REVIEW:
What is the intellectual merit of the proposed activity?
The work of Zeilberger has been enormously influential in
combinatorics. He has been a champion of using the computer to explore
a variety of combinatorial problems and to develop algorithms that
allows one to verify and prove various identities that arise in
combinatorics. He has had a large number of spectacular successes
including the first proof of the alternating sign matrix conjecture,
the development of the WZ method to prove holonomic identities, and
many contributions to the theory of $q$-series, permutation statistics,
and lattice paths. He continues to be extremely productive and he and his students
attack a wide variety of problems. In the period of his last grant, 6
students received their Ph.D. degree under his direction and he lists
over 30 papers that are the result of his research.
It should come as no surprise that the current proposal again
emphasizes the extensive use of computer methods to solve a variety of
combinatorial problems. Some of the more intriguing problem areas
include a series of interesting problems on higher dimensional lattice
paths, permutation statistics, the Ruzumov-Stroganov conjectures on
fully packed loops and alternating sign matrices, and extensions of the
WZ theory to prove formulas involving multi-sums and multi-integrals
like the famous Selberg integral formula.
What are the broader impacts of the proposed activity?
Zeilberger is tireless advocate to undergraduate students, graduate
students, and other researchers of using computer exploration to solve
mathematical problems. Thus he efforts will certainly have a broad
impact on the mathematical community.
Summary Statement
Based on his past performance, I have no doubt that many new and
exciting combinatorial results will be found by applying the type of
computer based methods described in this proposal. He is tireless
advocate to undergraduate students, graduate students, and other
researchers of using computer exploration to solve mathematical
problems. Thus he efforts will certainly have a broad impact on the
mathematical community. Thus I definitely think that Zeilberger should
be supported by the NSF and I would definitely rate this proposal in
the Excellent category.
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