Student Colloquium Series: Thursday, April 19 at noon in 268 Olin Science

### A Turing Celebration

### Felipe Perrone

Department of Computer Science

Peter Brooksbank

Department of Mathematics

Bucknell University

**Abstract: **This year marks the 100th birthday of Alan Turing, one of the most original and creative scientific minds of the 20th century. In his short life he laid the theoretical groundwork for the invention of the modern computer, he saved countless lives by cracking the Nazi Enigma machine in World War II, he was a pioneer of artificial intelligence, and he made influential contributions to logic and to mathematical biology. Please join us in a celebration of his extraordinary life.

PIZZA and DRINKS provided. All are welcome.

Distinguished Visiting Professor Seminar: Thursday, April 19, 4:00 pm in 371 Olin Science

### Applications of Orthogonal Polynomials

### Lilian Manwah Wong

School of Mathematics

Georgia Institute of Technology

**Abstract.** Orthogonal polynomials have a lot of applications, ranging from the Schroedinger equation to random processes. I will discuss the relation between orthogonal polynomials and spectral theory, and how orthogonal polynomials appear in the context of difference equations. The second half of the talk will focus on the discrete Schroedinger equation and one-dimensional random walks.

Distinguished Visiting Professor Colloquium: Tuesday, April 17, 4:00 pm in 371 Olin Science

### Orthogonal Polynomials, an introduction

### Lilian Manwah Wong

School of Mathematics

Georgia Institute of Technology

**Abstract.** Orthogonal polynomials appear in mathematics and physics literature and have a wide range of applications ranging from random matrices to random walks and stochastic processes. Well-known examples of orthogonal polynomials include Hermite polynomials, Chebyshev polynomials and Legendre polynomials. I will demonstrate how we obtain orthogonal polynomials given a measure on the real line (or on the unit circle) and the recurrence coefficients associated with the measure. This will be followed by a survey of the properties of orthogonal polynomials and the information carried by the recurrence coefficients.

Student Colloquium Series: Thursday, April 12 at noon in 268 Olin Science

### Lofty Leapfroggers

Sharon Garthwaite

Department of Mathematics

Bucknell University

**Abstract: **Let's play a game! Take a large square grid and draw a line down the middle. On one side of the line fill each square with a game piece (like a penny or paper clip). Your goal is to move a piece as far past the center line as possible in as few moves as possible. The catch? You can only move by "leapfrogging" over other pieces and then removing the one you leapt over. You can only leap left, right, up, or down. (This is like the peg game solitaire or jumping in the game checkers, but not diagonally.) Keep track of the minimum number of pieces you need at the start to move one space, top spaces, three spaces, and so on over the center line. It turns out that this is a very interesting sequence, and opens the door to some interesting results from Calculus and Number Theory! There is a board set up in Olin 383 if you'd like to try the game before the talk.

PIZZA and DRINKS provided. All are welcome.

Distinguished Visiting Professor Seminar: Thursday, April 5, 4:00 pm in 371 Olin Science

### Trace forms

### Raman Parimala

Department of Mathematics and Computer Science

Emory University

**Abstract.** The trace quadratic form in a finite extension is a quadratic form which is well-studied in literature. We shall explain how the invariants of trace forms are closely associated to classical arithmetic invariants for number fields like the discriminant of the extension and the number of real embedding’s.

Distinguished Visiting Professor Colloquium: Tuesday, April 3, 4:00 pm in 371 Olin Science

### Classical invariants for quadratic forms

### Raman Parimala

Department of Mathematics and Computer Science

Emory University

**Abstract.** We describe the classical invariants associated to quadratic forms, namely, the dimension, the discriminant, the signature and the Clifford invariant. Over number fields they classify quadratic forms up to isomorphism. We shall explain some that they also classify quadratic forms over fields of cohomological dimension two. This leads to more open questions and conjectures over such fields.

Distinguished Visiting Professor Colloquium: Thursday, March 29, 4:00 pm in 371 Olin Science

### Supporting students' use of formal mathematical

discourse

### Beth Herbel-Eisenmann

Department of Teacher Education

Michigan State University

**Abstract.** By considering the ways in which students use language in various “communication contexts” (i.e., interacting in small groups, reporting out to the whole class, producing individual written solutions, or making sense of the textbook), we can notice which students may need more support and set micro- and macro- level goals for helping students develop formal mathematical discourse practices. We often ask students to work in various communication contexts, but may not consider how these contexts shape the language students use. Participants will examine multiple student solutions to a high-level task in order to identify aspects of language use that are important to students’ articulation of mathematical solutions. These characteristics will be connected to a tool for considering how to scaffold students’ use of formal mathematical discourse, in relationship to communication contexts.

