Georgia Institute of Technology College of Sciences

Suppose f is a Laurent polynomial in n variables with degree d, exactly (n+2) monomial terms, and all its coefficients in {-H,...,H} for some positive integer H. Suppose further that the exponent vectors of f do not all lie in an affine hyperplane: Such a set of exponent vectors is referred to as a circuit. We prove that the positive zero set of f is isotopic to the real zero set of an explicit n-variate quadric q, and give a fast algorithm to explicitly compute q: The bit complexity is (log(dH))^O(n). The best previous bit-complexity bounds were of the form (dlog(H))^{\Omega(n)} (to compute a data structure called a roadmap). Our results also extend to real zero sets of n-variate exponential sums over circuits. Finally, we discuss how to approach the next case up: n-variate polynomials with exactly (n+3) terms.

Say you’ve got an (orientable) surface S and you want to do geometry with it. Well, the complex plane C has dimension 2, so you might as well try to model S on C and see what happens. The objects you get from following this thought are called complex structures. It turns out that most surfaces have a rich but manageable amount of genuinely different complex structures. I’ll focus in this talk on how to think about comparing and deforming complex structures on S. I’ll explain the remarkable result that there are highly structured “best” maps between (marked) complex structures, and how this can be used to show the right space of complex structures on S is a finite-dimensional ball. This is known as Teichmüller’s theorem, and I’ll be following Bers’ proof.

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.

The study of contact and symplectic manifolds has relied heavily on understanding Legendrian and Lagrangian submanifolds in them -- both for constructing the manifolds using these submanifolds, and for computing invariants of the ambient space in terms of invariants of the submanifolds. This thesis explores the construction of Legendrian submanifolds in high dimensional contact manifolds (greater than 3) in two directions. In one, using open book decompositions, we generalise a doubling construction defined by Ekholm and show that the Legendrians obtained are trivial. In the second, which is joint work in progress with Hughes, we use the doubling and twist spun constructions to build a large family of Legendrians, compute their sheaf-theoretic invariants to distinguish them using techniques of Casals-Zaslow, and study their exact Lagrangian fillability properties.

Zoom link:

https://gatech.zoom.us/j/93109756512?pwd=Skljb0tVdjZVNEUvSm9tNnFHZFREUT09