Tropical and combinatorial Hodge theory

Omid Amini

amini@math.tu-berlin.de


Fruitful interactions are emerging between combinatorics and Hodge theory, both in the applications of Hodge theory to address problems in combinatorics, and, in a combinatorial description of geometric phenomena appearing in asymptotic Hodge theory, in the study of variations of Hodge structures.
The aim of these lectures is to provide an introduction to these developments.

We start with a general introduction to tropical geometry and its underlying polyhedral geometry.
We then give a treatment of the topology of tropical varieties and establish Hodge theoretic properties of their cohomology groups.
Several applications to combinatorics and complex geometry will be discussed in the lectures and in the working groups.

References

  • Karim Adiprasito, June Huh, Eric Katz, Hodge theory for combinatorial geometries.
  • Omid Amini, Matthieu Piquerez, Homology of tropical fans.
  • Omid Amini, Matthieu Piquerez, Hodge theory for tropical varieties.
  • Omid Amini, Matthieu Piquerez, Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology class.
  • Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin, Ilia Zharkov, Tropical Homology.
  • Diane Maclagan, Bernd Sturmfels, Introduction to tropical geometry.
  • Richard Stanley, Enumerative combinatorics.

    Hours

  • Tuesday 2-4pm, Room MA649 (Maths Institute) TU-Berlin
  • Wednesday 2-4pm, Room H0106 (Main Building) TU Berlin

    • Starting date: October 18th.