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Research:
My research area lies in the intersection of algebraic geometry and combinatorics. Developing a bridge between these two distinct areas of mathematics stands to benefit both geometers and combinatorialists, as it allows researchers to answer geometric questions by looking at the combinatorics, and vice versa. For example, the study of toric varieties is a beautiful part of algebraic geometry where the dictionary between geometry and combinatorics is "perfect". This elegant structure makes it an invaluable tool in other areas of research such as coding theory, physics, and algebraic statistics.
In recent years, the theory of Newton-Okounkov bodies has attempted to enlarge this dictionary; it takes an arbitrary algebraic variety (geometric object), equipped with some auxiliary data, and attaches to it a combinatorial object (e.g. a polytope). By construction, this combinatorial object encodes some important geometric information about the original variety. The main question in this theory then is, `"What geometric data does this combinatorial object encode, and how?"
Since the theory of Newton-Okounkov bodies is relatively new, there is much that is still unknown. In particular, very few concrete and explicit examples of Newton-Okounkov bodies have been computed thus far. Therefore, it is an interesting problem to compute new concrete examples.
In my research, I have been working to explicitly compute the Newton-Okounkov bodies of certain algebraic varieties which have important ties to representation theory. In particular, I computed the Newton-Okounkov bodies of Bott-Samelson and Peterson varieties, for certain choices of auxiliary data.
Presentations:
Projects:
Professional Documents:
Lauren DeDieu - 2016
