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Abstracts for Conference on "Tropical aspects in Geometry and Topology"

Alternatively have a look at the program.

A tropical Hilbert-Chow morphism

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Speaker: 
Diane Maclagan
Date: 
Mon, 2013-09-02 10:00 - 11:15
Location: 
MPIM Lecture Hall

Tropical geometry is often regarded recording the cycle of a variety.  For example, the cohomology class of a subvariety of a toric
variety can be recovered from its tropicalization.  In the recent preprint arXiv.1308.0042 Jeff and Noah Giansiracusa introduced a
notion of scheme structure for tropical varieties.  In the affine case this is a congruence on the tropical semiring.  They show that the
tropical variety as a set is determined by these tropical scheme structure.  I will outline how to also recover the tropical cycle from

Abundance of projective subspaces on real algebraic hypersurfaces

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Speaker: 
Sergey Finashin
Date: 
Mon, 2013-09-02 11:15 - 12:00
Location: 
MPIM Lecture Hall

In a joint work with V.Kharlamov, we gave an estimate for the numbers of lines on real algebraic n-dimensional hypersurfaces of degree (2n-1) using evaluation of a certain signed count of these lines. The signs involved can be viewed as a generalized version of the Welschinger indices.  Our approach allows also a generalization to obtain a similar estimate for the numbers of projective subspaces on real algebraic  hypersurfaces (of certain dimensions and degrees).

Polyhedral complexes and topology of projective varieties

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Speaker: 
Michael Kapovich
Date: 
Mon, 2013-09-02 13:45 - 14:30
Location: 
MPIM Lecture Hall

We use Dirichlet polyhedral complexes of 3-dimensional real-hyperbolic orbifolds to show that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings.

Towards lower bounds for the number of real rational curves on K3 surfaces

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Speaker: 
Viatcheslav Kharlamov
Date: 
Mon, 2013-09-02 15:00 - 15:45
Location: 
MPIM Lecture Hall

Surprisingly, in a quite a number of real enumerative problems the number of real solutions
happens to satisfy high lower bounds. For the moment, such a phenomenon is rather deep
studied in the case of interpolation of real points on a real rational surface by real rational curves. In this talk,
based on a, joint with Rares Rasdeaconu, work in progress, I intend to show that a similar phenomenon should
hold in the case of counting rational curves on K3 surfaces. As in other similar situations, our lower bounds

Tropicalization of Classical Moduli Spaces

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Speaker: 
Bernd Sturmfels
Date: 
Mon, 2013-09-02 16:30 - 17:15
Location: 
MPIM Lecture Hall

Algebraic geometry is the study of solutions sets to polynomial equations.
Solutions that depend on an infinitesimal parameter are studied combinatorially by
tropical geometry. Tropicalization works especially well for varieties that are
parametrized by monomials in linear forms. Many classical moduli spaces (for curves
of low genus and few points in the plane) admit such a representation, and we here
explore their tropical geometry. Examples to be discussed include the Segre cubic,
the Igusa quartic, the Burkhardt quartic, and moduli spaces of marked del Pezzo

Singular tropical hypersurfaces

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Speaker: 
Eugenii Shustin
Date: 
Tue, 2013-09-03 10:00 - 10:45
Location: 
MPIM Lecture Hall

The study of singularities and singular varieties has been an important issue in tropical geometry from the very beginning, for example, the Mikhalkin's correspondence theorem deals with nodal plane tropical curves. In the talk we address the geometry of higher-dimensional singular tropical hypersurfaces and show that even in the case of one singularity one encounters
some unexpectedly interesting geometric phenomena involving combinatorics of lattice polytopes and metric geometry. Joint work with Hannah and Thomas Markwig.
 

Tropical multiwords and cluster structures on Plucker algebras

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Speaker: 
Gleb Koshevoy
Date: 
Tue, 2013-09-03 11:15 - 12:00
Location: 
MPIM Lecture Hall

A word is a subset of a finite alphabet. A multiword is a collection of words. We identify multiwords with integer arrays. Arrays form a cone and there is a bijection between  integer points of
this cone  and the set of semi-standard Young tableaux. We endow arrays with a structure of tropical semiring. There is a structure of cluster algebra on the Plucker algebra (cluster algebras were

Intrinsic features of the Newton polygon

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Speaker: 
Filip Cools
Date: 
Tue, 2013-09-03 13:45 - 14:30
Location: 
MPIM Lecture Hall

A sufficiently generic bivariate Laurent polynomial with given Newton polygon Delta defines an algebraic curve C, many of whose numerical invariants are encoded in the combinatorics of Delta. These include the genus (classical), the gonality, the Clifford index and the Clifford dimension (an enhancement of recent results by Kawaguchi), the scrollar invariants and, for sufficiently nice instances of Delta, certain secondary scrollar invariants that were introduced by Schreyer (new observation).

Complexity of tropical and min-plus linear prevarieties

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Speaker: 
Vladimir Podolskii
Date: 
Tue, 2013-09-03 15:00 - 15:45
Location: 
MPIM Lecture Hall

Abstract: A tropical (or min-plus) semiring is a set $\mathbb{Z}$ or $\mathbb{Z \cup \{\infty\}}$
endowed with two operations: $\oplus$, which is just usual minimum operation, and $\odot$, which is usual addition.
In tropical algebra a vector $x$ is a solution to a polynomial $g_1(x) \oplus g_2(x) \oplus \ldots \oplus g_k(x)$,
where $g_i(x)$'s are tropical monomials, if the minimum in $\min_i(g_{i}(x))$ is attained at least twice.
In min-plus algebra solutions of systems of equations of the form $g_1(x)\oplus \ldots \oplus g_k(x) = h_1(x)\oplus \ldots \oplus h_l(x)$

Roots of Trinomials from the Viewpoint of Amoeba Theory

Posted in
Speaker: 
Timo de Wolff
Date: 
Tue, 2013-09-03 16:30 - 17:15
Location: 
MPIM Lecture Hall

The behavior of the norms of roots of univariate trinomials z^{s+t} + pz^t+q in C[z] for fixed support A := {s+t, t, 0} ⊂f2  N with respect to the choice of coefficients p, q ∈f2  C∗f2  is a classical late 19th / early 20th century problem. Although algebraically described by P. Bohl in 1908, the geometry and topology of the corresponding space of coefficients C^A is unknown. We provide such a description yielded by a reinterpretation of this problem in terms of amoeba theory.
The talk is based on joint work with Thorsten Theobald.

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