Recent important results in Maths and Logic
Decidability and undecidability

The integers are first order definable by a universal formula in the field of rationals.
By Jochen Koenigsmann,
Survey
Published in Lecture Notes in Mathematics, vol. 2111
preprint
Knot theory

Explicit upper bounds for Reidemeister moves for recognizing unknots
and not equivalence.
unknots by Marc Lackenby, and
knotequivalence by Alexander Coward and Marc Lackenby.
slidesby A. Coward.
slidesby A. Lackenby.

A polynomial upper bound on Reidemeister moves Annals Math. (accepted)
http://people.maths.ox.ac.uk/lackenby/pol19514.pdf

(With Alexander Coward) An upper bound on Reidemeister moves Amer. J. Math. (accepted)
http://people.maths.ox.ac.uk/lackenby/rei09n11.pdf

(with Alexander Coward) Unknotting genus one knots Comment. Math. Helv. 86 (2011) 383399.
httP://PEOPle.maths.ox.ac.uk/lackenby/ugo05059.pdf