There is a strong relation between the existence of non-singular solutions for the normalized Ricci flow and the underlying smooth structure of a 4-manifold. We are going to discuss an obstruction to the existence of non-singular solutions and its applications. The main examples are connected sums of complex projective planes and complex projective planes with reversed orientation. The key ingredients in our methods are the Seiberg-Witten Theory and symplectic topology. This is joint work with M. Ishida and R. Rasdeaconu.

An even embedding of a graph on a closed surface is a fixed 2-cell embedding such that each face is bounded by a closed walk of even length. It is known that a complete graph on n (> 6) vertices has an even embedding on a closed surface F^2 with Euler characteristic chi <=n(5-n)/4.  Such embeddings are called minimum genus even embeddings.  It is known that there is an invariant of even embeddings of graphs, which is called "cycle parity''.  It divides non-bipartite even embeddings of graphs into three classes on a fixed surface.  In this talk, we introduce cycle parity and consider relationships between cycle parity and minimum genus even embeddings of the complete graphs.

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.

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

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).