Classification problems occur in all areas of mathematics. Descriptive set theory provides methods to assign complexity to such problems. Using a technique developed by Hjorth, Kechris and Sofronidis proved, for example, that the problem of classifying all unitary operators $\mathcal{U}(\mathcal{H})$ of an infinite dimensional Hilbert space up to unitary equivalence $\simeq_U$ is strictly more difficult than classifying graph structures with domain $\mathbb{N}$ up to isomorphism.

 
We present a game--theoretic approach to anti--classification results for orbit equivalence relations and use this development to reorganize conceptually the proof of Hjorth's turbulence theorem.  We also introduce a dynamical criterion for showing that an orbit equivalence relation is not Borel reducible to the orbit equivalence relation induced by a CLI group action; that is, a group which admits a complete left invariant metric (recall that, by a result of Hjorth and Solecki, solvable groups are CLI). We deduce that $\simeq_U$ is not classifiable by CLI group actions.

This is a joint work with Martino Lupini.  

This is a joint work with Piermarco Cannarsa and Wei Cheng. 

We study the properties of the set S of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation, for a Tonelli Hamiltonian. 

The main surprise is the fact that this set is locally arc connected—it is even locally contractible. This last property is far from generic in the class of semi-concave functions.

We also “identify” the connected components of this set S. 

This work relies on the idea of Cannarsa and Cheng to use the positive Lax-Oleinik operator to construct a global propagation of singularities (without necessarily obtaining uniqueness of the propagation).

This is a report of some recent progress and challenges we have made and encountered in modelling and numerical simulation of materially nonlinear beam structures with applications in micro-electrical-mechanical systems. For simplicity, the fully nonlinear DE’s and the associated initial/boundary value problems arising from modelling Hollomon’s power-law material structures are presented as our representative mathematical models. While for linear elastic materials, the principal operator in the equations appears to be the Laplacian or the bi-Laplacian operator, for the Hollomon’s materials, the principal operator is the p-Laplacian or the bi-p-Laplacian. The main results presented are centered around approximations of solutions to the nonlinear wave equation by lumped-parameter models and numerical methods. Similar, but more challenging models are also introduced for further investigations.

 We will discuss examples of singularity formation from smooth data for linear and quasilinear uniformly parabolic systems in the plane.