Date: Thurs., Jan. 30, 2020
Speaker: Doron Zeilberger, Rutgers University
Title: 0, 1, 8, 78, 944, 13800, 237432, 4708144, 105822432, 2660215680, 73983185000, 2255828154624, ...: "The sequence that started it all"
Abstract: OEIS sequence A1 (that starts: 0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4 ...) is not the chronologically first sequence in the amazing On-Line Encyclopedia of Integer Sequences. This honor goes to OEIS sequence A435, that according to Neil Sloane was the "sequence that started it all". In addition to its immense historical significance it is very interesting mathematically, and leads to new sequences. [Joint work with Yukun Yao]
Posted on Vimeo
Title: A Globally Convergent Newton Method for Polynomials
Abstract: See here
Posted on Vimeo
Date: Thurs., Feb. 13, 2020
Speaker: Marcus Michelen , University of of Illinois at Chicago.
Title: Using central limit theorems: balls in bins and more
Abstract: The multivariate central limit theorem yields a solution to a question of Behrouzi-Far and Zeilberger concerning the average maximal number of balls in a bin. An overview of the classical central limit theorem---both the univariate and multivariate version---will be presented along with the solution to the balls and bin problem as a worked example. Time permitting, recent work joint with Julian Sahasrabudhe will be discussed, in which we show that central limit theorems follow from zero-free regions of generating functions. No knowledge of probability theory will be assumed.
Posted on Vimeo
Date: Thurs., Feb. 20, 2020
Speaker: Matthew Hohertz, Rutgers University
Title: Collatz Polynomials: an introduction with bounds on their zeros
Abstract: In this talk we consider Collatz polynomials. In particular, we bound the moduli of the roots of these polynomials using a determinantal approach, prove theorems on when these polynomials have rational integer roots, and suggest further applications and avenues of research.
Date: Thurs., Feb. 27, 2020
Speaker: Doron Zeilberger, Rutgers University
Title: A New World Record for the Irrationality Measure of Pi
Abstract: Not all irrational numbers are born equal. The irrationality measure of a number tells you how `irrational' it is. It is a competitive sport to find better and better bounds for famous constants like Pi. V. Kh. Salikhov's 2008 upper bound of 7.606308 was the world record until Dec. 2019, when Doron Zeilberger and Wadim Zudilin proved a new world record of 7.10320533.
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Title: Packing in restricted permutations
Abstract: Let ρ be a permutation pattern and let S be a set of permutations. A permutation in S is ρ-optimal if it contains at least as many copies of ρ as any other member of S of the same length. In this talk, we focus on the structure of ρ-optimal permutations for some well-known sets of permutations such as when S is a classical pattern class and when S is the set of alternating permutations.
Posted on Vimeo
Date: Thurs., March 12 , 2020
GIVEN REMOTELY VIA VIDEO CONFERENCING (Due to University Policy regarding Corona)
Speaker: Douglas Iannucci, University of the Virgin Islands.
Title: The Hermite-Serret Algorithm
Abstract:
Hermite and Serret in 1848 independently devised an algorithm by which to find the two squares which add up to a given prime p of the form 4x+1.
It works as well if p is composite.
We generalize this algorithm to find representations of a natural number n as a binary quadratic form u2
+ kuv + v2 for integers k ≥ 0 .
Date: Tues., March 17 , 2020 , 3:00pm, Room TBD
NOT OPEN TO THE PUBLIC (Due to University Policy regarding Corona)
Speaker: Yukun Yao, Rutgers University
Title: An Experimental Mathematics approach to several combinatorial problems [Thesis defense]
Abstract:
With an experimental mathematics approach, this thesis deals with several combinatorial problems and demonstrates the methodology of experimental mathematics.
The first problem is related to the area statistic of of parking functions. Our methods are purely finitistic and elementary.
Next, we use two instructive case studies on spanning trees of grid graphs and "almost diagonal'' matrices, to show that often,
just like Alexander the Great before us, the simple, "cheating'' solution to a hard problem is the best.
We also apply experimental mathematics to algorithm analysis. Using non-linear difference equations, combined with symbolic computations,
we make a detailed study of the running times of numerous variants of the celebrated Quicksort algorithms.
Finally, we discuss, and make partial progress on, the peaceable queens problem, the protagonist of OEIS sequence A250000.
Note: The defense will be done with only the committee members. It will be videotaped and uploaded to the internet.
Posted on Vimeo
Date: Tues., March 24 , 2020, 4:00pm, HILL 425 [note special day, time, and room]
Via Rutgers WebEx (Due to University Policy regarding Corona)
Speaker: Yonah Biers-Ariel, Rutgers University
Title: Flexible Schemes and Beyond: Experimental Enumeration of Pattern Avoidance Classes [Thesis defense]
Abstract: We consider several applications of experimental math. First we introduce flexible schemes, an extension of the enumeration schemes of Zeilberger and Vatter that can enumerate any permutation class with a finite enumeration scheme or regular insertion encoding. Next we combine enumeration schemes with structural methods to give a new algorithm for counting 1342-avoiding permutations. Finally, we find a recurrence for the number of permutations avoiding four dashed patterns and generalize Lenormand's "raboter" operation.
Note: The defense was done via WebEx.
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Date: Thurs., April 2 , 2020 [REMOTE LECTURE via ZOOM]
Speaker:
Andrew Appel, Princeton University.
Title: Graph coloring and machine proofs in computer science, 1977-2017
Abstract: The Four Color Theorem of Kenneth Appel and Wolfgang Haken (1976) was proved and checked with the assistance of computer programs,
though much of the proof was written (and refereed) only by humans. Contemporaneously,
Edinburgh LCF (Logic for Computable Functions) was developed by Robin Milner--a system for proofs written by humans
(with computer assistance) but completely checked by computer; with particular application to proofs about computer programs.
