## Northfield Undergraduate Mathematics Symposium 2010

The Northfield Undergraduate Mathematics Symposium is an annual event sponsored jointly by Carleton and St. Olaf. At the 2010 Symposium nine students spoke on research they did around the country over the summer. The pictures at the right show the speakers in action, and below are the titles and abstracts of their talks, and in some cases links to their slides.

### Graphic Degree Sequences for Edge-Colored Graphs

**Amy Becker (Carleton)**

For an edge-colored graph, we can easily list the degrees of each color at each vertex. But under what conditions is there an edge-colored graph with a given candidate degree sequence? When such a graph exists we say that the given degree sequence is 'graphic.' We consider cases in which either the maximum degree or the number of colors is small and describe when we can decide, in polynomial time, whether a degree sequence is graphic.

### Bruhat Graphs and Pattern Avoidance

**Chris Conklin (St. Olaf) **

Our work this summer consisted of determining which permutations can generate a Bruhat graph that can be drawn both on a plane and on a torus. We have found a set of rules to determine which permutations generate such a graph and how many permutations generate such a graph. In determining this we have also discovered some generalizations about pattern avoidance in general.

### Modeling the Effect of Transmission-Blocking Vaccines on Malaria

**Gabe Davis (Carleton)**

New transmission-blocking vaccines (TBVs) which prevent the transmission of the parasite from vaccinated humans to susceptible mosquitoes, may offer new hope for the eradication of malaria in communities in Africa and elsewhere, but their epidemiological effects on large populations are still not well understood. I present an ODE model that can be generalized to show the effect of the vaccine on small isolated communities, and assess its effectiveness as an eradication tool. I conclude that TBVs alone are unlikely to be a solution, but may form part of a long-term, multi-prong eradication effort. A complementary stochastic model allows assessment of the long-term usefulness of the model's predictions by giving an estimate of the variance over time.

Modeling Malaria Vaccine Slides

### Equivalence Classes of Zero Divisor Graphs

**Cathryn Holm (St. Olaf)**

We study the graphs of equivalence classes of zero divisors of commutative Noetherian rings. In examining all possible graphs on four, five, and six vertices, we determine which graphs are realizable representations of rings and demonstrate examples of rings which have these graphs. Furthermore, we examine the finite fan graph and identify a method for constructing many fan graphs with *n ≥ 4*

vertices.

### Games of Cops and Robbers on Planar Graphs

**Aaron Maurer (Carleton)**

In the game of Cops and Robbers one or more cops attempt to catch a robber on a graph. The way this works is that each turn, each cop and the robber lie on a vertex in the graph. First, each cop either stays on his or her current vertex or moves to an adjacent vertex. Second, the robber makes a similar move. This is repeated until a cop lies on the same vertex as the robber, in which case the robber is caught. We say a graph has "Cop Number x" whenever x cops can, from any initial positioning of themselves and the robber, always capture the robber, and if x is the smallest value for which this is true. Aigner and Fromme have shown that all planar graphs have cop number at most three. Since cop number one graphs are easily identifiable, it's natural to ask which planar graphs are two cop, and which are three cop. In this talk we'll discuss this problem for several classes of planar graphs, including Archimedean Solids, maximal planar graphs, series parallel graphs, and outer planar graphs.

### Curve Fitting Variations and Neural Data

**Julie Michelman (Carleton)**

Curve fitting attempts to find a smooth function that describes the trend of data given in *(x _{i}, y_{i})* pairs. Cubic splines, of which smoothing splines are one type, are piece-wise cubic functions commonly used to fit such data. The smoothing spline fit

*f*minimizes the penalized sum of squares

*PSS(f) = ∑ (y*, where the first term is the residual sum of squares and the second is a global smoothing parameter λ multiplied by a penalty for curvature. Smoothing splines generally perform well unless the curvature of the data is highly variable, in which case they tend to overfit smoother areas and underfit areas with sharp curvature. We develop an algorithm to fit smoothing splines with a variable smoothing parameter, which accounts for local changes in curvature. Our automated algorithm finds the optimal smoothing parameter value at each point using cross validation in a local window and then uses this set of parameter values to calculate the fit. On data with trends that have highly variable curvature, our fits consistently have lower mean integrated squared error and better coverage for confidence intervals than ordinary smoothing spline fits.

_{i}- f(x_{i}))^{2}+ λ ∫(f''(x))^{2}dx### The Combinatorics of the Sandpile Model and its Generalizations

**Vladimir Sotirov (St. Olaf)**

A discrete dynamical system is basically a collection of states together with a rule that describes how one state can change into another state. Certain states then are of natural interest: we care about "fixed points", which are states from which no other state can be obtained (using the transition rule, that is), and we also care about "Gardens of Eden", which are states that cannot be obtained from any other state. In this talk, we will look at a relatively simple (to describe) dynamical system called the Sandpile model and also some of its generalizations (one of which is the product of my summer research project!), and specifically we will investigate the answers to the basic combinatorial question of "How many of these things (fixed points and Gardens of Eden) are there for these models?"

### Counting Modular Tableaux

**Bjorn Wastvedt (St. Olaf)**

In this talk we provide a bijection between all modular tableaux of size *kn* and all partitions of *n* into *k* colors. By using the generating function for the number of *k*-colored partitions of *n*, we can count the number of modular tableaux of size *kn*. We demonstrate an alternate proof of Stanton and White's bijection between *k*-rim hook tableaux of size *kn* and *k*-tuples of standard tableaux, which helps to establish our primary result. This is based on joint work with Nathan Meyer, Daniel Mork, and Benjamin Simmons.

Counting Modular Tableaux Slides

### On Subbarao's Conjecture on the Parity of the Partition Function

**McKenzie West (St. Olaf)**

Let *p(n)* denote the ordinary partition function. In 1966, Subbarao conjectured that in every arithmetic progression *r (mod t)* there are infinitely many integers *N = r (mod t)* for which *p(N)* is even (resp. odd). We prove Subbarao's conjecture for all moduli *t* of the form *m 2 ^{s}* where

*m є {1,5,7,17}*. To obtain this theorem we make use of recent results of Ono and Taguchi on the nilpotent action of Hecke algebras on certain spaces of modular forms modulo 2.