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Dr F Rincon
Applications accepted all year round
About the Project
The School of Mathematical Sciences of Queen Mary University of London invite applications for a PhD project commencing in September 2020 for self-funded students.
This project will be supervised by Dr. Felipe Rincon.
Tropical algebraic geometry provides a tool for transforming (tropicalising) a classical algebraic variety into a combinatorially defi ned polyhedral complex that retains important information about the variety. In this way, it allows for the use of combinatorial techniques in the study of algebro-geometric questions. Tropical geometry has been very successful in tackling problems in many different areas of pure and applied mathematics, which has led to a recent explosion of research in the field.
Tropical algebraic geometry has so far mostly been concerned with geometrically studying these polyhedral objects, called tropical varieties. However, the paper [GG16] opened the door for using algebraic techniques in the study of tropical varieties. It was quickly recognised in [MR] that this algebraic information was naturally encoded by a polynomial ideal over the tropical numbers, and also combinatorially modelled by a certain sequence of matroids.
All this led to the introduction of tropical ideals in [MR18], which generalise the class of ideals obtained by tropicalising a classical algebraic variety. They can be thought of as combinatorial objects that abstract the possible collections of supports of all polynomials in an ideal. They have a structure dictated by a sequence of matroids, and they have been shown to satisfy many desirable properties: the ascending chain condition, the fact that their varieties are fi nite polyhedral complexes, and the weak Nullstellensatz. Tropical ideals are expected to play a central role in the emergent theory of tropical schemes.
The goal of this project will be to learn about tropical geometry and tropical ideals, and push further our understanding of them. Possible directions involve tackling realisability questions for tropical ideals, studying their possible varieties, and studying connections to classical algebraic geometry or matroid theory.
The ideal PhD student working on this project will have a strong will to learn more about both (tropical) algebraic geometry and matroid theory, and preferably some familiarity with these or related topics.
The application procedure is described on the School website. For further inquiries please contact Dr Felipe Rincon at [Email Address Removed]@qmul.ac.uk.
This project will be supervised by Dr. Felipe Rincon.
Tropical algebraic geometry provides a tool for transforming (tropicalising) a classical algebraic variety into a combinatorially defi ned polyhedral complex that retains important information about the variety. In this way, it allows for the use of combinatorial techniques in the study of algebro-geometric questions. Tropical geometry has been very successful in tackling problems in many different areas of pure and applied mathematics, which has led to a recent explosion of research in the field.
Tropical algebraic geometry has so far mostly been concerned with geometrically studying these polyhedral objects, called tropical varieties. However, the paper [GG16] opened the door for using algebraic techniques in the study of tropical varieties. It was quickly recognised in [MR] that this algebraic information was naturally encoded by a polynomial ideal over the tropical numbers, and also combinatorially modelled by a certain sequence of matroids.
All this led to the introduction of tropical ideals in [MR18], which generalise the class of ideals obtained by tropicalising a classical algebraic variety. They can be thought of as combinatorial objects that abstract the possible collections of supports of all polynomials in an ideal. They have a structure dictated by a sequence of matroids, and they have been shown to satisfy many desirable properties: the ascending chain condition, the fact that their varieties are fi nite polyhedral complexes, and the weak Nullstellensatz. Tropical ideals are expected to play a central role in the emergent theory of tropical schemes.
The goal of this project will be to learn about tropical geometry and tropical ideals, and push further our understanding of them. Possible directions involve tackling realisability questions for tropical ideals, studying their possible varieties, and studying connections to classical algebraic geometry or matroid theory.
The ideal PhD student working on this project will have a strong will to learn more about both (tropical) algebraic geometry and matroid theory, and preferably some familiarity with these or related topics.
The application procedure is described on the School website. For further inquiries please contact Dr Felipe Rincon at [Email Address Removed]@qmul.ac.uk.
Funding Notes
This project can be undertaken as a self-funded project. Self-funded applications are accepted year-round for a January, April or September start.
The School of Mathematical Sciences is committed to the equality of opportunities and to advancing women’s careers. As holders of a Bronze Athena SWAN award we offer family friendly benefits and support part-time study.
The School of Mathematical Sciences is committed to the equality of opportunities and to advancing women’s careers. As holders of a Bronze Athena SWAN award we offer family friendly benefits and support part-time study.
References
[CLO07] David A. Cox, John Little, and Donal O'Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics). Springer-Verlag, Berlin, Heidelberg, 2007.
[GG16] Je rey Giansiracusa and Noah Giansiracusa. Equations of tropical varieties. Duke Math. J., 165(18):3379{3433, 2016.
[MR] Diane Maclagan and Felipe Rincon. Tropical schemes, tropical cycles, and valuated matroids. arXiv preprint. arXiv:1401.4654. To appear in J. Eur.
Math. Soc. (JEMS).
[MR18] Diane Maclagan and Felipe Rincon. Tropical ideals. Compos. Math.,154(3):640{670, 2018.
[MS15] Diane Maclagan and Bernd Sturmfels. Introduction to Tropical Geometry, volume 161 of Graduate Studies in Mathematics. American Mathematical
Society, Providence, RI, 2015.
[Oxl92] James G. Oxley. Matroid theory. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, 1992.
[GG16] Je rey Giansiracusa and Noah Giansiracusa. Equations of tropical varieties. Duke Math. J., 165(18):3379{3433, 2016.
[MR] Diane Maclagan and Felipe Rincon. Tropical schemes, tropical cycles, and valuated matroids. arXiv preprint. arXiv:1401.4654. To appear in J. Eur.
Math. Soc. (JEMS).
[MR18] Diane Maclagan and Felipe Rincon. Tropical ideals. Compos. Math.,154(3):640{670, 2018.
[MS15] Diane Maclagan and Bernd Sturmfels. Introduction to Tropical Geometry, volume 161 of Graduate Studies in Mathematics. American Mathematical
Society, Providence, RI, 2015.
[Oxl92] James G. Oxley. Matroid theory. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, 1992.
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