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I will explain how to combine tools of local tropical geometry and logarithmic geometry in order to study the structure of Milnor fibers of smoothings of isolated complex singularities, up to homeomorphisms. I will partly follow the paper “The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof”, written in collaboration with Marıa Angelica Cueto and Dmitry Stepanov.This course replaces a course on the same topic that should have been delivered by Angelica Cueto.
[-] I will explain how to combine tools of local tropical geometry and logarithmic geometry in order to study the structure of Milnor fibers of smoothings of isolated complex singularities, up to homeomorphisms. I will partly follow the paper “The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof”, written in collaboration with Marıa Angelica Cueto and Dmitry Stepanov.This course replaces a course on the same topic that ...
[+] 14B05 ; 14A21 ; 14M25 ; 14T90 ; 32S05 ; 32S55
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I will explain how to combine tools of local tropical geometry and logarithmic geometry in order to study the structure of Milnor fibers of smoothings of isolated complex singularities, up to homeomorphisms. I will partly follow the paper “The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof”, written in collaboration with Marıa Angelica Cueto and Dmitry Stepanov.This course replaces a course on the same topic that should have been delivered by Angelica Cueto.
[-] I will explain how to combine tools of local tropical geometry and logarithmic geometry in order to study the structure of Milnor fibers of smoothings of isolated complex singularities, up to homeomorphisms. I will partly follow the paper “The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof”, written in collaboration with Marıa Angelica Cueto and Dmitry Stepanov.This course replaces a course on the same topic that ...
[+] 14B05 ; 14A21 ; 14M25 ; 14T90 ; 32S05 ; 32S55
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These lectures will revolve around applications of Hrushovski and Kazhdan's theory of motivic integration. It associates motivic invariants to semi-algebraic sets in algebraically closed valued fields. Following the work of Hrushovski and Loeser, and in collaboration with Yin, we shall see that when applied to the non-archimedean Milnor fiber, the motivic volumes recover some classical invariants of the Milnor fiber. Finally, we will see how these methods can be applied to a singularity arising as the quotient of a smooth variety by a linear group. When the group is finite, the orbifold formula of Batyrev and Denef–Loeser provides a motivic version of the McKay correspondence. In collaboration with Loeser and Wyss, we establish a similar formula for a general linear group.
[-] These lectures will revolve around applications of Hrushovski and Kazhdan's theory of motivic integration. It associates motivic invariants to semi-algebraic sets in algebraically closed valued fields. Following the work of Hrushovski and Loeser, and in collaboration with Yin, we shall see that when applied to the non-archimedean Milnor fiber, the motivic volumes recover some classical invariants of the Milnor fiber. Finally, we will see how ...
[+] 03C98 ; 14B05 ; 14J17 ; 32S25 ; 32S55
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y
These lectures will revolve around applications of Hrushovski and Kazhdan's theory of motivic integration. It associates motivic invariants to semi-algebraic sets in algebraically closed valued fields. Following the work of Hrushovski and Loeser, and in collaboration with Yin, we shall see that when applied to the non-archimedean Milnor fiber, the motivic volumes recover some classical invariants of the Milnor fiber. Finally, we will see how these methods can be applied to a singularity arising as the quotient of a smooth variety by a linear group. When the group is finite, the orbifold formula of Batyrev and Denef–Loeser provides a motivic version of the McKay correspondence. In collaboration with Loeser and Wyss, we establish a similar formula for a general linear group.
[-] These lectures will revolve around applications of Hrushovski and Kazhdan's theory of motivic integration. It associates motivic invariants to semi-algebraic sets in algebraically closed valued fields. Following the work of Hrushovski and Loeser, and in collaboration with Yin, we shall see that when applied to the non-archimedean Milnor fiber, the motivic volumes recover some classical invariants of the Milnor fiber. Finally, we will see how ...
[+] 03C98 ; 14B05 ; 14J17 ; 32S25 ; 32S55
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y
I will explain how to combine tools of local tropical geometry and logarithmic geometry in order to study the structure of Milnor fibers of smoothings of isolated complex singularities, up to homeomorphisms. I will partly follow the paper “The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof”, written in collaboration with Marıa Angelica Cueto and Dmitry Stepanov.This course replaces a course on the same topic that should have been delivered by Angelica Cueto.
