Portland State University Discrete Mathematics Seminar

Winter 2014: Neuberger Hall 373, Mondays, 4pm-5pm (all quarters)

January 6, 2014:

Beth Malmskog, Colorado College
Classifying curves—How and why I study cyclic trigonal curves with good reduction away from p = 3

In this talk, I discuss my ongoing work with Chris Rasmussen which seeks to classify all curves of the form y3 = f(x), where deg(f(x)) is at most 6, having good reduction at all primes except p = 3. This work is part of a very long story (I start with the Greeks) about the ongoing quest to classify algebraic curves. The story connects many areas of and interesting problems in mathematics, and our work alone draws on the study of S-unit equations, Baker’s theorem on logarithmic forms, and the LLL lattice reduction algorithm.

January 13, 2014:

Damon Hochnadel, Portland State University
Cycles, chords, and planarity in graphs

The study of connectivity and planarity of graphs has a rich history that traces back to the early days of graph theory. Much of the early motivation was provided by the desire to prove the famous 4-color theorem, a feat that was finally accomplished in 1976 by Appel and Haken. Along the way, many related questions arose concerning the structure of planar graphs, and many of these questions continue to generate interesting research. In 1982 Carsten Thomassen conjectured that every longest cycle of a 3-connected graph has a chord. In 2008, Etienne Birmelé explored this conjecture for a certain class of 3-connected graphs. This talk will examine and expand upon Birmelé’s arguments.

January 20, 2014:

No talk due to Martin Luther King, Jr. Day.

January 27, 2014:

No seminar today!

February 3, 2014:

Clayton Petsche, Oregon State University
The Dynamical Mordell-Lang Problem

The Dynamical Mordell-Lang Problem combines dynamical systems, algebraic geometry, and number theory in interesting and exciting new ways. One starts in the simple setting of a map from a algebraic variety to itself; for example, one might consider a polynomial function from Euclidean n-space to Euclidean n-space. The goal is to find a simple law or pattern governing the distribution, in the Zariski topology, of forward orbits of points with respect to this map. The problem in full generality is still very much open, but we will survey interesting partial results, and we will give a new result whose proof combines methods from ergodic theory as well as the theory of Berkovich analytic spaces.

February 10, 2014 at the special time of 4:30pm:

James Mahoney, Portland State University
How to grasp graph groups and group graphs

I'll give an introduction to automorphism groups of graphs, Cayley graphs, Frucht's theorem, and related results.

February 17, 2014:

Derek Garton, Portland State University
Heuristics in number theory

Solving Diophantine equations often requires working in rings of integers that are not unique factorization domains. Ideal class groups are objects that measure the defectiveness of such rings; that is, how badly they fail to be unique factorization domains. At the moment, individual ideal class groups are difficult to compute. However, it is possible to guess properties of families of ideal class groups by studying simple heuristic models. I will describe heuristic models for ideal class groups in various settings, and show what kind results we can prove about these models.

February 24, 2014 (this week's talk is a Maseeh Colloquium, so there is a special day, time, and place)
February 21, 2014, 2:15pm, Neuberger Hall 347:

Colin Starr, Willamette University
Prime distance graphs

A graph G is a prime distance graph (respectively, a 2-odd graph) if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference between their labels is prime (either 2 or odd). We seek to characterize prime distance graphs as well as some generalizations of them. In this talk, I will make connections between this problem and several famous problems and theorems in number theory.

March 3, 2014:

John Caughman, Portland State University
HOW LOW CAN YOU GO? (Setting the bar low for rectangle visibility graphs) Part 1

Given a finite set of rectangles in the plane, we say two rectangles are visible to each other if there is an unobstructed vertical or horizontal line of sight between their boundaries. The associated graph, whose vertices are the rectangles, and where adjacency corresponds to visibility, is called a rectangle visibility graph (or RVG). In these two talks, we survey some recent results on RVGs and discuss a new algorithm for finding minimum height RVG representations of trees.

March 10, 2014:

John Caughman, Portland State University
HOW LOW CAN YOU GO? (Setting the bar low for rectangle visibility graphs) Part 2

Given a finite set of rectangles in the plane, we say two rectangles are visible to each other if there is an unobstructed vertical or horizontal line of sight between their boundaries. The associated graph, whose vertices are the rectangles, and where adjacency corresponds to visibility, is called a rectangle visibility graph (or RVG). In these two talks, we survey some recent results on RVGs and discuss a new algorithm for finding minimum height RVG representations of trees.