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SUMMARY:Andreas Bernig (Goethe-Universität Frankfurt)
DTSTART:20240202T150000Z
DTEND:20240202T160000Z
DTSTAMP:20260422T172556Z
UID:TGiZ/58
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TGiZ/58/">Ha
 rd Lefschetz theorem and Hodge-Riemann relations for convex valuations</a>
 \nby Andreas Bernig (Goethe-Universität Frankfurt) as part of Tropical Ge
 ometry in Frankfurt/Zoom TGiF/Z\n\n\nAbstract\nThe hard Lefschetz theorem 
 and the Hodge-Riemann relations have their origin in the cohomology theory
  of compact Kähler manifolds. In recent years it has become clear that si
 milar results hold in many different settings\, in particular in algebraic
  geometry and combinatorics (work by Adiprasito\, Huh and others). In a re
 cent joint work with Jan Kotrbatý and Thomas Wannerer\, we prove the hard
  Lefschetz theorem and Hodge-Riemann relations for valuations on convex bo
 dies. These results can be translated into an array of quadratic inequalit
 ies for mixed volumes of smooth convex bodies\, giving a smooth analogue o
 f the quadratic inequalities in McMullen's polytope algebra. Surprinsingly
 \, these inequalities fail for general convex bodies. Our proof uses ellip
 tic operators and perturbation theory of unbounded operators.\n
LOCATION:https://researchseminars.org/talk/TGiZ/58/
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