RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR
Archive of Speakers and Talks --- 2007
Jump to Fall 2007
Spring 2007
Date: Thursday, January 18, 2007
Speaker: Vince Vatter, St.
Andrews University
Title: Simple Permuatations
Abstract:
Simple permutations are permutations that do not map nontrivial intervals onto
intervals. I will discuss a result regarding the structure of simple
permutations, and its application to the study of restricted permutations. This
is joint work with Robert Brignall, Sophie Huczynska, and Nik Ruskuc.
Date: Thursday, January 25, 2007
Speaker: Thotsaporn "Aek" Thanatipanonda,
Rutgers
Title: Dancing with Toads and Frogs
Abstract:
Toads and Frogs is a combinatorial game introduced by Richard Guy. In this talk,
we will use the technique called "finite state method" to prove all the values
in each class of the game in one shot. We will also talk a bit about five
conjectures made by Jeff Erickson ten years ago.
Date: Thursday, February 1, 2007
Speaker: Konstantin Mischaikow, Rutgers
and Georgia Tech
Title: Building a Database of Dynamical Systems
Abstract:
In many application there are models for dynamics but specific parameters are
unknown or not directly measurable. This suggests the need to be able to search
through parameter space for specific types of dynamic behavior. Ideally, this
would be done numerically in some automated manner, and then later the
researcher would be able to query the results. To do this requires the
development of a database that contains useful rapidly identifiable information
about the types of dynamics simulated numerically.
I will discuss ongoing efforts to develop such a data base. I will emphasize
what I believe is the essential information that such a database must contain
and the open problems that must be resolved before such a database can be
efficiently constructed and queried.
Date: Thursday, February 8, 2007
Speaker: Andrew Baxter, Rutgers
Title: Periodic Billiards on an Equilateral Triangle
Abstract: A periodic billiard is a path followed by a ball on a
billiards table which retraces itself. The existence of periodic billiards on
arbitrary triangles remains an open problem, though many partial results are
known. The talk will be divided into two parts. In the first part, I will
summarize these partial results, including the recent experimental work of
Richard Schwartz. In the second part I will construct, classify, and count all
periodic billiards on the equilateral triangle.
Date: Thursday, February 15, 2007
Speaker: Aaron Robertson, Colgate University
Title: On Monochromatic Van der Waerden Triples
Abstract:
We show that the minimum number of monochromatic 3-term arithmetic progressions
that any 2-coloring of [1, n] can have is between 0.0511
n2 (1 + o(1)) and 0.0534 n2(1 +
o(1)). This disproves the conjectured answer of 0.0625
n2 (1 + o(1)) as well as a more general conjecture.
Date: Thursday, February 22, 2007
Speaker: Drew Sills, Rutgers and Georgia
Southern University
Title: On Dyson's q-Series Identities
Abstract: Before receiving his B.A. from Cambridge University,
Freeman Dyson served as referee for a pair of seminal papers by W. N. Bailey on
the derivation of identities of Rogers-Ramanujan type. Dyson wound up
contributing a number of Rogers-Ramanujan type identities of his own to Bailey's
papers, including a set of four identities related to modulus 27 in the same way
that the two Rogers-Ramanujan identities are related to the modulus 5. After
providing some mathematical and historical background, I will present a set of
identities related to the modulus 108, which I discovered experimentally by
playing around with variants on Dyson's mod 27 identities.
