[back]

# 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.
• Refined Iwasawa theory
• 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

• On anticyclotomic \mu-invariants of modular forms in families
in preparation
• On the refined conjectures of the Birch and Swinnerton-Dyer type" for elliptic curves with supersingular reduction, with Masato Kurihara
preprint
• On the refined conjectures of Kurihara and Mazur-Tate for elliptic curves with supersingular reduction
unpublished; the result here is incorparated in the joint work with Masato Kurihara above.
• On the indivisibility of derived Kato's Euler systems and the main conjecture for modular forms, with Myoungil Kim, Hae-Sang Sun
submitted
• An anticyclotomic Mazur-Tate conjecture for modular forms
submitted
• Overconvergernt quaternionic forms and anticyclotomic p-adic L-functions
Publicacions Matem\{a}tiques, to appear
• Variation of anticyclotomic Iwasawa invariants in Hida families (arXiv), with Francesc Castella and Matteo Longo
Algebra \& Number Theory, December 2017, Vol. 11, 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

## 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