Research Papers
Recent publications
Some of the manuscripts below may be different from the published versions.
Non-Abelian topological defects and strain mapping in 2D moiré materials (with Rebecca Engelke, Hyobin Yoo, Stephen Carr, Kevin Xu, Paul Cazeaux, Richard Allen, Andres Mier Valdivia, Mitchell Luskin, Efthimios Kaxiras, Minhyong Kim, Jung Hoon Han, Philip Kim ), 2022 (arXiv). (To be published in Physical Review B.)
Path Integrals and p-adic L-functions (with Magnus Carlson, Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, Jeehoon Park, Hwajong Yoo), 2022 (arXiv).
Linking emergent phenomena and broken symmetries through one-dimensional objects and their dot/cross products (with Sang-Wook Cheong, Fei-Ting Huang). Reports on Progress in Physics, Volume 85, Number 12, 2022 (link).
Learning algebraic structures: Preliminary investigations (with Yang-Hui He). International Journal of Data Science in the Mathematical Sciences 0, 1-20, 2022 (link).
A note on abelian arithmetic BF-theory (with Magnus Carlson). Bull. London Math. Soc., 54: 1299-1307, 2022 (link).
Arithmetic gauge theory: A brief introduction. Modern Physics Letters A, Volume 33, Issue 29, 2018 (link).
A non-abelian conjecture of Tate–Shafarevich type for hyperbolic curves (with J. Balakrishnan, I. Dan-Cohen, and S. Wewers). Mathematische Annalen, October 2018, Volume 372, Issue 1-2, pp 369-428, 2018 (link).
Principal bundles and reciprocity laws in number theory. Algebraic Geometry: Salt Lake City 2015, Proceedings of Symposia in Pure Mathematics, Volume 97, 2018 (link).
Abelian arithmetic Chern-Simons theory and arithmetic linking numbers (with H.-J. Chung, D. Kim, G. Pappas, J. Park, and H. Yoo). International Mathematics Research Notices, Published electronically, 2017 (link).
Arithmetic Chern-Simons Theory II (with H.-J. Chung, D. Kim, J. Park, and H. Yoo). Bhatt, B., Olsson, M. (eds) p-adic Hodge Theory, pp 81–128, Springer, Cham, 2020 (link).
Arithmetic Chern–Simons Theory I. Neumann, F., Schroll, S. (eds) Galois Covers, Grothendieck-Teichmüller Theory and Dessins d'Enfants. GGT-DE 2018. Springer Proceedings in Mathematics & Statistics, vol 330. Springer, Cham, 2016 (link).
Diophantine Geometry and non-abelian reciprocity laws I. Loeffler, David, Zerbes, Sarah Livia (Eds.) Elliptic Curves, Modular Forms and Iwasawa Theory: In Honour of John H. Coates' 70th Birthday, Cambridge, UK, 2015 (link).
A p-adic nonabelian criterion for good reduction of curves (with F. Andreatta and A. Iovita). Duke Math. J. 164, no. 13, 2597--2642, 2015 (link).
On the 2-part of the Birch-Swinnerton-Dyer conjecture for elliptic curves with complex multiplication (with J. Coates, Z. Liang, and C. Zhao). Muenster J. Math. 7, 83--103, 2014 (link).
Tangential localization for Selmer varieties. Duke Math. J. 161, no. 2, 173--199, 2012 (link).
A remark on fundamental groups and effective Diophantine methods for hyperbolic curves. Number Theory, Analysis, and Geometry, in memory of Serge Lang. D. Goldfeld et. al. (ed.), Springer-Verlag, 2012(link).
Selmer varieties for curves with CM Jacobians. (with John Coates) Kyoto J. Math. 50, no. 4, 827--852, 2010 (link).
Massey products for elliptic curves of rank 1. J. of Amer. Math. Soc. 23, 725--747, 2010 (link).
Appendix and erratum to "Massey products for elliptic curves of rank 1''. (with J. Balakrishnan and K. Kedlaya) J. Amer. Math. Soc. 24, no. 1, 281--291, 2011 (link).
p-adic L-functions and Selmer varieties associated to elliptic curves with complex multiplication. Annals of Math. 172, no. 1, 751--759, 2010 (link).
The unipotent Albanese map and Selmer varieties for curves. Publ. Res. Inst. Math. Sci. 45, no. 1, 89--133, 2009 (link).
The l-component of the unipotent albanese map (with Akio Tamagawa). Math. Ann. 340, no. 1, 223--235, 2008 (link).
The motivic fundamental group of the projective line minus three points and the theorem of Siegel. Invent. math. 161, no. 3, 629--656, 2005 (link).
The Hyodo-Kato theorem for rational homotopy types. (with Richard Hain) Math. Res. Lett. 12, no. 2-3, 155--169, 2005 (link).
Unpublished Preprints
Mirror symmetry, mixed motives and ζ(3)(with Wenzhe Yang), 2018 (arXiv).
A note on Szpiro's inequality for curves of higher genus, 2002 (arXiv).
A vanishing theorem for Fano varieties in positive characteristic, 2002 (arXiv).
Torsion points on modular curves and Galois theory (with Ken Ribet), 1999 (arXiv).
Expository Essays
Foundations of the nonabelian method of Chabauty (with Martin Lüdtke).
IUTT Workshop, Oxford, December, 2015: A Brief Summary, 2015
Brief superficial remarks on Shinichi Mochizuki's Interuniversal Teichmueller Theory, version 1, 2015.
On relative computability for curves. Asia Pac. Math. Newsl. 3, no. 2, 16--20, 2015.
Mathematical Journeys. Contribution to Korean edition of 'Les Dechiffreurs', 2011.
On number and space (lecture in memory of Yunsan Lee Won-Guk), 2010.
Non-abelian fundamental groups in arithmetic geometry (for INI administration), 2009.
Galois Theory and Diophantine geometry. Non-abelian fundamental groups and Iwasawa theory, 162--187, Cambridge Univ. Press, Cambridge, 2012.
Fundamental groups and Diophantine geometry Cent. Eur. J. Math. 8, no. 4, 633--645, 2010.
Mathematical vistas, 2007.
Diophantine geometry as Galois theory in the mathematics of Serge Lang. Notices Amer. Math. Soc. 54, no. 4, 476--497, 2007.
Motivic L-Functions in Autour des motifs-- Volume I, 1--25, Panor. Syntheses, 29, Soc. Math. France, Paris, 2009.
Why Everyone Should Know Number Theory, Graduate Student Colloquium, University of Arizona, 1998