Organizer(s) | Doron Zeilberger, Nathan H Fox | Archive | |
Website | http://www.math.rutgers.edu/~nhf12/expmath |
Past Talks
Thursday, April 27th |
Vince Vatter, University of Florida |
"Sorting with restricted containers " |
Time: 5:00 PM |
Location: Hill 705 |
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. |
Thursday, April 20th |
Joe Kileel, Univ. of California, Berkeley |
"Using Computational Algebra for Computer Vision " |
Time: 5:00 PM |
Location: Hill 705 |
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. |
Thursday, April 13th |
Aaron Robertson, Colgate University |
"Probability and Ramsey Theory " |
Time: 5:00 PM |
Location: Hill 705 |
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. |
Thursday, April 6th |
Evita Nestoridi, Princeton University |
"Shuffling large decks of cards and the Bernoulli-Laplace urn model (Rescheduled from February 9, 2017) " |
Time: 5:00 PM |
Location: Hill 705 |
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. |
Thursday, March 30th |
Nathan Fox, Rutgers University |
"An Exploration of Nested Recurrences Using Experimental Mathematics (thesis defense) " |
Time: 5:00 PM |
Location: Hill 705 |
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. |
Thursday, March 23rd |
Justin Semonsen, Rutgers University |
"On Oda's Strong Factorization Conjecture " |
Time: 5:00 PM |
Location: Hill 705 |
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. |
Thursday, March 9th |
Praneeth Vepakomma, Motorola Solutions |
"Combinatorics of Distance Covariance: Inclusion-Minimal Maximizers of Quasi-Concave Set Functions for Diverse Variable Selection " |
Time: 5:00 PM |
Location: Hill 705 |
Abstract: 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. |
Thursday, March 2nd |
Adi Ben-Israel , Rutgers University |
"Matrix volume and its applications " |
Time: 5:00 PM |
Location: Hill 705 |
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 |
Thursday, February 23rd |
Matthew Russell, Rutgers University |
"Fantastic identities, and where to find them " |
Time: 5:00 PM |
Location: Hill 705 |
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. |
Thursday, February 16th |
John Miller, Johns Hopkins University |
"The distribution of extremal values of linear recurrences modulo m " |
Time: 5:00 PM |
Location: Hill 705 |
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. |
Thursday, February 9th |
Evita Nestoridi, Princeton University |
"Shuffling large decks of cards and the Bernoulli-Laplace urn model " |
Time: 5:00 PM |
Location: Hill 705 |
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. |
Thursday, February 2nd |
John Conway, Princeton University |
"The Faulhaber triangle and the Bernoulli numbers and what they're good for " |
Time: 5:00 PM |
Location: Hill 705 |
Abstract: |
Thursday, January 26th |
Neil J. A. Sloane, Rutgers University and The OEIS Foundation |
"Winter Fruits: New Problems from OEIS " |
Time: 5:00 PM |
Location: Hill 705 |
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. |