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Our understanding of phases of matter has moved beyond those defined in terms of broken symmetries, such as simple solids or magnets, to include phases characterized by topological order, such as the recently discovered topological insulators. These topological quantum phases, which are being realized in various materials or nanostructures, can harbor quasi-particle states that behave as Dirac, Majorana, or Weyl fermions. These excitations in materials may have important potential applications for the future of conventional or quantum computation. I will describe experiments with scanning tunneling microscopes to probe these topological quasi-particle states using high-resolution atomic scale spectroscopic techniques. Among the experiments I will discuss are those that focus on the edge modes of topological insulators, and current efforts to realize a topological superconducting phase in nanometer-sized atomic chains fabricated on the surface of a superconductor. These experiments show strong signatures of Majorana fermions zero modes at the end of these chains and suggest possible novel ways in which these exotic topological excitations can be manipulated. |