MAA  Mathematical Association of America

Seaway Section

Location: Room 211, Hubbard Hall

1. Time:

   1:10 pm – 1:30 pm

   Title:

   A Gentle Introduction to Differential Geometry: From Curves to Curvature, with Linear Algebra Along the Way

   Speaker:

   Katie Wei (William Paterson University)

   Abstract

   Differential geometry studies curved rather than flat geometric objects, using ideas from calculus and linear algebra. This talk gives an accessible introduction to the subject, assuming only a background in calculus and linear algebra. We begin with parametrised curves $\gamma(t)$ and develop the Frenet frame $\{T,N,B\}$, curvature $\kappa$, and torsion $\tau$, including a worked computation for a circular helix. We then move to parametrised surfaces $\sigma(u,v)$ and view each tangent plane as an inner product space. The first fundamental form is represented by the Gram matrix \[ I=\begin{pmatrix}E&F\\F&G\end{pmatrix}, \] which encodes lengths and angles on the surface. The central object is the shape operator $S$, a self-adjoint linear map whose eigenvalues $\kappa_1,\kappa_2$ are the principal curvatures. From this viewpoint, Gaussian curvature and mean curvature are \[ K=\det(S)=\kappa_1\kappa_2,\qquad H=\frac{1}{2}\operatorname{tr}(S)=\frac{\kappa_1+\kappa_2}{2}. \] We work through a curvature computation for a paraboloid and discuss Gauss's Theorema Egregium, which shows that $K$ is intrinsic. Finally, we connect $H=0$ to the minimal-surface equation and briefly discuss applications to map projections, graphics, robotics, and general relativity.

2. Time:

   1:40 pm – 2:10 pm

   Title:

   Explicit computations of subspace arrangement homology and applications to graphs.

   Speaker:

   Jack Lamberg (Binghamton University)

   Abstract

   An arrangement is a collection of vector subspaces of a vector space. Day-Wade introduced subspace arrangement homology as a tool to understand the intersections within the arrangement, defining a list of Betti numbers. We created a simulation software, subspace arrangement Betti calculator, SAB-C, which can exactly calculate the Betti numbers of any finite arrangement and its dual arrangement. We used this software to study arrangements associated with graphs to understand what graphical features are encoded in these Betti numbers. This led us to conjecture and prove that if $G$ is a biconnected chordal graph, then the first Betti number of the dual arrangement induced by the minimal separators is exactly the number of simplicial vertices of $G$.

3. Time:

   2:20 pm – 2:40 pm

   Title:

   Differentiating Integers: Arithmetic Derivatives and Their Dynamics

   Speaker:

   Gavin Magill (Utica University)

   Abstract

   The arithmetic derivative of a number is the map that sends every prime number to 1 and satisfies the Leibnitz rule (the product rule). In this talk we will discuss the basic definition and properties of the arithmetic derivative and their connections with the Goldbach's conjecture. The main focus of the talk is a conjecture by Ufnarovski and Åhlander regarding the long term behavior of the sequence of higher derivatives. We will present a partial answer to the conjecture by Emmons and Xiao, and a result of ours that potentially gives a more general solution to the conjecture.

Location: Room 212, Hubbard Hall

Location: Room 213, Hubbard Hall