Questions about the structure of sums of Cantor sets, as well as related questions on properties of convolutions of singular measures, appear in dynamical systems (due to persistent homoclinic tangencies and Newhouse phenomena), probabilities, number theory, and spectral theory. We will describe the recent results (joint with D.Damanik and B.Solomyak) that claim that under some natural technical conditions convolutions of measures of maximal entropy supported on dynamically dened Cantor sets in most cases (for almost all parameters in a one parameter family) are absolutely continuous. This provides a rigorous proof of absolute continuity of the density of states measure for the Square Fibonacci Hamiltonian in the low coupling regime, which was conjectured by physicists more than twenty years ago.

The Nobel Prize-winning discovery of quasicrystals has spurred much work in aperiodic sequences and tilings. Here, we present numerical experiments conducted by undergraduates at the Summer Math Institute at Cornell under our supervision. Building on our previous work involving one-dimensional discrete Schrodinger operators with potentials given by primitive invertible substitutions on two letters, we present preliminary numerical data on the box-counting dimension and Hausdorff dimension of the spectrum of operators with potentials given by the Thue-Morse sequence and period doubling sequence. We also present preliminary numerical data on the spectrum of the discrete Laplacian on the Penrose tiling and octagonal tiling.

My talk will be devoted to a (quick and very brief) introduction to the domino tilings (intensively studied during the last fifty years), the subject that is very simple in the origin, while giving almost immediately very beautiful images. My goal will be to explain (roughly), where does the "arctic circle" effect in tilings come from, meanwhile mentioning asymptotic shape of Young diagrams, entropy, height function and variational problems. If the time permits, I will speak about computation of determinants and permanents.

Take a finitely-generated group of (analytic) circle diffeomorphisms. Since the times of Poincaré we know that any such action admits either a finite orbit, or a Cantor minimal set, or the action is minimal on all the circle. But what else can be said on such a group?

In this direction, there are well-known questions due to Sullivan, Ghys and Hector: assuming that such an action is minimal, is it necessarily Lebesgue-ergodic? If there is a Cantor minimal set, is it necessarily of a zero Lebesgue measure?

Our results provide a positive answer to the latter question, in some cases allow to resolve the former one and, more generally speaking, give some kind of understanding how a general characterization of an action can look like. This is a joint project with B. Deroin, D. Filimonov, and A. Navas.

As is well known, a class of one-dimensional lattice models, such as Ising models, Jacobi and CMV operators and others, are susceptible to renormalization analysis that can be carried out via the transfer matrix formalism. As a result, models on the one-dimensional lattice of a certain quasi-periodic type (namely, those generated by primitive substitutions) can be studied via dynamics of so-called trace maps, which are polynomial maps acting on the real (or complex) Euclidean space of appropriate dimension. A prominent example is the widely studied Fibonacci model. Much work has been done in this direction. At some point it became apparent that a model-independent framework, based on the dynamics of trace maps, can be built, that would cover essentially all models the relevant information of which is encapsulated in the traces of the associated transfer matrices (and, as experience has shown, this information is very difficult if not impossible to obtain via techniques other than the trace map). The purpose of this talk is to give a broad overview of past and very recent results on the dynamics of the Fibonacci trace map in a model-independent fashion, motivated by a class of models from physics, and with a view towards applications to those models. We shall cover hyperbolicity and partial hyperbolicity of the trace map and implications in spectral theory of Jacobi operators; some applications to Ising models; recent advances in understanding invariant measures on the invariant hyperbolic sets and implications for the density of states measures for the Jacobi operators; the Newhouse phenomenon and mixed behavior with large (in the sense of Hausdorff dimension) chaotic sea, and some connections with kicked two-level systems. Time permitting, we shall also state some open problems.