Georgia Institute of Technology College of Sciences

Some basic problems, notions and results of the Ergodic theory will be introduced. Several examples will be discussed. It is also a preparatory talk for the next day colloquium where finite time properties of dynamical and stochastic systems will be discussed rather than traditional questions all dealing with asymptotic in time properties.

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.

In this talk, we consider the structure of a real $n \times n$ matrix in the form of $A=JL$, where $J$ is anti-symmetric and $L$ is symmetric. Such a matrix comes from a linear Hamiltonian ODE system with $J$ from the symplectic structure and the Hamiltonian energy given by the quadratic form $\frac 12\langle Lx, x\rangle$. We will discuss the distribution of the eigenvalues of $A$, the relationship between the canonical form of $A$ and the structure of the quadratic form $L$, Pontryagin invariant subspace theorem, etc. Finally, some extension to infinite dimensions will be mentioned.