Eleny Ionel
Stanford University, Stanford, California, USA

There are several ways of counting holomorphic curves in Calabi-Yau 3-folds. Counting them as maps gives rise to the Gromov-Witten invariants. In general, these are not integer counts due to the presence of multiple covers with symmetries. But one can consider instead images of such maps (possibly with multiplicity), regarded either as subsets or as integral currents. Generically these images are smoothly embedded curves. In earlier joint work with Thomas Parker we constructed an integer count of embedded pseudo-holomorphic curves in symplectic Calabi-Yau 3-folds, and related it to the Gromov-Witten invariants. In recent work with Aleksander Doan and Thomas Walpuski we extended these arguments to also prove that the former invariants satisfy a finiteness property. The new ingredients are compactness (and regularity) results for pseudo-holomorphic cycles/currents without an a priori genus bound, instead of the Gromov compactness for pseudo-holomorphic maps. In this talk I will outline some of the key ideas involved in these constructions.