[-] I will explain how to combine tools of local tropical geometry and logarithmic geometry in order to study the structure of Milnor fibers of smoothings of isolated complex singularities, up to homeomorphisms. I will partly follow the paper “The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof”, written in collaboration with Marıa Angelica Cueto and Dmitry Stepanov.This course replaces a course on the same topic that ...
[+] 14B05 ; 14A21 ; 14M25 ; 14T90 ; 32S05 ; 32S55
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The Jacobian algebra, obtained from the ring of germs of functions modulo the partial derivatives of a function $f$ with an isolated singularity, has a non-degenerate bilinear form, Grothendieck Residue, for which multiplication by $f$ is a symmetric nilpotent operator. The vanishing cohomology of the Milnor Fibre has a bilinear form induced by cup product for which the nilpotent operator $N$, the logarithm of the unipotent part of the monodromy, is antisymmetric. Using the nilpotent operators we obtain primitive parts of the bilinear form and we compare both bilinear forms. In particular, over $\mathbb{R}$, we obtain signatures of these primitive forms, that we compare.
[-] The Jacobian algebra, obtained from the ring of germs of functions modulo the partial derivatives of a function $f$ with an isolated singularity, has a non-degenerate bilinear form, Grothendieck Residue, for which multiplication by $f$ is a symmetric nilpotent operator. The vanishing cohomology of the Milnor Fibre has a bilinear form induced by cup product for which the nilpotent operator $N$, the logarithm of the unipotent part of the ...
[+] 14B05 ; 32S65 ; 32S55
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In this talk we consider the Milnor fiber F associated to a reduced projective plane curve $C$. A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of $F$, also known as the Alexander polynomial of the curve $C$, is presented. This leads to an effective algorithm to detect all the roots of the Alexander polynomial and, in many cases, explicit bases for the monodromy eigenspaces in terms of polynomial differential forms. The case of line arrangements, where there are many open questions, will illustrate the complexity of the problem. These results are based on joint work with Morihiko Saito, and with Gabriel Sticlaru.
[-] In this talk we consider the Milnor fiber F associated to a reduced projective plane curve $C$. A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of $F$, also known as the Alexander polynomial of the curve $C$, is presented. This leads to an effective algorithm to detect all the roots of the Alexander polynomial and, in many cases, explicit bases for the ...
[+] 32S55 ; 32S35 ; 32S22
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The splice type singularities introduced in 2001 by Neumann and Wahl provide the largest class known so far of links of isolated complete intersection surface singularities which are integral homology spheres. These singularities are determined up to equisingularity by particular kinds of decorated trees, called splice diagrams. Neumann and Wahl formulated the so-called Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a kind of four-dimensional decomposition of the Milnor fiber of the associated singularity. The aim of this course is to explain the structure of a proof of this conjecture, obtained in collaboration with Maria Angelica Cueto and Dmitry Stepanov. lt uses a combination of toric, tropical and logarithmic geometry.
[-] The splice type singularities introduced in 2001 by Neumann and Wahl provide the largest class known so far of links of isolated complete intersection surface singularities which are integral homology spheres. These singularities are determined up to equisingularity by particular kinds of decorated trees, called splice diagrams. Neumann and Wahl formulated the so-called Milnor fiber conjecture, stating that any choice of an internal edge of a ...
[+] 32S50 ; 32S25 ; 32S55 ; 14T90 ; 14A21
Déposez votre fichier ici pour le déplacer vers cet enregistrement.
y
The splice type singularities introduced in 2001 by Neumann and Wahl provide the largest class known so far of links of isolated complete intersection surface singularities which are integral homology spheres. These singularities are determined up to equisingularity by particular kinds of decorated trees, called splice diagrams. Neumann and Wahl formulated the so-called Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a kind of four-dimensional decomposition of the Milnor fiber of the associated singularity. The aim of this course is to explain the structure of a proof of this conjecture, obtained in collaboration with Maria Angelica Cueto and Dmitry Stepanov. lt uses a combination of toric, tropical and logarithmic geometry.
[-] The splice type singularities introduced in 2001 by Neumann and Wahl provide the largest class known so far of links of isolated complete intersection surface singularities which are integral homology spheres. These singularities are determined up to equisingularity by particular kinds of decorated trees, called splice diagrams. Neumann and Wahl formulated the so-called Milnor fiber conjecture, stating that any choice of an internal edge of a ...
[+] 32S50 ; 32S25 ; 32S55 ; 14T90 ; 14A21
English
