Jeremy Martin's home page
Last updated Wed 11/14/07 2:00 PM
The KU Combinatorics Seminar will meet on Wednesdays, 3:00--4:00 PM, in Snow 408. Please contact Jeremy Martin if you are interested in speaking.
Abstract: Let n(r,d) be the maximum number of equal d-dimensional spheres of diameter 1 that can be packed into a d-dimensional cube whose edges have length r. We consider the problem of determination of n(r,d) for given r and d. Some exact values and estimates on n(r,d) are found. To get some lower bounds on n(r,d), one can use discrete mathematics and convex geometry as well. To get upper bounds, one can use polyhedral subdivisions of the cube and solve a corresponding optimization problem on few variables.
Abstract: The ith angle sum of a polytope counts the sum of the solid angles at i-dimensional faces of a polytope. We define the γ-vector of a polytope as a linear combination of the angle sums in a manner analogous to the h-vector as a linear combination of the f-vector. The Gram and Perles equations on angles give results analogous to the Euler and Dehn-Sommerville equations on the f- and h-vectors. We consider how the analogy between the face structure and angle structure of polytopes continues, proving that the γ-vector is non-decreasing for low-dimensional simplices and non-negative for low-dimensional simplicial polytopes. As time allows, we will also consider an angle analog of the Euler characteristic, showing that it is half the Euler characteristic for a large class of polytopal complexes.
Abstract: In this talk, I will begin by discussing chip-firing games on graphs, and how these yield a group structure on the set of spanning trees of a graph. In the second part, I will describe elliptic curves over finite fields, and how such objects also have group structures. For a family of graphs obtained by deforming the sequence of wheel graphs, the cardinalities of these groups satisfy a nice reciprocal relationship with the orders of elliptic curves as we consider field extensions. I will finish by discussing other surprising ways that these group structures are analogous. This research was completed as part of my dissertation work at the University of California, San Diego under Adriano Garsia's guidance.
Combinatorics seminar for current semester
Jeremy Martin's home page
Last updated Wed 11/14/07 2:00 PM