# Research

**RESEARCH INTERESTS**

- Differential Geometry, General Relativity, Partial Differential Equations.

- My research has been partly supported by the National Science Foundation, through grants DMS-0707317 and DMS-1207844. You can read a bit them here and here.

- My research was also supported in part by a Simons Collaboration Grant.

### **PUBLICATIONS**

A short note on the Bartnik mass. * Submitted. *

Localized deformation for initial data sets with the dominant energy condition. (Joint with Lan-Hsuan Huang). Calc. Var. Partial Differential Equations. *To Appear*. Preprint.

A Note on the Bartnik Mass. CMSA Series in Mathematics, Vol. 1, Nonlinear Analysis in Geometry and Applied Mathematics, International Press of Boston, (2017), 49-75.

On the *P*-scalar curvature. (Joint with Farhan Abedin). Journal of Geometric Analysis. ** 27 **, no. 2, 1589-1623 (2017)

Deformation of scalar curvature and volume. (Joint with Michael Eichmair and Penzi Miao). Math. Ann. ** 357 **, 551-584 (2013).

Construction of N-body initial data sets in general relativity. (Joint with Piotr Chrusciel and James Isenberg). Comm. Math. Phys.** 304 **, 637-647 (2011)

Scalar Curvature and the Einstein Constraint Equations. (Joint with Daniel Pollack). In: H. L. Bray, W. P. Minicozzi, II (eds.), * Surveys in Geometric Analysis and Relativity *, Adv. Lect. Math. ** 20 **, pp. 145-188 (2011)

Initial data for the relativistic gravitational N-body problem. (Joint with Piotr Chrusciel and James Isenberg). Classical and Quantum Gravity ** 27 ** (2010) 222002 (6pp)

Hamiltonian Stationary Tori in Kahler Manifolds. (Joint with Adrian Butscher). Calc. Var. Partial Differential Equations. **45**, no. 1-2, 63-100 (2012)

Constructions of time-symmetric N-body initial data sets in general relativity. (Joint with Piotr Chrusciel and James Isenberg). Cont. Math. ** 554 **, 83-92 (2011)

Construction of N-body initial data sets in general relativity. Oberwolfach Reports ** 6 **, no. 4, 2630-2633 (2009)

On Isoperimetric Surfaces in General Relativity, II. (Joint with Farhan Abedin, Shelven Kapita, Haotian Wu.) Journal of Geometry and Physics ** 59**, 1453-1460 (2009)

On the Center of Mass in General Relativity. (Joint with Haotian Wu). Classical and Quantum Gravity ** 25 ** (2008)

On Isoperimetric Surfaces and General Relativity. (Joint with 2005 REU group). Pacific J. Math. ** 231**, no. 1, 63-84 (2007)

On the Existence and Stability of the Penrose Compactification. Annales Henri Poincare’ ** 8**, 597-620 (2007)

Some Recent Results on the Einstein Constraint Equations. Oberwolfach Reports ** 3**, no. 1, 83-87 (2006)

On the Asymptotics of the Vacuum Einstein Constraint Equations. (Joint with Richard M. Schoen). Jour. Diff. Geom. **73**, no.2, 185-217 (2006)

A Note on Asymptotically Flat Metrics on R^3 which are Scalar-Flat and Admit Minimal Spheres. Proc. Amer. Math. Soc. **133**, 3669-3678 (2005)

Constructions of Vacuum Spacetimes Symmetric Near Spatial Infinity. In: V. G. Gurzadyan, *et. al.* (eds.) * Proceedings of the Ninth Marcel Grossmann Meeting on General Relativity* (Rome, 2000), pp. 993-994. Singapore: World Scientific, 2002

Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Comm. Math. Phys. ** 214 **, no. 1, 137–189 (2000)

**INVITED TALKS**

A Celebration of Mathematical Relativity in Miami (Conference in honor of G. J. Galloway on the occasion of his 70th birthday), December 14-16. * On Constraint Deformation and the Dominant Energy Condition. *

Workshop on Initial Data in General Relativity, University of Alaska, Fairbanks, May 21-23. * A survey of localized constraint deformation. *

20th School on Differential Geometry. Federal University of ParaĆba, Joao Pessoa, Brazil, February 27-March 3, 2018. * Deformation and Gluing Constructions for the Einstein Constraint Equations * (Plenary Talk)

Between Geometry and Relativity Summer School. Erwin Schrodinger Institute Vienna, Austria, July 2017. * Constructing initial data for the Einstein equations* (four lecture mini-course)

Banff International Research Station Workshop on Geometric Analysis and General Relativity , July 20, 2016, * Deformation and gluing for the Einstein constraints with the dominant energy condition. *

Center for Mathematical Sciences and Applications, Workshop on Aspects of General Relativity, May 24, 2016, Harvard University, Cambridge, MA. * Scalar curvature deformation and the Bartnik mass. *

