TCC Tropical Geometry
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Course DescriptionTropical geometry is an emerging area at the interface between algebraic geometry and combinatorics. At its most basic it is (algebraic) geometry over the tropical semiring, where multiplication is replaced by addition, and addition is replaced by minimum. This turns polynomials into piecewise-linear functions, and varieties into polyhedral complexes.In good situations this allows algebro-geometric invariants to be computed using simpler combinatorics. This has had success in enumerative geometry, mirror symmetry, Brill Noether theory and constructing compactifications, among others. See for example this picture for an illustration of computing Gromov-Witten numbers of the plane using tropical geometry. See also the first chapter of the book-draft mentioned below for many more different examples, pictures, and applications. In this module I will introduce the basics of tropical geometry. This will focus on the elementary algebraic geometry and computational aspects which are needed in many of the more technical applications. The first half of the module will not require any sophisticated algebraic geometry background. While the second half will require more, it will have more of a survey flavour. Tropical geometry is not only a subfield of algebraic geometry, but has deep links to applications. This module will focus on the algebraic geometry side, but will aim to be accessible to those whose interests are on the more applied side. | |||||||||
Lecturer
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Course Times and Location
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Recommended TextsThe main reference for this module is the draft book-in-progress I am writing with Bernd Sturmfels. A current draft is available here. This will be updated as the module progresses.PLEASE send me any typos you notice, no matter how minor. Include in your email the date of the version containing the typo. Further references, and a rough outline of the topics to be covered is available at the schedule page. | |||||||||
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AssessmentThere is no formal assessment for this module. Exercises will be given for each lecture, which will be available on the schedule page.Subject to logistics, there will also be a (virtual) seminar at the end of the term where participants give reports on tropical geometry papers they have read. This will let you tailor the module to your particular research interests, and expose you to aspects of tropical geometry not covered in the module. This will be announced early in the term. |