Colleen M Robles
Professor of Mathematics
Current Research Interests
My general interests are in geometric structures with symmetry, and in particular questions in which Lie theory plays a significant role, via an automorphism or symmetry group. Such structures arise throughout geometry, and those areas in which I have worked include Hodge theory, rational homogeneous varieties, Finsler geometry and calibrated geometry.
My current research focus is Hodge theory. Hodge theory associates to a family (or moduli space) of algebraic varieties a variation of Hodge structure, a.k.a. a period map. Period maps are powerful tools in the study of moduli spaces. The analysis of period maps involves both the geometry of the period domain (including curvature properties and a system of geometric PDE known as Griffiths' horizontality condition) and the representation theory of the associated automorphism group. My current work involves problems in this arena that are motivated by questions arising in the study of moduli spaces of algebraic varieties.
Office Hours
Current Appointments & Affiliations
- Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2020
Contact Information
- Physics Building, Room 023, 120 Science Drive, Durham, NC 27708
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robles@math.duke.edu
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Webpage
- Background
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Education, Training, & Certifications
- Ph.D., University of British Columbia (Canada) 2003
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Previous Appointments & Affiliations
- Director of Graduate Studies in the Department of Mathematics, Mathematics, Trinity College of Arts & Sciences 2018 - 2022
- Associate Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2015 - 2020
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Academic Positions Outside Duke
- FRIAS Senior Fellow and Marie Curie Fellow of the European Union, Freiburg Institute for Advanced Study, University of Freiburg. 2022
- Visiting Scholar, Korea Institute for Advanced Study. 2019
- Senior Simons Professor, Institute of Mathematics of the Polish Academy of Sciences. 2017
- Visiting Scholar, Korea Institute for Advanced Study. 2017
- Visiting Scholar, Korea Institute for Advanced Study. 2016
- Visitor, Simons Center for Geometry and Physics. 2016
- Member, Institute for Advanced Study. 2013 - 2014
- Visiting Scholar, University of Utah. 2012 - 2013
- Research
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Selected Grants
- Complex Geometric and Lie Theoretic Aspects of Hodge Theory awarded by National Science Foundation 2019 - 2023
- Complex Geometric and Lie Theoretic Aspects of Hodge Theory awarded by National Science Foundation 2016 - 2022
- FRG: Collaborative Hodge Theory and Representation Theory awarded by Texas A&M University 2015 - 2017
- Hodge Theory and Representation Theory awarded by National Science Foundation 2015 - 2016
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Fellowships, Supported Research, & Other Grants
- Marie Curie Fellow of the European Union awarded by European Union/European 2022
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External Relationships
- Freiburg Institute For Advanced Study https://www.frias.uni-freiburg.de/
- Publications & Artistic Works
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Selected Publications
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Academic Articles
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Green, M., Y. J. Kim, R. Laza, and C. Robles. “The LLV decomposition of hyper-Kähler cohomology (the known cases and the general conjectural behavior).” Mathematische Annalen 382, no. 3–4 (April 1, 2022): 1517–90. https://doi.org/10.1007/s00208-021-02238-y.Full Text
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Green, Mark L., Phillip Griffiths, and Colleen Robles. “Completions of period mappings: progress report,” June 8, 2021.Link to Item
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Green, Mark, Phillip Griffiths, and Colleen Robles. “Natural line bundles on completions of period mappings,” February 11, 2021.Link to Item
