# Bi-annual Algebraic and Tropical
Meetings of

Brown and YaLE

(BATMOBYLE)

Fall 2018 @ Yale

November 29, 2018

The BATMOBYLE is a vehicle for bringing together the algebraic and
tropical geometry groups of Brown and Yale for a biannual day of talks
alternating between the two universities.

Abstracts
Sara Lamboglia (Frankfurt) -- A tropical version of Fano schemes

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.

Junliang Shen (MIT) -- Perverse filtrations in Hyperkähler geometry

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.

Josephine Yu (Georgia Tech) -- Real tropicalization and analytification of semialgebraic sets

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.

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