Browsing by Author "Shustin, Eugenii"

Now showing 1 - 4 of 4
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    Real Enumerative Invariants Relative to the Anti-Canonical Divisor and their Refinement
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Itenberg, Ilia; Shustin, Eugenii
    We introduce new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from the appropriate enumeration of real elliptic curves. These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the rational case. We also construct tropical counterparts of the refined elliptic invariants under consideration and establish a tropical algorithm allowing one to compute, via a suitable version of the correspondence theorem, the above invariants.
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    Real Enumerative Questions in Complex and Tropical Geometry
    (Zürich : EMS Publ. House, 2011) Shustin, Eugenii; Walcher, Johannes; Welschinger, Jean-Yves
    The workshop Real Enumerative Questions in Complex and Tropical Geometry was devoted to a wide discussion and exchange of ideas between the best experts representing various points of view on the subject. Enumeration of real curves largely motivated the development of the tropical geometry and led to the discovery of new interesting geometric phenomena and deep links between this problematic and algebraic geometry, symplectic geometry, topology, and mathematical physics.
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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.