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Variations on a Theme of Schubert Calculus

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Recent Trends in Algebraic Combinatorics

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 16))

Abstract

In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear spaces. It is intended as a guide for readers with a combinatorial bent to understand and appreciate the geometric and topological aspects of Schubert calculus, and conversely for geometric-minded readers to gain familiarity with the relevant combinatorial tools in this area. We lead the reader through a tour of three variations on a theme: Grassmannians, flag varieties, and orthogonal Grassmannians. The orthogonal Grassmannian, unlike the ordinary Grassmannian and the flag variety, has not yet been addressed very often in textbooks, so this presentation may be helpful as an introduction to type B Schubert calculus. This work is adapted from the author’s lecture notes for a graduate workshop during the Equivariant Combinatorics Workshop at the Center for Mathematics Research, Montreal, June 12–16, 2017.

Supported by NSF MSPRF grant PDRF 1604262.

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Notes

  1. 1.

    See [49] for Schubert’s original work, or [45] for a modern exposition on Schubert’s methods.

  2. 2.

    A play on words that references the shared surname with musical composer Franz Schubert, who also lived in Germany in the nineteenth century.

  3. 3.

    Photograph of the train tracks downloaded from edupic.net.

  4. 4.

    Unfortunately, we could not find any photographs of parabolic train tracks.

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Acknowledgements

The author thanks Jennifer Morse, François Bergeron, Franco Saliola, and Luc Lapointe for the invitation to teach a graduate workshop on Schubert calculus at the Center for Mathematics Research in Montreal, for which these extended notes were written. Thanks also to Sara Billey and the anonymous reviewer for their extensive feedback. Thanks to Helene Barcelo, Sean Griffin, Philippe Nadeau, Alex Woo, Jake Levinson, and Guanyu Li for further suggestions and comments. Finally, thanks to all of the participants at the graduate workshop for their comments, questions, and corrections that greatly improved this exposition.

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Authors and Affiliations

  1. University of California, Davis, CA, USA

    Maria Gillespie

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  1. Maria Gillespie
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Correspondence to Maria Gillespie .

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Editors and Affiliations

  1. Mathematical Sciences Research Institute , Berkeley, CA, USA

    Hélène Barcelo

  2. Department of Mathematics, Pomona College, Claremont, CA, USA

    Gizem Karaali

  3. Department of Mathematics, Dartmouth College, Hanover, NH, USA

    Rosa Orellana

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Gillespie, M. (2019). Variations on a Theme of Schubert Calculus. In: Barcelo, H., Karaali, G., Orellana, R. (eds) Recent Trends in Algebraic Combinatorics. Association for Women in Mathematics Series, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-030-05141-9_4

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