Fall 2010

Distinguished Visiting Professor: December 5-December 10

**Kim Ruane (Tufts University), hosted by Adam Piggott.**

Professor Ruane will give two Faculty Colloquia during her visit.

Faculty Colloquium: Tuesday, December 7, 4:00pm in Olin 372

**"A New Way to Teach the Symmetric Group in Abstract Algebra", presented Kim Ruane**

Abstract: In this talk we will discuss how to view the symmetric group in a way that is analogous to the traditional approach for the dihedral groups. In particular, if the symmetric group on n objects is viewed as the isometry group of the n-1 simplex, some of the more elementary facts about this group become geometric and fun to compute!

Faculty Colloquium: Wednesday, December 8, 4:00pm in Olin 371

**"A Geometric Proof of Greenberg's Lemma for Free Groups", presented by Kim Ruane**

Abstract: Greenberg's lemma says that if N is a non-trivial, finitely generated normal subgroup of a free group F, then N has finite index in F. We give a geometric proof of this fact that generalizes appropriately to a much wider class of groups.

Distinguished Visiting Professor: November 29 - December 3

**Brett Wick (Georgia Tech), hosted by Pam Gorkin.**

Professor Wick will give two Faculty Colloquia during his visit.

Faculty Colloquium: Tuesday, November 30, 10:00 am in Olin 383

**"Bilinear Forms on the Dirichlet Space", presented by Brett Wick**

Abstract: The Dirichlet space is the set of analytic functions on the disc that have a square integrable derivative. In this talk we will discuss necessary and sufficient conditions in order to have a bilinear form on the Dirichlet space be bounded. This condition will be expressed in terms of a Carleson measure condition for the Dirichlet space. One can view this result as the Dirichlet space analogue of Nehari's Theorem for the classical Hardy space on the disc. This talk is bases on joint work with N. Arcozzi, R. Rochberg, and E. Sawyer.

Faculty Colloquium: Monday, November 29, 4:00 pm in Olin 372

**"Bilinear Forms on Spaces of Functions", presented by Brett Wick**

Abstract: Motivated by questions in operator theory and partial differential equations, one frequently encounters bilinear forms on various spaces of functions and then it is interesting to determine the behavior of this form (e.g., boundedness, compactness, etc.) in terms of function theoretic information about a naturally associated symbol of this operator. In this talk we will present numerous examples of this phenomenon and point out various interesting applications of these facts.

Student Colloquium: Thursday, December 2nd, 12:00 noon in Olin 268

**"Ellipses and Critical Points of Polynomials: A Blaschke Product Perspective", presented by Ueli Daepp**

We start with two ellipses, one inside the other, such that there is a polygon inscribed in the outer ellipse that circumscribes the inner ellipse. Can we describe such ellipses? Using some applets and a special type of function, finite Blaschke products, we can answer this question. We try to apply the concept to solve an open conjecture about the location of critical points of a polynomial. We will also show how the result of a recent Bucknell honors thesis relates to this.

Student Colloquium: Thursday, November 18th, 12:00 noon in Olin 268

**"Shapeshifting: Things you can and cannot do via cut and paste", presented by Stephen Wang**

How can we see that two shapes have the same area? One simple method would be to see if we can cut one of them into finitely many pieces, and then move the pieces around so that they exactly cover the second shape. If we have two polygons that have the same area, can we always tell that they have the same area by using this method? We'll consider this question and some of its generalizations.

MAA Event:Tuesday, November 16th, 6pm in Olin 383

**Outstounding Origami!**

Come to the First Annual Origami Event! Beginners and experts welcome---share your skills, cultivate your inner geometer, or just hang out and enjoy the Calzones, dr1nks, and Japanese snacks.

This event is sponsored by the MAA Math Club, the Department of Mathematics, and the Japanese Society.

Student Colloquium: Thursday, November 4th, 12:00 noon in Olin 268

**Student Panel: What I did with my summer vacation**

What will you do this summer? For some ideas on what is available and what some of Bucknell's mathematics students have done in the past, come to the student panel to hear students talk about their past summer experience. Students will speak about research experiences at Bucknell and other institutions, teaching experiences, and internships they received. A question and answer period will follow the session and, as always, there will be pizza and drinks.

Homecoming Academic Village: Saturday October 30th 11:00am-1:00pm, Academic & Science Quads

Spend time connecting with alumni and current students at the fourth annual Homecoming Academic Village. There will be complimentary tickets for the tailgate lunch before the football game, and free childcare is available (please include ages in RSVP). RSVP to Kay Heimbach at kheimbac@bucknell.edu. Questions should be directed to Jenna Tesauro, Program Director of Academic Interests, by emailing jtesauro@bucknell.edu or calling 7-2611.

Information Session: Tuesday, October 26th, 5pm-6pm in Olin 371-372

**Information Session for Prospective Math Majors/Minors**

This is an opportunity for all students (but particularly those in their first or second year at Bucknell) who are considering a mathematics major or minor to come along and find out more, from both faculty members and fellow students. Information on course requirements, careers, etc. will be presented. Food and drinks will be provided.

