Discrete Analogues in Harmonic Analysis
PhD Thesis, Princeton University (pdf)
Advisor: E. M. Stein
This thesis presents a number of results on discrete operators in harmonic analysis, ranging from discrete fractional integral operators along quadratic surfaces to a discrete analogue of fractional integration on the Heisenberg group. The techniques developed are motivated by the circle method and involve substantial analytic machinery.
The
3-Part of Class Numbers of Quadratic Fields
MSc Thesis, Oxford University, 2004 (pdf)
Advisor: D. R. Heath-Brown
This original thesis gives the first nontrivial bounds for the 3-part
of class numbers of quadratic fields, using techniques of analytic
number theory such as mean values of exponential sums, the square
sieve, and the
q-analogue of van der Corput's method.
The
Pair Correlation of the Zeroes of the Riemann Zeta Function
Undergraduate Senior Thesis, Princeton University, 2002 (pdf)
Advisor: E. M. Stein
An expository thesis giving a proof of Montgomery's original theorem,
the derivation of the GUE pair correlation function, and an examination
of the computational results of Odlyzko.
Hardy
Functions
Undergraduate Junior Paper, Princeton University, 2001 (pdf)
Advisor: E. M. Stein
An expository paper giving a proof of the Paley-Wiener theorem and
applications of Hardy functions as signal filters.