These two developments, and their convergence, have had significant impact on computer science, and my own research career:
graph-coloring algorithms for register allocation in compilers, functional programming languages,
fully machine-checked proofs of mathematical theorems, fully machine-checked proofs of software systems.
One result at the intersection of all these is a machine-checked proof of correctness of a program that
does register allocation by graph-coloring, using an algorithm related to one used in every four-color proof (and attempted proof) since 1879.
Here is the talk and here are the slides .
Speaker: No seminar due to the first day of Passover.
Date: Thurs., April 16 , 2020 [REMOTE LECTURE VIA ZOOM]
Speaker: Robert Scherer, Univeristy of California, Davis
Title: A criterion for asymptotic sharpness in the enumeration of simply generated trees
Abstract: We study the identity y(x)=xA(y(x)), from the theory of rooted trees, proving a conjecture of Greg Kuperberg made in 1996.
Here are the slides.
Date: Thurs., April 23 , 2020 [REMOTE LECTURE VIA ZOOM]
Speaker: Stephen Chen, Princeton University
Title: Mixed frequency data inputs for Recurrent Neural Networks
Abstract: Observations sampled at different frequencies provide multi-level details. How to effectively use them remains a challenging task. We can aggregate all the data to the lowest-frequency level, or we can fill all the missing data to the highest-frequency level. Is there a better way to utilize the information without eliminating the details or introducing noise? We here provide a recurrent neural network model for mixed frequency data, providing convergence proof, error estimation, and some numerical results.
Speaker: Darij Grinberg, Drexel University
Title: From generalized factorials to greedoids, or the unavoidability of the Vandermonde determinant
Abstract: A classical exercise in algebra asks to prove that the product of the pairwise differences between any given n + 1 integers is divisible by the product of the pairwise differences between 0, 1, ..., n. In the late 90s, Manjul Bhargava took this as a starting point for his theory of "generalized factorials", in particular giving a quasi-algorithm for finding the gcd of the products of the pairwise differences between any n + 1 integers in S, where n is a given number and S is a given set of integers.
In this talk, I will explain why this is actually a combinatorial question in disguise, and how to answer it in full generality (joint work with Fedor Petrov). The general setting is a finite set E equipped with weights (every element of E has a weight) and distances (any two distinct elements of E have a distance), where the distances satisfy the ultrametric triangle inequality. The question is then to find a subset of E of given size that has maximum perimeter (i.e., sum of weights of elements plus their pairwise distances). It turns out that all such subsets form a "strong greedoid" -- a type of set system particularly adapted to optimization. Finally, this greedoid will reveal itself as a "Gaussian elimination greedoid" -- which, roughly speaking, means that the problem reduces to linear algebra.
Apart from generalizing a result of Bhargava's, this turns out to be connected to a problem in phylogenetics studied by Moulton, Semple and Steel.
[Note: The original talk by George Szpiro will be hopefully given next Academic year.]
Date: Thurs., June 18 , 2020 [REMOTE LECTURE via ZOOM]
2:00pm DST (local time) (UTC 18:00), June 18, 2020, note special time .
Speaker: Noam Zeilberger, École Polytechnique.
Title: From lambda calculus to the four color theorem, via experimental mathematics
Abstract: Lambda calculus has its origins almost a century ago, in Alonzo Church's ambitious project to build a foundation for mathematics based on the concept of function. Although that project ultimately failed due to logical inconsistency, lambda calculus continues to serve as an important everyday tool for functional programmers, not to mention researchers in programming languages, type systems, and proof assistants.
In the talk, I will survey some surprising connections found over recent years between certain natural subsystems of lambda calculus and the study of graphs on surfaces, or "maps", and in particular the enumerative theory of maps initiated by Tutte in the 1960s. As with so many other such connections, this one owes its existence to the OEIS, and so I will present some practical background on how lambda terms may be efficiently enumerated, generated, and manipulated. Finally, as an illustration of one of the more amusing mashups of these explorations, I will explain how to give a reformulation of the Four Color Theorem as a simple statement about typing of lambda terms.
Here are the slides
Here is the talk: speaker's website or vimeo
Posted on vimeo.
Posted on vimeo.
Title: Risky Decisions: How Mathematical Paradoxes and Other Conundrums Have Shaped Economic Science
Abstract: At its core, economics is about making decisions. In the history of economic thought, great intellectual prowess has been exerted toward devising exquisite theories of optimal decision making in situations of constraint, risk, and scarcity. Yet not all of our choices are purely logical, and so there is a longstanding tension between those emphasizing the rational and irrational sides of human behavior. One strand develops formal models of rational utility maximizing while the other draws on what behavioral science has shown about our tendency to act irrationally.
In this talk I will give examples of mathematical paradoxes and psychological conundrums that have led to advances in economic science. I will challenge the audience with questions about how to make decisions, and thereby show how people who believe themselves to be rational can be led astray
Title: Algorithmically Distinguishing Irreducible Characters of the Symmetric Group
Abstract: Suppose χλ and χμ are two distinct irreducible characters of the symmetric group Sn. How hard is it to find a permutation π such that χλ(π) differs from χμ(π)? Surprisingly, this natural question seems not to have been considered before in the literature. One might expect that the problem is hard, since even determining whether χλ(π) is zero or not is in general NP-hard. A closely related problem is, given oracle access to a function that is promised to be an irreducible character of Sn, how many queries do we need to determine which irreducible character it is? We give an algorithm that solves this problem with polynomially many queries, and that also gives a polynomial time algorithm for the original problem. Coming up with our algorithm involved considerable experimentation. This is joint work with Jennifer Paulhus.
No previous knowledge of group characters will be assumed, all the necessary background will be reviewed.
Posted on vimeo