Distinguished Visiting Professor Seminar: Thursday, March 22, 4:00 pm in 371 Olin Science

### Density of the Polynomials in Bergman spaces

of Slit Domains

### John Akeroyd

Department of Mathematical Sciences

University of Arkansas

**Abstract.**

We review some well-known results concerning the density of the polynomials in the Bergman spaces of various classes of bounded domains in the complex plane, and then answer in the affirmative a rather long-standing question as to whether or not there is a a Jordan arc \(\Gamma\), with endpoints \(0\) and \(1\), such that \(\Gamma\setminus \{1\}\) is contained in \(\mathbb{D} := \{z: |z| < 1\}\) and the analytic polynomials are dense in the Bergman space \(\mathbb{A}^t(\mathbb{D}\setminus\Gamma)\); \(0 < t < \infty\). The proof is rather elementary and is given in some detail. The Hardy space case is also discussed.

### Notes on Compact Composition Operators

on H^{2 }

### John Akeroyd

Department of Mathematical Sciences

University of Arkansas

**Abstract.**

If \(\varphi\) is an analytic function in the (unit) disk \(\mathbb{D} : \{z: |z| < 1\}\) that maps the disk into itself, then \(\varphi\) is called an \(\textit{analytic self-map}\) of \(\mathbb{D}\), and \(\varphi\) gives rise to a bounded operator \(C_{\varphi}\) on the Hardy space \[H^2 := \{f: \mbox{$f$ is analytic in $\mathbb{D}$ and $||f||_{H^2}^2 := \sup_{0 < r < 1}\frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^2d\theta < \infty$}\};\] defined by \(C_{\varphi}(f) = f\circ \varphi\). A celebrated result of Joel Shapiro characterizes, in terms of \(\varphi\), the compact composition operators on \(H^2\); that is, those for which \(C_{\varphi}(\{f\in H^2: ||f||_{H^2} < 1\})\) has compact closure in \(H^2\). We examine an alternate characterization and use it to construct new examples of compact composition operators on \(H^2\) that are not in any of the Schatten classes; so, these examples are compact, but just barely so.

### The harmony of the prime numbers

The Riemann hypothesis

Gonzalo Tornaría

Universidad de la República (Uruguay)

**Abstract.** Hilbert, in his famous presentation at the ICM in Paris in 1900, included the Riemann hypothesis as one of the 23 problems with which he challenged mathematicians of the XX century. A hundred years later, the Clay Mathematics Institute posed it as one of the 7 millenium problems, offering a prize of a million dollars. This talk will be a historical introduction to the problem of understanding the distribution of the prime numbers, its relation with the zeros of the Riemann zeta function, and the Riemann hypothesis.

There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers belong to the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision.

Don Zagier (1975)

### Paramodular Forms and Böcherer's Conjecture

Central Values of twisted degree 4 L-functions

Gonzalo Tornaría

Universidad de la República (Uruguay)

**Abstract**. Conjectures about central values of L-functions abound; for example the Conjecture of Birch and Swinnerton-Dyer predicts that the order of vanishing of the central critical values of an elliptic curve L-function is equal to the rank of the elliptic curve's Mordell-Weil group. In the 1980s Böcherer formulated a conjecture relating the central value of the quadratic twists of the spinor L-function attached to a Siegel modular form F to the coeficients of F. Later Kohnen and Kuss gave numerical evidence for the conjecture in the case when F is a rational eigenform that is not a Saito-Kurokawa lift. In this talk I will introduce and motivate the problem of computing and understanding central values of twisted L-functions, survey some results and computations on this problem in the case of classical modular forms, and present joint work with Nathan Ryan extending Böcherer's conjecture to the case of paramodular forms.

### Congruences and Geometry

Alex Ghitza

Department of Mathematics

University of Melbourne

**Abstract.** We will discuss some of the (arithmetic) geometry that forms the modern framework for the study of congruences. The objective is to introduce and motivate some congruences conjectured by Harder, and which are the object of work in progress with Nathan Ryan.

