RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Archive of Speakers and Talks --- 2021


Spring 2021 Semester


Date: Jan. 28 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Victor S. Miller, Center for Communications Research, Princeton.

Title: Counting Matrices that are Squares

Abstract: On the math-fun mailing list (7 May 2013), Neil Sloane asked to calculate the number of n times n matrices with entries in {0,1} which are squares of other such matrices. In this paper we analyze the case that the arithmetic is in GF(2) We follow the dictum of Wilf (in the paper ``What is an answer?'') to derive a ``effective'' algorithm to count such matrices in much less time than it takes to enumerate them. The algorithm which we use involves the analysis of conjugacy classes of matrices. The restricted integer partitions which arise are counted by the coefficients of one of Ramanujan's mock Theta functions, which we found thanks to Sloane's OEIS (Online Encyclopedia of Integer Sequences).

lecture

slides


Date: Feb. 4 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Ron Adin, Bar Ilan University, Israel.

Title: Cyclic permutations, shuffles, and quasi-symmetric functions

Abstract: Richard Stanley proved that the distribution of descent number over all the shuffles of two permutations depends only on the descent numbers of the permutations.

We present an explicit cyclic analogue of this result. The tools used for the proof include a new cyclic counterpart of Gessels quasi-symmetric functions, an unusual ring homomorphism, and a mysterious binomial identity with an interesting history.

Based on joint work with Ira Gessel, Vic Reiner, and Yuval Roichman.

lecture

slides


Date: Feb. 11 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Colin Defant, Princeton University

Title: Cumulants and Stack-Sorting

Abstract: Cumulant sequences are numerical sequences that play a fundamental role in noncommutative probability theory. West's stack-sorting map is a combinatorially-defined operator that acts on permutations. In this talk, we will discuss how cumulants and stack-sorting, two topics from very disparate worlds, are actually very closely related. This unexpected connection allows us to use tools from noncommutative probability theory to prove difficult, surprising, and (occasionally) weird facts about the stack-sorting map. We will explore several applications of this method.

lecture

slides


Date: Feb. 18 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Nathan Fox, Canisius College

Title: Game Complexity: Between Geography and Santorini

Abstract: Santorini, a board game designed by mathematician Gordon Hamilton, is a two-player game of perfect information (a partizan combinatorial game) with simple rules and great depth of strategy. The structure of Santorini suggests a number of possible generalizations and simplifications that can be studied. We are interested in the difficulty of finding winning strategies in this family of games. One of our variants of Santorini turns out to be equivalent to a classical game called Geography, which is known to be "hard." In this talk, we present some background on Santorini, Geography, and game complexity, and we discuss what we can say about certain variants of Santorini.

Based on joint work with Carson Geissler.

lecture

slides   annotated slides


Date: Feb. 25 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Aaron Robertson, Colgate University

Title: Data Mining and Ramsey Theory

Abstract: Ramsey theory concerns itself with the emergence of patterns in sufficiently large structures. Data miners search for patterns in extremely large data sets. This is a cautionary tale for data miners about making inferences from large data sets wherein we provide a framework for testing whether or not a correlation is spurious.

lecture

slides  


Date: March 4 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: David Nacin, William Patterson University.

Title: Padovan, Pascal, and Proofs without Words

Abstract: What happens when we attempt to construct the Fibonacci spiral with triangles instead of squares? We get a new sequence, the Padovan sequence, which answers its own collection of unique and beautiful counting problems. In this talk we will show how this construction defines this sequence and then rediscover the same sequence hiding again in other surprising places. We then prove several identities without using either words or numbers, by considering triangles composed of colored dots.

