Moduli space of Henon maps
We will investigate the moduli space of Henon maps, which in one of most interesting examples in arithmetic dynamics. We will provides good geometric parametrizarion of conjugacy classes of Henon maps independent of the choice of coordinate and show the quadratic Henon maps are semistable in GIT sense.
Detecting election fraud by multiplication
Election fraud or vote manipulation can alter ballot counts by multiplying original data with a large real number. We call a set of ballot counts (i.e., a voting data set) $\{y_1,y_2,\dots,y_n\}$ \emph{actually multiplied} if $y_i$ is generated by $round(\alpha x_i)$ for a real number $\alpha\geq 2$, where $x_1, x_2, \dots, x_n$ are nonnegative integers representing original data before multiplication.
In this paper, we provide an algorithm that distinguishes between actually multiplied and unmultiplied voting data sets with high probability. Moreover, if a voting data set is actually multiplied, then the algorithm also gives a narrow interval containing the actual multiplicative factor $\alpha$ with high probability. Our mathematical analysis and experimental results give that the error probability decreases as $\alpha$ and $n$ increase. For example, after applying our algorithm to 40,000 voting data sets, we found that it gives perfectly correct answers if $\alpha\geq 16$ or $n\geq 40$.
This is joint work with Taejung Park and Hyunjoo Song.
Selmer near-companion curves for cyclic extensions
Let $E$ be an elliptic curve over a number field $K$ and let $p$ be a prime. To each cyclic extension $L/K$ of degree $p$, we attach the $L/K$-twist of $E$, which is an abelian variety of dimension $p-1$. Then one can study how Selmer ranks vary in the family of such twists. In this talk, I will introduce a certain condition on the Selmer ranks of these twists that determines the $p$-torsion Galois module of $E/K$.