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Han, X., and C. Robles. “Hodge Representations.” Experimental Results 1 (November 17, 2020). https://doi.org/10.1017/exp.2020.55.Full Text
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Green, Mark, Phillip Griffiths, and Colleen Robles. “The global asymptotic structure of period mappings,” October 13, 2020.Link to Item
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Green, Mark, Yoon-Joo Kim, Radu Laza, and Colleen Robles. “The LLV decomposition of hyper-Kaehler cohomology,” June 8, 2019.Link to Item
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Kerr, M., G. J. Pearlstein, and C. Robles. “Polarized relations on horizontal SL(2)'s.” Documenta Mathematica 24 (January 1, 2019): 1295–1360. https://doi.org/10.25537/dm.2019v24.1295-1360.Full Text
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Robles, C. “Characterization of Calabi–Yau variations of Hodge structure over tube domains by characteristic forms.” Mathematische Annalen 371, no. 3–4 (August 1, 2018): 1229–53. https://doi.org/10.1007/s00208-017-1594-3.Full Text
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Green, Mark, Phillip Griffiths, Radu Laza, and Colleen Robles. “Period Mappings and Ampleness of the Hodge line bundle,” August 30, 2017.Link to Item
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Kerr, M., and C. Robles. “Variations of Hodge structure and orbits in flag varieties.” Advances in Mathematics 315 (July 31, 2017): 27–87. https://doi.org/10.1016/j.aim.2017.05.013.Full Text
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Kerr, M., and C. Robles. “Classification of smooth horizontal Schubert varieties.” European Journal of Mathematics 3, no. 2 (June 1, 2017): 289–310. https://doi.org/10.1007/s40879-017-0140-x.Full Text
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Robles, C. “Degenerations of Hodge structure.” Proceedings of Symposia in Pure Mathematics 95 (January 1, 2017): 267–83. https://doi.org/10.1090/pspum/095/01627.Full Text
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Robles, C. “Classification of horizontal SL(2)s.” Compositio Mathematica 152, no. 5 (May 1, 2016): 918–54. https://doi.org/10.1112/S0010437X15007691.Full Text
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Brosnan, P., G. Pearlstein, and C. Robles. “Nilpotent cones and their representation theory,” January 31, 2016.Link to Item
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Robles, C. “Characteristic cohomology of the infinitesimal period relation.” Asian Journal of Mathematics 20, no. 4 (January 1, 2016): 725–58. https://doi.org/10.4310/AJM.2016.v20.n4.a7.Full Text
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Robles, C. “Nilpotent cones and adjoint orbits,” May 13, 2014.Link to Item
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Robles, C. “Singular loci of cominuscule Schubert varieties.” Journal of Pure and Applied Algebra 218, no. 4 (April 1, 2014): 745–59. https://doi.org/10.1016/j.jpaa.2013.08.014.Full Text
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Green, Mark, Phillip Griffiths, and Colleen Robles. “Extremal degenerations of polarized Hodge structures,” March 3, 2014.Link to Item
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Griffths, P., C. Robles, and D. Toledo. “Quotients of non-classical flag domains are not algebraic.” Algebraic Geometry 1, no. 1 (January 1, 2014): 1–13. https://doi.org/10.14231/AG-2014-001.Full Text
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Robles, C. “Schubert varieties as variations of Hodge structure.” Selecta Mathematica, New Series 20, no. 3 (January 1, 2014): 719–68. https://doi.org/10.1007/s00029-014-0148-8.Full Text
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Robles, C. “Principal Hodge representations,” 2014, 259–83. https://doi.org/10.1090/conm/608/12183.Full Text
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Coskun, I., and C. Robles. “Flexibility of Schubert classes.” Differential Geometry and Its Application 31, no. 6 (December 1, 2013): 759–74. https://doi.org/10.1016/j.difgeo.2013.09.003.Full Text
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Robles, C. “Schur flexibility of cominuscule Schubert varieties.” Communications in Analysis and Geometry 21, no. 5 (December 1, 2013): 979–1013. https://doi.org/10.4310/CAG.2013.v21.n5.a5.Full Text
