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REVIEW:
What is the intellectual merit of the proposed activity?
This proposal continues the PI's influential work on a new research
methodology that is loosely known as "rigorous experimental
mathematics". The main idea of this methdology is to program a computer
to discover a formula based on an assumed pattern (or "ansatz"), and
then to discover the proof based on induction. The PI is a pioneer of
this methodology, that has made him well-known, for instance, for the
Wilf-Zeilberger algorithm (relevant to hypergeometric summation and
integration).
The PI applies his theory to various sequences in enumerative
combinatorics, and distinguishes several ansatzes, as follows. Each of
them is subject for further research.
(1) The polynomial ansatz - applied to permutation statistics, in
particular to computing higher covariances of such statistics and their
asymptotics based on some clever recurrence relations.
(2) The C-finite ansatz - applied to sequences satisfying a
homogeneous linear recurrence relation with constant coefficients. This
is applied to approximations of seemingly intractable combinatorial
problems in statistical physics.
(3) The Schutzenberger ansatz - applied to sequences with
generating functions satisfying algebraic equations whose coefficients
are polynomials in x. This is applied to problems of tree and lattice
path enumeration.
(4) The holonomic ansatz - applied to sequences satisfying linear
recurrence relations with polynomial coefficients. This is applied to
lattice path counting (a beautiful example provided is the proof of a
conjecture of Gessel) and determinant evaluation.
(5) New ansatzes - a multi-basic generalization of the holonomic
ansatz, and WZ theory with arbitrarily many variables (for
integration).
Another research problem, based on a classical but promising
bijective approach, concerns the well-known Razumov-Stroganov
conjecture.
What are the broader impacts of the proposed activity?
The broader impacts of the proposal are related to the extensive
use of computers in mathematical research and freely available
software, as well as to the strong education component. The PI has been
very active in supervising graduate students, who went on to successful
careers.
Summary Statement
The PI is a very active promoter of his new research methodology
described above, which provides a powerful tool to mathematicians, and
which qualifies, in my opinion, as transformative research. The PI
lists 32 publications resulting from his previous NSF support
(2004-2008), appeared as online documents or in journals such as Annals
of Combinatorics, Advances in Applied Mathematics, Experimental
Mathematics, and J. Difference Equations and Applications. This work
deserves a lot of credit, opening a new direction in the evolution of
mathematics, that goes beyond experimental mathematics, but is
different from automated theorem proving. My only remark is that this
methodology needs to be complemented by a more conceptual understanding
of the discovered phenomena (cf. for instance, Gessel's formula in 4)
above).
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