RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR
Archive of Speakers and Talks --- 2019
Spring 2019
Date: Jan. 31, 2019
Speaker: Neil Sloane, the OEIS Foundation and Rutgers University
Title: Coordination Sequences, Planing Numbers, and Other Recent Sequences (II)
Abstract:
Take the graph of a periodic tiling of the plane. The coordination
sequence with respect to a node P gives the number of nodes that are n
edges away from P. Chaim Goodman-Strauss and I have a new simple method
for obtaining generating functions for such sequences. There are a
number of interesting open questions. I will also discuss Lenormand's
"raboter" operation that planes down numbers (e.g., 231 = 11100111_2
becomes 27 = 11011_2). This is an updated version of a talk given to
just 4 people during the blizzard of Nov 15 2018. There is also a lovely
new open problem: the knight's-move version of the Ulam-Warburton
cellular automaton (A319018).
Posted on Vimeo (2 parts):
Part 1
Part 2
Slides from talk
Date: Feb. 7, 2019
Speaker: Emily Sergel, University of Pennsylvania
Title: Some tools for proving asymptotic normality and an application to cores
Abstract:
An s,t-core partition is a partition with no hook length equal to s or
t. Recently Ekhad-Zeilberger used experimental methods to prove that the
asymptotic distribution size among s,t-cores (with s-t fixed) is not
normal. Following that, Zaleski gave strong evidence that (s,s+1)-cores
with distinct parts are asymptotically normally distributed. My
coauthors and I found both results surprising, and the proof we
discovered for the later one uses some beautiful, classical results. The
main purpose of this talk is to spread these tools. If time permits, I
will also discuss the possibility (and challenge) of applying these
tools to related core problems. Joint work with János Komlós and Gábor
Tusnády.
Posted on Vimeo (2 parts):
Part 1
Part 2
Date: Feb. 14, 2019
Speaker: Glenn Shafer, Rutgers Business School
Title: Game-Theoretic Foundations for Probability and Statistics
Abstract:
Fermat and Pascal's two different methods for solving the problem of
division lead to two different mathematical foundations for probability
theory: a measure-theoretic foundation that generalizes the method of
counting cases used by Fermat, and a game-theoretic foundation that
generalizes the method of backward recursion used by Pascal. The
game-theoretic foundation has flourished in recent decades, as
documented by my forthcoming book with Vovk, Game-Theoretic Foundations
for Probability and Finance. In this book's formulation, probability
typically involves a perfect-information game with three players, a
player who offers betting rates (Forecaster), a player who tests the
reliability of the forecaster by trying to multiply the capital he risks
betting at these rates (Skeptic), and a player who decides the outcomes
(Reality).
In this talk I will review the game-theoretic foundation for probability
as briefly as possible and then discuss its application to mathematical
statistics. The usual formulation for mathematical statistics begins
with the assumption that the statistician has only partial knowledge of
the probability measure that describes a phenomenon. The corresponding
game-theoretic move is to suppose that the statistician stands outside
the perfect-information game, seeing only some of the moves or some of
its consequences.
Posted on
Vimeo
Slides from talk
Date: Feb. 21, 2019
Speaker: Colin R. Defant, Princeton University
Title: Structure in Stack-Sorting
Abstract:
The study of permutation patterns began with Knuth's analysis of a certain
"stack-sorting algorithm" in 1968. In his 1990 PhD. thesis, West
investigated a deterministic variant of Knuth's algorithm, which we can
view as a function s that defines a dynamical system on the set of
permutations. He defined the fertility of a permutation to be the number
of preimages of that permutation under s. We will describe a colorful
method for computing the fertility of any permutation, answering a
question of Bousquet-Mélou. Applications of this method allow us to reprove and generalize several
known theorems and improve the best known upper bound for the number of
so-called "t-stack-sortable" permutations of length n when t=3 and when
t=4. The method also allows us to connect the stack-sorting map with free
probability theory, many well-studied combinatorial objects, and several interesting sequences. Finally, we will
consider two operators on words that extend the stack-sorting map.
Posted on
Vimeo
Slides from talk
Date: Feb. 28, 2019
Speaker: Chaim Even Zohar, University of California, Davis
Title: Patterns in Random Permutations
Abstract:
Every k entries in a permutation can have one of k! different relative
orders, called patterns. How many times does each pattern occur in a
large random permutation of size n? The distribution of this
k!-dimensional vector of pattern densities was studied by Janson,
Nakamura, and Zeilberger (2015). Their analysis showed that some
component of this vector is asymptotically multinormal of order
1/sqrt(n), while the orthogonal component is smaller. Using
representations of the symmetric group, and the theory of
U-statistics, we refine the analysis of this distribution. We show
that it decomposes into k asymptotically uncorrelated components of
different orders in n, that correspond to representations of Sk. Some
combinations of pattern densities that arise in this decomposition
have interpretations as practical nonparametric statistical tests.
