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Sem. Topologia i T. Mnogości


Property A and duality in linear programming

Prelegent: Andrzej Nagórko

2021-06-02 16:15

Property A was introduced in 2000 and turns out to be of great importance in many areas of mathematics. Perhaps the most striking example is the following implication.

"If group G has Property A then the Novikov conjecture is true for all closed manifolds with fundamental group G."

The Novikov conjecture asserts homotopy invariance of higher signatures of smooth manifolds. Is is one of most important unsolved problems in topology.

In my talk I'll show how duality theory from linear optimization can be applied to property A. It is very hard to find an example of a group without property A. The only known methods use probability techniques to construct groups that contain expanders in their Cayley graphs. Hence interest in invariants that imply that property A fails. I'll show a construction of a dual problem to property A and use it to show examples of graphs without property A. I'll fully characterize difference between groups without property A and groups with expanders in their Cayley graphs.

During the talk I'll show many fully calculated examples. I'll show computational tools, experimental results and insights from analysis of millions of finite graphs. I will not assume knowledge of duality theory.

This is joint work with G. C. Bell (UNCG).