Contact topology in dimensions 3 and higher

This research project will explore the geometry of spaces that play a central role in both mathematics and physics. It focuses on contact structures, which are special geometric structures defined on odd-dimensional spaces. These contact structures assign a hyperplane a subspace of one dimension less than the space itself to every point, in a way that twists maximally and never aligns smoothly. This maximally twisting condition gives contact geometry its rich and complex behavior. Such structures are central to describing constraints in physical systems and understanding the geometry of motion and interaction. In three dimensions, contact structures have been studied intensively for several decades and have led to deep insights into knot theory, topology, and dynamics. However, there are still important gaps in our understanding. This project aims to close some of these gaps, particularly by studying Legendrian knots special kinds of curves that are always tangent to the contact directions. These knots serve as sensitive probes for the geometry of the surrounding space. By using visual and combinatorial tools such as convex surfaces and decompositions of the space called open books, the project will develop new ways to classify and compare these knots and the contact structures they inhabit. In higher dimensions, the project will contribute to the foundations of contact geometry by developing techniques to understand when a contact structure is tight. A tight contact structure is one that avoids certain kinds of local flexibility a property that plays a central role in determining how rigid or structured the space is. The project will pursue two complementary approaches to identifying tightness: one based on convex hypersurfaces (the higher-dimensional analogue of curved surfaces), and one based on open book decompositions, a method that organizes a space like the pages of a book. Focusing especially on the five-dimensional case, where methods from lower dimensions can still be adapted, the project will construct explicit examples and techniques that can be used in further study. These results aim to lay the groundwork for a broader theory of contact structures in higher dimensions. The tools and insights developed through this project are expected to help lay the foundations for future work in contact geometry and related fields, particularly in higher dimensions where the landscape remains largely unexplored. The research will be carried out in collaboration with experts from Austria, Germany, India, Italy, and the United States, and includes support for a PostDoc researcher who will contribute to key components of the theory.