We give a quantitative proof that forsufficiently large N=N(c), every subset of$[N]^2$ of size at least $cN^2$ contains a square, i.e. four pointswith coordinates {(a,b),(a+d,b),(a,b+d),(a+d,b+d)}.

Carl Boyer, in the introduction of his 1949 calculus textbook,remarked that "Mathematics is as much an aspect of culture as it is acollection of algorithms." Indeed, any collection of human beings whichone might identify as having the rudiments of society -- a spokenlanguage, a sense of spirituality, a rule of law, etc. -- have also alwayshad some sense of number as a means of codifying the world around them.In this talk, we will examine how different cultures at different times havedeveloped their sense of number, space and form. We will also look therecurring roles basic tools --- such as rope, rods, and rocks --- haveplayed in their development and refinement --- be it ancient Incaspreadsheets in South America, stone altars built by rope in India, orbroken notched sticks on which the British treasury depended well into the1800s! This talk is aimed at anyone with an interest in culture,language, philosophy, history, ... or, of course, mathematics.REFRESHMENTS WILL BE SERVED!

Analogies between number fields and function fields have inspired newdevelopments in number theory for a long time. We will discuss asurprising non-analogy related to the distribution of primes. Forinstance, it is expected that any irreducible in ${mathbf Z}[t]$ havingat least two relatively prime values will take prime values infinitelyoften. (An example is $t^2+1$, while a nonexample is $t^2+t+2$, since$n^2+n+2$ is always even.) The analogue in ${mathbf F}_p[x][t]$ isfalse, e.g., $t^8+x^3$ is irreducible in ${mathbf F}_2[x][t]$ but$g(x)^8+x^3$ is reducible in ${mathbf F}_2[x]$ for every $g(x)$.

Whole-genome expression experiments provide the building blocks forunderstanding cellular control mechanisms underlying health anddisease, from cardiac function to learning and memory to oncogenesis.To put post-genomic information to work requires buildingbiophysically detailed, quantitative models for the biochemicalinteractions controlling cell behavior. Such biochemical signalingnetworks form a class of high-dimensional, stochastic nonlineardynamical systems, analytical and numerical techniques for the studyof which are only in the early stages of development. I will presentsome approaches to reducing the complexity of cellular signalingnetworks and to extending the classical deterministic approximationsfor these systems to include fluctuation effects. I will focus onapplications to biophysical systems currently under investigation at UCSD, Scripps and Salk Institute.