Date: Thursday, March 1, 2007
Speaker: Greg Chaitin, IBM Research and Rutgers
Title: Mini-course on Program-Size Complexity, Part I
Abstract:
Date: Thursday, March 8, 2007
Speaker: Greg Chaitin, IBM Research and Rutgers
Title: Mini-course on Program-Size Complexity, Part II
Abstract:
Date: Thursday, March 15, 2007
NO SEMINAR (SPRING BREAK)
Date: Thursday, March 22, 2007
Time: 12 Noon
Location: Hill 705
Speaker: Tewodros Amdeberhan, Tulane
University
Title: An arctangent dynamical system
Abstract:
Date: Thursday, March 22, 2007
Speaker: Greg Chaitin, IBM Research and Rutgers
Title: Mini-course on Program-Size Complexity, Part III
Abstract:
Date: Thursday, March 29, 2007
Speaker: Stavros Garoufalidis, Georgia
Tech
Title: An arithmetic ansatz for the singularities of
hypergeometric multisums
Abstract: A special term is a product of
binomials of linear forms in many variables. It is known that the generating
series of a special term is convergent near the origin and satisfies a linear
ODE with polynomial coefficients. There are three proofs of this property,
although they are all computationally costly, for multisums. In the talk we will
give a simple ansatz for the singularities of the generating series. We will
also give a proof of our ansatz in case the special term is positive, and an
illustration our ansatz with the Apery series, responsible for the irrationality
of zeta(3). Our ansatz is reminiscent of the Bethe ansatz, and has a motivic
interpretation using additive K-theory. If you don't know what these are, you
won't loose much!
Date: Thursday, April 5, 2007
Speaker: Michael Somos,
Georgetown University
Title: Somos sequences in context
Abstract: Somos sequences form a small part of the Elliptic Realm.
How do they relate to other parts? How unexpected are they? How could they be
discovered? What is their natural point of view? Are they related to classical
recurrence sequences? I will answer these questions as time permits.
Date: Thursday, April 12, 2007
Speaker: Michael Zieve, Rutgers
Title: Permutation Polynomials
Abstract: A polynomial f(x)
is a permutation polynomial (PP) over a field k if the map a-->f(a) permutes
the elements of k. I will present various results about PPs: for instance, it is
known that any PP over GF(q) of degree less than q^(1/4) will be a PP over
GF(q^n) for infinitely many n. Thanks to the combined efforts of several
mathematicians, we now have a conjecturally complete list of PPs of degree less
than q^(1/4), some of which are quite complicated. My talk will be accessible to
all, but I will touch on the methods used: the classification of finite simple
groups, the Riemann hypothesis for curves over finite fields, Galois cohomology,
higher ramification groups, bounds on automorphism groups of curves, etc.
Date: Thursday, April 19, 2007
Speaker: George Andrews, Pennsylvania State
University
Title: Partition Analysis and the Search for Modular Forms
Abstract: This talk will introduce P.A. MacMahon's method of
partition analysis. We will the discuss some of its applications including our
most recent discovery of certain partitions realted to directed graphs that have
modular forms as generating functions and consequently have a variety of
interesting related arithmetical problems and prospects. This work is joint with
Peter Paule and Axel Riese.
Date: Thursday, April 26, 2007
Speaker: Aaron Robertson, Colgate University
Title: An Off-diagonal Version of a Theorem of Rado
Abstract: An equation is called r-regular if every r-coloring of the
positive integers admits a monochromatic solution to the equation. Richard Rado
completely categorized those equations that are r-regular for all r. A lesser
known result of Rado's is that any 3 variable equation that has a solution in
the positive integers is 2-regular. We present an off-diagonal version of this
result as well as give the minimal n for certain equations for which every
2-coloring of [1,n] admits a monochromatic solution.
Fall 2007
Date: Thursday, July 26, 2007
Speaker:
Manuel Kauers, Research Institute for Symbolic Computation, Hagenberg, Austria
Title: A Solution to Exercise 95.
Abstract:
The 1994 edition of Concrete Mathematics (Graham/Knuth/Patashnik)
includes a research problem asking for extending "the Gosper-Zeilberger algorithm from
hypergeometric terms to terms that may involve Stirling numbers". In the talk, we propose
a solution to this problem. That is, we will (a) define a class of sequences that includes
Stirling numbers, Eulerian numbers and hypergeometric multiples of these, (b) give a
sufficient criterion for sums over such sequences to obey a recurrence equation, and
(c) present algorithms for computing such a recurrence equation efficiently.
