Lecture notes: Lectures 1-2, Lectures 3, Lectures 4, Lectures 5. Disclaimer: the "proofs" in these notes aim to convey the main idea as simply as possible. They often miss out some technical, but important, details. Please consult published sources for full proofs.
Matlab demonstrations: m1_RankOnePert.m, m2_BirmanSchwinger.m, m3_MagneticNodal.m, m4_FloquetBloch.m.
Main references:
In 1966 Kac asked the question "Can one hear the shape of a drum?", i.e. do the frequencies of a drum's vibration fully determine its shape? For plane domains with fixed boundary (which are the closest to the physical drums), this question was answered in the negative by Gordon, Webb and Wolpert (GWW) in 1992. Their construction based on the so-called Sunada method (1984) that uses representation theory. Various proofs, generalizations and applications of this construction exist; in particular one recent application is in the study of Dirac points in the spectrum of graphene, by A.Comech and myself. The above link illustrates a proof (due to Chapman) of the GWW example using foldable paper models.
This is a class file containing routines for setting up a quantum graph, computing its eigenvalue and computing and plotting its eigenfunctions. It was developed basing on the code by Phuongmai Truong and Ram Band with the primary aim of aiding in understanding the behavior of the zeros of the eigenfunctions.
It is part I of a 2-lecture attempt of explaining how Goedel's Incompleteness Theorem is proved. In these lecture notes a criterion of incompleteness is given and a non-enumerable set is constructed.
A short lecture prepared for 2007 Mini Math Fair at Texas A&M University (part of 2007 Math Awareness Month activities). It is aimed at school children and their parents.
A presentation on Buffon's needle problem for Aggieland Saturday 2012.
A talk on agreement between semiclassical expansions in chaotic transport and RMT predicitions.
This file was last modified on Tuesday, 11-Jan-2022 16:54:37 CST.