Georgia Institute of Technology College of Sciences

Systems biology focuses on modeling complex biological systems, such as metabolic and cell signaling networks. These biological networks are modeled with polynomial dynamical systems. Analyzing these systems at steady-state results in algebraic varieties that live in high-dimensional spaces. By understanding these varieties, we can provide insight into the behavior of the models. Furthermore, this algebro-geometric framework yields techniques for model selection and parameter estimation that can circumvent challenges such as limited or noisy data. In this talk, we will introduce biochemical reaction networks and their resulting steady-state varieties. In addition, we will discuss the questions asked by modelers and their corresponding geometric interpretation, particularly in regards to model selection and parameter estimation.

Given a plane field $dz-xdy$ in $\mathbb{R}^3$. A Legendrian knot is a knot which at every point is tangent to the plane at that point. One can similarly define a Legendrian knot in any contact 3-manifold (manifold with a plane field satisfying some conditions). In this talk, we will explore Legendrian knots in $\mathbb{R}^3$, $J^1(S^1)$, and $\#^k(S^1\times S^2)$ as well as a few Legendrian knot invariants. We will also look at the relationships between a few of these knot invariants. No knowledge of Legendrian knots will be assumed though some knowledge of basic knot theory would be useful.