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Hammond, C., and C. Robles. “Projective invariants of CR-hypersurfaces.” Complex Variables and Elliptic Equations 58, no. 11 (November 1, 2013): 1493–1516. https://doi.org/10.1080/17476933.2011.575464.Full Text
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Landsberg, J. M., and C. Robles. “Fubini-griffiths-harris rigidity of homogeneous varieties.” International Mathematics Research Notices 2013, no. 7 (January 1, 2013): 1643–64. https://doi.org/10.1093/imrn/rns016.Full Text
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Robles, C., and D. The. “Rigid Schubert varieties in compact Hermitian symmetric spaces.” Selecta Mathematica, New Series 18, no. 3 (August 1, 2012): 717–77. https://doi.org/10.1007/s00029-011-0082-y.Full Text
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Landsberg, J. M., and C. Robles. “Fubini-Griffiths-Harris rigidity and lie algebra cohomology.” Asian Journal of Mathematics 16, no. 4 (January 1, 2012): 561–86. https://doi.org/10.4310/AJM.2012.v16.n4.a1.Full Text
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Robles, C. “Parallel calibrations and minimal submanifolds.” Illinois Journal of Mathematics 56, no. 2 (January 1, 2012): 383–95. https://doi.org/10.1215/ijm/1385129954.Full Text
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Landsberg, J. M., and C. Robles. “Lines and osculating lines of hypersurfaces.” Journal of the London Mathematical Society 82, no. 3 (January 1, 2010): 733–46. https://doi.org/10.1112/jlms/jdq051.Full Text
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Landsberg, J. M., and C. Robles. “Fubini's theorem in codimension two.” Journal Fur Die Reine Und Angewandte Mathematik, no. 631 (June 1, 2009): 221–35. https://doi.org/10.1515/CRELLE.2009.047.Full Text
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Robles, C., and S. Salur. “Calibrated associative and Cayley embeddings.” Asian Journal of Mathematics 13, no. 3 (January 1, 2009): 287–306. https://doi.org/10.4310/AJM.2009.v13.n3.a1.Full Text
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Robles, C. “The adjoint variety of SLm + 1 C is rigid to order three.” Differential Geometry and Its Application 26, no. 6 (December 1, 2008): 683–96. https://doi.org/10.1016/j.difgeo.2008.04.017.Full Text
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Landsberg, J. M., and C. Robles. “Lines on hypersurfaces,” May 16, 2008.Link to Item
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Robles, C. “Geodesics in randers spaces of constant curvature.” Transactions of the American Mathematical Society 359, no. 4 (April 1, 2007): 1633–51. https://doi.org/10.1090/S0002-9947-06-04051-7.Full Text
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Robles, Colleen. “Geodesics in Randers spaces of constant curvature.” Transactions of the American Mathematical Society 359, no. 4 (January 1, 2007): 1633–51.Link to Item
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Bao, D., C. Robles, and Z. Shen. “Zermelo navigation on riemannian manifolds.” Journal of Differential Geometry 66, no. 3 (January 1, 2004): 377–435. https://doi.org/10.4310/jdg/1098137838.Full Text
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Bao, D., and C. Robles. “On Randers spaces of constant flag curvature.” Reports on Mathematical Physics 51, no. 1 (January 1, 2003): 9–42. https://doi.org/10.1016/S0034-4877(03)80002-2.Full Text
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- Teaching & Mentoring
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Recent Courses
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Advising & Mentoring
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- Panchali Nag, PhD Thesis: Differential Geometry Tools for Data Analysis, Duke U 2021 (co-advisor with Ingrid Daubechies)
- Xiayimei Han, Bachelors Thesis: Hodge representation of Calabi-Yau 3-folds, Duke U 2020
- Curtis Porter, PhD Thesis: The local equivalence problem for 7-dimensional, 2-nondegenerate CR manifolds whose cubic form is of conformal unitary type, Texas A&M 2016 (co-advisor with JM Landsberg & Igor Zelenko)
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- Scholarly, Clinical, & Service Activities
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Presentations & Appearances
