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Topics in Computational Algebraic Geometry and Deformation Quantization
Stockholm University, Faculty of Science, Department of Mathematics.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two parts, a first part on computations in algebraic geometry, and a second part on deformation quantization. More specifically, it is a collection of four papers. In the papers I, II and III, we present algorithms and an implementation for the computation of degrees of characteristic classes in algebraic geometry. Paper IV is a contribution to the field of deformation quantization and actions of the Grothendieck-Teichmüller group.

In Paper I, we present an algorithm for the computation of degrees of Segre classes of closed subschemes of complex projective space. The algorithm is based on the residual intersection theorem and can be implemented both symbolically and numerically.

In Paper II, we describe an algorithm for the computation of the degrees of Chern-Schwartz-MacPherson classes and the topological Euler characteristic of closed subschemes of complex projective space, provided an algorithm for the computation of degrees of Segre classes. We also explain in detail how the algorithm in Paper I can be implemented numerically. Together this yields a symbolical and a numerical version of the algorithm.

Paper III describes the Macaulay2 package CharacteristicClasses. It implements the algorithms from papers I and II, as well as an algorithm for the computation of degrees of Chern classes.

In Paper IV, we show that L-infinity-automorphisms of the Schouten algebra T_poly(R^d) of polyvector fields on affine space R^d which satisfy certain conditions can be globalized. This means that from a given L-infinity-automorphism of T_poly(R^d) an L-infinity-automorphism of T_poly(M) can be constructed, for a general smooth manifold M. It follows that Willwacher's action of the Grothendieck-Teichmüller group on T_poly(R^d) can be globalized, i.e., the Grothendieck-Teichmüller group acts on the Schouten algebra T_poly(M) of polyvector fields on a general manifold M.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2013. , 20 p.
Keyword [en]
Segre classes, Chern-Schwartz-MacPherson classes, topological Euler characteristic, computational algebraic geometry, numerical algebraic geometry, numerical homotopy methods, deformation quantization, polyvector fields, Fedosov quantization, Grothendieck-Teichmüller group
National Category
Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-87399ISBN: 978-91-7447-623-1 (print)OAI: oai:DiVA.org:su-87399DiVA: diva2:603248
Public defence
2013-03-11, room 14, house 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Accepted.

Available from: 2013-02-14 Created: 2013-02-05 Last updated: 2013-02-11Bibliographically approved
List of papers
1. A method to compute Segre classes of subschemes of projective space
Open this publication in new window or tab >>A method to compute Segre classes of subschemes of projective space
2013 (English)In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 12, no 2, 1250142- p.Article in journal (Refereed) Published
Abstract [en]

We present a method to compute the degrees of the Segre classes of a subscheme of complex projective space. The method is based on generic residuation and intersection theory. It has been implemented using the software system Macaulay2.

Keyword
Segre classes, computational algebraic geometry, numerical homotopy methods
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-87354 (URN)10.1142/S0219498812501423 (DOI)000316951500002 ()
Available from: 2013-02-04 Created: 2013-02-04 Last updated: 2017-12-06Bibliographically approved
2. An algorithm for computing the topological Euler characteristic of complex projective varieties
Open this publication in new window or tab >>An algorithm for computing the topological Euler characteristic of complex projective varieties
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We present an algorithm for the symbolic and numerical computation of the degrees of the Chern-Schwartz-MacPherson classes of a closed subvariety of projective space P^n. As the degree of the top Chern-Schwartz-MacPherson class is the topological Euler characteristic, this also yields a method to compute the topological Euler characteristic of projective varieties. The method is based on Aluffi's symbolic algorithm to compute degrees of Chern-Schwartz-MacPherson classes, a symbolic method to compute degrees of Segre classes, and the regenerative cascade by Hauenstein, Sommese and Wampler. The new algorithm complements the existing algorithms. We also give an example for using a theorem by Huh to compute an invariant from algebraic statistics, the maximum likelihood degree of an implicit model.

National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-87357 (URN)
Available from: 2013-02-04 Created: 2013-02-04 Last updated: 2013-02-11Bibliographically approved
3. A Macaulay2 package for characteristic classes and the topological Euler characteristic of complex projective schemes
Open this publication in new window or tab >>A Macaulay2 package for characteristic classes and the topological Euler characteristic of complex projective schemes
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The Macaulay2 package CharacteristicClasses provides commands for the computation of the topological Euler characteristic, the degrees of the Chern classes and the degrees of the Segre classes of a closed subscheme of complex projective space. The computations can be done both symbolically and numerically, the latter using an interface to Bertini. We provide some background of the implementation and show how to use the package with the help of examples.

National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-87358 (URN)
Available from: 2013-02-04 Created: 2013-02-04 Last updated: 2013-02-11Bibliographically approved
4. Globalizing L-infinity automorphisms of the Schouten algebra of polyvector fields
Open this publication in new window or tab >>Globalizing L-infinity automorphisms of the Schouten algebra of polyvector fields
(English)In: Differential geometry and its applications (Print), ISSN 0926-2245, E-ISSN 1872-6984Article in journal (Refereed) Accepted
Abstract [en]

Recently, Willwacher showed that the Grothendieck-Teichmuller group GRT acts by L-infinity-automorphisms on the Schouten algebra of polyvector fields T_poly(R^d) on affine space R^d. In this article, we prove that a large class of L-infinity-automorphisms on the Schouten algebra, including Willwacher's, can be globalized. That is, given an L-infinity-automorphism of T_poly(R^d) and a general smooth manifold M with the choice of a torsion-free connection, we give an explicit construction of an L-infinity-automorphism of the Schouten algebra T_poly(M) on the manifold M, depending on the chosen connection. The method we use is the Fedosov trick.

National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-87356 (URN)
Available from: 2013-02-04 Created: 2013-02-04 Last updated: 2017-12-06Bibliographically approved

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