Distinguished Visiting Professor: Thursday, October 21-25

**Samuel Antonio Lopes (Universidade do Porto, Portugal), hosted by Tom Cassidy.**

Professor Lopes will give two Faculty Colloquia during his visit.

Faculty Colloquium: Monday, October 25, 4:00pm in Olin 372

**"Generalized down-up algebras: their symmetries and arithmetic," presented by Samuel Antonio Lopes**

**Abstract: **Generalized down-up algebras were introduced by Cassidy and Shelton (J. Alg. 2004) as a generalization of down-up algebras, originally defined by Benkart and Roby (J. Alg. 1998). The latter were motivated by combinatorial operators on a (q, r)-differential poset. Those generalized down-up algebras which are Noetherian are part of a larger class of algebras known as generalized Weyl algebras (GWA). When the underlying base ring of a GWA is a polynomial ring in one variable, the automorphism group of such a GWA was studied in detail by Bavula and Jordan (Trans. Amer. Math. Soc. 2001). A Noetherian generalized down-up algebra is a GWA whose base ring is a polynomial algebra in two variables, and it has Gelfand-Kirillov three. In joint work with P. Carvalho (Comm. Alg. 2009) we determined the automorphism groups of these algebras, under certain restrictions. In this talk, I will introduce these algebras and discuss their automorphisms. Time allowing, I shall also refer ongoing related joint work with P. Carvalho, S. Launois and C. Lomp concerning the notion of a non-commutative Noetherian unique factorization ring (Chatters and Jordan 1986).

Faulty Colloquium: Thursday, October 21, 4:00pm in Olin 372

**"Quantized enveloping algebras, primitive ideals and representation theory: a survey of classical and quantum results," presented by Samuel Antonio Lopes**

**Abstract: **I will try to motivate quantum groups through a simple example from quantum mechanics. Focusing on the representation theory of these objects, I will proceed to illustrate the Dixmier correspondence and the orbit method of Kirillov in the classical setting. These will induce results leading to the construction of a class of quantum analogues of Weyl algebras. Turning to primitive ideals, I will discuss the stratification theory of Goodearl and Letzter and describe in detail the primitive spectra of some quantum algebras of small Gelfand-Kirillov dimension. Time allowing, I will also discuss the Andruskiewitsch-Dumas conjecture on the automorphism groups of these algebras and recent progress towards it.

Student Colloquium: Thursday, October 21st, 12:00 noon in Olin 268

**"A Glitch in the Model: Steiner Points and Spanning Trees," presented by Joan Reinthaler**

**Abstract: **In 1956, AT&T had consulted the Bell Labs Math Group about devising a model for the fair calculation of fees for unlimited telephone service among the offices of companies they served. Unsurprisingly, in the model Bell Labs came up with, a primary factor in determining the fee was the length of a minimum spanning tree with the offices as vertices. When Delta Airlines, a company that had hubs in Chicago, New York and Atlanta (cities lying roughly on an equilateral triangle 800 miles apart) opened a dummy office in a shed in the mountains of Kentucky, they shortened the length of the spanning tree and sent the mathematician of Bell Labs back to revise their model. A new criterion was added – that the price should be based on the shortest tree between existing offices and any possible additional offices. The Swiss mathematician Jakob Steiner had investigated this problem in the early 19th century. He had proven that, if the original graph had n vertices, then, at most, n-2 points could be added to reduce the length of the tree. These points are now known as Steiner Points. The construction of Steiner points and a proof of the defining property of “Steiner Trees” are among the topics of this talk.

Field Trip: Thursday, October 20th

**NCTM 2010 Regional Conference and Exhibition in Baltimore.**

Math 207 students Nate Frye, Jen Kokoska, Jin On, and Natalie Cetrulo took their professor, Lynn Breyfogle, along to the National Council of Teachers of Mathematics (NCTM) 2010 Regional Conference and Exhibition.

Distinguished Visiting Professor: Thursday, October 9-11

**Bruce E. Sagan (Michigan State University), hosted by Peter McNamara.**

Professor Sagan will give a Student Colloquium and a Faculty Colloquium during his visit.

Faulty Colloquium: Tuesday, October 12, 4:00pm in Olin 372

**"Combinatorial Proofs of Cyclic Sieving Phenomena," presented by Bruce E. Sagan**

**Abstract: **Let S be a set which admits an action of the cyclic group C_{n} of order n. Let ω_{d} denote a root of unity of order d in the group of roots of unity. Finally, let f(q) be a polynomial in q. Usually f(q) will be the generating function for some statistic on S.

We say that the triple (S,C_{n} ,f(q)) exhibits the *cyclic sieving phenomenon* (CSP) if, for every c in C_{n}, we have

f(ω_{d} )=the number of elements of S fixed by c,

where d is the order of c in C_{n} . This concept was first introduced and studied by Reiner, Stanton, and White, in part as a generalization of Stembridge's q=-1 phenomenon which is the case n=2.