### Integer Relation Algorithms

Alex Ghitza

Department of Mathematics

University of Melbourne

**Abstract**. In "experimental mathematics," a basic problem is that of searching for small integer relations between given real numbers. We will describe two algorithms (PSLQ and LLL) that are widely used for solving this problem and compare them from the points of view of theoretical complexity and typical performance in practice.

(This talk should be accessible to ungraduates.)

Student Colloquium Series: Thursday, March 29 at noon in 268 Olin Science

### Interactive Theorem Proving

Benoit Razet

Computer Science Department

Bucknell University

**Abstract: **This talk will introduce a software called an "Interactive Theorem Prover." This software is used to develop formal theories (usually in Computer Science or Mathematics). The theories are specifications, theorems, and proofs that are also validated by the software. During the last decade, this kind of software has advanced significantly, opening the door to new results in Mathematics, Computer Science and Software Engineering. The challenge of using an interactive theorem prover is the conversion of a traditional mathematical handwritten proof into a script that can be validated by the software. During this talk, I will show how to use an interactive theorem prover with one particular example and illustrate several interesting features.

PIZZA and DRINKS provided. All are welcome.

Mathematics Career Panel: Tuesday March 6, 5:30 - 6:30 in 232 O'Leary

### Mathematics Alumni Career Panel Discussion

Ryan Ward '11

PhD Student in Mathematics

Derrick Houck '10

Algebra Teacher

Nicole Falcaro '09

Operations Analyst

Kimberly Sacra

Mathematics Hiring Manager

**This panel is facilitated by Julee Bertsch from the Career Development Center**

**Bios of panelists**

Hear advice and perspectives from Bucknell alumni who will examine career paths that utilize the mathematics degree while discussing their work and available opportunities. The conversation will include a question and answer period and an opportunity to meet (and network with!) the alumni panelists.

**Nicole Falcaro '09, Operations Analyst, STELLAService**

Nicole graduated Bucknell in 2009 with a B.S. in Mathematics and Spanish. She was a member of the Cross Country and Track & Field teams, Vice President of Environmental Club, and President of SAAC (Student Athlete Advisory Committee). After graduating in the turbulent year of 2009, Nicole worked as a nanny for six months, one month of which was in Europe, while searching for a full-time job. Nicole joined STELLAService in its infancy after she met two '06 Bucknell Alums (John Ernsberger and Jordy Leiser) that December and moved to NYC in January 2010 to work for STELLA as an analyst, collecting data on the customer service of online retailers.

Since beginning her career at STELLAService, the company has grown from three Bucknellians to 11 full-time employees. Nicole has advanced to work in Operations and manages the inbound data as well as the scheduling of surveys conducted by various teams. She also organizes the summer and winter internship programs offered by the firm.

In addition to her full-time position at STELLA, Nicole continues to pursue her interests in running. Shortly after moving to NYC in early 2010, she joined a competitive club organization, Central Park Track Club. As a member, she travels and represents her team in high-level road races and track meets in Cross Country and Track. She visits Bucknell often to spectate track meets and represent STELLAService at the career and internship expos.

**Ryan Ward '11, Graduate Assistant, Mathematics, The Pennsylvania State University.**

Ryan is a first-year graduate student in mathematics at Penn State. Currently, he is preparing for qualifying examinations and teaching a course on ordinary and partial differential equations for engineers. Ryan graduated from Bucknell in the spring last year. His current interests are in computational mathematics.

**Derrick Houck '10, Algebra Teacher, Olney Charter High School, Philadelphia.**

Derrick began teaching in the 2010-11 academic year right after graduating from Bucknell. He taught at a public school in the School District of Philadelphia, Murrell Dobbins CTE High School, as part of the Teach for America program. Due to a budget crisis that forced the district to cut roughly 1500 teaching positions, he began teaching at Olney Charter High School in the 2011-12 academic year, where he has been teaching since. Derrick has taught Algebra I at both schools. Last year at Dobbins, he helped facilitate an after-school robotics club, and this year he helps run an after-school music club where students learn to play piano or guitar.

He has lived in South Philadelphia, two blocks from the famous Pat’s and Geno’s Steaks since he graduated from Bucknell. As part of the Teach for America, he has been pursuing a Master's Degree in Education from the University of Pennsylvania, which he expects to complete by May 2012.

