TH 4:00 – 5:30 FINE 314
Students who did a Junior Seminar in the fall have two options in the spring: either another Junior Seminar, or a Junior Paper.  Those who did a Junior Paper in the fall must sign up for a Junior Seminar in the spring.
SEMINARS TO BE OFFERED IN SPRING:
Zeev Dvir.  The seminar will be on line-point incidence theorems. These include modern variations on classic theorems such as Szemeredi-Trotter and Sylvester-Gallai as well as problems of Kakeya type.
Steven Sivek.  Morse Theory.  We can learn a lot about the topology of a manifold just by taking a real-valued function on it and studying where its derivative vanishes.  The goal of this seminar is to see how this is done, borrowing ideas from analysis and algebra to construct the homology of a manifold, prove several interesting applications, and possibly see how generalizations of these ideas appear in modern research in topology.
Vlad Vicol.  Singular integrals and applications. For the first half of the seminar we will roughly follow the first 2.5 chapters of the book "Singular Integrals and Differentiability Properties of Functions" by E. Stein. The goal is to understand the boundedness of Calderon-Zygmund operators (e.g. the Hilbert transform) on $L^p$ spaces, and become accustomed with the tools involved in the proof (e.g. decomposing an integrable function into its “good” and “bad” parts, Marcinkiewicz interpolation, $L^2$ bounds for Fourier multipliers). Since Calderon-Zygmund operators arise naturally in partial differential equations, for the second half we will cover a number of applications of the theory (e.g. BMO and Hardy spaces, Littlewood-Paley theory).