It is quite amazing that plugging a root of unity into a generating function would produce a nonnegative integer, much less that these integers would count something. But it appears that the CSP is quite wide spread and there is a growing literature on the subject.

Most proofs that a triple exhibits CSP use either algebraic manipulations involving roots of unity or representation theory. We will present the first completely combinatorial proof of such a result.

Student Colloquium: Thursday, October 14, 12:00 noon in Olin 268

**"Stalking the Wild Fibonomial," presented by Bruce E. Sagan**

**Abstract: **Let n,k be integers with 0≤k≤n. If one defines the binomial coefficients as C(n,k)=n!/k!(n-k)!, then it is not clear that this rational expression is always an integer. But if one shows that C(n.k) counts the number of k-element subsets on an n-element set, then integrality becomes clear.

The famous Fibonacci numbers are defined by F_{0} =0, F_{1}=1,and F_{n}=F_{n-1}+F_{n-2} for n≥2. Consider an analogous "factorial" F^{!}_{n} = F_{1}F_{2}^{...}F_{n} and the fibonomial coefficients C_{F}(n,k)= F^{!}_{n} / F^{!}_{k} F^{!}_{n-k} . Using tilings, we give a simple combinatorial interpretation for C_{F}(n,k) closely related to the one for C(n,k) which makes it clear that the fibonomials are always integers.

Right Angle Event: Thursday, October 7th, 12:00 noon in Olin 268

**An open discussion of Danica McKellar’s book: "Math Doesn’t Suck" (2007), with invited disucussants Leigh Arnold ‘13, Chris Boyatzis (Department of Psychology), Sharon Garthwaite (Department of Mathematics), Katharyn Nottis (Department of Education), Adam Piggott (Department of Mathematics). **

**Abstract: **Do you agree with Jimmy Buffet that “Math Suks” (see YouTube or iTunes)? Or know people, especially middle school girls, who think it sucks? Then come and join invited discussants in a lively discussion of merits and drawbacks from various perspectives of Danica McKellar’s book, *Math Doesn’t Suck: how to survive middle school math without losing your mind or breaking a nail*. Danica is best known for her roles on The Wonder Years and The West Wing, but is also a New York Times bestselling author, a mathematician, and an advocate for math education. The table of contents, introduction, and Chapter 7: “Is your sister trying to cheat you out of your fair share? (Comparing fractions)” are soon to be available in .pdf form on the Mathematics Department website. Whether you read the excerpt before or not, come join the discussion!

MAA Event:Tuesday, October 5th, 12:00 noon in Olin 383

**Make your own funny Venn Diagram and enter the contest, or just come along to the Judging Festival and enjoy the fun.**

To enter, send your submissions to Box# C-0924, or bring them to the Judging Festival. Pizza and drinks will be provided.

This event is sponsored by the MAA Club.

**"The title of this talk cannot be proven accurate," presented by John Bourke**

**Abstract: **Used carelessly, self-reference can lead to contradiction or absurdity, such as in the statement "This sentence is false." Properly harnessed, however, self-reference was used by Kurt Gödel to prove a beautiful and profound theorem that changed the nature of modern mathematics. His Incompleteness Theorem has been widely applied (and mis-applied) in fields as diverse as philosophy, theology, and theoretical physics. We will explore some of the appropriate applications of the theorem, and get a flavour of the ideas that underpin the result.

**"Mathematical Predictions for Aneurysm Treatment," presented by Dawn A. Lott '87**

**Abstract: **Do you watch CSI? CSI Miami? CSI New York? NUMB3RS? Crossing Jordan? ER?NCIS? Do you find yourself intrigued by Mathematics and find yourself excited by the solution of problems? In this presentation, mathematics and biology are intertwined. You will learn how researchers use advanced mathematics and biomechanics to improve the health of others through the art of solving partial differential equations. Techniques for creating healthier healed wounds and treating aneurysms will be explored. Mathematical treatments for aneurysm repair will be presented.

Find out more about Dawn Lott '87, Associate Professor of Mathematics at Delaware State University.

Are you even the slightest bit interested in teaching mathematics to middle or high school students?

Come to an informal lunch with **Professor Lynn Breyfogle**, a specialist in Mathematics Education. Meet her and others interested, learn about opportunities and come ask questions!! If you can’t make it, but you are still interested contact: Prof. Breyfogle at lynn.breyfogle@bucknell.edu or Olin 364

**"Where are the Switches on This Thing?” presented by Joe Tranquillo**

**Abstract:**From ecosystems, to the social scene at Bucknell, to the neural network you are using to read this abstract, networks are everywhere we look. A current mystery is how the individual units of a network can send and receive information to one another, while at the same time switch on and off entire portions of the network. For example, at the cellular level, information in the brain is transmitted from neuron to neuron through the propagation of electrical impulses. At the network level, however, there is precise coordination of when specific neural circuits are switched on and off. I will explain my recent investigations into the possible types of network switches that provide insight into seizures in the brain, the spread of rumors in social networks, and the genetic control of development. I will conclude the talk with a challenge to future mathematicians to build the analytic tools to begin proving general statements of all network switches.

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