RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Archive of Speakers and Talks --- 2017


Spring 2017

Date: January 26, 2017
Speaker: Neil J. A. Sloane, Rutgers University and The OEIS Foundation
Title: Winter Fruits: New Problems from OEIS
Abstract:
          New problems in number theory and combinatorics (probably not so hard) from Dec 2016 and Jan 2017. Crop circles, the number pau, and (more seriously) Fibonachos, Richard Guy, digital sums, carryless sums, Tisdale's sieve, square permutations, compact numbers, and 2-D counting.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 2, 2017
Speaker: John Conway, Princeton University
Title: The Faulhaber triangle and the Bernoulli numbers and what they're good for
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 9, 2017
TALK POSTPONED TO APRIL 6, DUE TO WEATHER


Date: February 16, 2017
Speaker: John Miller, Johns Hopkins University
Title: The distribution of extremal values of linear recurrences modulo m
Abstract:
          Consider the Fibonacci sequence modulo m, e.g. modulo 10:
1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8,...
We can also start with different initial values, e.g. (3,4):
3, 4, 7, 1, 8, 9, 7, 6, 3, 9, 2, 1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9, 2, 1,...
This sequence has a cycle of length 12. The cycle lengths of such sequences are a classical subject related to algebraic number theory. We may also consider the "extremal values" of this sequence. For example, the 2nd sequence has minimum 1 and maximum 9. In this talk we will discuss the properties of such extremal values as we vary the initial values and the modulus, for the Fibonacci recurrence and for other examples of linear recurrences. We will do some experiments, and we will connect the distribution of extremal values to questions of elementary geometry.

This talk arises out of work done in Professor Zeilberger's experimental mathematics class in the spring of 2012.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 23, 2017
Speaker: Matthew Russell, Rutgers University
Title: Fantastic identities, and where to find them
Abstract:
          In this talk, I will present a veritable cornucopia of new partition identities discovered over the past calendar year. Some of these have proofs, some are still conjectures; some should break new ground in the study of affine Lie algebra; all were discovered with the use of computers (in a surprising variety of ways). This is based on joint work in a handful of different groups, including with Terence Coelho and Jongwon Kim, with Shashank Kanade and Debajyoti Nandi, and with Shashank Kanade.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 2, 2017
Speaker: Adi Ben-Israel, Rutgers University
Title: Matrix volume and its applications
Abstract:
          The volume vol(A) of an mxn matrix A of rank r is: (a) the product of the r singular values of A, or (b) the square root of the sum of squares of all r x r subdeterminants of A, or (c) the volume of the image under A of a unit cube in the range of A transpose. Definition (b) is applicable to non-numerical matrices, in particular to rectangular Jacobians. Some representative applications will be discussed.

References:
[1] The matrix volume: http://benisrael.net/VOLUME.pdf
[2] Application to change-of-variables in integration: http://benisrael.net/INTEGRAL-AMS.pdf
[3] Application to probability: http://benisrael.net/MADISON-AMS.pdf
[4] Low-rank approximation In this paper we show that the negative sample distance covariance function is a quasiconcave set function of samples of random variables that are not statistically independent. We use these properties to propose greedy algorithms to combinatorially optimize some diversity (low statistical dependence) promoting functions of distance covariance. Our greedy algorithm obtains all the inclusion-minimal maximizers of this diversity promoting objective. Inclusion-minimal maximizers are multiple solution sets of globally optimal maximizers that are not a proper subset of any other maximizing set in the solution set. We present results upon applying this approach to obtain diverse features (covariates/variables/predictors) in a feature selection setting for regression (or classification) problems. We also combine our diverse feature selection algorithm with a distance covariance based relevant feature selection algorithm of [7] to produce subsets of covariates that are both relevant yet ordered in non-increasing levels of diversity of these subsets.of matrices: A. Deshpande, L. Rademacher et al, Matrix Approximation and Projective Clustering via Volume Sampling, Theory of Computing 2(2006), 225-247

Slides from talk
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 9, 2017
Speaker: Praneeth Vepakomma, Motorola Solutions
Title: Combinatorics of Distance Covariance: Inclusion-Minimal Maximizers of Quasi-Concave Set Functions for Diverse Variable Selection
Abstract:
          'Distance Covariance' is a popular measure of statistical dependence between random variables. In this talk I'd provide an algorithm to combinatorially choose subset(s) of random variables that are least statistically dependent on each other in terms of their distance covariances. Some applications of our framework to regression problems in statistics would also be discussed. Required conditions for our framework to be applicable to combinatorial optimization problems would also be shared. This is joint work done with Yulia Kempner.

