Turgay Bayraktar

Turgay Bayraktar
Sabanci University · Faculty of Engineering and Natural Sciences

PhD

About

28
Publications
1,035
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155
Citations
Introduction
Turgay Bayraktar currently works at the Faculty of Engineering and Natural Sciences, Sabanci University. Turgay does research in Complex Analysis and Complex Geometry and Probability Theory. Their current project is 'Asymptotic zero distribution of random polynomials and random holomorphic sections.'
Additional affiliations
July 2013 - July 2014
Indiana University Bloomington
Position
  • Visiting Assistant Professor
July 2012 - July 2013
Johns Hopkins University
Position
  • Lecturer

Publications

Publications (28)
Article
Full-text available
We study asymptotic distribution of zeros of random holomorphic sections of high powers of positive line bundles on projective homogenous manifolds. We work with a very general class of distributions that includes real and complex Gaussians. We prove that normalized simultaneous zero currents of i.i.d. random holomorphic sections, orthonormalized o...
Article
In this note, we prove, for instance, that the automorphism group of a rational manifold X which is obtained from CP^k by a finite sequence of blow-ups along smooth centers of dimension at most r with k>2r+2 has finite image in GL(H^*(X,Z)). In particular, every holomorphic automorphism $f:X\to X$ has zero topological entropy.
Article
Full-text available
We study asymptotic patterns of zeros of random Laurent polynomials whose support are contained in dilates of a fixed integral polytope P as their degree grow. We assume that the coefficients are i.i.d. random variables whose distribution law has bounded density and logarithmically decaying tails. Along the way, we develop a pluripotential theory f...
Article
In this note, we prove asymptotic normality of smooth linear statistics of zeros of random polynomials which are linear combinations of orthogonal polynomials with iid standard Gaussian coefficients. Along the way, we obtain Bergman kernel asymptotics for weighted $L^2$ space of polynomials for varying measures under a suitable assumption on the cu...
Article
Full-text available
In their seminal paper, Berman and Boucksom exploited ideas from complex geometry to analyze asymptotics of spaces of holomorphic sections of tensor powers of certain line bundles $L$ over compact, complex manifolds as the power grows. This yielded results on weighted polynomial spaces in weighted pluripotential theory in $\Bbb{C}^d$. Here, motivat...
Preprint
We study dynamical properties of asymptotically extremal polynomials associated with a non-polar planar compact set E. In particular, we prove that if the zeros of such polynomials are uniformly bounded then their Brolin measures converge weakly to the equilibrium measure of E. In addition, if E is regular and the zeros of such polynomials are suff...
Preprint
Full-text available
We generalize and strengthen some previous results on asymptotics of normalized zero measures and currents associated to random polynomials and random polynomial mappings in several complex variables.
Preprint
Full-text available
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 Kostl...
Article
Full-text available
The purpose of this note is to study asymptotic zero distribution of multivariate random polynomials as their degrees grow. For a smooth weight function with super logarithmic growth at infinity, we consider random linear combinations of associated orthogonal polynomials with subgaussian coefficients. This class of probability distributions contain...
Article
Full-text available
We work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body P in (R +) d. We define the logarithmic indicator function on C d : H P (z) := sup J∈P log |z J | := sup J∈P log[|z 1 | j1 · · · |z d | j d ] and an associated class of plurisubharmonic (psh) functions: L P := {u ∈ P SH(C d) : u(z) − H P (z...
Preprint
In this note, we obtain the growth order of Lebesgue constants for Fekete points associated with tensor powers of a positive line bundle. Moreover, by endowing the space of global holomorphic sections with a natural Gaussian probability measure we prove that Fekete points are sampling for random holomorphic sections.
Preprint
Full-text available
The purpose of this note is to study asymptotic zero distribution of multivariate random polynomials as their degrees grow. For a smooth weight function with super logarithmic growth at infinity, we consider random linear combinations of associated orthogonal polynomials with subgaussian coefficients. This class of probability distributions contain...
Preprint
Full-text available
We work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({\bf R}^+)^d$. We define the {\it logarithmic indicator function} on ${\bf C}^d$: $$H_P(z):=\sup_{ J\in P} \log |z^{ J}|:=\sup_{ J\in P} \log[|z_1|^{ j_1}\cdots |z_d|^{ j_d}]$$ and an associated class of plurisubharmonic (psh) func...
Preprint
We continue the study in a previous work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({\bf R}^+)^d$. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of $P-$pluripotential-theoretic notions. As an important preliminary step, w...
Preprint
Full-text available
We survey results on the distribution of zeros of random polynomials and of random holomorphic sections of line bundles, especially for large classes of probability measures on the spaces of holomorphic sections. We provide furthermore some new examples of measures supported in totally real subsets of the complex probability space.
Article
Full-text available
We survey results on the distribution of zeros of random polynomials and of random holomorphic sections of line bundles, especially for large classes of probability measures on the spaces of holomorphic sections. We provide furthermore some new examples of measures supported in totally real subsets of the complex probability space.
Article
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We prove almost sure invariance principle, a strong form of approximation by Brownian motion, for non-autonomous holomorphic dynamical systems on complex projective space P^k for Hölder continuous and DSH observables.
Article
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In this note, we obtain asymptotic expected number of real zeros for random polynomials of the form fn(z)=∑j=0najncjnzj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f...
Article
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In this work we prove an universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular Hermitian holomorphic line bundles on a compact K\"ahler complex space $X$. Namely, under mild moment assumptions, we show that the asymptotic distribution of zeros of random holomorphic sections i...
Article
Full-text available
In this work, we study asymptotic zero distribution of random multi-variable polynomials which are random linear combinations $\sum_{j}a_jP_j(z)$ with i.i.d coefficients relative to a basis of orthonormal polynomials $\{P_j\}_j$ induced by a multi-circular weight function $Q$ satisfying suitable smoothness and growth conditions. In complex dimensio...
Article
We study ergodic properties of evaluation processes generated by independent applications of holomorphic endomorphisms of the complex projective space chosen at random according to some probability distribution. Along the way, we construct positive closed currents which have good invariance and convergence properties. We provide a sufficient condit...
Article
In this note, we prove that every automorphism of a rational manifold which is obtained from $\Bbb{P}^k$ by a finite sequence blow-ups along smooth centers of dimension at most r with k>2r+2 has zero topological entropy.
Article
We study limiting distribution of the sequence of pull-backs of smooth $(1,1)$ forms and positive closed currents by meromorphic self-maps of compact K\"ahler manifolds.
Article
Full-text available
We consider the dynamics of meromorphic maps of compact K\"ahler manifolds. In this work, our goal is to locate the non-nef locus of invariant classes and provide necessary and sufficient conditions for existence of Green currents in codimension one.

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Projects

Projects (3)
Project
This project aims to develop a weighted pluripotential theory arising from polynomials associated to a convex body $P\subset (R^+)^d$.