The classical Fano scheme F_d(X) of a variety X parametrises d$-dimensional linear spaces contained in X. In this talk I am going to define a tropical analogue of the Fano scheme F_d( trop X) and I will show its relation with the tropicalization trop F_d(X) of the classical Fano scheme. In particular I will focus on the tropical version of Fano schemes of tropicalized linear spaces and tropicalized toric varieties.
For a Hyperkähler variety which admits a Lagrangian fibration, an increasing filtration is defined on its rational cohomology using the perverse t-structure. We will discuss the role played by this filtration in the study of the topology and geometry of Hyperkähler varieties. First, we will focus on the perverse filtration for the moduli of Higgs bundles with respect to the Hitchin fibration. We will discuss our recent proof of de Cataldo, Hausel, and Migliorini's P=W conjecture for parabolic Higgs bundles labelled by affine Dynkin diagrams. Then I will present a surprising connection between the perverse filtration for a projective Hyperkähler variety and the (pure) Hodge structure on itself. Based on joint work with Qizheng Yin and Zili Zhang.
Semialgebraic sets are those defined by polynomial inequalities over an algebraically closed field. In this talk I will discuss their tropicalization and analytification. We show that the real analytification is homeomorphic to the inverse limit of real tropicalizations, analogously to a result of Payne. We also show a real analogous of the fundamental theorem of tropical geometry. The talk will be accessible to graduate students with background on basic algebraic geometry. This is based on joint work with Philipp Jell and Claus Scheiderer.