This thesis explores the Combinatorial Polynomial Hirsch Conjecture.

]]>A partial answer comes through place cells, hippocampal neurons which

become associated to approximately convex regions of the world known

as their place fields. When an organism is in the place field of some place

cell, that cell will fire at an increased rate. A neural code describes the set

of firing patterns observed in a set of neurons in terms of which subsets

fire together and which do not. If the neurons the code describes are place

cells, then the neural code gives some information about the relationships

between the place fields–for instance, two place fields intersect if and only if

their associated place cells fire together. Since place fields are convex, we are

interested in determining which neural codes can be realized with convex

sets and in finding convex sets which generate a given neural code when

taken as place fields. To this end, we study algebraic invariants associated

to neural codes, such as neural ideals and toric ideals. We work with a

special class of convex codes, known as inductively pierced codes, and seek

to identify these codes through the Gröbner bases of their toric ideals.

]]>]]>In the setting of tropical mathematics, geometric objects are rich with inherent combinatorial structure. For example, each polynomial $p(x,y)$ in the tropical setting corresponds to a tropical curve; these tropical curves correspond to unbounded graphs embedded in $\R^2$. Each of these graphs is dual to a particular subdivision of its Newton polytope; we classify tropical curves by combinatorial type based on these corresponding subdivisions. In this thesis, we aim to gain an understanding of the likeliness of the combinatorial type of a randomly chosen tropical curve by using methods from polytope geometry. We focus on tropical curves corresponding to quadratics, but we hope to expand our exploration to higher degree polynomials.