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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2000-2001 MATHEMATICS COLLOQUIUM SERIES 10/26/00 Colloquium DR. RICHARD LUNDGREN DEPARTMENT OF MATHEMATICS UNIVERSITY OF COLORADO, DENVER AN APPLICATION OF BOOLEAN RANK OF (0,1) - MATRICES AND BICLIQUE COVERS OF BIPARTITE GRAPHS TO HUMAN LEUKOCYTE ANTIGEN SEROLOGY Abstract: We will start the talk by introducing and explaining the relationships between biclique covers and partitions of bipartite graphs and several matrix ranks including boolean rank, nonnegative integer rank, term rank, and real rank. We will then present and explore a comprehensive mathematical model for human leukocyte antigen serology, based on a mathematical formaliazation of the concept of specificity. The talk will include a little of the relevant immunological background, but will mostly focus on the analysis of the mathematical model. We will discuss how HLA reaction matrices are constructed and will give techniques and theorems which aid in reducing and analyzing these matrices. In particular, we will show how boolean rank and biclique covering numbers play a significant role in this analysis. We will also investigate how other matrix ranks or the biclique partition number might be helpful in this analysis. The model itself raises several new mathematical questions leading to some interesting open problems