Georgia Institute of Technology College of Sciences

One-dimensional discrete-time population models, such as Logistic or Ricker growth, can exhibit periodic and chaotic dynamics. Incorporating epidemiological interactions through the addition of an infectious class causes an interesting complexity of new behaviors. Here, we examine a two-dimensional susceptible-infectious (SI) model with underlying Ricker population growth. In particular, the system with infection has a distinct bifurcation structure from the disease-free system. We use numerical bifurcation analysis to determine the influence of infection on the types and appearance of qualitatively distinct long-time dynamics. We find that disease-induced mortality leads to the appearance of multistability, such as stable four-cycles and chaos dependent upon the initial condition. Furthermore, previous work showed that infection that alters the ability to reproduce can lead to unexpected increases in total population size. A similar phenomenon is seen in some models where an increase in population size with a decreased growth rate occurs, known as the ‘hydra effect.’ Thus, we examine the appearance and extent of the hydra effect, particularly when infection is introduced during cyclic or chaotic population dynamics.

Beginning in Spring 2020, we stepped away from traditional exams and collaboratively developed the concept portfolio assessment with the aim of creating a more equitable learning experience for students. Since then, we have implemented this model of assessment in courses from Pre-Calculus through Number Theory as faculty at a community college and a small liberal arts college. For the concept portfolio, students choose a subset of the topics covered in the course and synthesize the topics by providing a summary and annotated examples. The portfolio is completed iteratively where students submit rough drafts and engage in peer review. During this talk, we will share our motivation to design an equitable alternative to exams, compare and contrast our implementations of the concept portfolio assessment, and discuss student feedback.

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The talk is delivered in a hybrid format Everyone is welcome to join via zoom
https://gatech.zoom.us/j/94287395719?pwd=U216WTlIZHdMNVErZlFWUGlleDBiQT09
but we have also reserved 005 to attend the talk all together, hoping discussion will be easier.

An important recent topic is matrix completion, which is trying to recover a matrix from a small set of observed entries, subject to particular requirements. In this talk, we discuss results on symmetric tropical and symmetric Kapranov rank 2 matrices, and establish a technique of examining the phylogenetic tree structure obtained from the tropical convex hulls of their columns to construct the algebraic matroid of symmetric tropical rank 2 $n \times n$ matrices. This matroid directly answers the question of what entries of a symmetric $n \times n$ matrix needs to be specified generically to be completable to a symmetric tropical rank 2 matrix, as well as to a symmetric classical rank 2 matrix.

This is based on joint work with Cvetelina Hill and Kisun Lee.

The math-themed show at the Atlanta Science Festival will be less elaborate than in the last few years, but we are back to apearing live on stage!  We are also hoping to arrange for live-streaming.  Mathematics in Motion will use dance and circus arts to engage the public.   (Dan and Evans and several GT students are involved, but don't worry, mathematicians won't be doing the dancing!)

There will be two shows on Sunday the 13th, begininng at 2:00 and 5:00 pm.