Hodge Theory

Thursday, May 24, 2018 - 10:30am - 11:30am
Tingran Gao (University of Chicago)
We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. In this type of problems, the pairwise interaction between adjacent vertices in the graph is of a non-scalar nature, typically taking values in a group; the consistency among these non-scalar pairwise interactions provide information for the dataset from which the graph is constructed.
Tuesday, October 29, 2013 - 3:15pm - 4:05pm
Thomas Schick (Georg-August-Universität zu Göttingen)
Hodge theory is a beautiful synthesis of geometry, topology, and analysis which has been developed in the setting of Riemannian manifolds. However, many spaces important in applications do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure.

The goal here is to obtain a theory which is sensitive to features which are present at a given scale. Our approach is based on an Alexander-Spanier replacement of differential forms, with a localization with parameter a.
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