Friday, March 20, 2020 - 1:00pm to 2:00pm
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Bachir El Khadir
A convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few years later, Blekherman answered the question in the negative by showing through volume arguments that for a high enough number of variables, there must be convex forms of degree 4 that are not sums of squares. Remarkably, no examples are known to date. In this talk, we show that all convex forms in 4 variables and of degree 4 are sums of squares. We also show that if a conjecture of Blekherman related to the so-called Cayley-Bacharach relations is true, then the same statement holds for convex forms in 3 variables and of degree 6. These are the two minimal cases where one would have any hope of seeing convex forms that are not sums of squares (due to known obstructions). A main ingredient of the proof is the derivation of certain "generalized Cauchy-Schwarz inequalities" which could be of independent interest.
Bachir El Khadir is a PhD candidate at the Department of Operations Research and Financial Engineering (ORFE) of Princeton University, where he is advised by Prof. Amir Ali Ahmadi. Before joining Princeton, he graduated from École Polytechnique, France. He is broadly interested in the interplay between algebraic optimization techniques and their application to dynamical systems. He is also interested in bringing tools from dynamical systems theory to learning theory, optimization, and robotics.
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