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General

My mathematical research interest lies in algebraic number theory. I am now working on Iwasawa theory, which is regarded as the best way to understand the arithmetic meaning of zeta and L-values of various motives. Especially, Iwasawa main conjectures give the precise connection between the congruence properties of the special values of L-functions (p-adic L-functions) and the (dual of p-primary part of) Selmer groups of a given motive. Many interesting arithmetic properties are encoded in Selmer groups; for example, in the case of elliptic curves, Selmer groups contain the information of the rational points of elliptic curves (Mordell-Weil groups) and the failure of the local-global principle (Shafarevich-Tate groups).

Research Themes

  • Anticyclotomic Iwasawa theory for modular forms à la Bertolini-Darmon, Gross, and Vatsal.
  • Overconvergent construction of anticyclotomic p-adic L-functions.
  • A bit more explicit computation of Bloch-Kato's exponential and dual exponential maps.
  • Kato's Euler systems.
  • Integral periods of modular forms
  • Arithmetic applications of modularity lifting theorems.
  • Refined Iwasawa theory and the structure of Selmer groups.
  • Hida and Coleman families.
  • p-adic variation of Euler and Kolyvagin systems.
  • Kato's epsilon conjecture.
  • Applications of machine learning (any ``applied mathematics") to number theory (Hope I can do this soon..)

    Math Writings

    The arXiv version would be different from the published one. Please send me an email if you want to check the published version (and you do not have subscription).
  • On the Iwasawa invariants of Kato's zeta elements for modular forms
          with Jaehoon Lee and Gautier Ponsinet
          submitted
  • A proof of Perrin-Riou's Heegner point main conjecture
          with Ashay Burungale and Francesc Castella
          submitted (It's the new version of this paper.)
  • On the quantitative variation of congruence ideals and integral periods of modular forms
          with Kazuto Ota
          submitted; video lecture at Padova
  • Indivisibility of Kato's Euler systems and Kurihara numbers (survey)
          with an appendix by Alex Ghitza
          submitted to RIMS K\^{o}ky\^{u}roku Bessatsu; the relevant Sage code is available.
  • On the indivisibility of derived Kato's Euler systems and the main conjecture for modular forms
          with Myoungil Kim and Hae-Sang Sun
          under revision
  • An anticyclotomic Mazur-Tate conjecture for modular forms
          submitted
  • Remarks on Kato's Euler systems for elliptic curves with additive reduction (arXiv)
          with Kentaro Nakamura
          Journal of Number Theory, to appear
  • On the refined conjectures on Fitting ideals of Selmer groups of elliptic curves with supersingular reduction (arXiv)
          with Masato Kurihara
          International Mathematics Research Notices, rnz129, published online (5 July, 2019)
  • Overconvergent quaternionic forms and anticyclotomic p-adic L-functions (arXiv)
          Publicacions Matem\`{a}tiques, Vol. 63 (2019), No. 2, pp. 727--767
  • Variation of anticyclotomic Iwasawa invariants in Hida families (pdf)
          with Francesc Castella and Matteo Longo
          Algebra \& Number Theory, Vol. 11 (2017), No. 10, pp. 2339--2368
  • On the freeness of anticyclotomic Selmer groups of modular forms (pdf)
          with Robert Pollack and Tom Weston
          International Journal of Number Theory, July 2017, Vol. 13, No. 06, pp. 1443--1455
  • Anticyclotomic Iwasawa invariants and congruences of modular forms (pdf) (This is my thesis.)
          Asian Journal of Mathematics, June 2017, Vol. 21, No. 3, pp. 499--530

    Edited

  • Lecture note on local class field theory (99 pages)
          written by 9 Korean graduate students for the preparation of the instructional workshop on class field theory
          non-refereed and not intended for publication; available upon request

    In preparation (to push myself..)

  • (title undecided, something on eigencurves)
          in progress
  • (title undecided, something on control theorem)
          in progress
  • (title undecided, something on the simplest case of Selmer groups)
          in progress
  • (title undecided, something on Heegner and Gross points)
          in progress
  • (title undecided, something on Eisenstein series and theta series), with Ick Sun Eum
          in progress
  • (title undecided, something on the integral nature in Iwasawa theory)
          in preparation
  • (title undecided), with Jaehoon Lee
          in preparation

    List of talks I've given:

