The school is the final event of the PRA Project "Geometry, Algebra and Combinatorics of Moduli Spaces and Configurations" of the University of Pisa. There will be three 5-hours minicourses, some talks and ample time for discussion.
Talks:
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Fabrizio Caselli, University of Bologna
A classification of special matchings in lower Bruhat intervals [Slides]
Abstract:
Special matchings are the main tools in a proof of the combinatorial invariance of Kazhdan-Lusztig polynomials in lower Bruhat intervals. We show a new and simple classification of special matchings in arbitrary Coxeter groups and how the combinatorial invariance of Kazhdan-Lusztig polynomials can be deduced from it.
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Luca Migliorini, University of Bologna
Topological properties of projective maps: supports [Slides]
Abstract:
After discussing some foundational resuts on algebraic maps, in particular stratifications and constructibility, I will define, after Ngo, the notion of support of a map, and give a criterion, due to V. Shende and myself, relying on the notion of Higher discriminant, which turns out to be quite efficient to determine the supports. I will give some examples coming from families of planar curves, where higher discriminants can be determined by deformation theory.
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Short talks:
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Farhad Babaee
, Université de Fribourg
A tropical approach to a generalized Hodge conjecture for
positive currents [Slides]
Abstract:
Demailly (2012) showed that the Hodge conjecture is
equivalent to the statement that any (p, p)-dimensional closed current
with
rational cohomology class can be approximated by linear combinations
of integration currents. Moreover, the statement that all strongly
positive currents with rational cohomology class can be approximated
by positive linear combinations of integration currents can be viewed
as
a strong version of the Hodge conjecture (1982). In this talk, I will
explain the construction of a current which does not verify the latter
statement on a toric variety, where the Hodge conjecture is known to
hold. The example belongs to the family of `complex tropical
currents',
which we extend their framework to toric varieties, discuss their
extremality properties, and express their cohomology classes as
recession
fans of their underlying tropical varieties. Finally, the
counter-example will be the tropical current associated to a
2-dimensional balanced sub-fan of a 4-dimensional toric variety, whose
intersection form does not have the right signature in terms of the
Hodge index theorem. This is a joint work with June Huh.
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Lorenzo Guerra, Scuola Normale Superiore
On the cohomology of some reflection groups [Slides]
Abstract:
The talk will be about some aspects of the cohomology of the infinite families of finite reflection groups of type An, Bn and Dn. We discuss a certain Hopf ring structure on the ordinary cohomology of these groups, first studied by Giusti, Salvatore and Sinha in 2011 for the mod 2 cohomology of the symmetric group, and explain how we can use it as an organizational tool to obtain a nice description of the product structure on these cohomologies.
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Sara Lamboglia, University of Warwick
Computing toric degenerations of flag varieties [Slides]
Abstract:
Toric degenerations of flag varieties have been studied quite intensively in the last few decades and several methods have been applied.
In this talk I’m going to show results on the toric degenerations arising from the tropicalization of the full flag varieties of ℂ4 and ℂ5. These are obtained as Gröbner degenerations associated to the initial ideals corresponding to the maximal cones of the tropicalization of the flag variety.
I will then compare these degenerations with the ones
arising in representation theory from string polytopes and FFLV polytopes.
Moreover I will present a general procedure to find toric degenerations in the cases where the initial
ideal arising from a cone of the tropicalization of a variety is not prime.
This is joint work with L. Bossinger, K. Mincheva and F. Mohammadi.
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Yongqiang Liu, KU Leuven
The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties [Slides]
Abstract:
Given any connected topological space X, assume that there exists an epimorphism φ: π1(X) → Z. The deck transformation group Z acts on the associated infinite cyclic cover Xφ of X, hence on the homology group Hi(Xφ, C). This action induces a linear automorphism on the torsion part of the homology group as a module over the Laurent ring C[t,t−1], which is a finite dimensional complex vector space. We study the sizes of the Jordan blocks of this linear automorphism. When X is a compact Kähler manifold, we show that all the Jordan blocks are of size one. When X is a smooth complex quasi-projective variety, we give an upper bound on the sizes of the Jordan blocks, which is an analogue of the Monodromy Theorem for the local Milnor fibration. This is a joint work with Nero Budur and Botong Wang.
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Giovanni Paolini, Scuola Normale Superiore
Weighted discrete Morse theory [Slides]
Abstract:
After a brief introduction to discrete Morse theory, in the topological and in the algebraic context, we will talk about a "weighted" variant introduced by Salvetti and Villa. This machinery allows to make explicit computations of the twisted homology of Artin groups, as well as highlight interesting theoretical properties of homology groups in the finite and affine cases (joint work with Mario Salvetti).
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Piotr Pokora, Pedagogical University of Cracow
The Bounded Negativity Conjecture and Harbourne indices [Slides]
Abstract:
In the talk I would like to give an outline around the Bounded Negativity Conjecture. This conjecture tells us that for every smooth complex projective surface X there exists an integer b(X) such that for every reduced curve C ⊂ X one has C2 ≥ −b(X). It is know that this conjecture is true for some surfaces, for instance minimal models with Kodaira dimension 0. The main aim of the talk is to present a new tool to study the BNC, i.e., Harbourne indices which measure the local negativity. I will focus on some particular classes of configurations of curves in the complex projective plane providing effective bounds on their Harbourne indices.
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Elia Saini, Université de Fribourg
Combinatorics and topology of small arrangements [Slides]
Abstract:
We presents some results on the interplay between combinatorics and
topology in the study of the complements of complex hyperplane arrangements. In particular, we show that up to 7 hyperplanes the diffeomorphism type of the complement manifold of complex central essential hyperplane arrangements is combinatorially determined.
(Joint work with Matteo Gallet).
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University of Pisa - PRA