The Thompson extension is a construction in symplectic and contact geometry that extends a given Legendrian submanifold to a larger Legendrian submanifold. It plays a significant role in studying the topology and dynamics of Legendrian submanifolds. The construction involves the use of the symplectic form to define a flow, which is then used to extend the Legendrian submanifold along the flow lines. The Thompson extension has various applications, including finding new Legendrian submanifolds, studying the topology of Legendrian submanifolds, and understanding the dynamics of contact flows.
In the enigmatic realm of mathematics, there exist domains where manifolds, curves, and surfaces intertwine in intricate ways. Among these captivating realms are symplectic geometry and contact geometry, disciplines that explore the captivating interplays between differential forms and geometric structures. These geometries are not merely abstract concepts; they play a profound role in various fields, including Hamiltonian mechanics, low-dimensional topology, and geometric measure theory.
Symplectic geometry centers around symplectic manifolds, spaces endowed with a special differential form known as a symplectic form. This form captures the essential essence of symplectic geometry, governing the dynamics of smooth curves on the manifold. In the world of symplectic geometry, these curves exhibit remarkable properties, preserving their intrinsic “area” as they traverse the manifold.
Contact geometry, on the other hand, explores the fascinating world of contact manifolds, spaces that possess a contact form. This form plays a central role in contact geometry, providing a privileged direction on the manifold and enabling the definition of intriguing structures, such as Legendrian submanifolds and characteristic foliations.
The relationship between symplectic and contact geometry is a captivating interplay. Contact manifolds can arise as boundaries of symplectic manifolds, and symplectic manifolds can sometimes be constructed from contact manifolds. This interplay offers a rich tapestry of insights, connecting two seemingly distinct worlds.
To delve deeper into these captivating geometries, one can embark on a journey of exploration through the pages of renowned journals such as the “Journal of Symplectic Geometry” or “Contact and Symplectic Topology.” Prestigious conferences, like the “International Congress of Mathematicians” or the “International Colloquium on Symplectic Geometry,” provide platforms for scholars to exchange ideas and push the boundaries of these captivating fields. By engaging with these vibrant communities, one can unravel the secrets of entities with high closeness, uncovering the intricate relationships that govern the dance of curves, surfaces, and forms in the realm of symplectic and contact geometry.
Symplectic and Contact Geometry: Dive into the Realm of Manifolds and Foliations
In the tapestry of mathematics, symplectic and contact geometry emerge as vibrant threads, weaving together concepts from differential geometry, Hamiltonian mechanics, and topology. These geometries govern the behavior of symplectic manifolds and contact manifolds, captivating abstract objects that model the dynamics of various physical systems.
Symplectic Manifolds: Dancing with Closed Loops
Imagine a roller coaster winding through a landscape. The symplectic manifold, like its idealized counterpart, is a geometric surface where closed loops are abundant. These loops are akin to the rollercoaster tracks, tracing out paths that remain “parallel” to each other. The symplectic form embedded within the manifold embodies this notion, acting as a guiding force for these loops.
Contact Manifolds: The Touch of a Foliation
Shift your perspective to a smooth surface, this time endowed with a contact form. This form defines a preferred direction, like an invisible arrow pointing across the surface. The result is a contact manifold, where each point is graced by a tiny Legendrian submanifold. Think of these submanifolds as cross-sections of the surface, lying perpendicular to the direction indicated by the contact form.
Liouville and Reeb: Guiding Flows
Within the realms of symplectic and contact manifolds reside two exceptional vector fields: the Liouville vector field and the Reeb vector field. The Liouville vector field orchestrates the flow of closed loops, ensuring they maintain their interconnectedness. The Reeb vector field, on the other hand, induces an ever-rotating dance of Legendrian submanifolds around their parent contact manifold.
Foliations and Homology: Unraveling Hidden Patterns
Characteristic foliations, intricate webs of curves, adorn both symplectic and contact manifolds. These foliations are like hidden blueprints, revealing the underlying structure of these geometries. Contact homology, a sophisticated tool from algebraic topology, sheds further light on these foliations, probing their topological complexity and revealing insights into the dynamics of contact manifolds.
Connections between Symplectic and Contact Geometry and Other Fields
Hamiltonian Mechanics: Symplectic geometry is closely intertwined with Hamiltonian mechanics, a branch of physics that describes the motion of physical systems. Hamiltonian systems are characterized by the existence of a symplectic form, which encodes the energy of the system and allows for the formulation of conservation laws.
