Jugal Verma

Milnor numbers of hypersurface singularities, mixed multiplicities of ideals and volumes of polytopes

Date: 9/20/2018
Time: 4:00PM-5:00PM
Place: 315 Armstrong Hall

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If $H$ is an analytic surface defined by $f=0$ in $\mathbb C^{n+1}$ with an isolated singularity at the origin, then the colength of the Jacobian ideal $J(f)$ is called its Milnor number. B. Teissier refined this notion to a sequence of $\mu^*(H)$ Milnor numbers of intersections of $H$ with general linear spaces of dimension $i$ for $i=0,1,\dots, (n+1).$ J. J. Risler and Teissier showed that this sequence coincides with the mixed multiplicities of the maximal ideal and $J(f).$ They proposed conjectures about log-convexity of the $\mu^*(H)$ which were solved by B. Teissier, D. Rees-R. Y. Sharp and D. Katz. These give rise to Minkowski inequality and equality for the Hilbert-Samuel multiplicities of ideals. Mixed multiplicities are also connected with volumes of polytopes and hence to counting solutions to polynomial equations.

Jugal Verma

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