RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR
sponsored by the
Co-organizers:
Drew Sills
(asills {at} math [dot] rutgers [dot] edu)
and
Doron Zeilberger
(zeilberg {at} math [dot] rutgers [dot] edu)
Archive: Fall 2003
Spring 2004
Fall 2004 Schedule
Unless otherwise specified, seminars will be held
on the date indicated from 4:30 to 5:20, with the last five minutes
reserved for questions and answers. Professor Zeilberger has promised
to enforce the time limits.
This semester, the seminar will be held in several different rooms, so
please remember to check the listing each time!
Date: Thursday, September 9, 2004
Room: CoRE 431
Speaker: Neil Sloane (AT&T)
Title: From Packing Planes in 4-Space to Quantum Error-Correcting
Codes
Abstract:
I will describe the route that took us from experimental work
on a new packing problem (looking for "codes" in Grassmann
manifolds - e.g., how should you place 18 planes through the
origin in Euclidean 4-space so that they are as far apart
as possible?) to the construction of codes for quantum computers.
This work began as a project with Ron Hardin and John Conway,
but many others (Peter Shor, Rob Calderbank, Eric Rains,
Gabriele Nebe, ...) have since been involved.
There are also applications to medicine, to visualizing
multi-dimensional data, and to wireless communications.
Date: Thursday, September 16, 2004
No seminar.
Date: Thursday, September 23, 2004
Room: Hill 425
Speaker:Vince Vatter (Rutgers)
Title:Counting Restricted Permutations by Computer
Abstract: Restricted permutations arise in many
contexts, from sorting machines to
algebraic geometry. One of the most popular restricted permutation
activities is counting them, a topic about which dozens of papers using ad
hoc techniques have been written. I will talk about systematic approaches
to the problem that can be (and in fact, have been) taught to a computer,
and in particular, how to make one of Doron Zeilberger's algorithms work
in many more cases.
Date: Thursday, September 30, 2004
Room: Hill 425
Speaker: Kathy O'Hara (NSF)
Title: Some Matchings in Product Posets
Date: Thursday, October 7, 2004
Room Hill 425
Speaker: Arthur Benjamin (Harvey Mudd College and Brandeis Univ.)
Title: Counting the Sums of Cubes of Fibonacci Numbers
Abstract: We provide the first combinatorial proof for the sum of the cubes of
the first n Fibonacci numbers. Specifically, we prove that
åk=0n
(fk)3
= (f3n+4 + (-1)n
6 fn-1+5)/10
where fn is the nth Fibonacci number defined by
f0 = f1=1
and for n> 1,
fn = fn-1 +
fn-2.
Along the way, elegant combinatorial proofs are also given for other
Fibonacci identities. This is joint work with undergraduate Timothy
Carnes.
Date: Thursday, October 14, 2004
Room: Hill 425
Speaker: Mohamud Mohammed (Rutgers)
Title: The Sharpening of WZ Theory
Abstract:
A new proof of the Fundamental Theorem for Hypergeometric
(and q-Hypergeometric) summation/integration,
that does not depend on Sister Celine's method will be presented.
As a consequence, we get simplified versions of the Zeilberger,
q-Zeilberger, and Almkvist-Zeilberger algorithms. We also considerably
improve the upper bounds given, in 1992, by Wilf and Zeilberger,
for the orders of the recurrences
and differential equations outputted by these algorithms,
and prove sharp. More importantly, using the new approach,
we extend the above algorithms from one to
several dimensions. [Joint work with Doron Zeilberger].
Date: Thursday, October 21, 2004
Room: Hill 425
Speaker:Amitai Regev (Weizmann Institute, Israel)
Title: S¥ representations and combinatorial identities
Abstract: For various probability measures on the
space of the infinite standard
Young tableaux we study the probability that in a random tableau, the
(i,j)th entry equals a given number n. The analysis of these
probabilities leads to many explicit combinatorial identities, some of
which are related to hypergeometric series.
