\documentclass{article}
\begin{document}
Quadratic forms and their use in geometry and number theory have a  
long and distinguished history going back to ancient times. The  
algebraic theory is more recent. It originates from the work of Ernst  
Witt in the 1930s, who organized the non-degenerate quadratic forms  
over a field $K$ into what we now call the Witt ring of $K$. In the  
1960s Albrecht Pfister introduced the basic building blocks of this  
ring, which we now call Pfister forms.

The $d$-h Pfister number $\mathrm{Pf}_d(q)$ of a quadratic form $q$ is  
the smallest number of $d$-fold Pfister forms required to represent $q 
$.  (Here I am assuming that q is contained in the $d$-th power of the  
fundamental ideal.) This number is an important measure of the  
complexity of $q$.  In this talk I will define all of the above terms,  
including ``Witt ring", ``Pfister form" and ``fundamental ideal". I  
will then discuss a recent result, proved jointly with Patrick Brosnan  
and Angelo Vistoli, which shows that $\mathrm{Pf}_3($q$)$ can be  
surprisingly large.
\end{document}