Posted on
Vimeo
Date: Mar. 7, 2019
Speaker: Nathan Fox, College of Wooster
Title: Trees, Fibonacci Numbers, and Nested Recurrences
Abstract:
Conolly's sequence (A046699) is defined by the nested recurrence
relation C(n)=C(n-C(n-1))+C(n-1-C(n-2)) and the initial conditions
C(1)=C(2)=1. This sequence is monotone increasing with each positive
integer appearing at least once, a property known in the literature as slow.
Conolly's sequence and several other slow sequences generated by nested
recurrences are known to have combinatorial interpretations in terms of
enumerating leaves in infinite tree structures. For the Conolly sequence,
the tree-based interpretation illuminates an intimate connection between the
sequence and the powers of two. In fact, the Conolly sequence has an
alternate, purely number-theoretic definition based on powers of two.
Replacing powers of two with Fibonacci numbers in this construction yields a
different slow sequence (A316628). In this talk, we will describe the three
different ways of constructing the Conolly sequence (recurrence, number
theory, trees) and show why they all yield the same sequence. Then we will
construct this new sequence based on the Fibonacci numbers, which also is
slow, has a tree-based interpretation, and satisfies a nested recurrence.
If time permits, we will describe how to generalize the construction to
discover many new integer sequences.
Posted on
Vimeo
Date: Mar. 14, 2019
Speaker: Doron Zeilberger, Rutgers University
Title: What is Pi, and what it is not
Abstract:
The short answer to the second question of the title is: "It
is NOT a number".
For a longer answer, and for an answer to the first question, come to the talk.
Posted on
Vimeo
Date: Mar. 21, 2019
Spring Break (no talk)
Date: Mar. 28, 2019
Speaker: Craig Larson, Virginia Commonwealth University
Title:Automated Conjecturing in Mathematics - with the CONJECTURING Program
Abstract:
I will describe the ideas underlying the program CONJECTURING which
can be used to make conjectures about upper or lower bounds of
invariants (or necessary or sufficient conditions for properties) for
a wide variety of mathematical objects. The conjecturing heuristic is
the heuristic Fajtlowicz used in his Graffiti program; our program has
been broadly generalized to be useful in many domains - and the code
is open source, and can be installed as a Sage package.
We will give several examples of theorems conjectured by the program,
as well as a selection of open conjectures. We will also mention ways
to leverage the program to maximize its utility as a tool for
researchers. This is joint work with Nico Van Cleemput (Ghent
University).
Posted on
Vimeo
Slides from talk
Date: Apr. 4, 2019
Speaker: Jesús Guillera, University of Zaragoza
Title: When 1/pi^2 and Calabi-Yau meet
Abstract:
In this lecture, in memory of Gert Almkvist (1934-2018), I will describe some fascinating
connections he found between a family of series for 1/pi^2 and Calabi-Yau theory.
Posted on
Vimeo
Slides from talk
Date: Apr. 11, 2019
Speaker: Ein-Ya Gura, Hebrew University of Jerusalem
Title: Game Theory-An Alternative Mathematical Experience
Abstract:
Few branches of mathematics have been more influential in the social
sciences than game theory. In recent years, it has become an essential
tool for all social scientists studying the strategic behavior of
competing individuals, firms, and countries. However, the mathematical
complexity of game theory is often very intimidating for students who
have only a basic understanding of mathematics. The book: Insights into Game Theory: An Alternative Mathematical Experience
addresses this problem by providing students with an understanding of
the key concepts and ideas of game theory without using formal
mathematical notation.
In this talk I will present the main ideas of the book, what was our
motivation to write the book, and what we are doing now by using it.
Posted on
Vimeo
Slides from talk
Date: Apr. 18, 2019
Speaker: Harry Crane, Rutgers (Statistics)
Title: Some experimental observations and open questions about the alpha-permanent
Abstract:
The alpha-permanent is a matrix function that has a similar algebraic
form to the determinant but exhibits very different computational
behavior. The permanent (alpha=1) is known to be #P-complete, and is a
fundamental object in computational complexity theory. The more general
alpha-permanent appears in statistical models for point pattern data
and combinatorial data (e.g., partition and permutation data), where its
computational complexity limits its applied use in many cases.
