Oscillations and other patterns of neuronal activity arise throughout
the central nervous system. This activity has been observed in sensory
processing, motor activities, and learning, and has been implicated in
the generation of sleep rhythms, epilepsy, and parkinsonian tremor.
Mathematical models for neuronal activity often display an incredibly
rich structure of dynamic behavior. In this lecture, I describe how the
neuronal systems can be modeled, various types of activity patterns that
arise in these models, and mechanisms for how the activity patterns are
generated. In particular, I demonstrate how methods from geometric
singular perturbation theory have been used to analyze a recent model
for activity patterns in an insect's antennal lobe.

The set of multiplicative arithmetic functions over a ring R
(commutative with identity) can be given a unique functorial ring
structure for which (1) the operation of addition is Dirichlet
convolution and (2) multiplication of completely multiplicative
functions coincides with point-wise multiplication. This existence of
this ring structure can be derived from the existence of the ring of
``big'' Witt vectors, and it yields a ring structure on the set of
formal Dirichlet series that are expressible as an Euler product. The
group of additive arithmetic functions over R also has a naturally
defined ring structure, and there is a functorial ring homomorphism
from the ring of multiplicative functions to the ring of additive
functions that is an isomorphism if R is a Q-algebra. An application
is given to zeta functions of schemes of finite type over the ring
of integers.

In this talk we will present the foundation of a unified, object-oriented,
three-dimensional biomodelling environment, which allows us to integrate
multiple submodels at scales from subcellular to those of tissues and
organs [1]. Our current implementation combines a modified discrete model from
statistical mechanics, the Cellular Potts Model, with a continuum reaction
diffusion model [2] and a state automaton with well-defined conditions for
cell differentiation transitions to model genetic regulation. This
environment allows one to rapidly and compactly create computational models
of a class of complex-developmental phenomena. To illustrate model
development, we describe simulations of the simplified version of the
formation of the skeletal pattern in a growing embryonic vertebrate limb.

In the second half of the talk we will describe the first
three-dimensional
stochastic model [3,4] based on contact-mediated cell communication, for
studying myxobacteria fruiting body development. The myxobacteria under
starvation undergo several developmental stages including rippling,
streaming, jamming, aggregation, and, finally, developing mature fruiting
bodies. Combining contact signaling between cells and slime production
mechanisms, our model reproduces all different stages in a transition from
3D traffic jams to aggregates and demonstrates possible structure of cell
arrangement within the fruiting body.

Many remarkable classical questions about prime numbers have natural analogues in the context of elliptic curves. In this talk I will give a brief introduction to the theory of elliptic curves, and discuss how "higher dimensional" analogues of open questions about prime numbers appear naturally in the study of the reductions of an elliptic curve. In particular, I will discuss progress made towards the resolution of variations of a Lang-Trotter conjecture from 1976.