05C50 Online Seminar: Prateek Kumar Vishwakarma

Abstract: I will present a completion theory for hyperbolic distance data at the interface of matrix analysis, graph theory, and hyperbolic geometry, drawing particularly on representations of chordal graphs. In analogy with the positive semidefinite completion theorem of Grone, Johnson, Sá, and Wolkowicz and the Euclidean distance completion theorem of Bakonyi and Johnson, our existence theorem establishes chordality as a necessary and sufficient condition for locally consistent Lorentz-Gram data to always admit a global completion. We then give explicit constructions. 

For trees, these yield geodesic-rectification completions and recover the product-distance matrices of Bapat and Sivasubramanian. The latter construction extends to chordal graphs through matrix-valued transfers along clique trees. The resulting canonical completion is characterized by inverse sparsity via a formula of Johnson and Lundquist, and by a maximum-absolute-determinant principle. Finally, its metric distortion exhibits a dichotomy governed by clique separator size: distortion is uniformly bounded, by the square root of the graph’s diameter, precisely when the chordal graph is diamond-free. This talk is based on recent joint work with Mihai Putinar [arXiv:2609.10403].