[3][4] Alan Hoffman was born and raised in New York City, residing first in Bensonhurst, Brooklyn and then on the Upper West Side of Manhattan, with his sister Mildred and his parents Muriel and Jesse.
Although his coursework consisted primarily of mathematics, including small classes with luminaries in the field, he also studied philosophy, literature, and the history of governments.
Unable to draw, he carried in his head a vision of the configurations in space – points, circles and spheres – depicting phenomena analogous to the geometry of lines.
After additional Army training, Hoffman became an instructor at the anti-aircraft metrology [IL1] [2] school, teaching basic trigonometry used to track balloons to plot deduce winds aloft.
Following additional training in Electrical Engineering at the University of Maine and on the rudiments of long-lines telephony Hoffman was assigned to the 3186th Signal Service Battalion and sent to the European theatre in December 1944, as the war there was nearing its end.
During his time abroad he and others taught some mathematics in small self-organized courses and he recorded his forays into circular geometry to share with faculty back at Columbia.
During that academic year, he gained confidence and skills in his teaching, crystallized his ideas on axioms for circular geometry, and proposed marriage to Esther Walker, the sister of an Army friend.
Following successful completion of exams and defense of his doctoral dissertation on the foundations of inversion geometry in 1950, Hoffman spent a postdoctoral year at the Institute for Advanced Study in Princeton sponsored by the Office of Naval Research.
Hoffman learned linear programming from George Dantzig, who believed that their work would help organizations operate more efficiently through the use of mathematics – a concept that is now, 70 years later, continuing to be realized[IL4].
Through this work Hoffman became exposed to business concepts from management consulting, manufacturing, and finance, areas he enjoyed, but never felt fully at home in.
This early work at NBS, and Hoffman's continued interest using linear equalities to prove combinatorial theorems led to collaborations with Harold Kuhn, David Gale and Al Tucker and to the birth of a subfield that later became known as polyhedral combinatorics.
Hoffman also wrote about Lipschitz conditions for systems of linear inequalities, bounds on eigenvalues of normal matrices and the properties of smooth patterns of production.
He did mathematics across Europe, discovering on a train to Frankfurt a beautiful theorem (but a flawed proof, later corrected by Jeff Kahn) connecting a topic in algebra to his early work on the geometry of circles.
Hoffman choose the role in the larger, more established organization due to the location, the salary, and the opportunity to see if he, and the field of operations research could succeed in business.
In 1961 he accepted the invitation of Herman Goldstine, Herb Greenberg, and Ralph Gomory to join IBM Research, thinking that it would be a great place to work, but that it probably wouldn't last, and in a few years he would get a "real job" in academia.
Despite being a mere 11 years post-PhD, Hoffman quickly assumed the role of mentor to these young researchers, discussing their work and interest and providing guidance.
Summary of Mathematical Contributions (from his notes in Selected Papers of Alan Hoffman)[6] Hoffman's work in Geometry, beginning with his dissertation "On the foundations of inversion geometry," included proofs of properties of affine planes, and the study of points of correlation of finite projective planes, conditions on patterns of the union and intersection of cones (derived largely from his generalization of his earlier results on the rank of real matrices).
He produced an alternate proof, based on axioms for certain abstract systems of convex sets, of a result (by Scarf and others) on the number of inequalities required to specify a solution to an integer programming problem.
The paper on self-orthogonal Latin squares, with IBM co-authors Don Coppersmith and R. Brayton, was inspired by a request to schedule a spouse avoiding mixed doubles tournament for a local racquet club.
In the late 1990s he collaborated with Cao, Chvátal and Vince to develop an alternate proof, using elementary methods rather than linear algebra or Ryser's theorem about square 0-1 matrices.
A collaboration with Shmuel Winograd, also an IBM Fellow in the Mathematics department, produced an efficient algorithm for finding all shortest distances in a directed network, using pseudo-multiplication of matrices.
One such instance is the complete transportation problem, in the case where the cost coefficient exhibit a particular property discovered more than a century earlier by the French Mathematician Gaspard Monge.
His final paper on this topic "On greedy algorithms, partially ordered sets and submodular functions," co-authored with Dietrich, appeared in 2003.
Over his long career, Hoffman served on the editorial board of eleven journals and as the founding editor of Linear Algebra and its Applications.
In 1992, together with Phillip Wolfe (also of IBM) he was awarded the John von Neumann Theory Prize by ORSA and TIMS[6], predecessors of INFORMS[7].
In presenting the award George Nemhauser recognized Hoffman and Wolfe as the intellectual leaders of the mathematical programming group at IBM.