Alan J. Hoffman facts for kids

Alan Jerome Hoffman (born May 30, 1924 – died January 18, 2021) was an American mathematician. He was a top mathematician at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York.

He started the journal Linear Algebra and its Applications. Alan Hoffman also held several patents, which are special rights for inventions. He made important contributions to areas like combinatorial optimization and the eigenvalue theory of graphs.

With another mathematician, Robert Singleton, he created the Hoffman–Singleton graph. This is a special type of graph in mathematics. Alan Hoffman passed away on January 18, 2021, at 96 years old.

Early Life and School

Alan Hoffman was born and grew up in New York City. He lived in Brooklyn and later in Manhattan. From a young age, he knew he wanted to be a mathematician. He was good at all school subjects. He especially loved the clear, logical thinking in mathematics.

He finished George Washington High School in 1940. That fall, at age 16, he started at Columbia University. He attended on a special scholarship called the Pulitzer scholarship.

College Years and World War II

At Columbia, Hoffman joined the Debate Council. He wanted to get better at speaking in public. He also supported America helping the Allies in World War II. He wanted the U.S. to join the war directly.

Most of his classes were in mathematics. He learned from famous mathematicians. He also studied philosophy, literature, and history.

World War II stopped his studies for a bit. He joined the U.S. Army in February 1943. He served until 1946, spending time in Europe and the Pacific. He later said these three years were a very important time in his life.

While in basic training, he thought about new ideas for geometry. He imagined shapes like circles and spheres in his mind. These ideas later helped him write his doctoral paper. He often developed ideas in his head, which was unusual.

After more training, Hoffman taught trigonometry in the Army. He taught how to track balloons to figure out wind patterns. He also taught math to others while serving overseas.

Returning to Columbia

When he came back to Columbia in 1946, he taught a math course. This helped him improve his teaching skills. He also worked on his ideas for circular geometry. He felt very confident when he started his graduate studies in 1947.

Starting His Career

Alan Hoffman finished his doctoral paper in 1950. It was about the basics of inversion geometry. After that, he spent a year at the Institute for Advanced Study in Princeton. During this time, he focused on his work, believing "You are a mathematician, you do mathematics."

Working at the National Bureau of Standards

He couldn't find a university job he liked. So, Hoffman joined the National Bureau of Standards (NBS) in Washington D.C. in 1951. This turned out to be a very good choice for him. He said his whole career was shaped by his five years at NBS.

He was hired to work on "linear programming." This was a new field that helped organizations work more efficiently. Hoffman found it challenging and fun. He learned about it from George Dantzig, a pioneer in the field.

Hoffman learned about business ideas like management and manufacturing. He also worked with other important researchers. He wrote a paper showing that a method called the "simplex method" worked well. This paper was one of the first to test how well math methods worked on computers.

He also found the first example of "cycling" in the simplex method. This example is now in many math textbooks. Hoffman helped organize a big meeting about linear programming in 1955.

Important Discoveries at NBS

With a German mathematician, Helmut Wielandt, Hoffman used linear programming to compare matrices. This led to the Hoffman-Wielandt Theorem. This theorem is very well-known among mathematicians.

Hoffman also found a simple proof for the König-Egerváry Theorem. This theorem is about finding patterns in tables of numbers. His work at NBS helped start a new area of math called "polyhedral combinatorics." He also helped bring another important mathematician, Jack Edmonds, to NBS.

Hoffman and Joe Kruskal also showed why some math problems with whole numbers have whole number answers, and some don't. This was about a concept called "total unimodularity."

Time in Europe and IBM

In 1956, Hoffman moved to England with his family. He worked for the Office of Naval Research. His job was to reconnect American and European mathematicians. He traveled across Europe, doing math and making friends.

After a year, he looked for jobs in New York. He chose to work at General Electric (GE). He found the job interesting and was allowed to do his own math research.

In 1960, he attended a workshop at IBM Research. He loved the atmosphere there, with "people all around doing mathematics." In 1961, he joined IBM Research. He thought it would be a temporary job, but he stayed there until he retired in 2002. He also taught at several universities as a visiting professor.

Working at IBM

When Hoffman joined IBM, he was one of the older members of his department. He quickly became a mentor to the younger researchers. He helped them with their work and gave them advice. He was made an IBM Fellow in 1977, which is a very high honor.

He published over 200 academic papers during his career. More than a third of these were with other people. He worked on many different areas of math, including algebra and operations research. He collaborated with many colleagues, from top IBM Fellows to summer interns. People liked him for his humor, kindness, and love for math and music.

His Math Work at IBM (Simplified)

Hoffman's work covered many areas of mathematics:

• Geometry: He started with his doctoral paper on "inversion geometry." He also studied properties of shapes and sets of points.
• Combinatorics: He helped us understand different types of graphs. He worked on "comparability graphs" and "Latin squares." One paper was inspired by a request to schedule a tennis tournament!
• Partially Ordered Sets: He often studied these sets, which are like lists where some items are related. He used linear programming to prove new things about them.
• Matrix Inequalities and Eigenvalues: His work in these areas is important in matrix theory. He found a way to combine many different ideas into one general proof.
• Linear Programming: He continued to research how to solve problems using linear programming. He found elegant ways to prove results in this field.
• Network Flows: With Shmuel Winograd, he created a fast way to find the shortest paths in a network. He also generalized the "Max Flow – Min Cut" theorem.
• Greedy Algorithms: He studied problems that could be solved by making the "best" choice at each step. This is called a "greedy" approach. His work helped explain why some problems can be solved this way.
• Graph Spectra: He explored how the properties of a graph are related to its eigenvalues. This included work on the Hoffman–Singleton graph.

Awards and Recognition

Alan Hoffman received many honors for his work:

• He became an IBM Fellow in 1978.
• He was elected to the National Academy of Sciences in 1982.
• He became a Fellow of the American Academy of Arts and Sciences in 1987.
• He received an honorary doctorate from Technion – Israel Institute of Technology in 1986.
• In 1992, he won the John von Neumann Theory Prize with Philip Wolfe. This is a very important award in operations research.
• He was part of the first group of Fellows for INFORMS in 2002.

In 2000, he received the Founders Award from the Mathematical Programming Society. A mathematician named Uriel Rothblum wrote that Alan Hoffman loved everything he did. He enjoyed singing, ping pong, jokes, and especially doing mathematics.

Alan Hoffman's first wife, Esther, passed away in 1988. He married Elinor Hershaft in 1990. They later divorced. Elinor passed away in 2020. Alan spent his last years happily at a retirement community. He is survived by his two daughters, Eleanor and Elizabeth.

Awards

Alan Hoffman was a recipient of a number of awards.

• IBM Fellow, 1978–
• Member, National Academy of Sciences, 1982–
• Fellow, American Academy of Arts and Sciences, 1987–
• D. Sc. (Hon.) Technion – Israel Institute of Technology, 1986
• 1992 John von Neumann Theory Prize with Philip Wolfe
• 2002 class of Fellows of the Institute for Operations Research and the Management Sciences

Select publications

• Hoffman A. J. & Jacobs W. (1954) Smooth patterns of production. In Management Science, 1(1): 86–91.
• Hoffman A. J. & Wolfe P. (1985) History. Lawler E. L., Lenstra J. K., Rinnooy Kan A. H. G., & Shmoys D. B., eds. In The Traveling Salesman Problem. John Wiley & Sons: New York.