Songting Luo
Derrick Stolee
Steve Butler

# Fall 2014

November 4 - David Galvin (Notre Dame) Stirling numbers of graphs, and the normal ordering problem
Poster

November 11 -Chao Yang (LBNL)

November 18 -George Avalos (University of Nebraska)

December 2 - Sarah Reznikoff (Kansas State University)

December 9 -OPEN

December 16 -OPEN

# Abstracts

November 4 - David Galvin (Notre Dame) Stirling numbers of graphs, and the normal ordering problem

The Stirling number of the second kind ${n \brace k}$ counts the number of partitions of a set of size $n$ into $k$ non-empty blocks.  A graph theoretic interpretation of this quantity --- the number of partitions of the empty graph of order $n$ into $k$ non-empty independent sets --- admits a natural generalization to arbitrary graphs. A more analytic interpretation --- the coefficient of $x^k$ in the polynomial $p(x)$ defined by $\left(x \frac{d}{dx}\right)^ne^x = p(x)e^x$ --- also admits a natural generalization, with $\left(x \frac{d}{dx}\right)^n$ replaced by an arbitrary word in the alphabet $\{x,d/dx\}$. This latter generalization is the {\em Weyl algebra normal ordering problem}.

I'll show how these two generalizations are closely related, and give a simple graph theoretic answer to the normal ordering problem. In part joint work with J. Engbers and J. Hilyard.

October 28 -Garret Alston (U Oklahoma) Legendrian and Lagrangian invariants

A symplectic manifold is an even dimensional manifold M along with a closed, non-degenerate 2-form. It is the generalization of the notion of phase space (set of all positions and velocities of a system) from classical mechanics (in physics). The odd-dimensional analog is known as a contact manifold. One way to try to understand these types of manifolds is to study certain types of submanifolds called Lagrangians (in the symplectic case) and Legendrians (in the contact case). In this talk I will explain some invariants that are used to study these submanifolds. I will begin by explaining the simplest case: Chekanov's dga for Legendrian knots in R^3. This will involve knot diagrams and simple calculations that everyone can follow along with. After that, I will sketch the general picture a little bit and explain how these invariants help answer interesting questions in symplectic and contact geometry.

October 21 -Chao Zhu (UW-Milwaukee) On linear programing approach to inventory control problems

This work deals with inventory control problems under the discounted criterion. The objective is to minimize the discounted total holding and ordering costs. In contrast with the usual dynamical programming approach, this work first imbeds the inventory control problem into an infinite-dimensional linear program over a space of measures and then reduces the linear program to a simpler nonlinear optimization. This approach not only determines the value of the inventory control problem but also identifies an optimal impulse control policy within restricted classes of control policies. Additional auxiliary and dual linear programs are introduced to verify the optimality of the impulse control policy in the general class of control policies. This is a joint work with Kurt Helmes (Humboldt-Universitat zu Berlin) and Richard Stockbridge (University of Wisconsin-Milwaukee).

October 17 (Friday) - Lili Ju (U So Carolina) Joint with CAM Fast and stable compact explicit integration factor based methods for semilinear parabolic equations

In this talk, we present an explicit exponential integration factor based method and its fast implementation for the solution of a wide class of semilinear parabolic equations including the Allen-Cahn equation as a special case. The method makes use of a stable splitting scheme, and combines compact representations of spatial difference operators on the regular mesh with accurate exponential time differences and multistep approximations. It can deal with stiff nonlinearity and both homogeneous and inhomogeneous boundary conditions of different types.

Various numerical experiments demonstrate efficiency and effectiveness of the proposed method for both linear and nonlinear model problems.

October 14 -Alexander Roiterschtein (ISU) Local times of random walks on random media on Z and generalized Ray-Knight theorems

Generally speakingRay-Knight theorems describe the joint distribution of local times of Markov processes. For general one-dimensional nearest-neighbor random walks the theorem is in essence an encoding of the path of the random walk in terms of certain (non-standard) branching-type processes. This generic random walks-branching duality provides a very powerful tool for the study  of random walks in random media. The bulk of the talk will be focused on asymptotic results for the maximum local time of excited random walks.  I will also briefly discuss a work in progress on the number of most visited sites of a random walk in random environment.

September 30 - Marcus Bishop (ISU): Quiver presentations of algebras associated with finite Coxeter groups

We will discuss the methods used to calculate quiver presentations of three algebras associated to finite Coxeter groups, namely, the descent algebra, the Mantaci-Reutenauer algebra, and the zero-Hecke algebra. The talk will be appropriate for a general audience and will introduce quivers, Coxeter groups, and all of the algebras discussed.

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 non-commutative or non-distributive 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 computer-automated 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 current-best human-created proofs.

Mridul Nandi (Indian Statistical Institute, Kolkata) Minimum number of multiplication to compute a Delta-universal 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 Delta-universal function
Poster

September 2 - Derrick Stolee: Automated discharging arguments for density problems in grids
Poster

September 16 - John Harding (New Mexico State): Quantum structures
Poster

September 30 - Marcus Bishop (ISU): Quiver presentations of algebras associated with finite Coxeter groups
Poster

October 14 -Alexander Roitershtein (ISU) Local times of random walks on random media on Z and generalized Ray-Knight theorems
Poster

October 17 (Friday) - Lili Ju (U So Carolina) Joint with CAM Fast and stable compact explicit integration factor based methods for semilinear parabolic equations
Poster

October 21 -Chao Zhu (UW-Milwaukee) On linear programing approach to inventory control problems
Poster

October 21 -Chao Zhu (UW-Milwaukee) On linear programing approach to inventory control problems
Poster

October 28 -Garret Alston (U Oklahoma) Legendrian and Lagrangian invariants
Poster