I'll discuss some algebraic properties of the alpha-permanent which
suggest natural connections to familiar concepts in probability theory
and statistics. I'll also describe some immediate research problems
that arise out of these observations.
Posted on
Vimeo
Date: Apr. 25, 2019
Speaker: Eddy Chen, Rutgers (Philosophy)
Title:An Introduction to the Foundations of Quantum Theory
Abstract:
In this brief talk, we discuss the basic ingredients of quantum theory
and some steps towards understanding what it means. To that end, we
focus on a simple experiment (the double slit experiment), which nicely
illustrates the "quantum measurement problem". We suggest that the
standard textbook version of quantum mechanics, though making correct
predictions, lacks both conceptual and mathematical tools to
consistently analyze the situation. We introduce Bohmian mechanics as a
precise mathematical-physical theory that solves the measurement
problem. Time permitting, we may discuss some other solutions to the
quantum measurement problem and some other issues in the foundations of
quantum theory. [This is an introductory talk. We assume very little
knowledge of physics.]
Posted on
Vimeo
Fall 2019
Date: Thurs., Sept. 12, 2019. Note new title and speaker [the original talk is rescheduled to Jan. 30, 2020].
Speaker: Yukun Yao, Rutgers University
Title:
The Peaceable Queens Problem
Abstract:
One of the fascinating problems mentioned in a recent beautiful article
(Oct. 2018 issue of the Notices of the American Mathematical Society), by guru Neil Sloane,
is that of the maximal number of placing the same number of white queens and black queens on an n by n chess board so that no queen attacks
any queen of the opposite color. While the general soution is still open. We made some interesting progress.
[Joint work with Doron Zeilberger]
Posted on Vimeo
Date: Thurs., Sept. 19, 2019
Speaker: Adi Ben Israel Rutgers University (RUTCOR)
Title: Contour approximation of data, with applications
Abstract: Given a set of points S in Rn (the data), a contour approximation of S is a function that captures most points of S in its lower level sets,
A concrete application is the home range of an animal population, or the territory occupied by it, shown in 1980 by Dixon and Chapman to
involve the harmonic mean of certain distances, a result since then confirmed for many species. The harmonic mean of distances, or resistances,
also features in inverse distance weighted interpolation clustering, parallel circuits and multi-facility location.
This lecture gives an axiomatic framework, and a probabilistic optimization model that unifies the above results,
a model applied successfully to clustering and classification.Joint work with Tsvetan Asamov and Cem Iyigun.
[longer version]
[Here are the slides]
Posted on Vimeo
Date: Thurs., Sept. 26, 2019
Speaker: Yonah Biers-Ariel, Rutgers University.
Title: Flexible Schemes for Pattern-Avoiding Permutations
Abstract:
We present an extension of the enumeration schemes of Zeilberger and Vatter
so that they can efficiently enumerate many new classes of pattern-avoiding permutations including all such classes with a regular insertion encoding.
Posted on Vimeo
Date: Thurs., Oct. 3, 2019
Speaker: Yotam Smilansky, Rutgers University.
Title: Patterns and Partitions
Abstract:
A colored partition of a set in Rd is its representation as a disjoint union of subsets, referred to as tiles, where each tile is also assigned a color.
In the talk, we will consider sequences of colored partitions defined using multiscale substitution rules on finite collections of colored prototiles.
In the substitution process, which generalizes a construction first introduced by Kakutani, tiles of maximal volume in a given partition are replaced
by colorful patterns consisting of rescaled copies colored prototiles, thus defining the next partition in the sequence. Tiles that appear
in the process are modeled by a flow on a directed weighted graph, and distributional and statistical questions on sequences of partitions are
reformulated as questions on the distribution of paths on graphs. Under a natural incommensurability assumption, special properties of the poles
of the Laplace transforms of graph counting functions imply various explicit statistical results. In addition, computer experiments reveal
the beautiful patterns in which these poles appear in the complex plane, patterns which seem to be closely related to Diophantine properties
of the generating substitution rule.
Here are the slides
Posted on Vimeo
Date: Thurs., Oct. 10, 2019
Speaker: Neil Sloane, The OEIS Foundation and Rutgers University
Title: Old and New Problems from 55 Years of the OEIS
Abstract::
Some favorite old and new problems: Dissections; roots of theta series; the Recaman hypothesis; getting to zero by subtracting primes; van Eck's sequence;
the Forest Fire sequence; a strange property of 909; covering with geometric progressions; etc. I'll end with some remarks about keeping the OEIS running.
