We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. We establish a real analog of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data. As an application, we realize all possible diffeomorphism types of connected real Lagrangians in toric symplectic del Pezzo surfaces.

This thesis consists of four main chapters. Chapter 2, An introduction to the Chekanov torus, is an expository article introducing all the concepts necessary to define and under- stand the Chekanov torus. We give a detailed introduction to Hamiltonian circle actions, symplectic reduction, lifting techniques and versal deformations and show how those can be used to show that the Chekanov torus is exotic. Chapter 3, On the topology of real Lagrangians in toric symplectic manifolds, is a joint article with Joontae Kim and Jiyeon Moon focusing on constructing examples of real Lagrangian submanifolds in toric manifolds by lifting symmetries from the moment polytope. We also prove convexity and tightness for the examples we construct and give an analogue of the Delzant construction. Chapter 4, Real Lagrangian tori and versal deformations, is an article focusing on obstructions for a given Lagrangian submanifold to be real and is, in some sense, complementary to Chapter 3. We develop a general obstruction in terms of versal deformations and displacement energy and apply this to toric fibres and Chekanov tori in toric manifolds. Chapter 5, Squeezing via degenerations of the complex projective plane, is an appendix to the paper On certain quantifications of Gromov’s non-squeezing theorem by Kevin Sackel, Antoine Song, Umut Varolgunes and Jonathan J. Zhu. We prove that the symplectic four-ball can be squeezed after removing a subset of Minkowski dimension two.

This thesis consists of four main chapters. Chapter 2, An introduction to the Chekanov torus, is an expository article introducing all the concepts necessary to define and under- stand the Chekanov torus. We give a detailed introduction to Hamiltonian circle actions, symplectic reduction, lifting techniques and versal deformations and show how those can be used to show that the Chekanov torus is exotic. Chapter 3, On the topology of real Lagrangians in toric symplectic manifolds, is a joint article with Joontae Kim and Jiyeon Moon focusing on constructing examples of real Lagrangian submanifolds in toric manifolds by lifting symmetries from the moment polytope. We also prove convexity and tightness for the examples we construct and give an analogue of the Delzant construction. Chapter 4, Real Lagrangian tori and versal deformations, is an article focusing on obstructions for a given Lagrangian submanifold to be real and is, in some sense, complementary to Chapter 3. We develop a general obstruction in terms of versal deformations and displacement energy and apply this to toric fibres and Chekanov tori in toric manifolds. Chapter 5, Squeezing via degenerations of the complex projective plane, is an appendix to the paper On certain quantifications of Gromov’s non-squeezing theorem by Kevin Sackel, Antoine Song, Umut Varolgunes and Jonathan J. Zhu. We prove that the symplectic four-ball can be squeezed after removing a subset of Minkowski dimension two.

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