Interval translation maps (ITMs) are non-invertible generalizations of interval exchanges (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. The finiteness conjecture says that the subset of ITMs of finite type is open, dense, and has full Lebesgue measure. In my talk, I will prove the conjecture for the ITMs of three intervals and discuss some open problems.

We survey some results that employ renormalization ideas in the spectral analysis of suitable self-similar Hamiltonians. A guiding example will be given by the Laplacian on the Sierpinsky lattice. We describe results on the spectrum and the eigenfunctions of this operator and also talk about how they generalize to a wider class of models.

We will discuss recent results obtained for the one-dimensional discrete Schrodinger operator with potential given by a primitive invertible substitution sequence. This talk focuses on the methods used to obtain these results, similarities and differences from previous methods, and obstructions to further generalization.

In 1966, Marc Kac in his famous paper 'Can one hear the shape of a drum?' raised the following question: Is a bounded Euclidean domain determined up to isometries from the eigenvalues of the Euclidean Laplacian with either Dirichlet or Neumann boundary conditions? Physically, one motivation for this problem is identifying distant physical objects, such as stars or atoms, from the light or sound they emit.

The only domains which are known to be spectrally distinguishable from all other domains are balls. It is not even known whether or not ellipses are spectrally rigid, i.e. whether or not any continuous family of domains containing an ellipse and having the same spectrum as that ellipse is necessarily trivial.

In a joint work with Steve Zelditch we show that ellipses are infinitesimally spectrally rigid among smooth domains with the symmetries of the ellipse. Spectral rigidity of the ellipse has been expected for a long time and is a kind of model problem in inverse spectral theory. Ellipses are special since their billiard flows and maps are completely integrable. It was conjectured by G. D. Birkhoff that the ellipse is the only convex smooth plane domain with completely integrable billiards. Our results are somewhat analogous to the spectral rigidity of flat tori or the sphere in the Riemannian setting. The main step in the proof is the Hadamard variational formula for the wave trace. It is of independent interest and it might have applications to spectral rigidity beyond the setting of ellipses. The main advance over prior results is that the domains are allowed to be smooth rather than real analytic. Our proof also uses many techniques developed by Duistermaat-Guillemin and Guillemin-Melrose in closely related problems.

Thermodynamic formalism as a mathematical theory has its roots in one of the most successful theories of physics -- thermodynamics and statistical mechanics. In its core, the theory of thermodynamic formalism seeks to describe properties of observable "macroscopic" phenomena based on the average behavior of the "microscopic" constituents. In the language of dynamical systems: given a dynamical system $(X, f)$, with $X$ the phase space and $f$ the map defining the dynamics, one seeks to describe properties of functions defined on $X$ (the macroscopic observables) based on the (often averaged, in some well-defined sense) behavior of $f$. In particular, thermodynamic formalism leads to strong results in dimension theory of dynamical systems (e.g. describing fractal dimensions and measures of sets arising as invariant sets of some chaotic dynamical systems). In this first of a series of two talks, we shall present the main ingredients of thermodynamic formalism: topological entropy, metric entropy, topological pressure, and the variational principle for the pressure.