Session Index
Click a session title to jump to the abstracts.
Session 1
1:00pm--2:00pm, McGowan 104
Speakers: Isaac Craig, Kevin Byrnes, Wing Hong Tony Wong
Session 2
1:00pm--2:00pm, McGowan 106
Speakers: Christopher Micklewright, Craig M. Johnson, Cesar Martínez-Garza
Session 1, McGowan 104
1:00pm, Isaac Craig (Bryn Mawr College)
Knot Traces and Sliceness
View Abstract
The 4-manifold obtained by attaching a 0-framed 2-handle to the 4-ball along a knot is called the trace of the knot. Among its many applications to low-dimensional topology, this manifold can be used to obstruct when a knot is slice (bounds a smoothly embedded disk in the 4-ball), since any knot whose trace is diffeomorphic to the trace of a slice knot is necessarily slice. As a result, we are interested in methods which construct such pairs of knots. In this talk, we will repackage one such construction used in Lisa Piccirillo's 2018 paper “The Conway knot is not slice”.
Close Abstract1:20pm, Kevin Byrnes (Wilmington, DE)
The Maximum Length of Circuit Codes with Long Bit Runs
View Abstract
A circuit code of spread $k$ is a simple cycle $C$ in the graph of the $d$-dimensional hypercube $I(d)$ with the property that for any vertices $x,y\in C$, $d_{I(d)}(x,y)\ge \min \{d_C(x,y),k\}$. One application of circuit codes is as error-correcting codes, so it is of interest to find the maximum length of a circuit code in dimension $d$ with spread $k$, $K(d,k)$. However, finding closed form expressions for $K(d,k)$ for classes of $(d,k)$ combinations is extremely rare.
In this talk, suitable for a general audience, we provide an introduction to circuit codes and related concepts (e.g. transition sequences, bit runs). We discuss a recent result that gives an exact formula for $K(d,k)$ for an infinite class of symmetric circuit codes with long bit runs and a conjecture that this formula for $K(d,k)$ holds for an even larger class of circuit codes.
Close Abstract1:40pm, Wing Hong Tony Wong (Kutztown University of Pennsylvania)
Connected Permutation Graphs with Maximum Domination Number
View Abstract
Let $\pi=(\pi(1),\pi(2),\dotsc,\pi(n))\in S_n$, and let $G_\pi$ be a graph with the vertex set $V=\{v_1,v_2,\dotsc,v_n\}$ such that there is an edge between $v_i$ and $v_j$ if and only if $i<j$ and $\pi^{-1}(i)>\pi^{-1}(j)$. Such a graph $G_\pi$ is called a permutation graph. The domination number of a graph $G$ is the minimum cardinality of a subset $D\subseteq V$ such that every vertex in $V\setminus D$ is a neighbor of a vertex in $D$. It is well known that the domination number of a connected graph on $n$ vertices is at most $\frac{n}{2}$. In this talk, we will fully characterize all connected permutation graphs on $n$ vertices with domination number equal to $\frac{n}{2}$.
Close AbstractBack to index
Session 2, McGowan 106
1:00pm, Christopher Micklewright (Phoenixville, PA)
Mathematics, Philosophy, and Tolkien
View Abstract
One of the starting questions in the philosophy of mathematics asks whether new mathematical concepts are discovered or created. In this talk, we will briefly survey the most common philosophical perspectives on mathematics, taking note that the common positions very rarely value engagement with the world around us. We will then look at J.R.R. Tolkien's idea of sub-creation. Using this idea, we will explore a new perspective on mathematics, and its implications for the question of whether mathematics is discovered or created. These ideas were developed with the support of a grant given by Bridging the Two Cultures of Science and the Humanities II, a project run by Scholarship and Christianity in Oxford, the UK subsidiary of the Council for Christian Colleges and Universities, with funding by Templeton Religion Trust and The Blankemeyer Foundation.
Close Abstract1:20pm, Craig M. Johnson (Marywood University)
Group Theory through Music
View Abstract
The use of musical inversions was one of several techniques employed by some classical composers. A strict inversion changes the key of a piece of music in a computable way. The set of distinct inversions that result from using different pivot notes displays a structure that leads to identifying it as a coset of a particular subgroup of the dihedral group of order 24. This exploration provides a nice introduction of some key concepts of group theory to undergraduates.
Close Abstract1:40pm, Cesar Martínez-Garza (Penn State Berks)
The Victim is Linearity: A Quasi-report of a Semi-investigation on the Algebraic Crimes of First Semester Calculus Students
View Abstract
Several years ago, I embarked on a task to "try to fully understand (insert `snicker' here)" and catalogue the basic algebraic and trigonometric deficiencies of students in a first semester Calculus course. Armed with an internal mini-grant and no IRB (don't tell the peer-review police!), I designed a series of semiweekly assessments and an inventory to group major mathematical offenses. My results were very surprising to me, but most probably not surprising at all to the casual observer... after two double espressos. Everything in this presentation is real.
Close AbstractBack to index