The Mathematics Department is proud to introduce our new faculty for the 2013-14 academic year:Antonio Ache (Instructor and NSF Postdoctoral Fellow)
Ache’s research interests include the areas of Conformal Geometry, Differential Geometry, and Nonlinar Elliptic and Parabolic Partial Differential Equations.  He received his Ph.D. in 2012 from the University of Wisconsin-Madison and was awarded a 3-year NSF Postdoctoral Fellowship from 2012-2015.
Ache spent the 2012-2013 academic year as a Postdoctoral Research Fellow in our department.  He was fully supported by his NSF fellowship during that time and will be partially supported for the next two academic years. 
In his Ph.D. thesis, Ache obtained several optimal decay and singularity removal theorems for Asymptotically Locally Euclidean metrics, which are solutions of a class of degenerate elliptic systems of higher order of interest in conformal geometry.  One example in this class of systems is the equation satisfied by metrics whose Ambient Obstruction Tensor is zero, which is a conformally invariant tensor defined in even dimensional Riemannian manifolds that vanishes for conformally Einstein metrics.  Ache also obtained several gluing theorems for self-dual and anti-self dual metrics in dimension four.  
Ana Caraiani (Veblen Research Instructor and NSF Postdoctoral Fellow)
Caraiani’s field of research is in Algebraic Number Theory and Arithmetic Geometry.  She received her Ph.D. in 2012 from Harvard University and was awarded a 4-year NSF Postdoctoral Fellowship from 2012-2016.
Caraiani spent her first postdoctoral year appointed as an L.E. Dickson Instructor at the University of Chicago for the 2012-13 academic year.  She was fully-supported by her NSF fellowship during that time and will be partially supported for the next two academic years.
As a Veblen Research Instructor, Caraiani is jointly appointed between the Princeton Mathematics Department and the School of Mathematics at the Institute for Advanced Study.  She will spend the 2013-14 and the 2014-15 academic years at Princeton with teaching duties.  The last year of her appointment will be spent conducting research (without teaching duties) at the Institute for Advanced Study.
Caraiani’s Ph.D. thesis concerned the compatability between local and global Langlands correspondences for GLn.  She is also interested in geometric realizations of Langlands correspondences and in the geometry of Shimura varieties. More recently, she has become interested in the p-adic Langlands program and thinks about the connection between modularity lifting theorems and p-adic local Langlands. 
Javier Gomez Serrano (Instructor)
Gomez Serrano’s research centers on Harmonic Analysis and, specifically, the study of partial differential equations arising in fluid mechanics, computer-assisted proofs applied to those equations, and numerical analysis.  He received his Ph.D. in the Spring of 2013 from the Universidad Autonoma de Madrid in Spain.
Gomez Serrano has worked on the development of finite time singularities for incompressible fluids, especially for the two-dimensional free boundary incompressible Euler equations.  The free boundary is established between two irrotational fluids with different densities.  The singularities are developed by means of the interface self-intersecting into either one point (i.e., ‘splash singularities’) or along an arc (i.e., ‘splat singularities’).
Adam Levine (Assistant Professor) 
Levine’s field of research is in low-dimensional topology. He is interested in the properties and applications of Heegaard Floer homology, a package of powerful invariants for 3- and 4-dimensional manifolds developed a decade ago by Professors Zoltán Szabó and Peter Ozsváth of our department. Levine received his Ph.D. in 2010 from Columbia University and served as an NSF Postdoctoral Research Fellow at Brandeis University from 2010 to 2013.
Benoit Pausader (Assistant Professor)
Pausader’s field of research is in Analysis and Partial Differential Equations.  He received his Ph.D. in 2008 from the University of Cergy Pontoise in France.
Pausader was appointed as [Tamarkin] Assistant Professor at Brown University from 2008 to 2011 and as a Courant Instructor at the Courant Institute, NYU, from 2011 to 2012.  He was also a Chargé de recherché at the CNRS in the Université Paris 13 in 2012-2013.
His research focuses on the analysis of dispersive partial differential equations and their interactions with physics and geometry.  In his work he investigates global regularity and asymptotic behavior of semilinear (Nonlinear Schrodinger) or quasilinear systems (Euler-Maxwell model).  
Steven Sivek (Instructor and NSF Postdoctoral Fellow)
Sivek’s field of research is in low-dimensional topology with a special focus on contact and symplectic geometry, Floer homology theories, and knot theory.  He received his Ph.D. in 2011 from the Massachusetts Institute of Technology and was awarded a 3-year NSF postdoctoral fellowship from 2012-2015.
Sivek spent the 2012-13 academic year as an NSF postdoctoral fellow at Harvard University. He was fully supported by his NSF fellowship during that time and will be partially supported for the next two years.
Sivek's research applies techniques from Floer homology and its sutured variants, as well as Legendrian contact homology, to the study of knots and contact 3-manifolds.
Florian Sprung (Veblen Research Instructor)
Sprung’s area of research is in number theory and arithmetic geometry.  He received his Ph.D. in the Spring 2013 from Brown University.         
As a Veblen Research Instructor, Sprung is jointly appointed between the Princeton Mathematics Department and the School of Mathematics at the Institute for Advanced Study.  He will spend the 2014-15 and the 2015-16 academic years at Princeton with teaching duties.  The first year of his appointment will be spent conducting research (without teaching duties) at the Institute for Advanced Study.
Sprung’s Ph.D. thesis centers on the Iwasawa theory of elliptic curves.  He is conducting research in an area that has been highly scrutinized by many experts in the field over the past 30 years and is considered high level and highly technical.  He has displayed extensive knowledge of existing literature and techniques but has also developed new tools that undoubtedly will prove to be helpful beyond his Ph.D. work.