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Stoll, Robinorcid.org/0000-0002-2068-6228

Open this publication in new window or tab >>Newton's method in practice, II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees### Randig, Marvin

### Schleicher, Dierk

### Stoll, Robin

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_0_j_idt208_some",{id:"formSmash:j_idt204:0:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_0_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_0_j_idt208_otherAuthors",{id:"formSmash:j_idt204:0:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_0_j_idt208_otherAuthors",multiple:true}); 2024 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 437, article id 115427Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Root finding, Polynomial, Newton's method, Complexity, Algorithm, Iterated refinement
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:su:diva-223944 (URN)10.1016/j.cam.2023.115427 (DOI)001063015500001 ()2-s2.0-85166927506 (Scopus ID)
#####

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Available from: 2023-11-27 Created: 2023-11-27 Last updated: 2023-11-27Bibliographically approved

Stockholm University, Faculty of Science, Department of Mathematics.

We present an algorithm, based on Newton’s method, for finding all roots of univariate complex polynomials so that the observed complexity is linear in the degree, up to logarithmic factors. Unlike the usual Newton method, which finds at most one root at a time, it is global in the sense that it attempts to find all roots of polynomials simultaneously.

We demonstrate the feasibility of this algorithm by employing it to find all roots of several families of polynomials of degrees up to more than one billion (10^{9}). In all cases, the observed (empirical) complexity for finding all roots of a polynomial of degree* d* – measured either as the number of Newton iterations or computing time – was between O (*d*ln*d*) and O (*d*ln^{3}*d*), with small constants.

Open this publication in new window or tab >>Relative self-equivalences and graph complexes### Stoll, Robin

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_some",{id:"formSmash:j_idt204:1:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_otherAuthors",{id:"formSmash:j_idt204:1:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_otherAuthors",multiple:true}); 2024 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Stockholm: Department of Mathematics, Stockholm University, 2024. p. xxix
##### National Category

Geometry
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:su:diva-228529 (URN)978-91-8014-793-4 (ISBN)978-91-8014-794-1 (ISBN)
##### Public defence

2024-06-13, Lärosal 4, hus 1, Albano, Albanovägen 28, Stockholm, 14:00 (English)
##### Opponent

### Kupers, Alexander

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##### Supervisors

### Berglund, Alexander

### Arone, Gregory

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Available from: 2024-05-21 Created: 2024-04-21 Last updated: 2024-04-29Bibliographically approved

Stockholm University, Faculty of Science, Department of Mathematics.

This thesis consists of three papers.

In Paper I, we identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of S^{k} × S^{l}, where 3 ≤ k < l ≤ 2k - 2. We express the result in terms of Lie graph complex homology.

In Paper II, we construct a rational model for the classifying space Baut_{A}(X) of homotopy automorphisms of a simply connected finite CW-complex X relative to a simply connected subcomplex A. Using this model, we provide a purely algebraic description of the cohomology of this classifying space. This constitutes an important input for the results of Paper I.

In Paper III, we show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore we show that the Feynman transform corresponds to the cobar construction for modules of this kind. To make this precise, we extend the machinery of the bar and cobar constructions relative to a twisting morphism to modules over a general properad. As an application, we provide the foundations of a Koszul duality theory for modular operads.

Computer and Mathematical Sciences, University of Toronto at Scarborough, Toronto, Canada.

Stockholm University, Faculty of Science, Department of Mathematics.

Stockholm University, Faculty of Science, Department of Mathematics.

Open this publication in new window or tab >>The stable cohomology of self-equivalences of connected sums of products of spheres### Stoll, Robin

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##### Abstract [en]

##### National Category

Geometry
##### Identifiers

urn:nbn:se:su:diva-226075 (URN)10.1017/fms.2023.113 (DOI)001136559700001 ()
#####

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Available from: 2024-02-01 Created: 2024-02-01 Last updated: 2024-04-21Bibliographically approved

Stockholm University, Faculty of Science, Department of Mathematics.

We identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of S^{k}×S^{l}, where 3≤*k*<*l*≤*2k*−2. The result is expressed in terms of Lie graph complex homology.

Open this publication in new window or tab >>MODULAR OPERADS AS MODULES OVER THE BRAUER PROPERAD### Stoll, Robin

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##### Abstract [en]

##### Keywords

Modular operads, properads, Koszul duality
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:su:diva-214565 (URN)000904058600001 ()
#####

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Available from: 2023-02-06 Created: 2023-02-06 Last updated: 2024-04-21Bibliographically approved

Stockholm University, Faculty of Science, Department of Mathematics.

We show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore, we show that, in this setting, the Feynman transform corresponds to the cobar construction for modules of this kind. To make this precise, we extend the machinery of the bar and cobar constructions relative to a twisting morphism to modules over a general properad. This generalizes the classical case of algebras over an operad and might be of independent interest. As an application, we sketch a Koszul duality theory for modular operads.

Open this publication in new window or tab >>Equivariant algebraic models for relative self-equivalences### Berglund, Alexander

### Stoll, Robin

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##### National Category

Geometry
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:su:diva-228528 (URN)
#####

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Available from: 2024-04-21 Created: 2024-04-21 Last updated: 2024-04-21

Stockholm University, Faculty of Science, Department of Mathematics.

Stockholm University, Faculty of Science, Department of Mathematics.