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Explore advanced concepts in symplectic geometry and Hamiltonian dynamics through this 58-minute conference lecture delivered at the X Workshop on Conservative Dynamics and Symplectic Geometry. Delve into the HZ- and multiplicity conjectures for dynamically convex Reeb flows, examining cutting-edge research at the intersection of contact geometry and dynamical systems. Learn about the theoretical foundations and recent developments in understanding the behavior of Reeb flows on contact manifolds, with particular focus on dynamically convex cases and their implications for periodic orbit theory. Discover how these conjectures relate to broader questions in symplectic topology and Hamiltonian dynamics, including connections to Aubry-Mather theory and integrable systems. Gain insights into current research methodologies and open problems in this specialized area of mathematics, presented as part of IMPA's prestigious workshop series celebrating 20 years of bringing together international experts in conservative dynamics and symplectic geometry.
