Anders Buch

Enumerative Geometry of Grassmannians and Positivity Questions in Schubert Calculus

June 11 and 12, 2024

Source: Rutgers Current Trends in Mathematics

Poincaré Duality

If $M$ is an $n$-dimensional oriented closed manifold (compact and without boundary), then the $k$ th cohomology group of $M$ is isomorphic to the $(n-k)$ th homology group of $M$: $$ H^k(M)\cong H_{n-k}(M) $$

Quantum Cohomology

Two equivalent definitions of quantum cohomology $\text{QH}(X)$:

  1. the free $\mathbb{Z}[q]$-module with basis ${[X^\lambda]}$, or
  2. $H^*(X)\otimes_{\mathbb{Z}} \mathbb{Z}[q]$.

Gromov-Witten Invariant

Given a projective variety $X$ and $|\lambda| + |\mu| + |\nu| = \dim(X) + d$, the Gromov-Witten invariant is $$ \langle X^\lambda, X^\mu, X^\nu \rangle = # \left{ \phi: \mathbb{P}^1 \to X \mid \deg(\phi) = d\right}. $$