arXiv subjects

Eduardo Gonzalez

Publications and source records attributed to Eduardo Gonzalez.

11 recordsLinked to original sources

arXiv2026

Coulomb branch algebras via symplectic cohomology

Let $(\bar{M}, ω)$ be a compact symplectic manifold with convex boundary and $c_1(T\bar{M})=0$. Suppose that $(\bar{M}, ω)$ is equipped with a convex Hamiltonian $G$-action for some connected, compact Lie group $G$. We construct an action of the pure Coulomb branch of $G$ on the $G$-equivariant symplectic cohomology of $\bar{M}.$ Building on work of Teleman, we use this construction to characterize the Coulomb branches of Braverman-Finkelberg-Nakajima in terms of equivariant symplectic cohomology.

math.SG↗

arXiv2024

Quantum cohomology and toric minimal model programs

We give a quantum version of the Danilov-Jurkiewicz presentation of the cohomology of a compact toric orbifold with projective coarse moduli space. More precisely, we construct a canonical isomorphism from a formal version of the Batyrev ring to the quantum orbifold cohomology at a canonical bulk deformation. This isomorphism generalizes results of Givental, Iritani, and Fukaya-Oh-Ohta-Ono for toric manifolds and Coates-Lee-Corti-Tseng for weighted projective spaces. The proof uses a quantum version of Kirwan surjectivity and an equality of dimensions deduced using a toric minimal model program (tmmp). We show that there is a natural decomposition of the quantum cohomology where summands correspond to singularities in the tmmp, each giving rise to a collection of Hamiltonian non-displaceable tori.

math.AG↗

arXiv2023

A wall-crossing formula for Gromov-Witten invariants under variation of git quotient

We prove a quantum version of Kalkman's wall-crossing formula comparing Gromov-Witten invariants on geometric invariant theory (git) quotients related by a change in polarization. The wall-crossing terms are gauged Gromov-Witten invariants with smaller structure group. As an application, we show that the graph Gromov-Witten potentials of quotients related by wall-crossings of crepant type are equivalent up to a distribution in the quantum parameter that is almost everywhere zero. This is a version of the crepant transformation conjecture of Li-Ruan, Bryan-Graber, Coates-Ruan etc. in cases where the crepant transformation is obtained by variation of git.

math.AG↗

arXiv2022

Sparsely Activated Mixture-of-Experts are Robust Multi-Task Learners

Traditional multi-task learning (MTL) methods use dense networks that use the same set of shared weights across several different tasks. This often creates interference where two or more tasks compete to pull model parameters in different directions. In this work, we study whether sparsely activated Mixture-of-Experts (MoE) improve multi-task learning by specializing some weights for learning shared representations and using the others for learning task-specific information. To this end, we devise task-aware gating functions to route examples from different tasks to specialized experts which share subsets of network weights conditioned on the task. This results in a sparsely activated multi-task model with a large number of parameters, but with the same computational cost as that of a dense model. We demonstrate such sparse networks to improve multi-task learning along three key dimensions: (i) transfer to low-resource tasks from related tasks in the training mixture; (ii) sample-efficient generalization to tasks not seen during training by making use of task-aware routing from seen related tasks; (iii) robustness to the addition of unrelated tasks by avoiding catastrophic forgetting of existing tasks.

cs.LG↗

arXiv2016

Quantum Witten localization and abelianization for qde solutions

We prove a quantum version of the localization formula of Witten that relates invariants of a git quotient with the equivariant invariants of the action. Using the formula we prove a quantum version of an abelianization formula of S. Martin relating invariants of geometric invariant theory quotients by a group and its maximal torus, conjectured by Bertram, Ciocan-Fontanine, and Kim. By similar techniques we prove a quantum Lefschetz principle for holomorphic symplectic reductions. As an application, we give a formula for the fundamental solution to the quantum differential equation (qde) for the moduli space of points on the projective line and for the smoothed moduli space of framed sheaves on the projective plane (a Nakajima quiver variety).

math.SG↗

arXiv2015

Seidel elements and potential functions of holomorphic disc counting

Let M be a symplectic manifold equipped with a Hamiltonian circle action and let L be an invariant Lagrangian submanifold of M. We study the problem of counting holomorphic disc sections of the trivial M-bundle over a disc with boundary in L through degeneration. We obtain a conjectural relationship between the potential function of L and the Seidel element associated to the circle action. When applied to a Lagrangian torus fibre of a semi-positive toric manifold, this degeneration argument reproduces a conjecture (now a theorem) of Chan-Lau-Leung-Tseng relating certain correction terms appearing in the Seidel elements with the potential function.

math.SG↗

arXiv2015

Gauged Gromov-Witten theory for small spheres

For smooth projective G-varieties, we equate the gauged Gromov-Witten invariants for sufficiently small area and genus zero with the invariant part of equivariant Gromov-Witten invariants. As an application we deduce a gauged version of abelianization for Gromov-Witten invariants. In the symplectic setting, we prove that any sequence of genus zero symplectic vortices with vanishing area has a subsequence that converges after gauge transformation to a holomorphic map with zero average moment map.

math.SG↗

arXiv2011

Seidel elements and mirror transformations

The goal of this article is to give a precise relation between the mirror symmetry transformation of Givental and the Seidel elements for a smooth projective toric variety $X$ with $-K_X$ nef. We show that the Seidel elements entirely determine the mirror transformation and mirror coordinates.

math.AG↗

arXiv2010

Deformations of symplectic vortices

We prove a gluing theorem for a symplectic vortex on a compact complex curve and a collection of holomorphic sphere bubbles. Using the theorem we show that the moduli space of regular stable symplectic vortices on a fixed curve with varying markings has the structure of a stratified-smooth topological orbifold. In addition, we show that the moduli space has a non-canonical $C^1$-orbifold structure.

math.SG↗

arXiv2010

Classifying semi-free Hamiltonian $S^1$-manifolds

In this paper we describe a method to establish when a symplectic manifold $M$ with semi-free Hamiltonian $S^{1}$-action is unique up to isomorphism (equivariant symplectomorphism). This will rely on a study of the symplectic topology of the reduced spaces. We prove that if the reduced spaces satisfy a rigidity condition, then the manifold $M$ is uniquely determined by fixed point data. In particular we can prove that there is a unique family up to isomorphism of 6-dimensional symplectic manifolds with semi-free $S^1$-action and isolated fixed points.

math.SG↗

arXiv2003

Quantum cohomology and $S^1$-actions with isolated fixed points

This paper studies symplectic manifolds that admit semi-free circle actions with isolated fixed points. We prove, using results on the Seidel element due to McDuff and Tolman, that the (small) quantum cohomology of a $2n$ dimensional manifold of this type is isomorphic to the (small) quantum cohomology of a product of $n$ copies of $\bb{P}^1$. This generalizes a result due to Tolman and Witsman.

math.SG↗