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| 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, École Normale Superieure, Lyon, France | | Since 2016 | Bonn Junior Fellow, University of Bonn |
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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 p-adic interpolation, with arithmetic geometry (perverse sheaves, p-adic 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 p-adic local Langlands correspondence for  over a p-adic field and I showed that this construction recovers the p-adic local Langlands correspondence for  ; this led to proving some new cases of local-global 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 torsion-free 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 non-compact 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 non-compact Shimura varieties for the Galois representations corresponding to torsion classes in the cohomology of locally symmetric spaces. I am particularly interested in local-global 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.
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[ 1] Ana Caraiani, Bao V. Le Hung
On the image of complex conjugation in certain Galois representations
Compos. Math. , 152: (7): 1476--1488
2016
DOI: 10.1112/S0010437X16007016[ 2] Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskūnas, Sug Woo Shin
Patching and the p-adic local Langlands correspondence
Camb. J. Math. , 4: (2): 197--287
2016
DOI: 10.4310/CJM.2016.v4.n2.a2[ 3] Ana Caraiani
Monodromy and local-global compatibility for l=p
Algebra Number Theory , 8: (7): 1597--1646
2014
DOI: 10.2140/ant.2014.8.1597[ 4] Ana Caraiani
Local-global compatibility and the action of monodromy on nearby cycles
Duke Math. J. , 161: (12): 2311--2413
2012
DOI: 10.1215/00127094-1723706[ 5] Ana Caraiani
Multiplicative semigroups related to the 3x+1 problem
Adv. in Appl. Math. , 45: (3): 373--389
2010
DOI: 10.1016/j.aam.2010.01.009 |
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| 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 |
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| 2014 | MSRI-Evans lecture, Berkeley, CA, USA | | 2015 | Sophus Lie Days, Cornell University, NY, USA | | 2017 | Arizona Winter School, USA | | 2017 | Journees Arithmetiques |
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