Jump to Fall 2013

We explore a certain variation of the isoperimetric problem in which integer subsets take the role of geometric figures. In particular, after defining some simple notions of "perimeter" and "volume" for integer subsets, we ask the question "Among all subsets with volume n, what is the smallest possible perimeter?" For n=1, 2, 3, ..., this gives rise to an integer sequence, which will be the primary focus of the talk. We will also discuss the structure of these optimal subsets.

The talk will involve combinatorics, recurrence relations, algorithms, intricate fractal-type symmetries, a

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

I will discuss several examples of recent Internet research oriented math activities:

1) Polymath5 - Erdos discrepancy problem.

Background: please look at this MO problem

http://mathoverflow.net/questions/105383/the-behavior-of-a-certain-greedy-algorithm-for-erds-discrepancy-problem

(and the blog post linked there.)

2) Mobius randomness over blogs and MathOverflow.

We will talk only briefly about it. Here is one link:

MO posts: http://mathoverflow.net/questions/57543/walsh-fourier-transform-of-the-mobius-function

3) My debate with Aram Harrow on the feasibility of quantum computers.

It took place over the blog "Goedel's lost letter and NP=P" (The first post the last post )

I will try to give a little taste of the mathematical problems/issues and a little taste of this way of "doing mathematics".

To get in the mood for this brave new world, some of Dr. Z's rules will not apply (as far as I am concerned):

You can bring laptops tablets smartphones and paper to the lecture and do with them whatever you want. Comments and interuptions are welcome.

You are most welcome to look at the links here in the abstract before the lecture, after the lecture, during the lecture or even instead of the lecture.

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

This talk will discuss the history of the On-Line Encyclopedia of Integer Sequences (the OEIS), its present state, and our plans for the future. 1964: punched cards; 1973: A Handbook of Integer Sequences; 1995: The Encyclopedia of Integer Sequences and the email service; 1996: The OEIS launched; 2009: The OEIS Foundation; 2010: The OEIS Wiki; 2013: The OEIS Kiosk; 2014?: Paid editorial staff. I will also mention some highlights and favorite sequences, both solved and unsolved.

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

This talk was also recorded with a back-up camera. This is also available on YouTube (2 parts): Part 1 Part 2

I will discuss some of the formal nature of the Jacobi identity in a vertex operator algebra. Recurrences play a key role and also lead to a nontrivial example of a vertex operator. No prior knowledge of vertex operator algebras will be assumed.

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

A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials. When it does, we talk about the Laurent phenomenon. A large variety of examples of the Laurent phenomenon for commuting variables is supplied by the theory of cluster algebras. Much less is known in the noncommutative case. I will present a number of the noncommutative Laurent phenomenoma of a "geometric origin."

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

The "sup norm" on R^n is defined by ||x||:=max{x_i: 1<=i<=n}, where x:=(x_1,x_2,...,x_n). If D is a subset of R^n, a map T:D->R^n is called nonexpansive (with respect to the sup norm) if ||T(x)-T(y)||<=||x-y|| for all x and y in D. A point x in D is called a periodic point of T of period p if (T^j) (x) is defined for all positive j and (T^p)(x)=x, where p is minimal. (Here T^j denotes the jth iterate of T.) The 2^n conjecture asserts that p<=2^n, which, if true, would be an optimal upper bound. In this talk we shall explain why an analyst might be interested in this question and describe what results are known concerning the 2^n conjecture. Time permitting, we shall also discuss related questions for maps which are nonexpansive with respect to other "polyhedral norms" ||.||, where a norm is called polyhedral if {x in R^n: ||x||<=1} is a polyhedron.

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

It is well-known that the 321-avoiding permutations are counted by the Catalan numbers, and thus have an algebraic generating function. I will prove that every subclass of the 321-avoiding permutations which is defined by only finitely many additional restrictions has a rational generating function. The primary proof technique is the theory of formal languages applied to a restricted version of the ``staircase decomposition'' which every 321-avoiding permutation possesses. This is joint work with Michael Albert and Nik Ruskuc.

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

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

In this thesis defense, we will discuss two variations of the classical pattern avoidance problem in permutations. The first one is on the study of consecutive patterns in permutations, where an occurrence of a pattern must occur in consecutive terms of the permutation. In this case, we develop an automated approach for deriving recurrences and functional equations that can be used for enumerating the pattern-avoiding permutations. We will also mention a Wilf-equivalence result that is a by-product of this approach.