Date: Thursday, September 6, 2007
NO SEMINAR
Date: Thursday, September 13, 2007
NO SEMINAR (Rosh Hashanah)
Date: Thursday, September 20, 2007
Speaker:
Doron Zeilberger, Rutgers University
Title: Why Erev Yom Kippur Can Never Be On Thursday Evening?
Abstract:
The short answer is that that was G-d's will, but a longer answer
involves some beautiful mathematical properties of an Ancient Algorithm
called the Jewish Calendar.
Maple Package
Date: Thursday, September 27, 2007
Speaker:
Sandra Kingan, DIMACS
Title: Experimenting With Matroids
Abstract:
In this talk I will describe Oid, a software system for experimenting
with matroids. Several combinatorial objects such as graphs,
matrices, linear spaces, and lattices can be viewed as matroids. Oid
accepts matroids in different formats and can check them for
isomorphism. It can list the bases, circuits, flats, connectivity
function, Tutte polynomial, etc. For matroids represented as matrices,
it has a finite field engine that handles structural computations such
as deletions, contractions, extensions, and coextensions. Code from
Oid's library can be used to make stand-alone programs for computing
splitters, stabilizers, and minimal excluded minors. I will describe
how we use Oid to automate case analyses in structural results. We
also use Oid to generate data and discover patterns that lead to
conjectures, which we then prove by completely theoretical methods.
Date: Thursday, October 4, 2007
Speaker:
Walter Stromquist, Swarthmore College
Title: Cakes, and Pies, and Fair Division
Abstract:
Mathematicians enjoy cakes for their own sake and as a metaphor for more
general fair division problems. We describe the state of the art of
cake cutting, including some new results on computational complexity.
Suppose that a cake is to be divided by parallel planes into n pieces,
one for each of n players whose preferences are defined by separate
measures. We show that there is always an "envy-free" division, meaning
that no player prefers another player's piece, and that such a division
is always Pareto optimal. Alas, such a division cannot always be found
by a finite procedure. Even assuring each player 1/n of the cake
seems to require n log n steps.
Pies, of course, have their own attractions. We cut them radially into
wedges. It turns out that pie cutters, unlike cake cutters, may be
forced to choose between envy-freeness and Pareto optimality.
Slides from Walter Stromquist's talk
Date: Thursday, October 11, 2007
Speaker:
Dan Cranston, DIMACS
Title: Antimagic labelings of regular bipartite graphs: An application of the Marriage Theorem
Abstract:
A labeling of a graph is a bijection from its edge set to the set {1,
2,... , |E(G)|}.
A labeling is antimagic if for every pair of distinct vertices
u and v, the sum of the labels on edges incident to u is different
from the sum of the labels on edges incident to v. We say a graph is
antimagic if it has an antimagic labeling. In 1990, Ringel conjectured
that every connected graph other than K2 is antimagic. The most
significant progress was been made by Alon et al. (in 2004), who showed
there exists a constant C, such that if an n-vertex graph G has
δ(G)≥ Cn, then G is antimagic.
In this paper, we show that every regular bipartite graph (with degree at
least 2) is antimagic. Our technique relies heavily on the Marriage
Theorem.
Date: Thursday, October 18, 2007
SEMINAR CANCELLED due to anticipated football-related mayhem.
Date: Thursday, October 25, 2007
Speaker:
Debbie Yuster, DIMACS
Title: Geometry of Hyperdeterminants
Abstract:
Hyperdeterminants are the analogs of determinants for
higher-dimensional matrices. Hyperdeterminants arise in many contexts in
mathematics and other sciences, and have some very interesting associated
geometry. For example, the 2x2x...x2-hyperdeterminant has a strong
relationship to triangulations of the n-cube. In this talk we will explore
the geometry of the 2x2x2- and 2x2x2x2-hyperdeterminants. I will also
discuss how we exploited symmetry in order to compute the
2x2x2x2-hyperdeterminant (a challenge issued by I.M. Gelfand). This is
joint work with Peter Huggins, Bernd Sturmfels, and Josephine Yu.