Static Metrics and Bartnik’s Quasi-local Mass Conjecture, Tubingen, Germany, May 19, 2016, * Scalar curvature deformation and the Bartnik mass. *

AMS Sectional Meeting, Special Session on Mathematical General Relativity, Stony Brook, March 21, 2016. * Deformation of the Einstein constraints with the dominant energy condition. *

Tsinghua-Sanya International Mathematics Form on Mathematical Relativity, January 7, 2016, Sanya, China. * Deformation and gluing for the Einstein constraints with the dominant energy condition.*

40th Ravello Summer School on Mathematical Physics , September 21-26, 2015, Ravello, Italy. * Geometric Analysis and the Einstein Constraint Equations. * (six-lecture mini-course)

Warwick-Imperial-Cambridge Seminar, July 17, 2015, Imperial College, London. * Localized scalar curvature deformation. *

General Relativity and Gravitation Einstein Centennial Conference, Penn State University, June 2015. Parallel Session on Mathematical GR. *Density and Deformation Theorems for the Constraint Equations.*

Geometric Analysis in Geometry and Topology 2014 Workshop, Tokyo University of Science, Tokyo, Japan, October 28-31, 2014. *Deformation and gluing constructions for scalar curvature, with applications* (three lectures).

ESI-EMP-IAMP Summer School on Mathematical Relativity. Erwin Schrodinger Institute, Vienna, Austria, July 28-August 1, 2014. * The Einstein Constraint Equations * (five-lecture mini-course).

University of Connecticut Colloquium. Storrs, CT. April 17, 2014. * On the scalar curvature and volume of Riemannian manifolds. *

2013 Taiwan International Conference on Geometry. National Taiwan University, Taipei. December 17, 2013. * Deformation, gluing and density theorems for the Einstein constraint equations. *

2013 Taiwan International Conference on Geometry. National Taiwan University, Taipei. December, 2013. Co-taught invited mini-course joint with Pengzi Miao (Miami) *Scalar Curvature and the Einstein Constraint Equations *

MSRI Workshop: Initial data and Evolution Problems in General Relativity. Berkeley, CA. November 22, 2013. * Density and deformation theorems for the Einstein constraint equations. *

New York Area General Relativity Seminar. Stony Brook, NY. March 29, 2013. * Asymptotics and gluing for the Einstein constraint equations. *

ICTP Conference on Geometric Analysis. Trieste, Italy. June 29, 2012. * Deformation and gluing constructions for scalar curvature.*

University of Pennsylvania Geometry and Topology Seminar. March 22, 2012. * Deformation and gluing constructions for scalar curvature. *

Lehigh University Colloquium. October 26, 2011. * Deformation of Scalar Curvature and Volume. *

Beijing Summer Program in Mathematical Relativity. Peking University (Beijing, China). June 10, 2011. * Deformation of Scalar Curvature and Volume.*

CADS V Workshop on General Relativity and the Einstein Equations (Akko, Israel). May 23, 2011. *Deformation of Scalar Curvature and Volume*.

PRIMA Conference on Geometric Analysis. Pacific Institute for the Mathematical Sciences, Vancouver. July 25, 2010. * Localized Scalar Curvature Gluing. *

Columbia University Geometry and Analysis Seminar. November 5, 2009. * Geometric Perturbation and Gluing Constructions.*

Lehigh University Colloquium. October 29, 2009. * Geometric Perturbation and Gluing Constructions. *

Oberwolfach Meeting on Mathematical Aspects of General Relativity. October 12, 2009. * Construction of N-body initial data sets in general relativity.*

Princeton University Differential Geometry and Geometric Analysis Seminar. October 2, 2009. * Hamiltonian Stationary Tori in Kahler Manifolds. *

CADS IV Workshop on General Relativity and the Einstein Equations (Nahariya, Israel). May 21, 2009. * Constructions of N-body solutions of the Einstein constraint equations.*

Mittag-Leffler Institute Seminar. December 4, 2008. * The center of mass of isolated systems from a geometric point of view. *

Banff Workshop. December 13, 2007. * Geometry and the Einstein Constraint Equations.*

Lehigh University Geometry Seminar. November 28 and December 5, 2006. * On the Einstein Constraint Equations.*

Oberwolfach Meeting on Mathematical Aspects of General Relativity. January 9, 2006. * Some Recent Results on the Einstein Constraint Equations.*

Isaac Newton Institute Conference on the Einstein Constraint Equations. December 15, 2005. * Some Applications of Scalar Curvature Deformation in General Relativity.*

Lafayette College Physics Club/Sigma Xi World Year Physics Seminar. April 15, 2005. * On the Structure of Isolated Systems in General Relativity. *

Haverford Colloquium. November 8, 2004. * Curvature, Topology and General Relativity.*

Lafayette-Lehigh Geometry-Topology Seminar. October 19, 2004. * The Einstein Constraint Equations.*