A series of informal talks (January-February, 2013), following the book by Peter J. Olver.

In this talk I will discuss how strong forms of non-amenability are reflected in the asymptotic behavior of a group's Bernoulli action. Central to the discussion will be the notion of shift-minimality: A countable group G is called shift-minimal if every non-trivial measure preserving action weakly contained in the Bernoulli shift of G is free. I will discuss the connection between shift-minimality and certain properties of the reduced C*-algebra of G, and indicate the proof that if G admits a free pmp action of cost >1 then there is a finite normal subgroup N of G such that G/N is shift-minimal.

Fourier series provide a way of writing almost any signal as a superposition of pure tones, or musical notes. Unfortunately, this representation is not local, and it does not reflect the way that music is actually generated by instruments playing individual notes at different times. We will discuss time-frequency representations, which are a type of local Fourier representation of signals. While such representations have limitations when it comes to music, they are powerful mathematical tools that appear widely throughout mathematics (e.g., partial differential equations and pseudodifferential operators), physics (e.g., quantum mechanics), and engineering (e.g., time-varying filtering). We ask one very basic question: are the notes in this representation linearly independent? This seemingly trivial question leads to surprising mathematical difficulties. This talk is intended to be introductory and accessible to beginning graduate students. Tea at 3:30 pm in SC 1425.

Frames can be seen as generalized bases, that is, over-complete collections, which are used for stable representations of signals as linear combinations of basic building atoms. It is very useful when we can use locally adapted atoms, which in addition behave as elements of local bases. We explore the possibility of using localized parts of frames and bases when building a customized frame. After a review on Gabor frames and Wilson bases, we consider the question of combining parts of these collections into a multi-frame set and look at its properties.

Ever wonder how you can safely send your credit card number over the internet? The answer is RSA, the first widely used public-key cryptographic communications system. Using only elementary techniques from
number theory, RSA allows you to send secure communications over public channels without a pre-arranged code. In this talk, we discuss the difference between public-key and private-key cryptography, and cover some
basic ideas from number theory. Then we will show how to use RSA to encode and decode messages, and explain why this process works and why it is so difficult to crack.

We will show that the reverse mathematical strength of the statement "every commutative ring with identity has a prime ideal" is equivalent to WKL (Weak K\"onig's Lemma) over RCA (Recursive Comprehension Axiom).

This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. Tea at 3:30 pm in 1425 Stevenson Center

The Four Color Theorem is easy to state:  If you have a (nice enough) map of countries, and you want to color them so that no two adjacent countries are the same color, then you never need more than four colors to do it!  This theorem was proved in the 70's, but its proof is controversial among mathematicians due to its heavy use of a computer.  We will discuss the history of this theorem, and then prove the Five Color Theorem, which states you never need more than five colors to color our map, which is much easier than four!