On the extreme rays of the cone of 3 times 3 quasiconvex quadratic forms: Extremal determinants vs extremal and polyconvex forms.

Abstract: Quasiconvexity is a central notion in the Calculus of Variations and is tied with lower-semicontinuity and existence of minimizers of integral functionals. This talk is concerned with characterization of all 3 by 3 quasiconvex quadratic forms. The problem has a long story and arises naturally in several questions in Materials Science. We study the extreme rays of the convex cone of 3 by 3 quasiconvex quadratic forms by providing a link between the extremality of a form and the extremality of its acoustic tensor determinant. The problem is also closely related to the problem of "Sums of Squares" in Real Algebraic Geometry, where in the language of positive biquadratic forms, quasiconvex quadratic forms correspond to nonnegative biquadratic forms. Our results recover all previously known results (to our best knowledge) on examples of extreme rays of the cone. The proofs are all established by means of several classical results from Linear Algebra, Convex Geometry, Real Algebraic Geometry, and the Calculus of Variations.

This is joint work with Narek Hovsepyan

ArmenianMathematics

Audience: general audience

Comments: The talk will be chaired by Hayk Mikayelyan (Univ. Nottingham Ningbo, China)

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Yerevan Mathematical Colloquium

Series comments: "Yerevan Mathematical Colloquium" invites survey talks aimed at a general mathematical audience, that emphasize proof methods, relations between branches of mathematics, possible applications, and open problems.

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