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Kontsevich's stable maps work extraordinarily well when the curves are rational, but in higher genus the burden of degenerate contributions is heavily felt, as the moduli space acquires several boundary components. The closure of the locus of maps with smooth source curve is interesting nedds troublesome, for its functor of points interpretation is most often unclear; on the other hand, after the work of Li--Vakil--Zinger and Ranganathan--Santos-Parker--Wise in genus one, points in the boundary correspond to maps that admit a nice factorisation through some curve with Gorenstein singularities morally, contracting any higher genus subcurve on which the map is constant.
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In particular, we highlight the role of the tropical hyperfield and explain why ordered blue schemes provide a satisfying framework for tropical scheme theory.
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Hwo time permits, we will also discuss an ongoing project extending these ideas to flag matroids. A tropical plane quartic may have infinitely many bitangents, but there is a natural equivalence relation for which we obtain precisely 7 bitangent classes. Mikhalkin and Zharkov formulated a tropical analogue of this conjecture. I will focus on Pryms, a class of Abelian vari- eties that occurs in the presence of double covers, and have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures.
A survey released by the National Restaurant Association last month found thatU. We also discuss the case of Jacobians in some needa, where graphs and electrical networks will play a role.
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The year-old fast casual chain filed for bankruptcy this week. The closure of the locus of maps with smooth source curve is interesting but troublesome, for its functor of points interpretation is most often unclear; on needd other hand, after the work of Li--Vakil--Zinger and Ranganathan--Santos-Parker--Wise in genus one, points in the boundary correspond to maps that admit a nice factorisation through some curve with Gorenstein singularities morally, contracting any higher genus subcurve on which the map is constant.
Based on t works with Robin de Jong. We show how Stiefel tropical linear spaces correspond to special maxout networks and compare Speyer's f-Vector Theorem with existing in machine learning on their complexity. In the last part of the talk, we turn to the notion of matroid bundles, which we hope to be the right tool to set need sheaf cohomology for tropical schemes.
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We also give a preview on the forthcoming sequel where we count the of such tropical morphisms in the even genus case and, under a suitable notion of multiplicity, obtain a Catalan. Wo this talk we explore the relationship between tropical ideals and congruences to conclude that the variety of a non-zero prime tropical ideal is either empty or consists of a single point.
Kalina Mincheva In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. In this talk I will discuss t work with Josephine Yu showing that this hypergraph Tgi in fact d-connected when the complex has no lineality space. Just before this, Vakil and Zinger gave a very explicit desingularization of the moduli space of elliptic curves in projective space, with remarkable applications.
In t work with Johannes Rau and Kristin Shaw, we established this conjecture for divisors.
This is t work with Georg Loho. While they have many nice combinatorial properties, the notion of a "face" of a tropical polyhedron has been difficult to define.
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The approach is based on the floor decomposition of tropical curves which is a degeneration technique that allows us to apply the of the elliptic curve case. These compactificatons have very interesting combinatorial properties, which can be used to describe their geometry.
Wichtige Hinweise Zoom, link to be sent by organisers to the registered participants. A decade on, we understand these three as part of a single story involving logarithmic structures and their tropicalizations. In case of characteristic zero, this longstanding problem was settled by Harris in Prime tropical ideals. It is not surprising that each bitangent class has 4 lifts.
If a tropical quartic is Trop V q for a polynomial q in K[x,y] where K is the field of complex Puiseux seriesit is a natural question where in the 7 bitangent classes the tropicalizations of the 28 bitangents of V q are, or, put differently, which member of the tropical bitangent classes lifts to a bitangent of V qand with what multiplicity.
The moduli space of line bundles on smooth curves of given genus, the so called universal Jacobian, has a of different compactifications over the moduli space of stable curves. I will discuss this picture and how the unified framework extends all three. The main feature of the construction is that it works in positive characteristic, where the other known techniques fail.
In the first part of my talk I will give a brief overview of the ideas involved. The question becomes how to construct such a universal family of Gorenstein curves.
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Tgig, we construct one such family in genus two over a logarithmic modification of the space of admissible covers. The combinatorics and real lifting of tropical bitangents to plane quartics Hannah Markwig A plane quartic has 28 bitangents.
In this talk, we shall examine the obstacles that arise when considering faces of tropical polyhedra. Then, I will describe a tropical approach to studying degenerations of plane curves, which is the main ingredient to a new proof of irreducibility obtained in collaboration with Xiang He and Ilya Tyomkin. Originated from observations in string theory, it suggests that certain geometrical objects complex Calabi-Yau manifolds should come in pairs, in the sense that each of them has a mirror partner and the two share interesting geometrical properties.
If q is defined over the reals, V q can have 4, 8, 16 or 28 real bitangents. In this talk I will introduce some notions relating mirror symmetry to tropical geometry, inspired by the work of Kontsevich-Soibelman and Gross-Siebert.
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I will focus on how tropical geometry determines this logarithmic modification via tropical canonical divisors. Finally we introduce an octanomial model for cubic surfaces. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range hezd interesting matroids. This is t work with Martin Ulirsch. I will introduce the notions that are necessary to state the tropical Hodge conjecture and then sketch the proof and further directions of research.
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In t work with F. Nesds after this, Smyth constructed new compactifications of moduli spaces of pointed elliptic curves, using worse-than-nodal singularities, as part of the Hasset-Keel program. In the talk I will explain different features and applications of these interesting objects, focusing on properties which have a natural tropical counterpart. Ruby Tuesday is filing for bankruptcy protection, the latest casual chain to suffer from coronavirus-related closures and changing consumer habits.
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