Experimental Mathematics Seminar - Spring 2013
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Organizer(s) | Doron Zeilberger, Brian Nakamura, Edinah Gnang | Archive | |
Website | http://www.math.rutgers.edu/~bnaka/expmath/index.html |
Past Talks
Thursday, May 9th |
Ron Adin, Bar-Ilan University (Israel) |
"Characters, descents and matrices" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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. |
Thursday, May 2nd |
Jim Lepowsky, Rutgers University |
" A motivated proof of Gordon's identities" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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. |
Thursday, April 25th |
Frank Garvan, University of Florida |
"The Dyson Rank of Partitions" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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. |
Thursday, April 18th |
Melkamu Zeleke, William Paterson University |
" On Subsets of Ordered Trees Enumerated by a Subsequence of Fibonacci Numbers" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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 n coins in the bottom row is equal to the number of directed column convex polyominoes with n cells. I will also discuss Shapiro’s generating function identity related to these combinatorial objects. |
Thursday, April 11th |
Marc Chamberland, Grinnell College |
"A Feast of Experimental Mathematics" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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. |
Thursday, April 4th |
Art DuPre, Rutgers University |
"A New Yoga for Constructing Tensegrities" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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. |
Thursday, March 28th |
Brian Nakamura, Rutgers University |
"Computational Methods in Permutation Patterns (Ph.D. Thesis Defense)" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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 r occurrences of a pattern. In this case, we derive functional equations for certain families of patterns and use these to enumerate the desired permutations. We will also mention how this approach can be extended to handle multiple patterns simultaneously as well as refine by the number of inversions. Finally, we will give a brief example on how certain existing techniques can be automated so that a computer can derive rigorous results (beyond what is possible by purely human means). |
Thursday, March 14th |
Doron Zeilberger, Rutgers University |
"How I need a drink, alcoholic of course, after the heavy lectures involving ..." |
| Time: 5:00 PM |
| Location: Hill 705 |
Thursday, March 7th |
Vince Vatter, University of Florida |
"321-avoiding permutations" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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. |
Thursday, February 28th |
Roger Nussbaum, Rutgers University |
"The 2^n Conjecture and Related Questions" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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. |
Thursday, February 21st |
Vladimir Retakh, Rutgers University |
"Noncommutative Laurent phenomenon - A geometric approach" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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." |
Thursday, February 14th |
Thomas Robinson, Rutgers University |
"Recurrences in the Jacobi identity of a vertex operator algebra" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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. |
Thursday, February 7th |
Neil Sloane, The OEIS Foundation |
"The On-Line Encyclopedia of Integer Sequences: The First Hundred Years" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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. |
Thursday, January 31st |
Gil Kalai, Hebrew University and Yale University |
"Open collaborative mathematics over the Internet - three examples" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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. |
Thursday, January 24th |
Patrick Devlin, Rutgers University |
" Integer Subsets with High Volume and Low Perimeter" |
| Time: 5:00 PM |
| Location: Hill 705 |
| Abstract: 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 wee bit of analysis, and (of course) experimental math will ultimately come to the rescue. No background knowledge whatsoever is required (or assumed). The driving questions explored in the talk were first posed in a paper by Miller, Morgan, Newkirk, Pedersen and Seferis in 2011, and the talk itself will be based on a 2012 article in Integers by the same name. |