- Period mappings at infinity. Classical Elegance: the Geometry of Algebraic Varieties, in honour of Rita Pardini’s birthday. June 2022 2022
- Variations of Hodge structure: representation theoretic aspects, and an open question. Representation Theory and Geometry. June 2022 2022
- Period mappings at infinity. Conference on the occasion of Claire Voisin's 60th birthday. Institut Henri Poincare. May 2022 2022
- Completions of period mappings. Geometry/Topology Seminar. Stony Brook. April 2021 2021
- Lie group structures in Hodge theory. Recent Developments in Hodge theory. IMSA. March 2021 2021
- Completions of period mappings. IMSA. February 2021 2021
- Linear structures in Hodge theory. Introductory Workshop: Combinatorial Algebraic Geometry. ICERM. February 2021 2021
- Completions of period mappings. Essener Seminar für Algebraische Geometrie und Arithmetik. Fakultät für Mathematik. December 2020 2020
- Completions of period mappings. Geometry Seminar. KTH. December 2020 2020
- Completions of period mappings. Brussels–Oxford–Warwick–London. October 2020 2020
- Completions of period mappings. Komplex Analysis: algebraicity and transcendence. Oberwolfach. August 2020 2020
- Refinement of Lefschetz decomposition for hyper-Kahler manifolds. Algebraic Geometry Seminar. U Colorado. February 2020 2020
- Characterization of the Gross CY-VHS. Legacy of Elie Cartan. Tsinghua Sanya International Mathematics Forum. December 19, 2019 2019
- Refinement of Lefschetz decomposition for hyper-Kahler manifolds. Yaufest: Recent Advances in Mirror Symmetry. Tsinghua Sanya International Mathematics Forum. December 19, 2019 2019
- Gauss-Bonnet Theorem. Senior Women in Science Lecture. Senior Women in Science at Duke. November 2019 2019
- Identification of Hodge Domains. Algebraic Geometry Seminar. Washington University in St Louis. September 2019 2019
- Refinement of Lefschetz decomposition for hyper-Kahler manifolds. Discrete Groups and Moduli. Nagoya University. June 2019 2019
- Refinement of Lefschetz decomposition for hyper-Kahler manifolds. Cartan Geometry. Korea Institute for Advanced Study. May 2019 2019
- Refinement of Lefschetz decomposition for hyper-Kahler manifolds. Hodge Theory, Arithmetic and Moduli. Pacific Institute for the Mathematical Sciences. May 2019 2019
- Refinement of Lefschetz decomposition for hyper-Kahler manifolds. Baltimore-Washington Metro Area Differential Geometry Seminar. Howard U. April 2019 2019
- Refinement of Lefschetz decomposition for hyper-Kahler manifolds. Women in Geometry Workshop. Rice U. April 2019 2019
- The Orbit Theorems and Applications. Algebraic Geometry and Complex Geometry. Centre International de Rencontres Mathématiques . December 17, 2018 - December 21, 2018 2018
- Hodge Theory and Algebraic Moduli for Beginners. Department Colloquium. Georgia Southern U, Mathematics Department. November 30, 2018 2018
- Degeneration of Hodge structures. New trends and open problems in Geometry and Global Analysis. Philipps-Universität Marburg. August 27, 2018 - August 31, 2018 2018
- Representation theory as a Hodge theoretic tool. Algebraic Geometry Seminar. Università di Pisa, Mathematics Department. June 28, 2018 2018
- Completion of Period Mappings. Modern Geometry. University of Miami. March 1, 2018 - March 4, 2018 2018
- Compactifications of moduli spaces. Department Colloquium. Wichita State University, Mathematics Department. February 9, 2018 2018
- Generalizing the Satake-Baily-Borel compactification. Simons Workshop on Homological Mirror Symmetry and Hodge Theory. Harvard. January 10, 2018 - January 13, 2018 2018
- Generalizing the Satake-Baily-Borel compactification. Polish Academy of Science, Institute of Mathematics. November 2017 2017
- Generalizing the Satake-Bailey-Borel compactification. Department Colloquium. Brandeis U. October 2017 2017
- Generalizing the Satake-Bailly-Borel compactification. Second Mathematical Congress of the Americas. July 2017 2017
- Hodge Theory and Representation Theory. Mathematics Seminar. KIAS. May 2017 2017
- Generalizing the Satake-Baily-Borel compactification. Algebraic Geometry Seminar. U of Utah. April 2017 2017
- Generalizing the Satake-Bailey-Borel compactification. Workshop on Algebraic Varieties. Fields Institute. March 2017 2017