The Fibonacci sequence is connected to the golden ratio which arises from a simple question about rectangles and proportion. A slightly different natural question leads to a new ratio and yet another method for defining our sequence. We then observe the uses of this sequence and its ratio in architecture and discuss the history behind the patterns we've uncovered. We conclude with a counterexample to a conjecture about this sequence that leads us to a final construction involving copies of the Fibonacci sequence itself.

lecture

slides  


Date: March 11 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Michael Kiessling, Rutgers University

Title: A Maple-assisted study of a Schroedinger-Newton, a.k.a. Schroedinger-Poisson, a.k.a. Choquard, a.k.a. Pekar, a.k.a. ... equation

Abstract: In several different contexts (mathematical) physicists have proposed a nonlinear system of PDEs which can be recast into a single Schroedinger equation with a Schroedinger potential that is the solution to a Poisson equation with the square of the solution of the Schroedinger equation as source term; as such it is known under a variety of names (see above). For instance, Roger Penrose proposed it in his theory of gravity-induced quantum-mechanical wave function collapse, but also down-to-earth condensed matter theories, by Pekar and later by Choquard, produce this equation without invoking gravity. Interestingly, the question of the asymptotic large-distance behavior has received several plausible but conflicting answers which cannot all be true simultaneously. Recently I managed to settle the issue of the leading order term --- partly rigorously, partly with the help of Maple. Subsequently Andrey Yudin from Moskow managed to tickle Maple to produce an intriguing formula for all the putative asymptotic correction terms to the leading order term which are of power law type. Numerically this formula seems pretty accurate, but it has yet to be proved; moreover, there are correction terms beyond all orders of powers, and a formula for these terms has yet to be found

lecture

slides  


Date: March 25 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Alejandro Morales, UMass, Amherst, and William Shi, Northview highschool, Johns Creek, GA. [starting Harvard, Fall 2021]

Title: Refinements and symmetries for volumes of flow polytopes

Abstract: Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations. The Chan-Robbins-Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which Mészáros gave an interpretation as the volume of a collection of flow polytopes. We introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes. Our results generalize Mészáros's construction and a recent flow polytope interpretation of the Morris identity by Corteel-Kim-Mészáros. We prove the product formula of our refinement following the strategy of the Baldoni-Vergne proof of the Morris identity.

lecture

slides (w/o annotations)   slides (w annotations)  


Date: April 1 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Doron Zeilberger, Rutgers University

Title: Proofs of the Riemann Hypothesis, the P ≠ NP conjecture, the Goldbach Cojecture, and the Irrationality of γ

Abstract: We present proofs of these conjectures, and time permitting, a few other ones.

[Joint work with Shalosh B. Ekhad]

lecture


Date: April 8 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Eric Sundberg, Occidental College

Title: Pairing Strategies for Tic-Tac-Toe on the Boolean Hypercube

Abstract: We consider a tic-tac-toe-style game on the vertices of the n-dimensional Boolean hypercube {0,1}n with k-dimensional subcubes as winning sets. We describe a pairing strategy which allows the second player to force a draw when k = n/4 +1 in the case where n is a power of 4. Our results arose from significant experimentation using Mathematica.
(Based on joint work with Klay Kruczek and Ramin Naimi)

lecture

slides  


Date: April 15 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Igor Pak, UCLA.

Title: What is a combinatorial interpretation

Abstract: The question in the title is deceptively simple, as the answers tend to be the number of certain trees, lattice paths, Young tableaux, and other friendly combinatorial objects. However, the question lies in the heart of connections between enumerative/algebraic combinatorics and computer science. I will survey what is known about the subject, and discuss some of my recent results.

lecture

slides  


Date: April 22 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Toufik Mansour, University of Haifa

Title: On Stanley-Wilf limit of the pattern 1324

Abstract: We present an explicit formula for the generating function for the number of permutations of length $n$ that avoid 1324 in terms of generating functions for permutations that have a kernel shape of length m, m ≥2. This allows us to write down a systematic procedure for finding a lower bound for approximating the Stanley-Wilf limit of the pattern 1324.
Joint work with Christian Nassau

lecture


Date: April 29 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Victor S. Miller, IDA Center for Communications Research, Princeton

