Gershon Zeilberger is an Israeli-American mathematician known for his contributions to combinatorics, especially the development of algorithmic methods for proving hypergeometric identities. His work bridges classical mathematics and computational approaches, and he has been a vocal advocate for using computers in mathematical research.
Zeilberger was born in 1950 in Israel. He earned his PhD from the Weizmann Institute of Science in 1976, under the supervision of Harry Dym. He held positions at several institutions before joining the faculty at Temple University in Philadelphia, where he has been a professor of mathematics since the 1990s.
Algorithmic Proofs and the WZ Method
Zeilberger's most famous contribution is the WZ method, developed in collaboration with Herbert Wilf in the early 1990s. This method provides a systematic, algorithmic way to prove identities involving sums of hypergeometric terms. The method reduces such proofs to checking a finite number of cases, making it amenable to computer implementation.
The WZ method has been implemented in software packages, most notably in the Maple and Mathematica systems. Zeilberger's own package, called EKHAD, is widely used by researchers. This work has had a significant impact on the field of combinatorics, enabling the automated verification of many identities that were previously proven by hand.
Views on Mathematical Rigor
Zeilberger is known for his provocative and often controversial opinions about the nature of mathematical proof. He has argued that traditional, human-readable proofs are not always necessary, and that computer-assisted proofs are equally valid. In a famous 1993 essay, he proposed a "semi-rigorous" approach to mathematics, suggesting that proofs need not be fully formal if they can be verified by a computer.
His views have sparked debate within the mathematical community. Some mathematicians embrace his computational approach, while others argue that it undermines the aesthetic and conceptual aspects of mathematics. Despite this, Zeilberger's work has been widely cited and has influenced a generation of researchers in combinatorics and related fields.
Contributions to Combinatorics
Beyond the WZ method, Zeilberger has made numerous contributions to enumerative combinatorics, including work on partitions, q-series, and the theory of plane partitions. He has also explored connections between combinatorics and other areas, such as Artificial intelligence and Machine learning, using computational tools to discover new patterns and conjectures.
His research often involves generating large amounts of data with computer programs and then analyzing it to find underlying structures. This approach has led to the discovery of several new identities and has helped to clarify existing ones. He has published over 200 papers and has been an editor for several major mathematical journals.
Awards and Recognition
Zeilberger has received several awards for his work. In 2004, he was awarded the Leroy P. Steele Prize for Seminal Contribution to Research, jointly with Herbert Wilf, for their development of the WZ method. He is also a fellow of the American Mathematical Society, a recognition he received in 2012.
His work has been supported by grants from the National Science Foundation, and he has been invited to speak at numerous international conferences, including the International Congress of Mathematicians. He continues to be an active researcher and a prominent figure in the mathematical community.
Legacy and Influence
Zeilberger's influence extends beyond his own research. He has mentored many students and postdoctoral researchers, and his ideas about algorithmic mathematics have helped to shape the field of experimental mathematics. His advocacy for computer-assisted proof has become more mainstream as computational tools have become more powerful.
His work is often cited in discussions about the role of Deep learning and Neural network methods in mathematics, as his approach to proof is analogous to algorithmic reasoning. While he remains a polarizing figure, his contributions have undeniably advanced the field and opened new avenues for research.
Personal Life
Zeilberger is known for his eccentric personality and his love of wordplay. He maintains a personal website where he posts his opinions, essays, and even a "Opinion" column that has been running for decades. He is also an avid chess player and has written about the connections between chess and mathematics.
He has been married to his wife, a fellow mathematician, and they have children. He continues to work at Temple University, where he is an active member of the department and a frequent speaker at seminars and colloquia.