  • 2019/10/09, Kyushu University, Fukuoka, Japan
  • 2019/08/08, KIAS
  • 2019/05/31, Padova school on Serre conjectures and the $p$-adic Langlands program, Padova, Italy
  • 2019/05/16, Korea University, Seoul, Korea
  • 2019/05/14, Yonsei University, Seoul, Korea
  • 2019/03/15, SNU, Seoul, Korea
  • 2019/02/12, National Taiwan University, Taipei, Taiwan
  • 2018/12/04, Keio-Yonsei Joint Workshop, Keio, Yokohama, Japan
  • 2018/11/27, Algebraic number theory and related topics 2018, RIMS, Kyoto, Japan
  • 2018/10/05, Advances on automorphic forms and related topics, COEX, Seoul
  • 2018/06/26, Workshop on Number Theory and Algebra, Yeosu, Korea
  • 2018/04/09, KAIST, Daejeon, Korea
  • 2018/03/22, Shanghai Center for Mathematical Sciences, Fudan University, China
  • 2018/02/12, The 7th Number Theory Festival, UNIST, Korea
  • 2018/02/07, The 7th East Asia Number Theory Conference, TIMS, Taiwan
  • 2017/11/03, UNIST, Ulsan, Korea
  • 2017/11/01, Postech, Pohang, Korea
  • 2017/09/14, Workshop on $p$-adic $L$-functions and algebraic cycles, Taipei, Taiwan
  • 2017/04/19, UNIST
  • 2017/04/17, UNIST
  • 2017/04/13, KIAS
  • 2017/02/21, The 6th Number Theory Festival, Sungkyunkwan University, Korea
  • 2017/01/20, Tokyo Institute of Technology, Japan
  • 2017/01/13, Waseda University, Japan
  • 2016/11/14, Keio University, Japan
  • 2016/09/21, UNIST
  • 2016/04/28, Shanghai Center for Mathematical Sciences, Fudan University, China
  • 2016/02/15, The 5th Number Theory Festival, Kyungnam University, Korea
  • 2016/02/02, 2016 Korea-Japan Joint Number theory Seminar, Postech, Korea
  • 2016/01/29, Sendai International Conference on Arithmetic Geometry in 2016, Tohoku University, Japan
  • 2015/11/27, Industrious Number Theory, KIAS
  • 2015/09/21, Postech
  • 2015/08/25, Arithmetic of Euler systems, Centro de Ciencias de Benasque Pedro Pascual, Benasque, Spain
  • 2015/07/28, KAIST
  • 2015/07/23, Pan Asian Number Theory 2015, Sanya, China
  • 2015/04/09, Caltech
  • 2015/03/16, Universit\'{e} Laval, Qu\'{e}bec, Canada
  • 2014/12/23, KIAS
  • 2014/11/04, University of Chicago
  • 2014/11/03, Northwestern University
  • 2014/08/29, Postech
  • 2014/04/17, UCSD
  • 2014/03/03, UCLA
  • 2013/10/01, UC Irvine
  • 2013/09/12, Qu\'{e}bec-Vermont Number Theory Seminar, Montreal, Canada
  • 2013/01/08, Workshop on modular forms and Galois representations, KIAS
  • 2012/12/17, Keio University, Japan
  • 2012/11/05, BU

    Interesting math and non-math stuff on the internet

  • Thinking Space
  • How to study spectral sequences
  • Geometric vs. Arithmetic Frobenii
  • The topology of the Robba ring
  • BSD, p-adic BSD, and Iwasawa main conjecture
  • Flatness in algebraic geometry
  • Almost purity theorem
  • Vanishing cycles
  • Computing hypercohomology
  • Monodromy
  • Computing monodromy
  • Monodromy and global cohomology
  • Pontryagin dual and mu invariants
  • Ring homomorphisms
  • *Ordinary* Galois representations
  • Ordinary local Galois invariants
  • Well-definedness of modular Jacobians
  • On the module structure of local systems
  • Characteristic complexes in Iwasawa theory
  • The difference between an etale finite group scheme and a finite group
  • Why does the Section Conjecture exclude curves of genus 1?
  • Prime 2 and 3
  • Prime 2
  • Various topologies in algebraic geometry
  • Flat topology
  • Local properties of schemes
  • Mayer-Vietoris
  • Timeline of class field theory
  • Finding the Zariski closure of a set
  • Around locally ringed spaces. See also *Enlightning Exercise 4.3.A* in Vakil's book
  • Intuition for etale morphisms
  • 5/8 bound in group theory
  • Are there Maass forms where the expected Galois representation is l-adic?
  • Reference book for Galois Representations
  • Frobenius weights on etale cohomology and purity
  • Semisimplicity of Frobenius on *integral* Tate module
  • Psi operator on Phi-Gamma modules
  • Refereeing a Paper
  • Demonstrating that rigour is important
  • What is the motivation for a vertex algebra?
  • Heuristic argument that finite simple groups _ought_ to be "classifiable"?
  • Reading the mind of Prof. John Coates (motive behind his statement)
  • etale cohomology of an abelian variety and its dual
  • Why do we care about dual spaces?
  • Counter-examples for the quasi-unipotence of monodromy over an annulus?
  • Torsors in Algebraic Geometry?
  • What does the Lefschetz principle (in algebraic geometry) mean exactly?
  • Avoiding Minkowski's theorem in algebraic number theory
  • Do Tamagawa numbers of Galois representations stabilise in the cyclotomic tower?
  • How to avoid any wrong elementary ``proofs" of Fermat's last theorem