Title: Gopakumar-Vafa invariants of local curves 
Abstract: In the 1990s, two physicists, Gopakumar and Vafa, proposed an ideal way to count curves in a Calabi-Yau threefold, which is conjecturally equivalent to other curve counting theories such as Gromov-Witten theory. It is very recent that Maulik and Toda gave a mathematically rigorous definition of GV invariants. In this talk,…

In higher dimensional algebraic geometry, the following three types of varieties are considered to be the building blocks: Fano varities, Calabi-Yau varieties, and varieties of general type. In the study of varieties of general type, one usually works on "good models" inside birtationally equivalent classes. Minimal models and canonical models are natural choices of good models.
In the…

Abstract: I will discuss three recent combinatorial theorems about hyperplane arrangements:  the top-heavy conjecture, log concavity of the characteristic polynomial, and non-negativity of the Kazhdan-Lusztig polynomial.  Each of these results is proved by studying the cohomology of a projective algebraic variety associated with the arrangement.