Quadratic Forms, Lattices, and the Fifteen Theorem -- Neeraj ("Nee") Pradhan, July
6, 2004

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The Conway-Schneeberger Fifteen Theorem is a powerful result that gives a 
sufficient condition that classifies quadratic forms as either universal or 
nonuniversal. A direct proof eluded C-S, and consequently their results 
were not published. The problem was not satisfactorily understood, however 
later Bhargava discovered a direct, elegant solution that not only solved 
the conjecture, but also shed more light on the intrinsic nature of 
quadratic forms and lattices. We will present his proof and the related 
mathematical machinery upon which its foundations rest.