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 Lvalues of various motives. Especially, Iwasawa main conjectures give the precise connection between the congruence properties of the special values of Lfunctions (padic Lfunctions) and the (dual of pprimary 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 (MordellWeil groups) and the failure of the localglobal principle (ShafarevichTate groups).
Research Themes
Anticyclotomic Iwasawa theory for modular forms à la BertoliniDarmon, Gross, and Vatsal.
Overconvergent construction of anticyclotomic padic Lfunctions.
A bit more explicit computation of BlochKato's exponential and dual exponential maps.
Refined Iwasawa theory
Hida and Coleman families
padic 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 \muinvariants of modular forms in families
in preparation
On the refined conjectures of the ``Birch and SwinnertonDyer type" for elliptic curves with supersingular reduction, with Masato Kurihara
preprint
On the refined conjectures of Kurihara and MazurTate 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unpublished; the result here is incorparated in the joint work with Masato Kurihara above.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, HaeSang Sun
submitted
An anticyclotomic MazurTate conjecture for modular forms
submitted
Overconvergernt quaternionic forms and anticyclotomic padic Lfunctions
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. 23392368
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. 14431455
Anticyclotomic Iwasawa invariants and congruences of modular forms (pdf) (This is my thesis.)
Asian Journal of Mathematics, June 2017, Vol. 21, No. 3, pp. 499530
Interesting math and nonmath stuff on the internet
Thinking Space
How to study spectral sequences
Geometric vs. Arithmetic Frobenii
The topology of the Robba ring
BSD, padic 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
Welldefinedness 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
MayerVietoris
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 ladic?
Reference book for Galois Representations
Frobenius weights on etale cohomology and purity
Semisimplicity of Frobenius on *integral* Tate module
Psi operator on PhiGamma 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?
Counterexamples for the quasiunipotence 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