Title: Locality preserving hash functions, a partial order and tiles in binary space

Abstract: A "tile" in the space B^n of bit vectors of length n, is a subset S of B^n, such that there is another subset A of B^n so that every element of B^n can be written uniquely in the form a + s, where a in A and s in S. A particular class of tiles are the subsets of minimum weight elements in the cosets of a linear code over GF(2). In systematically investigating locality preserving hash functions, we generated the list of all possible tiles of cardinality <=64 satisfying a certain optimality condition. All but 10 of them turned out to be the sets of minimum weight elements described above. Attempts to prove the same for the remaining 10 remained elusive. Instead we found two computational criteria -- one using linear programming, and the other using combinatorial bin packing, which showed that remaining 10 could not be tiles.

[Joint with Don Coppersmith, Dan Gordon and Peter Ostapenko]

lecture

slides  


Date: May 6 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Adi Ben Israel, Rutgers University (RUTCOR).

Title: Data analysis in high-dimensional spaces

Abstract:

1. The unreliability of the Euclidean distance in high-dimension, making a proximity query meaningless and unstable because there is poor discrimination between the nearest and furthest neighbor [3], see also [4].

2. The uniform probability distribution on the n-dimensional unit sphere S_n, and some non-intuitive results for large n. For example, if x is any point in S_n, taken as the "north pole", then most of the area of S_n is concentrated in the "equator".

3. The advantage of the ℓ1-distance, which is less sensitive to high dimensionality, and has been shown to "provide the best discrimination in high-dimensional data spaces," [1, p. 427].

4. Clustering high-dimensional data using the ℓ1 distance, [2].

References

[1] C.C. Aggarwal et al, On the surprising behavior of distance metrics in high dimensional space, Lecture Notes in Computer Science, vol 1973(2001), Springer, https://doi.org/10.1007/3-540-44503-X_27

[2] T. Asamov and A. Ben-Israel, A probabilistic ℓ1 method for clustering high-dimensional data, Probability in the Engineering and Informational Sciences, 2021, 1-16

[3] K. Beyer et al, When is "nearest neighbor" meaningful?, Lecture Notes in Computer Science, vol 1540(1999), Springer, https://doi.org/10.1007/3-540-49257-7_15

[4] J.M. Hammersley, The distribution of distance in a hypersphere, The Annals of Mathematical Statistics 21(1950), 447-452.

lecture

slides


Fall 2021 Semester


Date: Sept. 9 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Neil Sloane, The OEIS Foundation and Rutgers University

Title: Lovely New Problems From the Past Months

Abstract: Tired of the sequence [Covid, drought, fires, floods, war, repeat]? Then come and hear about the astonishing Inventory Sequence 0; 1, 1, 0; 2, 2, 2, 0; ...; a new counting problem based on the complete graph; the Stepping Stones puzzle; Problem of the Week 1321; the Ennesrem primes; Gerrymandering; Vicious Watermelons; and other highlights from the past few months of the OEIS. tbd

lecture

slides



Date: Sept. 16 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Carsten Schneider, RISC-Linz (Austria)

Title: Difference Ring Algorithms for Symbolic Summation and Challenging Applications

Abstract: A major breakthrough in symbolic summation was Doron Zeilberger's creative telescoping method to compute linear recurrences of definite hypergeometric sums. In this talk I will illustrate the algorithmic framework in the setting of difference rings in which one can extend Z's method for summands that are built by indefinite nested sums and products. In combination with a sophisticated recurrence solver one obtains a rather general machinery to simplify big classes of definite multi-sums to expressions in terms of indefinite nested sums and products. The underlying difference ring algorithms implemented in the summation package Sigma will be illustrated by non-trivial applications coming, e.g., from combinatorics and particle physics.

lecture

slides



Date: Sept. 23 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Noga Alon, Princeton University

