Biographic Statement, Andrejs Treibergs

Andrejs Treibergs, Professor of Mathematics at the University of Utah, is interested in geometric analysis. Recently he has studied various applications of geometric analysis to applied mathematics. He has written papers and remains interested in harmonic maps, prescribed mean and Gauss curvature, eigenvalues of the Laplacian, geodesics, isoperimetric inequalities and isometric embedding problems. He has recently become interested in curvature evolution problems that arise from applications as well as Ricci flow solution of the Geometrization Conjecture.

Treibergs has been interested in geometric problems, which occur in physics and engineering. Currently, with A. Cherkaev and Krtolica, he studies compatibility conditions for discrete structures. These are conditions for the solvability of the prescribed length equations and their linearization for networks. He wishes to relate these to continuum compatibility for elasticity and linearized elasticity equations. In Riemannian Geometry, these are integrability conditions for prescribing a metric isometric to Euclidean space (vanishing of Riemannian curvature) and its linearization.   With Briane and Milton, he studied which electric fields are realizable. The solution boils down to analyzing solutions of a first order PDE.

Together with J. Ratzkin, he resolved the missing case of a conjecture of Bramson and Griffeath about random pursuit that has been open twenty years. They have shown that if four predators chase one prey that has a head start on the real line, each doing independent standard Brownian motion, then the expected capture time is finite. In a second paper, they have found a conceptual argument for the eigenvalue estimates of the previous paper and generalized an estimate of Payne and Weinberger to spherical domains. Ratzkin has begun extending this to higher dimensions. A second interest is the question from general relativity involving the realization of wormhole spaces. Treibergs, together with his former Ph. D. student H. Chan have generalized Chan’s result that assuming square integrable second fundamental form and embeddedness of the end, that there are no isometric immersions of a slice of Misner’s wormhole initial data into Euclidean space. They show that any one-ended nonpositively curved surface embedded in Euclidean three space in this manner must lie between two parallel planes.  Chan and Treibergs have investigated the infinitesimal rigidity of complete noncompact nonpositively curved surfaces of Euclidean space. He proved the stability of flow in J. Zhu’s model of a moving interface between two reacting chemicals in a gravitational field in the regime when combustion dominates. The curve evolves by combustion driven curvature flow plus a nonlocal hydrostatic boundary integral.