George D. Nasr's Math Website - Research Interests

Research Interests

Broadly, I enjoy thinking about any problems involving combinatorics. The majority of my current work revolves around matroids, which abstract the combinatorial interactions between bases of a vector space and spanning trees of a graph. I like working with a special subclass of matroids called paving matroids due to their nice structure and the conjectured belief that they make up the majority of matroids. However, I am currently working on some projects involving other types of matroids.

One can witness a matroid as a geometric object called a polytope. Polytopes can be decomposed into smaller polytopes, and sometimes this can be done in a way that is meaningful on the level of matroids, so that the smaller polytopes correspond to smaller matroids. This allows one to study something called a polytope valuation for matroids, which informally are functions that are compatible with any decomposition of a polytope.

Two such valuations for matroids are called Kazhdan-Lusztig polynomials and Ehrhart polynomials, both of which I have worked on in the context of paving matroids. For the former, I even have an interpretation involving skew Young tableaux.

You can find my specific work on the Publications and Preprints page.

You may also read a version of my research statement which goes into a bit more detail about all the above mentioned ideas.

Skew Young Tableaux

Similar to standard Young tableaux, you fill a skew tableaux diagram with numbers so that the rows (resp. columns) increase from left to right (resp. top to bottom).

The entries range from 1 to the number of cells in the diagram, as in the above diagram.

Matroids

Matroids generalize the shared combinatorial structure arising from the linearly independent subsets of a collection of vectors and the acyclic edge sets of a graph.

The collection of acyclic edge sets for the above graph happens to form a sparse paving matroid, a class of matroids that are conjectured to be almost all matroids (and play an essential role in one of my papers).

Polytopes

One can think about polytopes as the convex hull of a collection of points in R^n, for any n.

The triangle above can be thought of as the convex hull of the points (1,1,0), (1,0,1), and (0,1,1). Note the convex hull of these points also includes the interior of the triangle, not just the boundary.