Here are the slides
[Some slides have been changed since the talk]
Posted on Vimeo
Date: Thurs., Oct. 17, 2019
Speaker: Robert Doughtery-Bliss
Title:: The Ergonomics of Computer Algebra
Abstract:
Popular computer algebra systems are rightly praised for automating away the
"trivial" parts of mathematical life. These good deeds have, frankly, enabled
them to coast along without much critical examination. No longer! Borrowing
some paradigms from software engineering, we will discuss the "ergonomics" of
computer algebra systems, the importance of language design, and show that not
all tools are created equal.
Here are the slides
Posted on Vimeo
Date: Thurs., Oct. 24, 2019
Speaker: Amita Malik, Rutgers University
Title: On the parity of restricted partition functions
Abstract:
The ordinary partition function is believed to take even values approximately half the time. However, the best results known in this regard are far from confirming this belief. In fact, it was not until 1959 when it was shown that this function takes even (odd) values infinitely often. We discuss results in this spirit for certain restricted partition functions.
Posted on Vimeo
Date: Thurs., Oct. 31, 2019
Speaker: Michael Kiessling, Rutgers University.
Title:: The classical radiation reaction problem
Abstract:
Physics folklore says that a classical point charge, when accelerated by some electromagnetic
field, produces electromagnetic radiation through which it loses energy and momentum to the field degrees
of freedom. Many physicists (in particular: Abraham, Lorentz, Dirac, Landau and Lifshitz, ...) tried to compute
the associated `radiation-reaction' force on the point charge. Alas, all proposals so far failed the litmus test:
their fomulas produce a vanishing radiation-reaction when a point charge is accelerated by a uniform electric
field (such as occurs between the plates of a charged capacitor). Recently I combined ideas of Poincare' and
of Bopp, Lande'-Thomas, and Podolsky, and managed to compute the exact expression for the radiation-reaction
force in a classical electrodynamics model. Jointly with Shadi Tahvildar-Zadeh we also proved local well-posedness
of the joint initial-value problem for fields and point charges. Since the self-force expression is very complicated,
MAPLE was then used to study the motion of a point charge accelerated by a constant electric field. Radiation-reaction
effects showed very clearly and dramatically: unexpectedly, a dynamical phase transition was found. Subsequently
a proof of this dynamical scenario was supplied. I will explain all this without getting into the nitty-gritty of the
technical details.
Here are the slides
Posted on Vimeo
Date: Thurs., Nov. 7, 2019
Speaker: Vladimir Retakh, Rutgers University
Title::
Diamond operations on lattices and factorizations of noncommutative polynomials
Abstract:
We introduce and study new operations on lattices and
directed graphs and use them to explain rational relationships among
pseudo-roots of noncommutative polynomials. This is a joint work with
Michael Saks.
Posted on Vimeo
Date: Thurs., Nov. 14, 2019
Speaker: Mingjia Yang, Rutgers University
Title:
Systematic counting of pattern-avoiding partitions and some new partition identities
Abstract: See here.
Here are the slides
Posted on Vimeo
Date: Thurs., Nov. 21, 2019
No talk (due to the
Dimacs 30 conference
held at the Heldrich hotel, New Brunswick)
Date: Thurs., Nov. 28, 2019
No talk (Thanksgiving)
Date: Dec. 5, 2019
Speaker: Shashank Kanade, University of Denver.
Title: Searching for modular companions
Abstract:In this talk, I will explain that there are (most likely) no other "modular" companions to certain mod-9 partition
identities conjectured jointly with Matthew C. Russell. The experiments crucially hinge on Kursungoz's analytic sum-sides
for these conjectures and a result of Vlasenko-Zwegers on analytic properties of such sum-sides.
I'll have to let q be an actual number, but hopefully it will be fun!
Posted on Vimeo
Date: Dec. 12, 2019
Speaker: Fernando Chamizo, Universidad Autónoma de Madrid.
Title: Where is the spiral?
Abstract:
It is a known fact that the partial sums of some trigonometric series generate appealing patterns when plotted as points in the complex plane. The usual explanation is based on variants of the stationary phase approximation employed in analytic number theory. In a simple case, the theory predicts an Archimedean spiral, but the computer shows a very different experimental truth. It turns out that at least for integer values of a parameter the computer plot can be explained solving a curious recurrence relation.
[Joint work with D. Raboso]
Posted on Vimeo