Georgia Institute of Technology College of Sciences

How densely can one pack spheres in $d$-dimensional space? It is not too hard to show a lower bound of $2^{-d}$. (The only known upper bounds are exponentially larger.) Various proofs of lower bounds of the form $cd2^{-d}$ have been given; recently, Campos, Jenssen, Michelen, and Sahasrabudhe gave the first asymptotic improvement on such bounds in 75 years. I will discuss an extension of this improvement to packing other shapes in high dimensions, along with some connections to log-concave probability.

We present a lemma, inspired by dependent random choice and sampling procedures from statistical physics, for finding dense structure in arbitrary $d$-partite $d$-uniform hypergraphs. We will then discuss how this lemma leads to the concept of local rank, a notion of tensor rank which is instrumental in proving a "structure vs. randomness" result for tensors (and by extension, polynomials): namely, a relation between the partition and analytic ranks of tensors over finite fields. This is joint work with Guy Moshkovitz.

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.