We use the extent to which the combinatorial principle "weak
square" holds to quantify the influence of Martin's Maximum
on the combinatorics of singular cardinals.
(joint work with Menachem Magidor)

Which consistent statements can be forced to be true?
It is shown that "resectionable" \Sigma_1 statements about parameters
in H_{\omega_2} which are "honestly consistent" can be forced
to be true in a stationary set preserving extension, and we also show
that a strong form of BMM, according to which all "honestly consistent"
\Sigma_1 statements about parameters in H_{\omega_2} are true,
is consistent. We also give some applications.

Finding a point on a variety amounts to finding a solution to a system of
polynomials. Finding a "rational point" on a variety amounts to finding a
solution with coordinates in a fixed base field. (Warning: our base field
will not be the field of rational numbers Q.) We will present some
theorems about when it is possible to find such a rational point. We will
state Tsen's theorem and the Chevalley-Warning Theorem. We will also
state some more recent results of Hassett-Tschinkel and
Graber-Harris-Starr, which rely on the notion of a "rationally connected
variety". This notion is an analogue of the notion of "path
connectedness" in topology.