Ilia Itenberg
Refined invariants for real elliptic curves.
The talk is devoted to several real and tropical enumerative problems. We suggest new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from appropriate enumeration of real elliptic curves.These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the rational case. We discuss the combinatorics of tropical counterparts of the elliptic invariants under consideration and establish a tropical algorithm allowing one to compute them. This is a joint work with Eugenii Shustin.
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Patrick Popescu-Pampu
Combinatorics of real analytic morsifications.
I will present a work done in collaboration with A. Bodin, E. Garc\'{\i}a Barroso and M.-\c{S}. Sorea. We study a large class of morsifications of germs of univariate real analytic functions. We characterize the combinatorial types of the associated Morse functions in terms of planar contact trees built from the real Newton-Puiseux roots of the polar curves of the morsifications. This relates archimedean and non-archimedean aspects of morsifications.
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Daniel Corey
The tropical Ceresa class Abstract
The Ceresa cycle of an algebraic curve is a null-homologous algebraic cycle associated to a curve, and as Ceresa proved in the 1980s, it is not algebraically equivalent to zero for a very general curve. This cycle has garnered much interest in the last several years from the arithmetic and tropical geometry communities. In this talk, I will review recent developments in our understanding of this cycle. I will focus on my work with Jordan Ellenberg and Wanlin Li on the "tropical Ceresa class" and its relation to the image in étale cohomology of the Ceresa cycle of a curve defined over C((t)).
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Martin Möller
The boundary complex of the tropical Hodge bundle
The moduli space of tropical curves is the dual complex of the boundary divisor of the moduli space of curves. As such, its cohomology carries information about the top weight cohomology of the moduli space of curves. In this talk we report on progress about the analogous questions for the tropical Hodge bundle in comparison with the strata of moduli spaces of differentials.
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Rohini Ramadas
Intersections of tropical psi classes in genus zero, with non-trivial valuations
I will present joint work with Sean Griffin, Jake Levinson and Rob Silversmith, in which we expand the notion of tropical psi classes (including psi classes pulled back along forgetful maps) by considering tropicalizations of carefully chosen effective representatives defined over non-trivially valued fields. We study their intersections in this modified setting, and establish several attractive features. I will contrast with the Kerber-Markwig tropical intersections, and describe how the two versions encode dual information.
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Enrica Mazzon
(Non-archimedean) SYZ fibrations for Calabi-Yau hypersurfaces
The SYZ conjecture is a conjectural geometric explanation of mirror symmetry. Based on this, Kontsevich and Soibelman proposed a non-archimedean approach, which led to the construction of non-archimedean SYZ fibrations by Nicaise-Xu-Yu. A recent result by Li relates explicitly the non-archimedean approach to the classical SYZ conjecture.
In this talk, I will give an overview of this subject and I will focus on families of Calabi-Yau hypersurfaces in P^n. For this class, in collaboration with Jakob Hultgren, Mattias Jonsson and Nick McCleerey, we solve a non-archimedean conjecture proposed by Li and deduce that classical SYZ fibrations exist on a large open region of CY hypersurfaces.
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Diane Maclagan
Tropical vector bundles
In this talk I will describe a new definition, joint with Bivas Khan, for a tropical vector bundle on a subvariety of a tropical toric variety. This builds on the tropicalizations of toric vector bundles. I will discuss when these bundles do and do not behave as in the classical setting.
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Martin Ulirsch
Principal bundles in logarithmic and tropical geometry
In recent years the tropical and, subsequently, the logarithmic Picard group have seen significant attention, e.g. in the combinatorial study of limit linear series and the construction of natural tautological cycles on various moduli spaces of curves. Generalizations of this story to vector bundles of higher rank have, so far, been limited by our lack of understanding of tropical linear algebra. In this talk I will outline a program to alleviate this situation. This approach proceeds via principal bundles and the combinatorics of the extended affine building associated to their structure groups.