Speaker: Gorapada Bera
Date: December 15, 2025
Affiliation: Stony Brook University, U.S.A.

Abstract: $G_2$-manifolds are $7$-dimensional Riemannian manifolds with special holonomy, analogous to Calabi--Yau $3$-folds, and associative submanifolds are a special class of volume-minimizing $3$-dimensional submanifolds, analogous to holomorphic curves or special Lagrangians. Inspired by the counting of holomorphic curves or special Lagrangians in Calabi--Yau $3$-folds, Joyce, Doan, and Walpuski have proposals about defining enumerative invariants of $G_2$-manifolds by counting closed associative submanifolds. In this talk, I will focus on three topics connected to this enumerative theory. First, I will discuss a key ingredient of these proposals: the appearance of conically singular associative submanifolds and their desingularizations. Afterwards, I will describe a method for constructing associative submanifolds in a class of $G_2$-manifolds known as twisted connected sums. Finally, I will discuss Donaldson’s program on the adiabatic limit of $K3$-fibered $G_2$-manifolds, along with conjectures regarding associative submanifolds and potential directions for future research.