In addition to his math major at Bucknell, he also completed a minor in Education. He also worked as a math tutor in the Writing Center and at the Math Department's Calculus Help Sessions. Moreover, he volunteered at an after-school homework help program in a local low-income housing development.

**Kim Cocoros Sacra '04, Mathematics Hiring Manager, National Security Agency**

Kim currently works as the Mathematics Hiring Manager at the National Security Agency [NSA]. NSA is the single largest employer of mathematicians in the United States. Its mission is to collect information from foreign signals and to prevent foreign adversaries from gaining access to national security information.

Prior to her position as Mathematics Hiring Manager, Kim worked as a Cryptographic Vulnerability Analyst, evaluating commercial cryptographic products and protocols. Previously, Kim worked as a high school mathematics teacher in Baltimore County, Maryland. In 2009, Kim earned her Master's degree in Applied and Computational Mathematics from Johns Hopkins University. In addition to her position at NSA, Kim is currently an adjunct faculty member at Hartford Community College.

Light refreshments will be offered.

Distinguished Visiting Professor Seminar: Tuesday, February 21, 4:00 pm in 372 Olin Science

Distinguished Visiting Professor Seminar: Wednesday, February 22, 4:00 pm in 372 Olin ScienceStudent Colloquium Series: Thursday, February 23 at noon in 268 Olin Science

### Mathematical Models for Medical Decision Making

### Matthew Bailey

School of Management

Bucknell University

and Geisinger Health System

**Abstract: **In this talk I discuss the use of mathematical modeling within the realm of medical decision making. In particular, I focus on the question of when to initiate HIV therapy. The question of when to initiate HIV treatment is perhaps the most important question in HIV care. We model this problem as an optimal stopping problem: for certain health states patients may choose to wait one more month, whereas for others they may decide to accept the expected remaining lifetime associated with initiating therapy. We prove conditions under which there exist structural properties of the optimal solution and compare them to our data and results. Using data from a set of 25,000 HIV patients, we model and solve the problem as a Markov decision process. If time permits, I will discuss a broader class of operations research models for healthcare delivery.

PIZZA and DRINKS provided. All are welcome.

Distinguished Visiting Professor Seminar: Thursday, February 9, 4:00 pm in 371 Olin ScienceDistinguished Visiting Professor Seminar: Tuesday, February 7, 4:00 pm in 371 Olin ScienceStudent Colloquium Series: Thursday, February 9, 12:00 noon in 268 Olin Science

### The Mystic Hexagon Theorem: Polynomials and Geometry

### Karen Chandler

Department of Mathematical Sciences

Susquehanna University

**Abstract: **Consider a hexagon as a six-sided figure with sides not necessarily of equal lengths. When Blaise Pascal (1623-1662) was sixteen, he exhibited an amazing fact on hexagons. We will see how this extends to concepts in algebraic geometry: comparing polynomials with the geometry of their (common) sets of solutions. In particular, we shall see how Pascal's theorem extends to the theorem of Etienne Bezout (1730-1783) on curve intersections. We will then examine precisely how to remove all mystery on hexagons "from scratch." I will then describe how these concepts relate to my own research.

PIZZA and DRINKS provided. All are welcome.

Student Colloquium Series: Thursday, January 26, 12:00 noon in 268 Olin Science

### Sums of Squares: An Introduction to Quadratic Forms

Jodi Black

Department of Mathematics

Bucknell University

**Abstract: **The 2-square identity gives that products of sums of 2 squares are sums of 2 squares. More precisely:

(x_{1}^{2}+x_{2}^{2})(y_{1}^{2}+y_{2}^{2})=(x_{1}y_{1}-x_{2}y_{2})^{2}+(x_{2}y_{1}+x_{1}y_{2})^{2}

One of the nice properties of the expression above is that it is bilinear in the x_{i} and y_{j}. By 1898 bilinear 4-square and 8-square identities had been discovered and Hurwitz had shown that no other bilinear n-square identities exist. But what if we don't require bilinearity? For which positive integers n is a product of sums of n squares a sum of n squares? A complete answer would not come for more than half a century until Pfister's groundbreaking work on quadratic forms which not only resolved this longstanding open problem but also revolutionized the study of quadratic forms. We will discuss the sums of squares question as motivation for a brief foray into quadratic forms.

PIZZA and DRINKS provided. All are welcome.