Martin Nitsche

I am a postdoctoral research fellow at the Institute of Geometry, in the working group of Andreas Thom. In my research I study discrete groups with tools from geometry and/or algebraic topology.

Projects

• Property (T) via semidefinite programming: Kazhdan's Property (T) is a rigidity property for groups with applications in geometric group theory, ergodic theory, representation theory and operator algebras. Following a novel approach by Ozawa, Property (T) was recently proved for the groups Aut(F_n), n=5 and n>=6, by solving a convex optimization problem with the computer. In "Computer proofs for Property (T), and SDP duality" I treat the remaining case, n=4. Here I give a simplified version of this computer proof.
• Equations over groups: The question whether a given equation can be solved over a group is usually studied with combinatorial methods, but can also be attacked with algebraic topology if the coefficient group is Connes-embeddable (no counter-examples are known!). In "Universal solvability of group equations" (joint with Andreas Thom) we give a strong sufficient criterion for solvability of equation systems over Connes-embeddable groups.
• Positive scalar curvature: The Rosenberg index of the Dirac operator on a Spin manifold provides an obstruction against the existence of metrics with positive scalar curvature. In my PhD thesis I studied this obstruction for certain circle bundles and submanifolds with the tools of Spin-bordism, spectral sequences and KK-theory. In "Transfer maps in generalized group homology via submanifolds" (joint with Rudolf Zeidler und Thomas Schick) we construct, for certain submanifolds N<M, an associated transfer from KO_*(Bπ₁M) to KO_*-k(Bπ₁N).
• Bounded cohomology: Brandenbursky and Marcinkowski constructed a transfer map in order to obtain non-trivial elements in the third-degree bounded cohomology of certain measure-preserving transformation groups on manifolds. Currently, I am working on generalizing these results.

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Dr. Martin Nitsche

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Institute of Geometry

Institute of Geometry

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Bürogebäude Willersbau, Raum C14 Zellescher Weg 12-14

01062 Dresden

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TUD Dresden University of Technology Faculty of Mathematics
Institute of Geometry

01062 Dresden

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