##### Department of Mathematics,

University of California San Diego

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## Thomas Walpuski

#### Humbolt University

## Gopakumarâ€“Vafa finiteness: an application of geometric measure theory to symplectic geometry

##### Abstract:

The purpose of this talk is to illustrate an application of the powerful machinery of geometric measure theory to a conjecture in Gromovâ€“Witten theory arising from physics. Very roughly speaking, the Gromovâ€“Witten invariants of a symplectic manifold (X,Ï‰) equipped with a tamed almost complex structure J are obtained by counting pseudo-holomorphic maps from mildly singular Riemann surfaces into (X,J). It turns out that Gromovâ€“Witten invariants are quite complicated (or â€œhave a rich internal structureâ€). This is true especially for if (X,Ï‰) is a symplectic Calabiâ€“Yau 3â€“fold (that is: dim X = 6, c_1(X,Ï‰) = 0). In 1998, using arguments from Mâ€“theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabiâ€“Yau 3â€“folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromovâ€“Witten invariants by a transformation of the generating series. The Gopakumarâ€“Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition. The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromovâ€™s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that Gromovâ€™s compactness theorem can (and should!) be replaced with the work of Federerâ€“Flemming, Allard, and De Lellisâ€“Spadaroâ€“Spolaor. This upgrade of Ionel and Parkerâ€™s cluster formalism proves both the integrality and finiteness conjecture. This talk is based on joint work with Eleny Ionel and Aleksander Doan.

### December 1, 2022

### 1:00 PM

APM 7321

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