We characterize which Borel functions are decomposable into
a countable union of functions which are piecewise continuous on
$\Pi^0_n$ domains, assuming projective determinacy. One ingredient of
our proof is a new characterization of what Borel sets are $\Sigma^0_n$
complete. Another important ingredient is a theorem of Harrington that
there is no projective sequence of length $\omega_1$ of distinct Borel
sets of bounded rank, assuming projective determinacy. This is joint
work with Adam Day.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.