- Characterization of Gross's CY-VHS by characteristic forms. Lie Theory Workshop. UC Berkeley. February 2017 2017
- Characterization of Gross's CY-VHS over tube domains. Geometry Seminar. Johannes Gutenberg Universität. December 2016 2016
- Characterization of Gross's CY-VHS over tube domains by characteristic forms. Algebraic Geometry Seminar. Purdue U. November 2016 2016
- Characterization of Gross's canonical CY-VHS over tube domains by characteristic forms. Algebraic Geometry Northeastern Series. November 2016 2016
- Characterization of Gross's canonical CY-VHS over tube domains by characteristic forms. Algebraic Geometry Seminar. UC San Diego. October 2016 2016
- Characterization of Gross's canonical CY-VHS over tube domains by characteristic forms. Geometry Seminar. NC State U. September 2016 2016
- Differential Invariants of Curves. Smath Conference. Smith College. September 2016 2016
- Degenerations of Hodge structures. Algebraic Cycles and Moduli. Centre de Recherches Mathématiques. June 2016 2016
- Degenerations of Hodge structures. Algebraic Geometry Seminar. Standford U. May 2016 2016
- Role of representation theory in Hodge theory. Korea Institute for Advanced Study. April 2016 2016
- Analog of the Satake-Baily-Borel compactification for period domains. Focused Research Group on Hodge Theory, Moduli and Representation Theory: Workshop VI. Simons Center for Geometry and Physics. March 2016 2016
- Degenerations of Hodge structures. Geometry Seminar. U Miami. March 2016 2016
- Degenerations of Hodge structure. Department Colloquium. Washington U in St. Louis. January 2016 2016
- Horizontal Schubert Varieties. Geometric Methods in Representation Theory Seminar. U North Carolina. October 2015 2015
- Nilpotent orbits in Hodge theory. Algebraic Geometry Summer Research Institute Bootcamp. July 2015 2015
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Service to the Profession
- Participant. Faculty Curriculum on Anti-Racism. Duke Office of Faculty Advancement. January 11, 2021 - January 14, 2021 2021
- co-organizer. Duke Mathematical Journal Conference. Duke University. April 2018 2018
- co-organizer. Hodge Theory, Moduli and Representation Theory. Simons Center for Geometry and Physics. August 2017 2017
- co-organizer. Hodge Theory, Moduli and Representation Theory. Simons Center for Geometry and Physics. August 2017 2017
- co-organizer. Flag Domains and Cycle Spaces. Korea Institute for Advanced Study. May 2017 2017
- co-organizer. Flag Domains and Cycles Spaces. Korea Institute for Advanced Study. May 2017 2017
- Editor. Differential Geometry and its Applications. April 2017 - 2021 2017 - 2021
- co-organizer. Workshop on Algebraic Varieties. Fields Institute. March 2017 2017
- co-organizer. Algebraic Cycles and Moduli. Centre de Recherches Mathématiques. June 2016 2016
- co-organizer. Women in Geometry. Banff International Research Station. November 2015 2015
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Service to Duke
- Duke. Provost's Ad Hoc Working Group on Guaranteed 12-months Funding for Duke PhD Students. July 2021 - October 2021 2021
- SPIRE Fellows Mentor. Duke. August 2020 2020
- Lecturer. Summer Workshop in Mathematics. Duke University, Mathematics Department. June 2019 2019
- Lecturer. Summer Workshop in Mathematics. Duke University, Mathematics Depatment. June 2018 2018
- Faculty Champion. UCEM. Duke. June 2017 2017
- Lecturer. Summer Workshop in Mathematics. Duke University, Mathematics Depatment. June 2017 2017
- Duke. Graduate Student Selection Committee for the 2018 Deans Awards for Excellence in Mentoring. 2017 2017
- Mathematics. Duke Mathematical Sciences Sabbatical Fellows Committee. 2017 2017
- Gergen Speaker Selection Committee. 2016 - 2018 2016 - 2018
- Mathematics. Graduate Admissions Committee. 2016 - 2018 2016 - 2018
- Organizer. Women In Math Lunches. Mathematics Department, Duke. 2016 2016
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Academic & Administrative Activities
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Director of Graduate Studies
Mathematics Department
Duke University
Jan 2019 -- present.
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Director of Graduate Studies
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