Much of my early graduate work concerned properties of quadratic forms. I wrote several notes along the way which are linked below. I also took extensive notes on T. Y. Lam's excellent book Introduction to Quadratic Forms Over Fields, but those are not available online.
In addition to the notes here, most of the problems discussed on my computational number theory page relate to quadratic forms.
Inside I list several problems I was particularly interested in, including background and references. While mostly my research has switched to different questions, this is my primary reference when I want to return to problems more purely about quadratic forms.
This is a longer list of questions that cropped up earlier in my studies. I have not spent much time thinking about many of them, so some may be embarrassingly easy.
This note surveys a few different proofs of the strong and weak Hasse principles for quadratic forms. I was particularly interested in the degree to which the arguments are effective.
Following up on the previous note, this note elaborates on one proof of the "weak" principle that is fully effective.
In this note I worked out two exercises that were particularly interesting to me: the classification of odd, unimodular, indefinite, integral lattices and the fact that the positive definite unimodulars have genus 1 up to dimension 7.
It is well known that over F2, for any given finite dimension, there are at most 2 (isometry classes of) inner product spaces. What about infinite-dimensional spaces? In this note I prove that when the dimension is countably infinite there are exactly 4. I abandoned such questions, though, when I discovered that the problem was well-studied.
This note proves some technical results in the algebraic theory about Pfister neighbors and the structure of the Witt ring.
In this note I review the proof of a theorem of Greenberg generalizing a result of Lang about C1 Laurent series fields. While not inherently about quadratic forms, this theorem has implications for the u-invariant.