Title: Coloring subsets with r-wise intersecting color classes

Abstract: What is the minimum number of colors required in a coloring of all k-subsets of an n-set so that every color class is r-wise intersecting? We suggest a conjectured answer for all r, k and n, note that for r=2 this is Kneser's conjecture proved by Lovasz, and prove the conjecture for any r which is either a prime or a power of 2.

lecture

slides



Date: Sept. 30 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Doron Zeilberger, Rutgers University

Title: An Experimental (yet fully rigorous!) Study of a certain "Measure Of Disarray" that 12-year Noga Alon Proved was always Even

Abstract: In a beautiful new "cofee table book", "Do not Erase", by the very talented artistic photographer Jessica Wynne, there are pictures of more than one hundred blackboards by a very diverse set of mathematicians. One of them is Noga Alon's blackboard. Each blackboard photo is accompanied by a short essay by the creator of the blackboard, where they often describe how they decided to become mathematicians. According to Noga Alon, the "epiphany" occurred when he was 12 years old, when he settled a heated argument in a "Eurovision watching party" that his parents threw, where he conclusively proved that a certain "measure of disarray" must always be even, in particular causing one of the guests, "a grown-up engineer", to concede that he was wrong in claiming that it was a coincidence that the scores turned out to be all even. According to Noga, the fact that a mere 12-year-old can win an argument against a grown-up (especially one who is an engineer) shows the objectivity of mathematical truth, and reinforced his decision to become a mathematician.

While I do agree that mathematical knowledge is more objective than most other kinds, it is not as objective as it seems. But the point of the present talk is to investigate, in a purely empirical (yet fully rigorous!) way, that same measure of disarray, that turned out to be called "Spearman's footrule", going far beyond just proving that it is always even, considerably extending a 1977 paper by Persi Diaconis and Ron Graham, debunking yet-another myth: that mathematics is always deductive.
Joint work with Shalosh B. Ekhad.

lecture

paper (instead of slides)



Date: Oct. 7, 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Yochay Jerby, Holon Institute of Technology, Israel

Title: A dynamic approach for the zeros of the Riemann zeta function - Collision and repulsion

Abstract: he Riemann hypothesis is a question regarding the solutions of the transcendental equation ζ(s)=0, that is the zeros of the Riemann zeta function. The starting point of our talk is a remarkable connection between the zeros of zeta and the solutions of the much simpler equation χ(s)+1=0, whose non-trivial solutions could be completely described in closed form and all lie on the critical line. We will explain that the two equations are related by a sequence of functions ζN(s) called $N$-th sections of zeta. For N=1 one has ζ1(s)=1/2(1+χ(s)) while for N=[Im(s)/2] the section ζN(s) is approximately ζ(s), up to a small error. Studying the dynamical change of the zeros of ζN(s) with respect to N we will show that zeros can go off the critical line only if a process of collision occurs between two consecutive zeros at a certain stage. We will also suggest a method of re-arranging the dynamics so that collisions could be avoided altogether, that is, in a way expected to keep the zeros on the critical line for any N, including the final stage where they essentially coincide with the zeros of zeta. If time permits we will also discuss how the suggested viewpoint relates to various classical topics such as: Gram's law, the Davenport-Heilbronn function and the Montgomery pair correlation conjecture.

-lecture

slides



Date: Oct. 14 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Stoyan Dimitrov, University of Illinois Chicago

Title: Moments of permutation statistics and central limit theorems

Abstract: We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of Zeilberger. We use an approach of Chern, Diaconis, Kane and Rhoades, previously applied on set partitions and matchings. In addition, we give a new proof of the central limit theorem (CLT) for the number of occurrences of classical patterns which uses a lemma of Burstein and Hasto. We give a simple interpretation of this lemma and an analogous lemma that would imply the CLT for the number of occurrences of any vincular pattern. Furthermore, we obtain explicit formulas for the moments of the descents and the minimal descents statistics. The latter is used to give a new direct proof of the fact that we do not necessarily have asymptotic normality of the number of pattern occurrences in the case of bivincular patterns tbd

slides



back