Dr. Coskunuzer will undertake research in Differential Geometry and Geometric Topology. He will investigate the problems in minimal surface theory in hyperbolic space by using topological techniques. There are five main problems in the project. PROBLEM 1: (Universal Cover Conjecture) The first one is a famous classical 3-manifold topology problem, namely The Universal Cover Conjecture. The Conjecture asserts that the universal cover of any irreducible 3-manifold with infinite fundamental group is a 3-ball. This problem is one of the most famous problems in geometric topology. PROBLEM 2: (Intersections of Least Area Planes in Hyperbolic 3-space) The second main problem is about Least Area Planes in Hyperbolic 3-space. In recent years, Dr. Coskunuzer showed very strong generic uniqueness results on the subject. Now, he is studying the intersections of different least area planes with non-transverse asymptotic boundary. He is aiming to prove that these planes are also disjoint. Such a result will have a very wide range of applications in geometric topology. PROBLEM 3: (Properly Embedded Least Area Planes in Hyperbolic 3-space) Dr. Coskunuzer is aiming to prove another ambitious conjecture about the least area planes in hyperbolic 3-space in this project. The conjecture is that any least area plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve is properly embedded. He already got very strong partial results about the conjecture. PROBLEM 4: (Hyperbolic 3-manifolds with Minimal Foliation) The next problem in the project is the existence of a hyperbolic 3-manifold with foliation by minimal surfaces. This is also a famous question in geometric topology. PROBLEM 5: (Embedded Plateau Problem) The aim is to prove the following conjecture: For any nullhomotopic curve C in a 3-manifold M, there is an embedded disk which minimizes the area among the embedded disks in M with boundary C.
Call for proposal
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