Monday (17:15-18:30)

Speaker: Dante Luber (TU Berlin)
Title: Generalized Permutahedra and Positive Flag Dressians
Abstract: We study valuated matroids, their tropical incidence relations, flag matroids and total positivity. Our techniques employ the polyhedral geometry of the hypersimplices, the regular permutahedra and their subdivisions.

Speaker: Victoria Schleis (Eberhard Karls Universität Tübingen)
Title: Tropical linear degenerate flag matroids
Abstract: Grassmannians and flag varieties are important moduli spaces in algebraic geometry. Their linear degenerations arise in representation theory as they describe quiver representations and their irreducible modules. As linear degenerations of flag varieties are difficult to analyze algebraically, we describe them in a combinatorial setting and further investigate their tropical counterparts. In this talk, I will introduce matroidal, polyhedral and tropical analoga and descriptions of linear degenerate flags and their varieties obtained in joint work with Alessio Borzì. Using techniques from matroid theory, polyhedral geometry and tropical geometry, we use the correspondences between the different descriptions to gain insight on the structure of linear degeneration. Further, we analyze the structure of linear degenerate flag varieties in all three settings, and provide some cover relations on the poset of degenerations.

Speaker: Tomasz Mańdziuk (University of Warsaw)
Title: On the Slip component of a multigraded Hilbert scheme
Abstract: I consider the multigraded Hilbert scheme associated with the Hilbert function of points in general position in a projective plane. It has an irreducible component called Slip, whose points are related to the notion of the border Waring rank. I will present a sufficient condition for a point of that scheme to belong to Slip.

Tuesday (16:45-18:30)

Speaker: Jyoti Dasgupta (Indian Institute of Science Education and Research, Pune)
Title: Seshadri constants of equivariant vector bundles on toric varieties
Abstract: Seshadri constants measure the local positivity of an ample line bundle. They were introduced by Demailly. Later, Hacon generalized the notion of Seshadri constants to vector bundles. In this talk, we consider torus equivariant vector bundles on toric varieties. Assuming certain conditions on the vector bundle, we compute precise values of Seshadri constants at arbitrary points on projective spaces and Bott towers of height at most 3. This talk is based on joint work with Bivas Khan and Aditya Subramaniam.

Speaker: Robert Crumplin (Imperial College London and LSGNT)
Title: Fano Mirror Superpotentials & Broken Lines
Abstract: I will give a brief overview of what the statement of mirror symmetry for Fano varieties is and how the mirror potentials W can be computed via a GHK/GS style procedure, which can be computed using tropical geometry techniques using so-called "scattering diagrams". I will illustrate this with some particular examples, e.g. Toric Fanos, dP_5.

Speaker: Anna Katharina Bot (University of Basel)
Title: The ordinal of dynamical degrees of birational maps of the projective plane
Abstract: For any birational map of the projective plane, one can consider its dynamical degree, which describes the dynamical behaviour of this map. Taking all dynamical degrees of birational maps together, we obtain a well-ordered subset of the real line, and we may ask: what is the ordinal of this set? In this talk I will give an introduction to the dynamical degree and ordinals, and prove that the set of dynamical degrees of all birational maps of the plane is of order type \omega^\omega, where \omega is the first infinite ordinal.

Speaker: Patience Ablett (University of Warwick)
Title: Gorenstein curves of codimension four
Abstract: Gorenstein ideals are well understood in codimension three and below, but less is known about codimension four. Recent work of Schenck, Stillman and Yuan describes all possible Betti tables for Artinian Gorenstein algebras of regularity and codimension four. We explain how to interpret these Betti tables as a recipe book for constructing Gorenstein codimension four varieties, and give an example construction utilising the Tom and Jerry formats of Brown, Kerber and Reid.

Thursday (17:15-18:30)

Speaker: Andres Jaramillo Puentes (University Duisburg-Essen)
Title: Enriched Tropical Intersection
Abstract: Tropical geometry has been proven to be a powerful computational tool in enumerative geometry over the complex and real numbers. In this talk we present an example of a quadratic refinement of this tool, namely a proof of the quadratically refined Bézout's theorem for tropical curves. We recall the necessary notions of enumerative geometry over arbitrary fields valued in the Grothendieck-Witt ring. We will mention the Viro's patchworking method that served as inspiration to our construction based on the duality of the tropical curves and the refined Newton polytope associated to its defining polynomial. We will prove that the quadratically refined multiplicity of an intersection point of two tropical curves can be computed combinatorially. We will use this new approach to prove an enriched version of the Bézout theorem and of the Bernstein–Kushnirenko theorem, both for enriched tropical curves. Based on a joint work with S. Pauli.

Speaker: Felix Roehrle (Goethe University Frankfurt)
Title: A tropical version of Donagi's $n$-gonal construction.
Abstract: The $n$-gonal construction associates to a double cover $\widetilde C \to C$ of an $n$-gonal algebraic curve $C$ a $2^n$-gonal curve $C'$. For $n = 2,3,4$ there are theorems relating the Prym variety of the input to the Prym or Jacobian variety of the output. In this ongoing project we give a version of the construction for tropical curves and study the relation of tropical Prym and tropical Jacobian varieties of the input and output of the construction. This is joint work in progress with Dmitry Zakharov.

Speaker: Alheydis Geiger (MPIM Leipzig)
Title: Towards tropically counting binodal surfaces
Abstract: As tropical geometry is a successful tool for solving enumerative problems of curves, it is natural to investigate the possibilities of using tropical geometry for higher dimensional enumerative questions, like counting nodal surfaces. The first tool for tackling this problem are tropical floor plans, introduced by Markwig et al. They count all cases where the tropicalizations of the nodes are far apart and can be easily generalized to so-called separated nodes. This, however, does not suffice to recover even the count of binodal cubic surfaces completely. We make first steps towards counting binodal surfaces, where the nodes are not separated, in providing a full analysis of the extreme case where the nodes tropicalize to the same vertex of the tropical surface. This talk presents joint work with Madeline Brandt.