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Fall 2014
September 16  John Harding (New Mexico State): Quantum structures
Poster
Abstracts
September 16  John Harding (New Mexico State): Quantum structures
Quantum structures is broad term for a wide range of areas. The common theme is the study of mathematical structures motivated by quantum mechanics. Such structures are studied both for their own interest, and for their application to quantum mechanics. They are frequently noncommutative or nondistributive versions of more classical objects.
In this talk I'll discuss some of my results in a number of different parts of quantum structures. This is a bit of a tour through several areas in the field, some quite old, others very current. We will cross a range of areas from operator algebras, to ordered structures and universal algebra, to category theory. Hopefully we provide a view of at least a portion of the topics of interest in this broad field.
Derrick Stolee: Automated discharging arguments for density problems in grids
Motivated by problems in wireless sensor networks, we consider minimizing the density of an identifying code in the hexagonal grid. An identifying code is a set of vertices where every vertex in the grid is uniquely identified by its adjacent code elements. While the minimum density of an identifying code is known for the square and triangular grids, there is still a gap between the upper and lower bounds for the hexagonal grid. Most lower bounds are found using discharging, which is a method to demonstrate the interaction between local structure and global averages.
Verifying a discharging proof is straightforward but usually very tedious. However, the creation of a discharging argument can be very mysterious. We will present a new computerautomated approach to not only verify discharging arguments, but also to generate them from scratch. A critical component to this method is solving a linear program that will assign value to the specified discharging rules, resulting in the best possible proof using those rules. Using this method, we find a new lower bound of 23/55 (approximately 0.4181818) on the density of an identifying code in the hexagonal grid, improving on the currentbest humancreated proofs.
Mridul Nandi (Indian Statistical Institute, Kolkata) Minimum number of multiplication to compute a Deltauniversal function
Delta universal function is a family of (multivariate polynomial) function with small differential probability. It is an important combinatorial object which has application in many areas of computer science. Any multivariate polynomial can be computed by a sequence of multiplication and addition. As multiplications are usually costlier operations than addition, we study the lower bounds on the number of multiplication to compute a delta universal hash function. In this talk, we obtain a concrete form of the lower bound. We also discuss memory requirement of computing such functions and how to tweak an optimum construction to reduce the memory constraint.
Archived
August 26  Mridul Nandi (Indian Statistical Institute, Kolkata) Minimum number of multiplication to compute a Deltauniversal function
Poster
September 2  Derrick Stolee: Automated discharging arguments for density problems in grids
Poster