January 26,

** Abstract: ** Sudoku puzzles have appeared all over the world in the last year and are currently very popular in newspapers. What are they, how does one solve them, and what are some interesting mathematical questions associated with them? This is a low level fun talk with some open mathematical questions.

**"On generalized Schwarz-Pick estimates"**

** Abstract:** Pick's invariant form of Schwarz's lemma gives a best possible estimate of the derivative of an analytic function which is bounded by one on the unit disk. A Hilbert space method will be used to obtain a similar estimate of higher order derivatives.

February 9,

**A Day in the Life of a Cryptanalyst **

** Abstract:** What does a cryptananalyst do? What is it like to be a mathematician at NSA? What are the summer internship possibilities for college students at NSA? What are the job prospects for mathematicians as a cryptananalyst?

*Find out from a Bucknell mathematics alumnus with twenty plus years experience at NSA*

February 12-18-

Colloquium Talk

February 14: 4:00 OLIN 372**"How far from normal are you?"**

*Abstract :* If you are a matrix A and somewhat un-normal, can you find your closest normal neighbor? In this talk we will discuss different ways of measuring how far you are from being normal! Of course, if you are a matrix, then you live in the world of matrices and your normal neighbors are normal matrices. So, for a matrix A, can you find the closest normal matrix to A? Or how about the closest symmetric matrix to A or the closest diagonal matrix to A? We will discuss these types of questions and give some estimates on finding the distance from a matrix A to the set of normal matrices. There are several open problems in this area, even problems about 3x3 matrices!

Seminar Talk

February 16: 4:00 OLIN 372**"The distance to the Normal Matrices: Part II"**

** Abstract**: In this talk we will discuss some proofs of some of the results mentioned in part I of this talk, as well as discuss some infinite dimensional versions of these questions.

The first talk is accessible to students who have had linear algebra. The second talk will be a little more in depth, but may be still meaningful for some students.

February 23,

**"A Taste of Complexity"**

**ABSTRACT:** Underlying many modern information security systems is the notion that users can quickly create instances of a particular mathematical problem that would be extremely difficult for an evesdropper to solve. (Without the required ''key'', an unauthorized recipient of a message must solve the mathematical problem in order to decipher the message...remember last week's talk?)

To impose a practical standard on the computational difficulty of a problem, one might require that a typical instance of the problem will take several years to solve, even with the assistance of a very powerful computer. (This is true of the system that protects your e-mail, for example.) In this talk I will introduce a more theoretical measure of computational ease and difficulty, thereby providing a mathematical platform for investigating computational tractibility. This leads to a wealth of interesting problems, including several fascinating open questions.

Mathematical Competition

**The Bucknell Mathematics Department invites any high school who wishes to participate as a team of three. The competition is intended to discover and encourage mathematical talent. The exam itself is designed to be a collection of challenging and fun-filled problems. You must contact the Math Dept to register. ****(570-577-1343)**

*9:30 AM Registration -Vaughn Auditorium10:00-12:00 Exam12:00 - Lunch "Bostwick Cafeteria" Langone Center*

March 23,

**Winning at Combinatorial Games**

Abstract: Attend this talk and afterward impress your friends with your winning ways in games such as Kayles, Binary Nim, Fibonacci Nim, and Nim itself.

April 6

**Title: A card trick (and some coding theory)Speakers: Linda Scalici ('06) and George Exner**

Abstract: We'll begin with a card trick. We'll then turn to an area of mathematics called algebraic coding theory, which concerns the sending of messages over a "noisy" channel that may produce errors in what is received compared to what was sent. How can we arrange our messages so the receiver can detect that an error or errors were made? How can we form our messages so that the receiver can correct back to what was actually sent? The techniques used are everywhere in our data driven society, ranging from credit card numbers and ISBN book numbers to music CD's and pictures from Mars. All are invited, since no mathematical background is needed for the talk.

April 8-15-

**Colloquium 1 - April 10 - 4:00 - Olin 372**

** Abstract:** This talk will be somewhat accessible to undergraduates and will represent a survey of many known results. If G is a finite group of order n then it is well known that if H is a subgroup of G then |H| divides the order of G. This result, Lagrange’s Theorem, does not have a converse. However there are a number of important theorems asserting that certain subgroups of certain orders of a group exist. Principal among these are the theorem of Sylow asserting that if p is a prime and |G| = prm, where p,m are relatively prime then G contains a subgroup of order pr. These theorems of Sylow were generalized by P. Hall in 1928 (and even further in the early 1960’s). We shall first discuss these results and their consequences and then see how this relates to infinite groups.

**Colloquium 2 - April 13 - 4:00 - Olin 372**

**LINEAR GROUPS WITH RANK RESTRICTIONS ON THE SUBGROUPS OF INFINITE CENTRAL DIMENSION**

** Abstract:** The subject of this talk represents recent joint work with Leonid Kurdachenko. Let F be a field, let A be a vector space over F and let GL(F,A) denote the group of all automorphisms of A. As usual, a group G that is isomorphic to a subgroup of GL(F,A) is called a linear group. If dimFA, the dimension of A over F, is finite, then G is often called a finite dimensional linear group. Such groups have been well-studied, partly because of the well-known identification of GL(F,A) with the group of n × n matrices with entries in F where n= dimFA In this talk we study the case when A is infinite dimensional over the field F. Let H be a subgroup of GL(F,A) and note that H acts on the quotient space A/CA(H) in a natural way. We define dimFH to be dimF (A/CA(H)) and say that H has finite central dimension if dimFH is finite. A group G is said to have finite 0-rank (or finite torsion-free rank), r0(G) = r, if G has a finite subnormal series with exactly r infinite cyclic factors, all other factors being periodic. In this talk we discuss groups which are of infinite central dimension and infinite torsion-free rank, but in which every proper subgroup either has finite central dimension or has finite torsion-free rank. We shall sketch the proofs of some recent results.

George Andrews Penn State University

** "Fibonacci numbers, Ramanujan and Continued Fractionsor Why I Took Zeckendorf's Theorem Along On My Last Trip To Canada"**

** Abstract:** This talk focuses on the famous Indian genius, Ramanujan. We shall try to lead gently from some simple problems involving Fibonacci numbers to a discussion of some of Ramanujan's achievements. All of the talk should be easily understood by anyone who has taken calculus, and much of it can be understood by students currently taking calculus.

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