REPRESENTATION THEORY AND AUTOMORPHIC FORMS

By ABIGAIL PEREZ

CONTENTS INTRODUCTION THE MOONSHINE THEOREM A BRIEF HISTORY THE THEOREM BORCHERDS-KAC-MOODY LIE SUPERALGEBRAS VECTOR VALUED MODULAR FORMS BORCHERDS-KAC-MOODY LIE ALGEBRAS AND MODULAR FORMS ?-GRADED VERTEX ALGEBRAS A CONSTRUCTION OF A CLASS OF BORCHERDS-KAC-MOODY LIE (SUPER)ALGEBRAS DE?NITIONS AND ELEMENTARY PROPERTIES BILINEAR FORMS THE ROOT SYSTEM UNIQUENESS OF THE GENERALIZED CARTAN MATRIX A CHARACTERIZATION OF BKM SUPERALGEBRAS CHARACTER AND DENOMINATOR FORMULAS SINGULAR THETA TRANSFORMS OF VECTOR VALUED MODULAR FORMS LATTICES ORDINARY MODULAR FUNCTIONS VECTOR VALUED MODULAR FUNCTIONS THE SINGULAR THETA CORRESPONDENCE ?-GRADED VERTEX ALGEBRAS THE STRUCTURE OF ?-GRADED VERTEX ALGEBRAS ?-GRADED LATTICE VERTEX ALGEBRAS FROM LATTICE VERTEX ALGEBRAS TO LIE ALGEBRAS LORENTZIAN BKM ALGEBRAS AUTOMORPHIC FORMS ON GRASSMANNIANS VECTOR VALUED MODULAR FORMS AND LBKM ALGEBRAS AN UPPER BOUND FOR THE RANK OF THE ROOT LATTICES OF LBKM ALGEBRAS? A CONSTRUCTION OF LBKM ALGEBRAS FROM LATTICE VERTEX ALGEBRAS