The study of polyhedra began in antiquity, yet they play a prominent role throughout modern mathematical research. This session will focus on the ubiquity of polyhedra in mathematics by introducing various contexts in which they arise, including number theory, applied analysis, computational mathematics, algebra, combinatorics, and math education.
The study of partitions and compositions (i.e., ordered partitions) of integers goes back centuries and has applications in various areas within and outside of mathematics. Partition analysis is full of beautiful--and sometimes surprising--identities, starting with Euler's classic theorem that the number of partitions of an integer k into odd parts equals the number of partitions into distinct parts.
Motivated by work of G. Andrews et al from the last 1 1/2 decades, we will show how one can shed light to certain classes of partition identities by interpreting partitions as integer points in polyhedra. Our approach yields both "short" proofs of known results and new theorems.
This is joint work with Ben Braun, Ira Gessel, Nguyen Le, Sunyoung Lee, and Carla Savage.
The volumes of and integrals over polyhedra are perhaps the most fundamental and basic concept in the history of mathematics. Already ancient civilizations worried about it (e.g., Egyptian, Babylonian) and we teach formulas for volumes of pyramids and cubes to K-6 students. Yet, volumes and integrals of convex polytopes are properties to be computed, from algebraic geometry to computer graphics, from combinatorics to probability and statistics.
So, how does one go about actually computing an integral over a convex polytope if one cares to compute the number exactly? In this talk we survey why exact integral computation is relevant in everyone's life, why calculus techniques fail miserably for the goal, what is currently known about efficient computation of integrals of polynomials over convex polytopes. Software may be used in this talk.
We will discuss examples of where polyhedra have appeared in the K--16 curriculum, and then consider some ideas for additional topics and activities involving polyhedra that might be propitiously incorporated into the curriculum.
Motivated by the problem of p-value calculations in statistical hypothesis testing, we study the projection volumes of polyhedral cones. Namely, given a polyhedral cone C in n-dimensional real space, which fraction of the unit sphere is occupied by points x such that the orthogonal projection of x onto C lies in the interior of a k-dimensional face of C?
For cones arising from reflection arrangements, the answer is given by the coefficients of the characteristic polynomial associated to the hyperplanes which bound the cone. The proof of this result is achieved by considering the geometry and combinatorics of angle sums of zonotopes, a particularly nice class of polyhedra.
(No knowledge of any of the above terms will be assumed for the talk.)
Joint work with Mathias Drton and Ed Swartz.
Classically, a typical measured signal will be sparse after an appropriate transformation, meaning that most of its components will be very close to zero. After transforming a signal, one compresses it by setting all of these small values to zero. In Compressed Sensing, a recent twist in the theory of digital signal processing, the opposite approach is taken. It is assumed that the signal being measured is sparse, and that far fewer measurements are taken than the number of components in the signal. This leads to an underdetermined system of linear equations, which can have infinitely many solutions, so recovering the actual signal we measured among this infinite set of solutions seems like a lost cause.
Surprisingly, we can use the sparsity assumption to our advantage, and can develop tractable algorithms to recover our original signal more often that you might think. Further, the techniques of convex optimization used for finding this sparest solution lead to the study of the neighborliness of a polytope associated with the problem. In this talk, we will discuss this relationship and similar relationships to related problems.
The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes. We relax the regularity conditions to show the cd-index exists for non-regular CW complexes and extend the notion of a graded poset to that of a quasi-graded poset.
This is work-in-progress with Richard Ehrenborg and Mark Goresky.