arXiv subjects
Publications and source records attributed to Ali Ulaş Özgür Kişisel.
In this note we study asymptotic isotopy of random real algebraic plane curves. More precisely, we obtain a Kac-Rice type formula that gives the expected number of two-sided components (i.e.\ ovals) of a random real algebraic plane curve winding around a given point. In particular, we show that expected number of such ovals for an even degree Kostlan polynomial is $\frac{\sqrt{d}}{2}$ and independent of the given point.
We develop a variant of the barrier method in order to address questions about topology of Kostlan random real algebraic plane curves. In particular we prove that the expected number of connected components of the curve of length at least $\displaystyle{O\left(\sqrt{d^{-1}\log \log d}\right)}$ grows to infinity with $d$, and likewise, the expected number of nests of the curve of depth at least $\displaystyle{O\left(\log\log d\right)}$ grows to infinity with $d$. In another direction, we adapt an $L^{\infty}$-norm bound result of Shifmann and Zelditch to subspaces and employ it to obtain a lower bound for the probability that a finite number of points remain all in different components of the complement of a large degree random curve.
We prove that the expected area of the amoeba of a complex plane curve of degree $d$ is less than $\displaystyle{3\ln(d)^2/2+9\ln(d)+9}$ and once rescaled by $\ln(d)^2$, is asymptotically bounded from below by $3/4$. In order to get this lower bound, given disjoint isometric embeddings of a bidisc of size $1/\sqrt{d}$ in the complex projective plane, we lower estimate the probability that one of them is a submanifold chart of a complex plane curve. It exponentially converges to one as the number of bidiscs grow to $+\infty$.
We compute the expected multivolume of the amoeba of a random half dimensional complete intersection in $\mathbb{CP}^{2n}$. We also give a relative generalization of our result to the toric case.
Let $\mathbb F_a$ denote the Hirzebruch surfaces and $\mathcal{T}_{α,α^{\prime}}(\mathbb{F}_{a})$ denotes the set of positive, closed $(1,1)$-currents on $\mathbb{F}_{a}$ whose cohomology class is $αF+α^{\prime} H$ where $F$ and $H$ generates the Picard group of $\mathbb F_a$. $E^+_β(T)$ denotes the upper level sets of Lelong numbers $ν(T,x)$ of $T\in \mathcal{T}_{α,α^{\prime}}(\mathbb{F}_{a})$. When $a=0$, ($\mathbb F_a=\mathbb P^1\times \mathbb P^1$), for any current $T\in \mathcal T_{α,α'}(\mathbb P^1\times \mathbb P^1)$, we show that $E^{+}_{(α+α')/3}(T)$ is contained in a curve of total degree $2$, possibly except $1$ point. For any current $T\in \mathcal T_{α,α'}(\mathbb F_a)$, we show that $ E^{+}_β(T)$ is contained in either in a curve of bidegree $(0,1)$ or in $a+1$ curves of bidegree $(1,0)$ where $β\geq (α+ (a+1)α^{\prime})/(a+2)$.
Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on $\mathbb P^2$ and $V_α(T)$ be the upper level sets of Lelong numbers $ν(T,x)$ of $T$. For any $α\geq \frac{1}{3}$, we show that $|V_α(T)\setminus C|\leq 2$ for some cubic curve $C$.