Original Abstract (more technical than the talk will be):
          In this talk we show that the negative sample distance covariance function is a quasiconcave set function of samples of random variables that are not statistically independent. We use these properties to propose greedy algorithms to combinatorially optimize some diversity (low statistical dependence) promoting functions of distance covariance. Our greedy algorithm obtains all the inclusion-minimal maximizers of this diversity promoting objective. Inclusion-minimal maximizers are multiple solution sets of globally optimal maximizers that are not a proper subset of any other maximizing set in the solution set. We present results upon applying this approach to obtain diverse features (covariates/variables/predictors) in a feature selection setting for regression (or classification) problems. We also combine our diverse feature selection algorithm with a distance covariance based relevant feature selection algorithm of Kong, Wang, and Wahba to produce subsets of covariates that are both relevant yet ordered in non-increasing levels of diversity of these subsets.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 23, 2017
Speaker: Justin Semonsen, Rutgers University
Title: On Oda's Strong Factorization Conjecture
Abstract:
          In this talk we will translate a conjecture by Oda about birational maps of toric varieties into a conjecture about fans of simplicial cones. Further combinatorial techniques reduce the conjecture to an algorithmic form suitable for computation and experimentation. Experimental results gotten via Java provide insight into how one might prove the conjecture via a much simpler mechanism.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 30, 2017
Speaker: Nathan Fox, Rutgers University
Title: An Exploration of Nested Recurrences Using Experimental Mathematics (thesis defense)
Abstract:
          Nested recurrence relations, such as the Hofstadter Q-recurrence Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2)), have no general theory. Solutions are highly dependent on the initial conditions, and many sequences they generate are not even known to be infinite. In this talk, we will see a variety of results pertaining to sequences arising from nested recurrences. These results include a method of automatically discovering solutions of a particular form, some sequences exhibiting never-before-seen behavior, and methodology for analyzing entire families of initial conditions simultaneously.
Slides from talk
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 6, 2017 (rescheduled from February 9, 2017)
Speaker: Evita Nestoridi, Princeton University
Title: Shuffling large decks of cards and the Bernoulli-Laplace urn model
Abstract:
          In board games, in Casino games with multiple decks and cryptography, one is sometimes faced with the practical problem: how can a human (as opposed to the computer) shuffle big decks of cards. One natural procedure (used by casinos) is to break the deck into several reasonable size piles, shuffle each thoroughly, assemble, do some simple deterministic thing (like a cut) and repeat. G. White and I introduce variations of the classical Bernoulli-Laplace urn model (the first Markov chain!) involving swaps of big groups of balls. A coupling argument and spherical function theory allow the original problem to be solved.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 13, 2017
Speaker: Aaron Robertson, Colgate University
Title: Probability and Ramsey Theory
Abstract:
          We will find a threshold function f(k;r) such that almost all r-colorings of more than f(k;r) consecutive integers admit a monochromatic k-term arithmetic progression while almost no r-colorings do if we have less than f(k;r) consecutive integers. We will then move on to investigate the distribution of the random variable X = number of monochromatic k-term arithmetic progressions in [1,n] under random coloring of each integer. It is known that X tends to a Poisson distribution as k tends to infinity. We investigate what X is for small k.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 20, 2017
Speaker: Joe Kileel, Univ. of California, Berkeley
Title: Using Computational Algebra for Computer Vision
Abstract:
          Scene reconstruction is a fundamental task in computer vision: given multiple images from different angles, create a 3D model of a world scene. Nowadays self-driving cars need to do 3D reconstruction in real-time, to navigate their surroundings. In this talk, we will explain how key subroutines in reconstruction algorithms amount to solving polynomial systems. We will quantify the "algebraic complexity" of systems that engineers have been hoping to solve quickly and reliably for some while. Our approach combines symbolic and numerical methods from computational algebra. Those wondering "if algebraic geometry is good for anything practical" are especially encouraged to attend.
Slides from talk
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 27, 2017
Speaker: Vince Vatter, University of Florida
Title: Sorting with restricted containers
Abstract:
          Restricted containers are a new data structure generalizing stacks, queues, and deques. I will describe how this viewpoint rapidly leads to functional equations for the classes of sortable permutations. These functional equations can sometimes be solved by the kernel method, but, much more easily, by the computer via guessing and checking.

