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Topic: the Milnor fibration. See Singular Points on Complex Hypersurfaces.

Sasha: covering 6 and 7: isolated singularities, Milnor numbers. Skip 1 and 2. State curve selection lemma. Focus on setting up topologically locally trivial fibration.

Vanishing cycles: attachments/Pasted%20image%2020220324235106.png attachments/Pasted%20image%2020220324235131.png

In formal geometry

attachments/Pasted%20image%2020220325101019.png

Motivations

attachments/Pasted%20image%2020220410190108.png attachments/Pasted%20image%2020220410190135.png attachments/Pasted%20image%2020220410190502.png attachments/Pasted%20image%2020220410205453.png # Summary

attachments/Pasted%20image%2020220410200559.png attachments/Pasted%20image%2020220410200617.png attachments/Pasted%20image%2020220410190221.png attachments/Pasted%20image%2020220410190232.png attachments/Pasted%20image%2020220410190314.png attachments/Pasted%20image%2020220410190337.png attachments/Pasted%20image%2020220410190351.png attachments/Pasted%20image%2020220410190430.png attachments/Pasted%20image%2020220410190520.png attachments/Pasted%20image%2020220410190854.png See plumbing.

Fillings: attachments/Pasted%20image%2020220410191352.png

attachments/Pasted%20image%2020220410191814.png attachments/Pasted%20image%2020220410191827.png attachments/Pasted%20image%2020220410200709.png attachments/Pasted%20image%2020220410200835.png

Intro info

attachments/Pasted%20image%2020220410210251.png attachments/Pasted%20image%2020220410210325.png attachments/Pasted%20image%2020220410210357.png attachments/Pasted%20image%2020220410210456.png attachments/Pasted%20image%2020220410210538.png attachments/Pasted%20image%2020220410210643.png

Topological degree

attachments/Pasted%20image%2020220410210949.png attachments/Pasted%20image%2020220410211013.png attachments/Pasted%20image%2020220410211527.png attachments/Pasted%20image%2020220410211700.png attachments/Pasted%20image%2020220410211720.png attachments/Pasted%20image%2020220410212155.png attachments/Pasted%20image%2020220410220122.png attachments/Pasted%20image%2020220410220142.png attachments/Pasted%20image%2020220410220231.png attachments/Pasted%20image%2020220410220306.png

What are ADE singularities and du Val singularities?

Example using motivic homotopy

Example using motivic homotopy: attachments/Pasted%20image%2020220410220922.png attachments/Pasted%20image%2020220410220854.png

attachments/Pasted%20image%2020220410221210.png

Paper notes

attachments/Pasted%20image%2020220410163241.png attachments/Pasted%20image%2020220410163258.png attachments/Pasted%20image%2020220410163538.png

attachments/Pasted%20image%2020220410163509.png

Milnor class

attachments/Pasted%20image%2020220410165029.png attachments/Pasted%20image%2020220410165154.png attachments/Pasted%20image%2020220410165208.png attachments/Pasted%20image%2020220410165355.png attachments/Pasted%20image%2020220410165452.png attachments/Pasted%20image%2020220410165922.png attachments/Pasted%20image%2020220410165941.png

attachments/Pasted%20image%2020220410170555.png attachments/Pasted%20image%2020220410170715.png attachments/Pasted%20image%2020220410170801.png attachments/Pasted%20image%2020220410170833.png

Chern-Schwartz-MacPherson class

Relation to Schubert calculus attachments/Pasted%20image%2020220410181954.png attachments/Pasted%20image%2020220410181919.png attachments/Pasted%20image%2020220410181912.png attachments/Pasted%20image%2020220410171334.png attachments/Pasted%20image%2020220410171343.png

Singularities

Example singular subscheme

attachments/Pasted%20image%2020220410171731.png attachments/Pasted%20image%2020220410172211.png

attachments/Pasted%20image%2020220410172708.png

Main theorem

attachments/Pasted%20image%2020220410192724.png attachments/Pasted%20image%2020220410192733.png

Picard-Lefschetz theory

attachments/Pasted%20image%2020220410193004.png attachments/Pasted%20image%2020220410193436.png attachments/Pasted%20image%2020220410193606.png attachments/Pasted%20image%2020220410194703.png # Orbifolds

attachments/Pasted%20image%2020220410193256.png

Exotic spheres

attachments/Pasted%20image%2020220410200443.png attachments/Pasted%20image%2020220410200622.png attachments/Pasted%20image%2020220410200646.png attachments/Pasted%20image%2020220410200950.png attachments/Pasted%20image%2020220410201003.png

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Link

attachments/Pasted%20image%2020220410204228.png attachments/Pasted%20image%2020220410204307.png attachments/Pasted%20image%2020220410204323.png attachments/Pasted%20image%2020220410204344.png attachments/Pasted%20image%2020220410204553.png attachments/Pasted%20image%2020220410204603.png

Vanishing cycles

attachments/Pasted%20image%2020220410204748.png [](attachments/Pasted%20image%2020220410204803.png

?

attachments/Pasted%20image%2020220410224653.png attachments/Pasted%20image%2020220410224732.png attachments/Pasted%20image%2020220410224816.png attachments/Pasted%20image%2020220410224842.png

attachments/Pasted%20image%2020220410225423.png attachments/Pasted%20image%2020220410225439.png

Hodge structures

attachments/Pasted%20image%2020220410230349.png attachments/Pasted%20image%2020220410230358.png attachments/Pasted%20image%2020220410230418.png attachments/Pasted%20image%2020220410230438.png

\({\mathbf{E}}_8\) as a Milnor lattice: attachments/Pasted%20image%2020220411140636.png

Resulting open book decomposition: attachments/Pasted%20image%2020220411141512.png

attachments/Pasted%20image%2020220411141631.png

attachments/Pasted%20image%2020220411141846.png attachments/Pasted%20image%2020220411141901.png

Fillability

attachments/Pasted%20image%2020220411141942.png attachments/Pasted%20image%2020220411142002.png attachments/Pasted%20image%2020220411142023.png attachments/Pasted%20image%2020220411142101.png

#projects/my-talks