A key phenomenon in the study of cell-to-cell communication and
protein regulation is the all-or-none, ultrasensitive dose response, which
transforms a continuous input into a digital output. Multisite systems are
often used in conjunction with allosteric effects to create such a behavior.
In this talk I describe a non-allosteric mechanism for a multisite system to
present strongly ultrasensitive behavior. Applications are given to protein
activation through multisite phosphorylation, clusters of receptors and DNA
regulation through histone modifications.

In this lecture, we will talk about a recent joint
work of Gordon Heier and myself about curvature characterizations
of uniruledness and rational connectivity of projective manifolds. A
result on projective manifolds with zero total scalar curvature will
also be discussed.

Our research focus on the mathematical modeling and numerical simulations of vascular tumor growth and chemotherapy. We have developed a model by coupling a discrete angiogenesis model and a continuous tumor growth model accounting for vascular and interstitial fluid dynamics (vascular flow, vascular fluid extravasation, interstitial fluid flow and lymphatic drainage), which affects the delivery of nutrients and therapeutical agents during tumor growth/treatment. The talk will discuss over (1) overview of vascular tumor growth, (2) the tumor vascular/interstitial pressure/flow and the related physiological factors, (3) traditional drug transportation, chemotherapy and the physical barrier therein (4) the novel treatment strategy.

We investigate a series of related problems in the area of incomplete Weil sums where the sum is run over a set of points that produces the image of the polynomial. We establish a bound for such sums, and establish some numerical evidence for a conjecture that this sum can be bounded in a way similar to Weil's bounding theorem.

To aide in the average case, we investigate the problem of the cardinality of the value set of a positive degree polynomial (degree $d > 0$) over a finite field with $p^m$ elements. We show a connection between this cardinality and the number of points on a family of varieties in affine space. We couple this with Lauder and Wan's $p$-adic point counting algorithm, resulting in a non-trivial algorithm for calculating this cardinality in the instance that $p$ is sufficiently small.

What does it mean for a collection to be finite? On the one hand, we have our preschool notion that a collection is finite when can be counted with natural numbers in a way that terminates. On the other hand, there is a definition due to Dedekind that a set is finite if and only if it cannot be put in one-to-one correspondence with a proper subset. Intuitively these two notions should be equivalent, but can we prove it? I will argue that to avoid a circular argument, one direction requires more care than one would initially think. Further, the other direction is true only by virtue of the Axiom of Choice. To outline the proof of this fact, we will examine formal notions of definiabilty and the set-theoretic technique of forcing.