Browsing by Author "Markwig, Hannah"

Now showing 1 - 6 of 6
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    Counting curves on toric surfaces tropical geometry & the Fock space
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Cavalieri, Renzo; Johnson, Paul D.; Markwig, Hannah; Ranganathan, Dhruv
    We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams, and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are shown to coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttche, Cooper–Pandharipande, and Block–Gathmann–Markwig.
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    A graphical interface for the Gromov-Witten theory of curves
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Cavalieri, Renzo; Johnson, Paul; Markwig, Hannah; Ranganathan, Dhruv
    We explore the explicit relationship between the descendant Gromov–Witten theory of target curves, operators on Fock spaces, and tropical curve counting. We prove a classical/tropical correspondence theorem for descendant invariants and give an algorithm that establishes a tropical Gromov–Witten/Hurwitz equivalence. Tropical curve counting is related to an algebra of operators on the Fock space by means of bosonification. In this manner, tropical geometry provides a convenient “graphical user interface” for Okounkov and Pandharipande’s celebrated GW/H correspondence. An important goal of this paper is to spell out the connections between these various perspectives for target dimension 1, as a first step in studying the analogous relationship between logarithmic descendant theory, tropical curve counting, and Fock space formalisms in higher dimensions.
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    Polynomiality, wall crossings and tropical geometry of rational double Hurwitz cycles
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Bertram, Aaron; Cavalieri, Renzo; Markwig, Hannah
    We study rational double Hurwitz cycles, i.e. loci of marked rational stable curves admitting a map to the projective line with assigned ramification profiles over two fixed branch points. Generalizing the phenomenon observed for double Hurwitz numbers, such cycles are piecewise polynomial in the entries of the special ramification; the chambers of polynomiality and wall crossings have an explicit and “modular” description. A main goal of this paper is to simultaneously carry out this investigation for the corresponding objects in tropical geometry, underlining a precise combinatorial duality between classical and tropical Hurwitz theory.
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    Tropical Aspects in Geometry, Topology and Physics
    (Zürich : EMS Publ. House, 2015) Markwig, Hannah; Mikhalkin, Grigory; Shustin, Eugenii
    The workshop Tropical Aspects in Geometry, Topology and Physics was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject. The development of tropical geometry is based on deep links between problems in real and complex enumerative geometry, symplectic geometry, quantum fields theory, mirror symmetry, dynamical systems and other research areas. On the other hand, new interesting phenomena discovered in the framework of tropical geometry (like refined tropical enumerative invariants) pose the problem of a conceptual understanding of these phenomena in the “classical” geometry and mathematical physics.
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    Tropical Geometry: new directions
    (Zürich : EMS Publ. House, 2019) Markwig, Hannah; Mikhalkin, Grigory; Shustin, Eugenii
    The workshop "Tropical Geometry: New Directions" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject, notably, to new phenomena that have opened themselves in the course of the last 4 years. This includes, in particular, refined enumerative geometry (using positive integer q-numbers instead of positive integer numbers), unexpected appearance of tropical curves in scaling limits of Abelian sandpile models, as well as a significant progress in more traditional areas of tropical research, such as tropical moduli spaces, tropical homology and tropical correspondence theorems.
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    Tropical Methods in Geometry
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Itenberg, Ilia; Markwig, Hannah; Shaw, Kris; Tyomkin, Ilya
    The workshop "Tropical methods in geometry" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject including tropical methods in symplectic and Lagrangian geometry, topology of real algebraic varieties and tropical homology, tropical methods in algebraic, Berkovich analytic and log geometries, refined tropical enumerative geometry and enriched counting, and algebraic geometry and matroids.