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--- fables [2023/02/12 13:10] – kalinin0
+++ fables [2023/02/28 16:53] – kalinin0
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 ====== Séminaire "Fables Géométriques". ======
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+  Monday, March 6, 2023
+  room 6-13
+**15h00 — Ali Ulaş Özgür Kişisel**
+**Expected measures of amoebas of random plane curves**
+There are several natural measures that one can place on the amoeba of an algebraic curve in the complex projective plane. Passare and Rullgård prove that the total mass of the Lebesgue measure on the amoeba of a degree $d$ curve is bounded above by $π^{2} d^{2} / 2$, by comparing it to another Monge-Ampère type measure, which is dual to the usual measure on the Newton polytope of the defining polynomial via the Legendre transform. Mikhalkin generalizes this upper bound to half-dimensional complete intersections in higher dimensions, by considering another measure supported on their amoebas; their multivolume. The goal of this talk will be to discuss these measures in the setting of random plane curves. In particular, I’ll first present our results with Bayraktar, showing that the expected multiarea of the amoeba of a random Kostlan degree $d$ curve is equal to $π^2 d$. For Lebesgue measure, it turns out that the expected asymptotics are much lower: I’ll describe our results with Welschinger, showing that the expected Lebesgue area of the amoeba of a random Kostlan degree $d$ curve is of the order $(\log d)^2$.
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