# Jerusalem Mathematics Colloquium

Thursday, 7th May 2009, 4:00 pm

Mathematics Building, Lecture Hall 2

##

Dan Romik

(Hebrew University)

"Alternating sign matrices"

** Abstract: **

Alternating sign matrices (ASMs) were discovered by Robbins and Rumsey in
the early 1980's in connection with their study of Charles Dodgson's
19th-century condensation algorithm for computing matrix determinants.
They have since been found to have fascinating properties and connections to
representation theory, statistical physics and various combinatorial objects
such as domino tilings and plane partitions. Two early conjectures of Robbins,
Mills and Rumsey about the enumeration of ASMs of order N - both the total
enumeration and the so-called refined enumeration with respect to a natural
parameter indexing the first row of the matrix - became famous open problems
and were resolved in the mid-1990's by
Zeilberger.
In this talk I will give an overview of this fascinating field and describe
some recent results, obtained in joint works with Ilse Fischer and with
Matan Karklinsky, on a more detailed "doubly-refined"
enumeration of ASMs with respect to the first two rows of the matrix.
I will conclude with some speculation on how an extension of these results
might be used to attack some famous open problems on the behavior of
randomly chosen ASMs of large order.

Light refreshments will be served in the faculty lounge at 3:30.

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