Four dimensions is unique in many ways. For example $n$-dimensional Euclidean space has a unique smooth structure if and only if $n$ is not equal to four. In other words, there is only one way to understand smooth functions on $R^n$ if and only if $n$ is not 4. There are many other way that smooth structures on 4-dimensional manifolds behave in surprising ways. In this talk I will discuss this and I will sketch the beautiful interplay of ideas (you got algebra, analysis and topology, a little something for everyone!) that go into proving $R^4$ has more that one smooth structure (actually it has uncountably many different smooth structures but that that would take longer to explain).
The Institute for Defense Analyses - Center for Computing Sciences is a nonprofit research center that works closely with the NSA. Our center has around 60 researchers (roughly 30 mathematicians and 30 computer scientists) that work on interesting and hard problems. The plan for the seminar is to begin with a short mathematics talk on a project that was completed at IDA-CCS and declassified, then tell you a little about what we do, and end with your questions. The math that we will discuss involves symbolic dynamics and automata theory. Specifically we will develop a metric on the space of regular languages using topological entropy. This work was completed during a summer SCAMP at IDA-CCS. SCAMP is a summer program where researchers from academia (professors and students), the national labs, and the intelligence community come to IDA-CCS to work on the agency's hard problems for 11 weeks.