The second case is a generalization to the classical pattern avoiding problem, where we want to enumerate permutations with exactly

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

or

Strutting and Stringing Polyhedrally Scaffolded Tensegrities

Tensegrities are ethereal structures invented by Kenneth Snelson and expropriated and named by Buckminster Fuller. The mathematics of their statics and dynamics have been studied extensively. All the descriptions I have seen about making them seemed to be somewhat mysterious. I will remove the mystery by constructing a simple three-strut tensegrity during the talk. The method of construction is new to me, and at least, if not new, is certainly not that well-known.

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

The use of computer packages has brought us to a point where the computer can be used to discover new mathematical patterns and relationships, create impressive graphics to expose mathematical structure, falsify conjectures, confirm analytically derived results, and perhaps most impressively for the purist, construct formal proofs. This talk will give some examples from my research concerning geometry, integrals, binomial sums, dynamics and infinite series.

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

The 3x+1 Problem is a long-standing conjecture. Let T be a map from the positive integers into itself, where T(x)=x/2 if x is even and T(x) = (3x+1)/2 if x is odd. The conjecture asks whether, under iteration of the map T, any positive integer eventually reaches the value one. This talk gives a survey of the various approaches and results, intersecting areas such as number theory, dynamical systems, and functional equations.

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

Herb Wilf and Andrew Odlyzko provided a bijection between fountains of coins and partitions of integers studied by Szekeres in connection with a combinatorial interpretation of Ramanujan’s continued fraction. In this talk, I will provide a direct bijection between subsets of ordered trees where no two vertices at the same level have different parents (a.k.a. Skinny Trees) and ordered trees with height at most three (a.k.a. Emeric’s Trees) thereby showing the number of contiguous stacking of coins in which there are

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

I show how I used MAPLE to discover and prove new identities for the generating functions of the Dyson Rank and the Andrews SPT functions.

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

In joint work with Minxian Zhu, we generalize the "motivated proof" of the Rogers-Ramanujan identities given by G. E. Andrews and R. J. Baxter to provide an analogous "motivated proof" of B. Gordon's generalization of the Rogers-Ramanujan identities. Our main purpose is to provide insight into certain vertex-algebraic structure being developed.

A certain family of square matrices plays a major role in character formulas for the symmetric group and related algebras. These matrices are non-symmetric relatives of Hadamard matrices, and have some fascinating properties (including sign patterns and determinants) which may be explained by use of Moebius inversion. They provide a handy tool for translation of statements about permutation statistics to results in representation theory, and vice versa. We shall describe some of these properties and connections.

Joint work with Yuval Roichman.

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

In this talk we discuss a polynomial encoding which provides a unified framework for discussing the algebra and the spectral analysis of matrices and tensors. In addition to describing some algorithms for performing orthogonalization and spectral analysis of tensors, we discuss some computational aspects, more specifically the important role of symmetries in Alon's Combinatorial Nullstellensatz method for solving combinatorial problems.

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

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

See here (http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qdyson.html)

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

Pattern-avoiding trees have appeared in various computational contexts since at least the 1980s. A more recent topic of interest is the exact enumeration of trees that avoid some other tree pattern. In 2010, Rowland considered this enumeration problem for rooted, ordered binary trees where tree T contains tree pattern t if and only if T contains t as a contiguous rooted, ordered subtree. In this talk we consider Rowland's contiguous tree patterns as well as non-contiguous tree patterns. We also consider implications of tree enumeration results for pattern-avoiding permutations.

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

In the past decade there have been several papers studying number theoretic properties of fundamental combinatorial sequences such as the Catalan and Motzkin numbers. For the most part, the proofs use ad hoc methods particular to the combinatorics of each sequence. I will talk about a general method for producing congruences for sequences whose generating functions are algebraic. This method gives completely automatic proofs of existing theorems and has produced many new theorems. Main ingredients include finite automata and diagonals of formal power series. Joint work with Reem Yassawi.

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

In this talk I will discuss some old and some new results and open problems on point-line configurations that one can find in finite projective planes.

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

See the paper at the Electronic Journal of Combinatorics

For integers g >= 2, k >= 2, call a number N a (g,k)-reverse multiple if the reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are the two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089 and 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to study the problem of finding all (g,k)-reverse multiples. By using modified versions of her trees, which we call Young graphs, we determine the possible values of k for bases g = 2 through 100, and then show how to apply the transfer-matrix method to enumerate the (g,k)-reverse multiples with a given number of base-g digits. These Young graphs are interesting finite directed graphs, whose structure is not at all well understood. The talk will mention William The Conquerer, G. H. Hardy, The Unabomber, Lara Pudwell, and others.

