Northfield Undergraduate Mathematics Symposium 2011
The Northfield Undergraduate Mathematics Symposium is an annual event sponsored jointly by Carleton and St. Olaf. This year thirteen students spoke at the symposium 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.
t-Factorization in the Integers
Nathan Bishop (St. Olaf)
This talk will serve as an introduction to the Theory of t-Factorization, a generalization of Factorization Theory. The theory hinges on the relation t: let D be an integral domain, U(D) be the units of D, and D# be defined as D \ {0}ÈU(D). Then t is any relation defined on D# x D#. For the purposes of this talk, we will consider t to be the relationship of modular equivalence. After laying the foundation of the theory, we will demonstrate several results, including properties of the integers under a t-relation.
The Isoperimetric Inequalities on Constant Gauss Curvature Surfaces
Xin Chen (Carleton)
We give a new proof for the isoperimetric inequalities on spheres and hyperbolic planes using metacalibration. Unlike the classical optimization approach calculus of variations, metacalibration compares competitors directly to the proposed minimizer via vector fields and the divergence theorem. It paves the way to solve open problems such as multiple bubbles and isoperimetric problems with boundary on constant Gauss curvature surfaces.
Enumerating Partitions of Generalized Stars
Robert Crandall (St. Olaf)
In graph theory, a topic of study is graph partitions of a graph G: graphs which can be obtained from G by removing vertices and/or edges. We wish to enumerate the number of partitions of certain classes of graphs. Interesting sequences arise in this study, particularly since the partitions of a path Pn on n vertices correspond to the integer partitions of n. Counting these partitions is an NP-hard problem, particularly since many partitions of G will be isomorphic. We will examine two types of generalized stars, graphs obtained by identifying the end vertices of any number of paths of specific lengths, and will present and demonstrate formulas for the number of partitions of these graphs.
The Dirichlet Process Prior in a Hierarchical Catch-Effort Model for Animal Abundance
Prasit Dhakal and Jun Young Park (Carleton)
The Dirichlet Process Prior (DPP) in Bayesian Statistics offers useful insight in studying animal abundance if heterogeneity of animal abundance has been unobserved. Consider Ni, the animal abundance in region i, follows a Poisson distribution with mean of Ai φi, where Ai is the area of region i. Then it would be one way to assume φi to come from one single distribution in the model, such as N(μφ, σφ), but the model with the DPP would not make this assumption and estimate the animal abundance better. This talk includes a brief review of previous work (without the DPP) and its comparison with the model with the DPP.
Extremal Graphs Without 4-Cycles or, Why It's Hip To Have No Squares
Frank Firke (Carleton)
Extremal graph theory deals with two main questions: what is the maximum number of edges a graph on n vertices can have before it must contain a given subgraph, and what graphs achieve that maximal condition? In this talk we will examine the question of extremal graphs when we forbid 4-cycles. While the problem remains unsolved in general, we will consider a result that answers the question for infinitely many n. The problem, while graph theoretic in nature, also has a significant connection to finite geometry that will be touched on briefly.
Classifying f-vectors of Manifolds with Boundary
Jonathan Hahn (Carleton)
A 3-dimensional manifold with boundary, such as a ball or solid torus, can be represented by sets called simplicial complexes made of faces of various dimension—points, line segments, triangles, and tetrahedrons. For a given simplicial complex, we can count the number of faces in each dimension, and encode this information in its f-vector. So far, we know all the f-vectors for very few 3-dimensional manifolds with boundary. In this talk, I will discuss how to characterize f-vectors for the simplest case, a ball.
Restricted Symmetric Signed Permutations
Andy Hardt (Carleton)
The symmetry group D4+Z2 acts on the set of signed permutations by rotations, reflections, and bar operations (flip the sign of each entry). Following Egge's work on unsigned permutations, we enumerate the signed permutations that, given a symmetry subgroup H and a set R of length-2 signed patterns, are invariant under H and avoid R. Mansour and West began this work by enumerating the signed permutations that avoid $R$, not taking symmetries into account. Dukes and Mansour continued by enumerating signed involutions that avoid R. This talk considers the remaining subgroups of D4+Z2, thus completing the enumeration. The resulting sequences include the Catalan numbers and the central binomial coefficients, and some of them are given recursively. We present some of these results in the talk, and the rest are in our accompanying paper.
Restricted Symmetric Signed Permutations Slides
Non-negativity of Generalized Immanants of Monomial Positive Matrices
Vladimir Sotirov (St. Olaf)
On one hand, a monomial positive matrix is a (square) matrix whose entries are polynomials in some number of indeterminates, satisfying the property that the determinant of every square submatrix is a polynomial in those indeterminates with non-negative coefficients (perversely, we say that the determinant is monomial positive function on the square submatrices). Monomial positive matrices arise naturally as the weight matrices of planar networks with indeterminate weights. On the other hand, generalized immanants are functions on matrices which generalize the determinant by replacing the sign function (in the expansion of the determinant as sum) with an arbitrary function on the symmetric group. Tautologically, the determinant of a monomial positive matrix is monomial positive; little is known, however, about monomial positive immanants of monomial positive matrices. By brute force computation in Maple, it is known that for up to n=5 the cone of monomial positive immanants of n x n monomial positive matrices is finitely generated (unexpectedly, the number of generators for n=5 is 121, rather than 120=5!). Since brute force becomes computationally infeasible for n=6, I will describe in this talk a more systematic approach, which reduces the computational complexity to the point where a human could not only verify in an hour or two that there are finitely many generators for n=6, but also determine in a day or two of careful computation by hand (or a few seconds of execution on a computer after a week of careful programming) exactly how many generators there actually are.
Thompson’s group F: A New Generating Set
Amelia Stonesifer (St. Olaf)
Thompson’s group F was introduced by Richard Thompson in the 1960’s in connection with questions in logic. It has since found applications in many areas of mathematics including algebra, logic and topology, and its metric properties with respect to the standard generating sets, Xn, have been studied heavily. In this talk, we introduce a new family of generating sets, which we denote as Zn, use “wave diagrams” as tools to establish a length formula for the word metric with respect to Z1 and apply the word length formula to demonstrate that F is not almost convex and F has a dead end of depth at least 1 with respect to Z1.
Thompson Group Generator Slides
A Method of Word Recognition
Reid Whitaker (Carleton)
Automated word recognition is extremely critical in a wide variety of human machine interactions. Some examples include automated voice answering systems, automated dialing, and direct voice input in airplanes. A method of word recognition was developed using Fourier and wavelet analysis to determine the error of an unknown word compared to the known words in a small library. Semi-reliable results were achieved at identifying an unknown word from a speaker not included in the library.
Triangle Puzzles and Quantum Cohomology
Erik Wyatt (St. Olaf)
Combinatorial representations of geometric objects can be used to find their intersections. We will show how to compute a variation on Littlewood-Richardson coefficients that describes all lines passing through two varieties.