Date: Thursday, November 1, 2007
Speaker:
Pierre Lalonde, Universite du Quebec a Montreal
Title: On the Minors of the Paths Matrix in a Forest
Abstract:
Consider an acyclic graph (a forest) with edges labeled by a formal
"weight". The paths matrix gives the weight of the path (product of
the labels of its edges) between any two vertices of the graph. We
will have a combinatorial interpretation of the minors of this matrix
that will lead, via a sequence of bijections and sign-reversing
involutions, to an enumerative formula. The result generalizes the
Graham-Pollak theorem on the "distance-matrix" and many of its
variants, which can often be recovered by linearization.
Slides from Pierre Lalonde's talk
Date: Thursday, November 8, 2007
Speaker:
Neil J. A. Sloane, AT&T
Title: The On-Line Encyclopedia of Integer Sequences: New Sequences (Solved and Unsolved), Music, etc.
Abstract:
This talk will discuss some highlights from the 10,000 sequences
that were added to the OEIS in the past year - some that have
been analyzed (including some joint work with David Applegate
and Marc LeBrun), and many others that are crying out to be
analyzed. Some new ways for displaying sequences, both
graphically and musically, will be demonstrated.
Date: Thursday, November 15, 2007
Speaker:
Dimitrije Kostic, Drew University
Title: How to Experiment with Logjams
Abstract:
In this talk, we present the notion of an x-logjam, where x is a
vector of n integers. The set of x-logjams can be thought of as a structure
similar to the order ideal (under the partial order of majorization) of x. We
explore some combinatorial and number theoretic aspects of these structures,
and how some random experimentation led to an interesting conjecture.
Date: Thursday, November 22, 2007
NO SEMINAR (Thanksgiving)
Date: Thursday, November 29, 2007
Speaker:
Victor Moll, Tulane University
Title: p-adic pictures from irresistible sequences
Abstract:
Legendre's formula νp(n!)=(n-sp(n))/(p-1) is the simplest instance of the p-adic valuation for a sequence defined by a first order recurrence tn=Q(n)tn-1. Here Q is a polynomial with integer coefficients and sp(n) is the sum of the digits of n in base p.
In this talk we describe the asymptotics of νp(tn) as n → ∞
The extension to the case tn=Q1(n)tn-1+Q2(n)tn-2 will be illustrated with the p-adic valuation of the Stirling numbers.
Joint work with T. Amdeberhan, Dante Manna and Luis Medina.
Click for properly typeset abstract in pdf.
Date: Thursday, December 6, 2007
Speaker:
Aviezri Fraenkel, Weizmann Institute, Israel [joint talk with Mathematics Colloquium]
Title: Polynomial Approximation of Computationally-Hard Sequences
Abstract:
The question whether an m-tuple (x1,...,xm) in Z≥ 0m is in (a1,...,am), where the ai are given integer sequences, can sometimes be decided efficiently
(in polynomial time). More often, the answer is unknown, the best
known algorithms being exponential. We present a polynomial approximate
algorithm for deciding this question for some sequences ai. Specifically, we consider the complementary sequences an=mex{ai, bi : 0 ≤ i < n}, b=an
+ |_ n/k _| (sequences A102528/9 in Sloane's encyclopedia for k=2).
Using Fekete's Lemma we show that the polynomially computable sequences sn=|_n α_|, tn=|_n β_| where α = (√(17)+3)/4, β=α+1/2 are very good approximations, namely, sn-1 ≤ an ≤ sn, tn-1 ≤ bn ≤ tn for all n ≥ 1 (k=2). We conjecture that the percentage of n for which an=sn-1 is about 73%, an=sn, 19%, an=sn-2, 8%. Similar results for bn, tn.
Analogous results for every fixed k>1. Existence of a limiting
distribution, with one of the percentages dominant, would lead to first
probabilistic algorithms for determining the winning positions of
certain impartial games, where the (a1, ..., am) are the second player winning positions.
Joint work with Udi Hadad.
Click for properly typeset abstract in pdf.