Georgia Institute of Technology College of Sciences

We look at two combinatorial problems which can be solvedusing careful estimates for the distribution of smooth numbers. Thefirst is the Ramsey-theoretic problem to determine the maximal size ofa subset of of integers containing no 3-term geometric progressions.This problem was first considered by Rankin, who constructed such asubset with density about 0.719. By considering progressions among thesmooth numbers, we demonstrate a method to effectively compute thegreatest possible upper density of a geometric-progression-free set.Second, we consider the problem of determining which prime numberoccurs most frequently as the largest prime divisor on the interval[2,x], as well as the set prime numbers which ever have this propertyfor some value of x, a problem closely related to the analysis offactoring algorithms.

Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.

Diabetes is a disease of poor glucose control. Glucose is controlled by two hormones that work in opposite directions: insulin and glucagon. Pancreatic beta-cells release insulin when blood glucose is high, while pancreatic alpha-cells secrete glucagon when blood glucose is low. Both insulin and glucagon secretion are disregulated in people with diabetes. In these people, not enough insulin is secreted in response to elevated glucose levels, while the problem with glucagon secretion is two-fold: too much glucagon is secreted at high glucose levels, while not enough is secreted at low glucose levels. So far, the treatment of diabetes has focused solely on increasing insulin secretion from beta-cells. Therefore, understanding glucose regulated glucagon secretion may lead to new therapies for those with diabetes.There is an ongoing debate as to whether glucose suppresses glucagon secretion directly through an intrinsic mechanism, within the alpha-cell, or indirectly through an extrinsic mechanism. I developed a mathematical model of glucagon secretion in alpha-cells and use it to show that they can control their own secretion. However, experimental evidence shows that factors secreted by pancreatic beta- and delta- cells can also affect glucagon secretion. Therefore, I created the BAD model for pancreatic islets which contains one representative cell of each type and the cellular interactions between them. I use this model to show that these paracrine effects suppress alpha-cell heterogeneity and suggest that delta-cells play a more important role in this than beta-cells.