MAA  Mathematical Association of America

Seaway Section

Location: Room 211, Hubbard Hall

1. Time:

   1:20 pm – 1:40 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:

   1:50 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 $G$.

Location: Room 212, Hubbard Hall

1. Time:

   1:20 pm – 1:40 pm

   Title:

   Abacus

   Speaker:

   Ashim Pandey (Utica University)

   Abstract

   Long before digital calculators transformed modern computing, the abacus served as humanity’s primary instrument for numerical calculation. Originating thousands of years ago in ancient Mesopotamia and evolving through Chinese and Japanese designs, the abacus relies on a systematic, bead-based place-value layout. Calculations are executed through a series of foundational visual formulas that map directly to place value, allowing users to perform addition, subtraction, multiplication, and division by manipulating physical beads. Beyond its historical role in trade and commerce, the modern abacus evolved into a powerful pedagogical tool for mental arithmetic. In Nepal, this methodology was formally structured and taught by the United Calculation Mental Arithmetic System. Through a rigorous 10-semester curriculum delivered via weekend modules, students build abstract spatial visualization, transitioning from tactile bead movements to rapid mental calculations. This enduring cognitive framework highlights the abacus not merely as an ancient relic, but as a timeless foundation for mental agility and mathematical thinking.

2. Time:

   1:50 pm – 2:10 pm

   Title:

   An Agent-Based Model of C. difficile Transmission Between Hospitals and Communities

   Speakers:

   Nora Hart (Amherst College), Hayden Curfman (Williams College)

   Abstract

   $\textit{Clostridioides difficile}$ ($\textit{C. difficile}$) has traditionally been viewed as a nosocomial infection, but community-associated infections have increased in recent years. We developed an agent-based model using NetLogo to investigate $\textit{C. difficile}$ transmission between a hospital and its local community and identify major sources of colonization. The model represents 10,000 individuals moving among a hospital, houses, workplaces, schools, and public restrooms. Individuals can move between four disease classes: resistant, susceptible, colonized (asymptomatic), and diseased (symptomatic). To obtain baseline results, we ran 100 one-year simulations in our community. Then, we implemented various mitigation strategies to see which most decreased incidence of colonization and disease. Preliminary results show that decreasing hospital length of stay is the most effective mitigation strategy for reducing incidence of colonization and disease in both the community and the hospital. Within the hospital, contaminated environmental surfaces are the primary source of colonization, while workplace interactions are the most common cause in the community. Additionally, the mean time for $\textit{C. difficile}$ to infiltrate the community from a contaminated hospital is approximately 12 times longer than the mean time for the reverse. Our model demonstrates the potential for hospital-based interventions to reduce the incidence of $\textit{C. difficile}$ beyond the hospital setting and highlights the importance of effective cleaning protocols and adherence to these protocols.

3. Time:

   2:20 pm – 2:40 pm

   Title:

   Clothesline Mathematics (NEWLY ADDED TALK!)

   Speaker:

   Justin Beatty (Binghamton University)

   Abstract

   In the late 1950s, Samuel Eilenberg, working in module theory, and Barry Mazur, working in geometric topology, independently proved unrelated theorems using strikingly similar techniques—methods that, at first glance, seem almost too good to be true. Yet, despite their apparent mathematical sleight of hand, both proofs are entirely rigorous. In this talk, I will explore what has come to be known as the "Eilenberg-Mazur Swindle" and examine related 'swindles' that cleverly exploit infinite series in seemingly inappropriate ways.