# Antonio Ache

### Visiting Assistant Professor

Ph.D in Mathematics, University of Wisconsin-Madison. 2012

B.Sc. in Mathematics, Universidad Simón Bolívar, Caracas, Venezuela. 2005

Email:aache@nd.edu

Office: 283 Hurley Hall

Phone: (574) 631-6391

Fax: (574) 631-6579

For additional information see Antionio Ache’s Personal Page.

#### Research Interests

My research area is geometric analysis, specifically the study of critical metrics in dimension four and its applications to topology and geometry. I am particularly interested in problems in conformal geometry that involve looking at higher order (fourth order and higher) degenerate elliptic equations. One example of these problems is given by the study of Bach-flat metrics, which are metrics that are critical to a functional closely related to the Gauss-Bonnet formula in dimension four. Bach-flat metrics satisfy a fourth order system of equations that has conformal properties.

I am also interested in applications of methods in modern conformal geometry to the study of partial differential equations. For example, I have worked on using critical metrics in conformal geometry to prove Sobolev type inequalities of order four on manifolds with boundary.

#### Selected Publications

- Obstruction-flat asymptotically locally Euclidean metrics (with Jeff Viaclovsky), Geometric and Functional Analysis, Volume 22, 2012, Number 4, Pages 832-877
- Asymptotics of the Self-Dual deformation complex (with Jeff Viaclovsky), 45 pages, Journal of Geometric Analysis, 2013, DOI 10.1007/s12220-013-9452-3
- Metrics with non-negative Ricci curvature on convex three-manifolds (with Davi Maximo and Haotian Wu) 14 pages, \emph{Geom. and Top.} to appear, 2016
- Trace Sobolev Inequalities of order four (with Sun-Yung Alice Chang), preprint, 25 pages, preprint, arXiv:1509.06069v1, 2015
- Ricci Curvature and the Manifold Learning Problem (with Micah Warren), 41 pages, preprint, arXiv:1410.3351, 2015

Please direct questions and comments to: aache@nd.edu