Symplectic Geometry: Contact geometry is a natural extension of symplectic geometry, which studies the behavior of smooth manifolds equipped with closed differential forms. Symplectic manifolds possess a symplectic form, which governs the dynamics of Hamiltonian systems. Contact manifolds, on the other hand, have a contact form, which constrains the dynamics to a specific submanifold called the Legendrian submanifold.
Low-Dimensional Topology: Symplectic and contact geometry have深い connections to low-dimensional topology, which explores the topological properties of spaces with a small number of dimensions. Contact topology, in particular, has played a crucial role in understanding the topology of three-manifolds, yielding important insights into their properties and classification.
Geometric Measure Theory: Symplectic and contact geometry also intersect with geometric measure theory, a branch of mathematics that studies the geometry of sets and measures. Symplectic and contact forms can be used to define volume elements and characterize the geometric properties of various structures, such as submanifolds and foliations.
Mirror Symmetry: One of the most profound connections in symplectic and contact geometry is with mirror symmetry, a duality between certain types of symplectic and contact manifolds. Mirror symmetry relates the symplectic structure of one manifold to the contact structure of its mirror partner, leading to remarkable insights into their geometry and topological properties.
Mirror Symmetry: A Twisted Tale of Shapes and Reflections
Imagine a world where shapes have doppelgangers that are seemingly different but mysteriously connected. This is the realm of mirror symmetry in symplectic and contact geometry, a fascinating dance of mathematical objects that has revolutionized our understanding of geometry.
The Essence of Mirror Symmetry
In mirror symmetry, two worlds collide: the world of symplectic geometry, characterized by its symplectic forms (mathematical objects that measure the area of surfaces), and the world of contact geometry, defined by its contact forms (objects that measure the “contact” between surfaces).
A mirror symmetry transforms a symplectic manifold (a curved shape defined by a symplectic form) into a contact manifold (a shape with a contact form). The twist is that these reflections aren’t perfect. The symplectic manifold curves in one way, while its contact counterpart curves in a seemingly contradictory way.
Significance in Geometry
Mirror symmetry has profound implications in geometry. It unveils hidden connections between symplectic and contact manifolds, revealing that these seemingly disparate worlds are intertwined in a dance of complementary forms. This has opened up new avenues for exploring the structure and behavior of complex shapes.
Applications in Physics and Beyond
Beyond geometry, mirror symmetry has applications in physics, particularly in string theory. It offers a bridge between different string theories, potentially providing a unified framework for understanding the fundamental forces of nature.
Exploring the Mirror Universe
The study of mirror symmetry is a vibrant and ongoing endeavor, with conferences and symposia dedicated to its intricacies. Leading journals, such as the Journal of Symplectic Geometry and Contact and Symplectic Topology, publish groundbreaking research on this fascinating topic.
As we continue to unravel the mysteries of mirror symmetry, we gain deeper insights into the hidden connections between geometrical shapes, revealing the intricate tapestry of the universe we inhabit.
Publications and Conferences: Where the Symphony of Symplectic and Contact Geometry Plays
In the intricate world of mathematics, symplectic and contact geometry dance gracefully together like two sides of a captivating coin. These fields have captivated the minds of mathematicians for decades, giving rise to a wealth of publications and conferences that serve as vibrant platforms for sharing research and fostering collaboration.
One of the most respected journals dedicated solely to this field is the Journal of Symplectic Geometry. This esteemed publication showcases groundbreaking research papers that push the boundaries of knowledge in both symplectic and contact geometry. Another notable journal is Contact and Symplectic Topology, which specializes in the interplay between these two geometries and their applications in other areas of mathematics.
Beyond journals, conferences provide a forum for experts to gather, exchange ideas, and witness the latest advancements in the field. The International Congress of Mathematicians (ICM), held every four years, is the grandest stage for mathematical discourse, and symplectic and contact geometry feature prominently in its program.
The International Colloquium on Symplectic Geometry (ICMS) is another renowned event that brings together leading researchers in the field. It fosters an atmosphere of intense intellectual exchange and stimulates collaborations that can lead to groundbreaking discoveries.
Another key gathering is the Contact and Symplectic Topology Conference (CSTC). This conference focuses specifically on the interplay between these two geometries, showcasing recent developments and providing opportunities for researchers to connect and explore new directions.
These publications and conferences are essential touchpoints for researchers in symplectic and contact geometry. They serve as conduits for knowledge sharing, inspiration, and the advancement of this captivating field.