Date: Thursday, October 28, 2004
Room: CoRE 431
Speaker:Etienne Rassart (Institute for Advanced Study, Princeton)
Title: Partitioning the Permutahedron
Abstract: A permutahedron is a polytope obtained by taking the convex hull of the
orbit of a point under the action of the symmetric group. Permutahedra
appear in algebra because the weight diagram for a certain group
representation is the set of all the lattice points inside a
permutahedron, together with a function that associates an integer (called
multiplicity) to each lattice point. I will explain how this function
partitions the permutahedron into subpolytopes over which it is expressed
by polynomials. With the help of the computer, in three dimensions, we can
count these regions, not only for a given permutahedron but for all
permutahedra at once. The multiplicity function has a continuous analogue
called the Duistermaat-Heckman function, and the computer proves (with a
little help) that these two functions partition the permutahedron in the
exact same way in 3D.
Date: Thursday, November 4, 2004
Room: Hill 425
Speaker: Sujith Vijay (Rutgers)
Title: Expected Number of Spins in Dreidel
Abstract: We show that the expected number of spins in the popular
Chanukah game dreidel where each player starts with n tokens each is
O(n^2), confirming a conjecture of Doron Zeilberger. This is joint work
with Thomas Robinson.
Date: Thursday, November 11, 2004
Room: CoRE 431
Speaker: Cilanne Boulet (Massachusetts Institute of Technology)
Title: A new combinatorial proof of the generalized Rogers-Ramanujan identities
Abstract:We give a combinatorial proof of the first Rogers-Ramanujan identity by
definining a new generalization of Dyson's rank and presenting two related
symmetries. These symmetries are established by direct bijections. We
will also show how to extend this proof to Andrews' generalization of the
Rogers-Ramanujan identities.
This is joint work with Igor Pak.
Date: Thursday, November 18, 2004
Room: CoRE 431
Speaker: Diane Maclagan (Rutgers)
Title: Experiments on the Hilbert scheme
Abstract: The Hilbert scheme parameterizes all ideals with a given Hilbert
polynomial. Studying just the monomial ideals introduces combinatorics,
and gives us information about the structure of the schemes. I will
introduce these concepts, and explain how to compute all monomial ideals
on the (multigraded) Hilbert scheme, and how this lets us conduct
experiments about these objects.
Date: Thursday, November 25, 2004
No seminar; Thanksgiving holiday
Date: Thursday, December 2, 2004
Room: Hill 425
Speaker: Holly Swisher (University of Wisconsin)
Title: Stanley's partition funciton and its relation to p(n)
Abstract: Recently Richard Stanley formulated a new partition
fuction t(n). This function counts the number of partitions π for
which the number of odd parts of π is congruent to the number of
odd parts in the conjugate partition π' modulo 4. G.E. Andrews has
recently proven a nice generating function for t(n) in terms of
the generating function for p(n), the usual partition function. He
also showed that the mod 5 Ramanujan congruence for p(n) also
holds for t(n). In light of these results, it is natural to ask
the following questions: What is the size of t(n)? Are there
other congruences satisfied by both t(n) and p(n)? We
will address both of these questions.
Date: Thursday, December 9, 2004
Room: CoRE 431
Speaker: Jim Haglund (University of Pennsylvania)
Title: A Combinatorial Model for the Macdonald Polynomials
Abstract: We discuss a recent result of M. Haiman, N. Loehr, and the
speaker, which gives a combinatorial formula, involving generalizations
of the permutation statistics maj and inv, for the coefficient of a monomial
in the modified Macdonald polynomial. Consequences of the formula include
a new, short proof of Lascoux and
Schützenberger's cocharge description for
Hall-Littlewood polynomials. The formula was first discovered by
the speaker using experimental methods, and we describe the sequence of steps
which led
to the statistics.
This page is maintained by Drew Sills.
Send comments to asills {at} math [dot] rutgers [dot] edu.