Özer Öztürk

On Schur function expansions of Thom polynomials,

arXiv:1111.6612, in: "Contributions to Algebraic Geometry", EMS Series of Congress Reports, 2012, 443-479 (with P. Pragacz).

Abstract: We discuss computations of the Thom polynomials of singularity classes of maps in the basis of Schur functions. We survey the known results about the bound on the length and a rectangle containment for partitions appearing in such Schur function expansions. We describe several recursions for the coefficients. For some singularities, we give old and new computations of their Thom polynomials.

Thom polynomials and Schur functions: the singularities $III_{2,3} (−)$,

Ann. Polon. Math. 99.3 (2010), 295–304.

Abstract: We give a closed formula for the Thom polynomials of the singularities $III_{2,3}(−)$ in terms of Schur functions. Our computations combine the characterization of the Thom polynomials via the "method of restriction equations" of Rimányi et al. with the techniques of Schur functions.

Toric varieties and the diagonal property,

in "Arrangements, Local Systems and Singularities", "Progress in Mathematics", Birkhauser, 2010, 191–207 (with A. U. Ö. Kişisel).

Abstract: A smooth variety $X$ of dimension $n$ is said to satisfy the diagonal property if there exists a vector bundle $E$ of rank $n$ on $X \times X$ and a section $s$ of $E$ such that the image $Δ(X)$ of the diagonal embedding of $X$ into $X \times X$ is the zero scheme of $s$. A study of varieties satisfying the diagonal property was begun by Pragacz, Srinivas and Pati, in [8]. Even though there are many cases where the answer is affirmative, only in a few examples an explicit description of the vector bundle is known. After an exposition of toric varieties, we discuss this question in the particular case when X is a toric surface, in search for such examples.

On Thom polynomials for $A_4 (−)$ via Schur functions,

Serdica Math. J. 33 (2007), 301–320.

Abstract: We study the structure of the Thom polynomials for $A_4(-)$ singularities. We analyze the Schur function expansions of these polynomials. We show that partitions indexing the Schur function expansions of Thom polynomials for $A_4(-)$ singularities have at most four parts. We simplify the system of equations that determines these polynomials and give a recursive description of Thom polynomials for $A_4(-)$ singularities. We also give Thom polynomials for $A_4(3)$ and $A_4(4)$ singularities.

Notes and Translations

P. Pragacz, Miscellany on the zero schemes of sections of vector bundles,

(notes by Ö. Öztürk), in “Algebraic cycles, sheaves, shtukas, and moduli”, “Trends in Mathematics”, Birkhauser, 2007, 105-116.

Abstract: This purely expository article is a summary of the author's lectures on topological, algebraic, and geometric properties of the zero schemes of sections of vector bundles. These lectures were delivered at the seminar Impanga at the Banach Center in Warsaw (2006), and at the METU in Ankara (December 11-16. 2006). A special emphasis is put on the connectedness of zero schemes of sections, and the "point" and "diagonal" properties in algebraic geometry and topology. An overview of recent results by V.Srinivas, V. Pati, and the author on these properties is given.

P. Pragacz, Józef Maria Hoene-Wroński’nin Hayatı ve Çalışmaları Üzerine Notlar,

(çeviren: Ö. Öztürk), Oyun Dergisi (Türk Zeka Vakfı), Haziran 2008, pp. 4-7 ve 52-58.

Öz: Bu makale bilim tarihinin en orijinal figürlerinden birisi olan Hoene-Wroński (1776–1853) hakkındadır. 12–13 Ocak 2007’de Varşova’da, Polonya Bilimler Akademisi Matematik Enstitüsü’nde yapılan Józef Hoene-Wroński’ye bir anma isimli Impanga oturumunda yapılan iki konuşma temel alınarak yazılmıştır.