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

Smale's 7th problem concerns N-point configurations on the 2-sphere which minimize the logarithmic pair-energy V(r) = - ln r averaged over the pairs in a configuration; here, r is the chordal distance between the points forming a pair. More generally, V(r) may be replaced by the standardized Riesz pair-energy. In a recent paper with Brauchart and Nerattini we inquired into the concavity of the map from the integers >1 into the minimal average standardized Riesz pair-energies of the N-point configurations on the sphere. It is known that this map is strictly increasing and in some Riesz parameter range bounded above, hence ``overall concave.'' It is (easily) proved that for the Riesz parameter -2 it is even locally strictly concave. By analyzing computer-experimental data of putatively minimal average Riesz pair-energies for the Riesz parameters -1,0,1,2,3 and N up to 200 we found that the map in question is locally strictly concave for parameter -1, while not always locally strictly concave for the other parameter values. It is found that the empirical map from the Riesz parameter into the set of convex defect-N is set-theoretically increasing; moreover, the percentage of odd numbers in the range is found to increase with the Riesz parameter. They form a curious sequence of numbers for the logarithmic kernel, reminiscent of the ``magic numbers'' in nuclear physics; it is conjectured that the ``magic numbers'' in Smale's 7th problem are associated with optimally symmetric optimal-energy configurations. The talk emphasizes the role of computer experiments, in particular also of Maple, in our investigation.

Posted on Vimeo: Link

See the applet referenced in the talk.

If a(n) is the sum of the cubes of the entries on the nth row of Pascal'a triangle, then (n+1)^2 a(n) = (7n^2 - 7n + 2)a(n-1) + 8(n-1)^2a(n-2). It seems challenging for a human to find a bijective proof of this, but a computer can do it, with a little help. I'll show you a real live bijection, implemented of course by the computer, that proves this identity, and describe a method that might help computers bijectify other difficult identities.

Posted on Vimeo: Link

The study of sequences extends in a wide range from the golden ratio to the current financial trading algorithm. Although not realized by many, both the Fibonacci and Lucas sequences are incorporated into the algorithm today. In addition, among the far-reaching applications of these numbers are the Fibonacci search method and the Fibonacci heap data structure in computer science. Hence, mathematicians seek to find more results and further studies into these sequences for the purpose of greater potential benefits. In the area of mathematics, Fibonacci and Lucas numbers are used in connection to efficient primality testing of Mersenne numbers; the method revolves around the fact that if F(n) is prime, then n is prime with the exception for F(4)=3. Some of the largest known primes were discovered via this process. Yet, despite the extensive literature on Fibonacci and Lucas numbers, there are still many questions, which are still unanswered. Recently, several authors considered the problem of estimating expressions of the classical Fibonacci and Lucas sequences given by F(0) = 0, F(1) =1,F(n) =F(n-1)+F(n-2) and L0 =2,L1 =1,Ln =L(n-1)+L(n-2), respectively. In this paper, we find the exact asymptotic behavior for a class of infinite partial sums whose general term is a negative integer power of L(k). Most of the previous works focused on the relatively simple cases s = 1 and s = 2 where s represents the power. For this paper, we find the general idea in which expressions could be found for cases of s from 1 to 6.

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This talk will describe several finiteness theorems for quadratic forms, and progress on the question: "Which positive definite integer-valued quadratic forms represent all positive integers?". The answer to this question depends on settling the related question "Which integers are represented by a given quadratic form?" for finitely many forms. The answer to this question can involve both arithmetic and analytic techniques, though only recently with the use of computers has the analytic approach become practical. We will describe the theory of quadratic forms as it relates to answering these questions, its connections with the theory of modular forms, and give an idea of how one can obtain explicit bounds to describe which numbers are represented by a given quadratic form.

Posted on Vimeo (2 parts): Link

We exhibit a family of sequences of noncommutative variables, recursively defined using monic palindromic polynomials in Q[x], and show that each possesses the Laurent phenomenon. This generalizes a conjecture by Kontsevich.

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

One of the greatest combinatorialists and number theorists of our time is George Andrews, who made partitions an active and hot mathematical area, and helped make Ramanujan a household name. But all this dwarfs compared with his PIONEERING use of Symbolic computation, starting with very creative use of the early computer algebra system SCRATCHPAD, and continuing to this day with the still-going-strong saga, joint with the RISC-Linz gang, on implementing and beautifully applying MacMahon's Omega calculus. Alas, George, being a tie-and-jacket wearing traditionalist, does not like revolutions, and hence even he does not realize the full impact of his mathematical legacy, that is FAR larger than the sum of its many impressive parts.

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