# Lagrangians from Algebra and Combinatorics

Lead Research Organisation:
University of Edinburgh

Department Name: Sch of Mathematics

### Abstract

Mirror symmetry is a connection between three seemingly unrelated kinds of geometry.

This link, first observed by physicists in string theory, has led to an flurry of surprising predictions, powerful tools, and fascinating questions for geometers.

The first kind of geometry, algebraic geometry, studies rigid objects defined by equations such as lines, circles, and ellipses. Geometers have studied enumeration problems in algebraic geometry since antiquity. Starting at common knowledge (e.g., how many points do two lines intersect at), questions in enumerative geometry rapidly approach the limits of our mathematical capability.

Symplectic geometry is defined by the laws of motion, like the trajectory of a planet around the sun. While historically used to calculate the mechanics of constrained systems, such as a pendulum attached to a metal rod, the language of symplectic geometry has become the natural setting for mathematical physics as a whole.

The third kind of geometry --- called tropical geometry for its historical connection to Brazil --- studies optimization and maximization. Shapes here describe the problems in linear optimization, such as efficiently assigning workers to jobs in a factory.

On a small scale, algebraic and symplectic geometry look incredibly different: algebraic geometry seems very orderly, while symplectic geometry frequently handles chaotic scenarios such as a table full of billiard balls.

However, when one zooms out to view the large-scale behaviour, both geometries can be approximated by tropical geometry. If an algebraic space and symplectic space tropically look the same, they are called ``mirror spaces''.

In the last two decades, the study of ``Complex to Tropical correspondences'' has produced exciting results by viewing problems in algebra through the lens of tropical geometry.

This proposal will look at recently developed ``Tropical to Lagrangian correspondences,'' where symplectic structures (Lagrangians) are built from tropical data.

The research agenda within the scope of mirror symmetry includes finding new mirror spaces, developing new computational methods in symplectic geometry, and investigating exciting applications of symplectic geometry in tropical curves. The project will also look at interfaces outside of geometry, specifically to dimer models (a description of domino tilings) and mutations (a modification process occurring in symplectic geometry, cluster algebras, and triangulated categories).

This link, first observed by physicists in string theory, has led to an flurry of surprising predictions, powerful tools, and fascinating questions for geometers.

The first kind of geometry, algebraic geometry, studies rigid objects defined by equations such as lines, circles, and ellipses. Geometers have studied enumeration problems in algebraic geometry since antiquity. Starting at common knowledge (e.g., how many points do two lines intersect at), questions in enumerative geometry rapidly approach the limits of our mathematical capability.

Symplectic geometry is defined by the laws of motion, like the trajectory of a planet around the sun. While historically used to calculate the mechanics of constrained systems, such as a pendulum attached to a metal rod, the language of symplectic geometry has become the natural setting for mathematical physics as a whole.

The third kind of geometry --- called tropical geometry for its historical connection to Brazil --- studies optimization and maximization. Shapes here describe the problems in linear optimization, such as efficiently assigning workers to jobs in a factory.

On a small scale, algebraic and symplectic geometry look incredibly different: algebraic geometry seems very orderly, while symplectic geometry frequently handles chaotic scenarios such as a table full of billiard balls.

However, when one zooms out to view the large-scale behaviour, both geometries can be approximated by tropical geometry. If an algebraic space and symplectic space tropically look the same, they are called ``mirror spaces''.

In the last two decades, the study of ``Complex to Tropical correspondences'' has produced exciting results by viewing problems in algebra through the lens of tropical geometry.

This proposal will look at recently developed ``Tropical to Lagrangian correspondences,'' where symplectic structures (Lagrangians) are built from tropical data.

The research agenda within the scope of mirror symmetry includes finding new mirror spaces, developing new computational methods in symplectic geometry, and investigating exciting applications of symplectic geometry in tropical curves. The project will also look at interfaces outside of geometry, specifically to dimer models (a description of domino tilings) and mutations (a modification process occurring in symplectic geometry, cluster algebras, and triangulated categories).

## People |
## ORCID iD |

Jeffrey Hicks (Principal Investigator / Fellow) |

Description | Project on Rouquier Dimensions of Categories via Lagrangian Cobordisms |

Organisation | University of Massachusetts Boston |

Country | United States |

Sector | Academic/University |

PI Contribution | Andrew Hanlon, Oleg Lazarev and I are studying applications of Lagrangian cobordisms and other techniques in symplectic geometry to obtain bounds for the Rouquier dimension of the Fukaya category. An immediate mathematical application is to study the Rouquier dimension of a toric variety. My contribution to the project has been to handle some of the combinatorial manipulations that bound the Rouquier dimension, and to understand how Lagrangian cobordisms enter the bound. |

Collaborator Contribution | Professor Oleg Lazarev is an expert in partially wrapped Fukaya categories and Weinstein domains. He has suggested a strategy for understanding the more general setting of cotangent bundles of a manifold via studying the Lusternik-Schnirelmann category. Dr. Andrew Hanlon is a frequent collaborator who brings expertise on homological mirror symmetry for toric varieties. In particular, he has related several of the constructions we've discussed to the toric map of Frobenius, which answers a conjecture of Bondal. |

Impact | Our collaboration has produced one preprint, "Resolutions of toric subvarieties by lines bundles and applications". Hanlon. Lazarev, and Ihave given several conference talks on our approach to this problem |

Start Year | 2022 |