

2003  2007  B.A. in Mathematics, Summa cum Laudae, Princeton University, NJ, USA  2007  2012  PhD in Mathematics, Harvard University, Cambridge, MA, USA (advisor: Richard Taylor)  2012  2013  L.E. Dickson Instructor and NSF Postdoctoral Fellow, University of Chicago, IL, USA  2013  2016  Veblen Research Instructor and NSF Postdoctoral Fellow, Princeton University and Institute for Advanced Study, Princeton, NJ, USA  2014  Research Visit, Hausdorff Center for Mathematics, Bonn  2014  Research Visit, New Geometric Methods in Number Theory and Automorphic Forms, Mathematical Sciences Research Institute, Berkeley, CA, USA  2015  Research Visit, Ă‰cole Normale Superieure, Lyon, France  Since 2016  Bonn Junior Fellow, University of Bonn 


My research is at the interface of the Langlands correspondence with arithmetic algebraic geometry. At the heart of the Langlands program lies reciprocity, the conjectural correspondence between Galois representations and automorphic forms. Progress in the field has historically required the combined forces of many areas of mathematics and continues to do so. In my work, I approach the correspondence from several different directions, combining number theoretic techniques, such as padic interpolation, with arithmetic geometry (perverse sheaves, padic Hodge theory), harmonic analysis (the trace formula) and representation theory (the theory of types). In joint work with Emerton, Gee, Geraghty, Paskunas and Shin, I constructed a candidate for the padic local Langlands correspondence for over a padic field and I showed that this construction recovers the padic local Langlands correspondence for ; this led to proving some new cases of localglobal compatibility. In joint work with Levin, I showed that the moduli space of Kisin modules with tame descent data is smoothly equivalent to a local model of a Shimura variety with parahoric level structure. In joint work with Scholze, I showed that the cohomology of compact unitary Shimura varieties is torsionfree and concentrated in the middle degree, when localized at a sufficiently generic system of Hecke eigenvalues.
My future research is focused on extending, in joint work with Scholze, the results about torsion in the cohomology of compact unitary Shimura varieties to the noncompact case. This presents significant difficulties and requires new ingredients, both from the point of view of arithmetic geometry (one needs to work with both minimal and toroidal compactifications of Shimura varieties) and from the point of view of the trace formula. I am also working on understanding consequences of such a result about noncompact Shimura varieties for the Galois representations corresponding to torsion classes in the cohomology of locally symmetric spaces. I am particularly interested in localglobal compatibility at and applications to modularity or potential modularity of Galois representations. Furthermore, I am interested in discovering other consequences of the fact that Shimura varieties with infinite level at p are perfectoid and in studying moduli spaces of Galois representations via the moduli space of Kisin modules.


[ 1] Ana Caraiani, Bao V. Le Hung
On the image of complex conjugation in certain Galois representations Compos. Math. , 152: (7): 14761488 2016 DOI: 10.1112/S0010437X16007016[ 2] Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Pa\v skūnas, Sug Woo Shin
Patching and the padic local Langlands correspondence Camb. J. Math. , 4: (2): 197287 2016[ 3] Ana Caraiani
Monodromy and localglobal compatibility for l=p Algebra Number Theory , 8: (7): 15971646 2014 DOI: 10.2140/ant.2014.8.1597[ 4] Ana Caraiani
Localglobal compatibility and the action of monodromy on nearby cycles Duke Math. J. , 161: (12): 23112413 2012 DOI: 10.1215/001270941723706[ 5] Ana Caraiani
Multiplicative semigroups related to the 3x+1 problem Adv. in Appl. Math. , 45: (3): 373389 2010 DOI: 10.1016/j.aam.2010.01.009



2001  Silver Medal, International Mathematical Olympiad  2002, 2003  Gold Medal, International Mathematical Olympiad  2003, 2004  Putnam Fellow and Elizabeth Lowell Putnam Prize  2006  Andrew H. Brown Prize, Department of Mathematics, Princeton University, NJ, USA  2006  Member of 1st place Putnam team  2007  Alice T. Schafer Prize, Association for Women in Mathematics  2007  Middleton Miller Prize, Department of Mathematics, Princeton University, NJ, USA  2007  George B. Covington Prize, Department of Mathematics, Princeton University, NJ, USA  2007  2009  William Lowell Putnam Fellowship for Graduate Study, Harvard University, Cambridge, MA, USA  2007  2010  James Mills Peirce Fellowship for Graduate Study, Harvard University, Cambridge, MA, USA  2014  Research Member, Fall, Mathematical Science Research Institute, Berkeley, CA, USA 


2014  MSRIEvans lecture, Berkeley, CA, USA  2015  Sophus Lie Days, Cornell University, NY, USA  2017  Arizona Winter School, USA  2017  Journees Arithmetiques 


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