Ali Ulas Ozgur Kisisel

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• PhD
• Professor at Middle East Technical University

We prove that the expected area of the amoeba of a complex plane curve of degree d is less than 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/ √ d in the complex projective plane, we lower estimate the...

Let \({\mathbb {F}}_a\) denote the Hirzebruch surfaces and \({\mathcal {T}}_{\alpha ,\alpha ^{\prime }}({\mathbb {F}}_{a})\) denotes the set of positive, closed (1, 1)-currents on \({\mathbb {F}}_{a}\) whose cohomology class is \(\alpha F+\alpha ^{\prime } H\) where F and H generates the Picard group of \({\mathbb {F}}_a\). \(E^+_{\beta }(T)\) deno...

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 $\mathcal{T}_{\alpha,\alpha^{\prime}}(\mathbb{F}_{a})$ denotes the set of positive, closed $(1,1)$-currents on $\mathbb{F}_{a}$ whose cohomology class is $\alpha F+\alpha^{\prime} H$ where $F$ and $H$ generates the Picard group of $\mathbb F_a$. $E^+_{\beta}(T)$ denotes the upper level sets of Le...

Let T be a positive closed current of bidimension (1, 1) with unit mass on P2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb P^2$$\end{document} and Vα(T)\docum...

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...

Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on $\mathbb P^2$ and $V_{\alpha}(T)$ be the upper level sets of Lelong numbers $\nu(T,x)$ of $T$. For any $\alpha\geq \frac{1}{3}$, we show that $|V_{\alpha}(T)\setminus C|\leq 2$ for some cubic curve $C$.

It has been conjectured that the only nets realizable in $\mathbb{CP}^2$ are 3-nets and the Hesse configuration (up to isomorphism). We prove this conjecture.

Let X be a closed algebraic subset of A^n(K) where K is an algebraically closed field complete with respect to a nontrivial non-Archimedean valuation. We show that there is a surjective continuous map from the Berkovich space of X to an inverse limit of a certain family of embeddings of X called linear tropicalizations of X. This map is injective o...

Let $X$ be a proper algebraic scheme over an algebraically closed field. We assume that a torus $T$ acts on $X$ such that the action has isolated fixed points. The $T$-graph of $X$ can be defined using the fixed points and the one dimensional orbits of the $T$-action. If the upper Borel subgroup of the general linear group with maximal torus $T$ ac...

Let $X$ be a closed algebraic subset of $\mathbb{A}^{n}(K)$ where $K$ is an algebraically closed field complete with respect to a nontrivial non-Archimedean valuation. We show that there is a surjective continuous map from the Berkovich space of $X$ to an inverse limit of a certain family of embeddings of $X$ called linear tropicalizations of $X$....

A net is a special configuration of lines and points in the projective plane. There are certain restrictions on the number of its lines and points. We proved that there cannot be any (4,4) nets in $\mathbb{C}P^2$. In order to show this, we use tropical algebraic geometry. We tropicalize the hypothetical net and show that there cannot be such a conf...

Let where is an integer. We prove that is never squarefull, and in particular never a square, using arguments similar to those in J. Cilleruelo (2008) [2], where it is proven that is not a square for . In T. Amdeberhan et al. (2008) [1], among many other results, it is claimed that is not a square if μ is an odd prime and . However, we have found a...

Let Ωμ(n)=(μ1+1)(μ2+1)⋯(nμ+1)Ωμ(n)=(1μ+1)(2μ+1)⋯(nμ+1) where μ⩾2μ⩾2 is an integer. We prove that Ω3(n)Ω3(n) is never squarefull, and in particular never a square, using arguments similar to those in J. Cilleruelo (2008) [2], where it is proven that Ω2(n)Ω2(n) is not a square for n≠3n≠3. In T. Amdeberhan et al. (2008) [1], among many other results,...

A smooth variety X of dimension n is said to satisfy the diagonal property if there exists a vector bundle ε of rank n on X × Xand a section s of ε such that the image Δ(X) of the diagonal embedding of X into X × Xis the zero scheme of s. A study of varieties satisfying the diagonal property was begun by Pragacz, Srinivas and Pati, in [8]. Even tho...

A new linkage type for resizing polygonal and polyhedral shapes is proposed. The single degree-of-freedom planar linkages considered mainly consist of links connected by revolute joints. It is shown that the group of mechanisms obtained realize Cardanic Motion. The polyhedral linkages proposed are constructed by implementing the proposed planar lin...

Using the conformally invariant Cotton tensor, we define a geometric flow, the "Cotton flow", which is exclusive to three dimensions. This flow tends to evolve the initial metrics into conformally flat ones, and is somewhat orthogonal to the Yamabe flow, the latter being a flow within a conformal class. We define an entropy functional, and study th...

Polyhedral linkages using Cardan motion along radial axes are synthesized. The resulting single degree-of-freedom linkages are compared with some existing designs. The classification and the transformation characteristics of the linkages are given.

Polyhedral Linkages are used as spatial deployable structures. Although the present designs seem to be individual inventions, a number of these inventions have a common basic motion characteristic. In this paper, the polyhedral linkages with Cardanic motion characteristics are investigated. With this new approach, it is possible to make analysis an...

This is an expository paper with an aim of explaining some of the main ideas relating completely integrable systems to Gromov-Witten theory. We give a self-contained introduction to integrable systems and matrix integrals, and their relation to Witten’s original conjecture (Kontsevich’s theorem). The paper ends with a brief discussion of further de...

In this paper, we present a method for constructing large families of quadratic Poisson brackets on a manifold using more elementary brackets on a different manifold. The method is then applied to several examples of completely integrable systems. One can recover several known brackets for systems such as the Toda lattice or the open discrete KP hi...

This paper studies a certain completely integrable discretization of the KP hierarchy. This was constructed by Gieseker in \cite{Gie1}, from certain algebro-geometric data. This paper has the dual aim of showing that this construction is generically invertible, and obtaining explicit expressions for the flow equations. A subsequent article will dis...

This paper investigates Hamiltonian properties of the algebro-geometric discretization of KP hierarchy introduced in \cite{Gie1}. A Poisson bracket is introduced. The system is related to the periodic band matrix system of \cite{vM-M}. It is shown that the bracket descends to the latter and endows it with bi-Hamiltonian structure together with the...

We define a convolution operation on the set of polyominoes and use it to obtain a criterion for a given polyomino not to tile the plane (rotations and translations allowed). We apply the criterion to several families of polyominoes, and show that the criterion detects some cases that are not detectable by generalized coloring arguments.