I will also present several general results about the rationality, algebraicity, and the existence of Wilfian formulas for certain classes of permutations sortable by restricted containers and present several examples of relatively small permutation classes which, although we can generate thousands of terms of their enumerations via this viewpoint, appear to not have D-finite generating functions.

This is joint work with Michael Albert, Cheyne Homberger, Jay Pantone, and Nathaniel Shar.
Posted on Vimeo (2 parts): Part 1 Part 2


Fall 2017

Date: September 14, 2017
Speaker: Amita Malik, Rutgers University
Title: Sporadic Apéry-like numbers modulo primes
Abstract:
          At the ICM in 1978, R. Apéry's proof of the irrationality of ζ(3) was presented. In this proof, he introduced a sequence of integers, now known as Apéry numbers. Apéry-like numbers are special integer sequences, studied by Beukers and Zagier, which are modeled after Apéry numbers. Among their remarkable properties are connections with modular forms, Calabi-Yau differential equations, and a number of p-adic properties, some of which remain conjectural.
A result of Gessel shows that Apéry's sequence satisfies Lucas congruences. We prove corresponding congruences for all sporadic Apéry-like sequences. While, in some cases, we are able to employ approaches due to McIntosh, Samol-van Straten and Rowland-Yassawi to establish these congruences, there are few others for which we require a finer analysis. As an application, we investigate modulo which numbers these sequences are periodic. In particular, we show that the Almkvist-Zudilin numbers are periodic modulo 8, a special property which they share with the Apéry numbers. This is joint work with Armin Straub.

Posted on Vimeo (3 parts): Part 1 Part 2 Part 3


Date: September 21, 2017
NO SEMINAR FOR ROSH HASHANA


Date: September 28, 2017
Speaker: Doron Zeilberger, Rutgers University
Title: CNF-DNF and All That
Abstract:
          The acronyms CNF and DNF feature prominently in Norbert Blum's brave attempt at proving the most important open problem of our time (with the possible exception of the much more intractable problem of establishing world peace). But there are many other aspects of CNFs, and their duals, DNFs, worth pursuing for their own sake. These other problems won't earn you a million dollars, but they are even more fun.
[Joint work with Anthony Zaleski]

Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 5, 2017
Speaker: Neil Sloane, The OEIS Foundation and Rutgers University
Title: Three (No, 8) Lovely Problems from OEIS
Abstract:
          I'll discuss problems from geometry, number theory, and the theory of computing.
(i) Poonen and Rubinstein famously counted the intersection points in a regular n-gon with all diagonals drawn. But what if we start with n points on a line rather than a circle? (A6561, A290447).
(ii) Mysterious things happen when you iterate arithmetic functions, for example n -> (φ(n)+σ(n))/2. Although it is hard to believe, the orbit of 270 seems to be integral and ever-increasing (A291789). John Conway recently lost a $1000 wager on the iteration of another arithmetic function (A195264).
(iii) Back in the 1930s Emil Post studied "tag systems", which in general are now known to be universal Turing machines. However, Post's simple 3-shift tag system is still open, 80 years later. In the last month there has been some progress, but this is ongoing work and the topic will be postponed until a later talk.

Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 12, 2017
Speaker: Art DuPre, Rutgers University
Title: Wittgenstein versus Gödel
Abstract:
          Because of a "Notorius" paragraph in Wittgenstein's 'Remarks on the Foundations of Mathematics' and the controversy it caused among mathematicians, logicians and philosophers, Hao Wang, among others, lay this controversy squarely at the feet of the fact that Godel was a Platonist, while Wittgenstein was a strong anti-Platonist. A Platonist believes that new mathematics is discovered, while a creationist believes that new mathematics is created. These and other ideas will be explored.

Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 19, 2017
Speaker: Edinah Gnang, Johns Hopkins University
Title: Growing graceful trees
Abstract:
          In this talk we will describe and motivate the graceful labeling conjecture. We will discuss enumerative aspects of the conjecture and describe how it serves as a wonderful test case for the polynomial method as well as for symbolic algebraic experimental methods.

Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 26, 2017
Speaker: Harry Gingold, West Virginia University
Title: Problems in Celestial Mechanics
Abstract:
          At the beginning of the 20th century, G.D. Birkhoff called the N body problem of celestial mechanics the most celebrated problem in mathematics. Although this perspective has changed after an explosive expansion of mathematics in the past decades, numerous important problems remain. I will discuss a selected number of these problems.
The talk will be accessible to graduate students.

Posted on Vimeo (2 parts): Part 1 Part 2
Accompanying Slides


Date: November 2, 2017
Speaker: Anthony Zaleski, Rutgers University
Title: Boolean Satisfiability
Abstract:
          Given a logical formula in n Boolean variables, how can we determine whether there is an assignment to the variables that makes it true? This is the task of Boolean satisfiability (SAT), the first problem proved to be NP-complete. Here, we shall talk about our ideas for constructing SAT solvers. Then, we shall discuss some computer-generated results and conjectures regarding generalizations of SAT inspired by integer covering systems. Joint work with Dr. Zeilberger.

Posted on Vimeo (2 parts): Part 1 Part 2


Date: November 9, 2017
Speaker: Alexander Ryba, Queens College (CUNY)
Title:
Abstract:
          A well known example of an incidence theorem is Pappus' Hexagon Theorem that:
                If the six vertices of a hexagon lie alternately on two straight lines, then the three intersection points of opposite sides are collinear.
We shall consider a simple computer program that outputs incidence theorems of a similar nature. Some of its theorems are easily understood in terms of standard human geometrical concepts, but others are decidedly strange. However, all of its proposed theorems are undoubtedly true (although this particular program does not find a proof) and more importantly are theorems rather than trivialities in the sense that we feel they do require some sort of a proof.

Posted on Vimeo (2 parts): Part 1 Part 2



Date: November 16, 2017
Speaker: Richard Lyons, Rutgers University
Title: Remarks on the classification of the finite simple groups
Abstract:
          Remarks on the classification of the finite simple groups

Posted on Vimeo (2 parts): Part 1 Part 2



Date: November 23, 2017
NO SEMINAR FOR THANKSGIVING


Date: November 30, 2017
Speaker: Peter Winkler, Dartmouth College
Title: Sleeping Beauty and Other Probability Conundrums
Abstract:
          Probability theory rests on seemingly firm axioms, yet simple questions continue to confound philosophers and intrigue the public. We'll examine some of these questions and try to determine whether they uncover real issues, or just challenges to our flawed human intuition.
Accompanying Slides


Date: December 7, 2017
Speaker: Michael Kiessling, Rutgers University
Title: Hilbert's Monkey Saddle and other Curiosities in the Riesz Equilibrium Problem for 3 Points on a Circle
Abstract:
          This talk reports the results of a research project with undergraduate student Renna Yi this past Summer. What was meant as a warm-up problem for understanding what `Steve Smale's 7th problem for the 21st century' is all about turned out to hold some surprises. In particular, two mathematical constructions which are usually cooked up for the purpose of illustrating some calculus (II and III) concepts show up naturally in the Riesz Equilibrium Problem of 3 Points on a Circle: Hilbert's Monkey Saddle is one of those, the other one will be revealed at the talk.

Posted on Vimeo (2 parts): Part 1 Part 2
Accompanying Slides