We shall review recent progress obtained in the understanding of localization
properties of random Schrodinger operators. The hard issue of the Anderson
transition is stated in terms of the spreading and of the non spreading of a
wave-packet initially located at the origin. It particular it is shown that
slow transport cannot happen for ergodic random operators. As an application,
we study quantum Hall systems, that is the Hamiltonian of an electron confined
to a two dimensional plane and subjected to a constant transverse magnetic
field. We prove delocalization around each Landau level, and localization
outside a small neighborhood of these levels.

Number theory and algebraic geometry have numerous applications,
including to cryptography. Cryptography is concerned with encrypting
and decrypting secret messages. This talk will give an elementary
introduction to elliptic curve cryptography and pairing-based
cryptography, and will discuss some interesting open problems. Only
undergraduate algebra will be assumed.

Motivated by several physical systems, we study a simple model of driven, two-species lattice gases. Our system consists of only two types of NON-interacting (apart from an excluded volume constraint) particles, diffusing on a periodic lattice and with biased moves in opposite directions. On a square system, increasing the overall particle density leads to a transition - from a homogeneous phase with high particle current to one with spatial structure and minimal current. For rectangular cases, several structures can appear, with relative frequency depending on the aspect ratio of the system. Using a simple continuum theory, we are able to describe much of the novel transitions. Variations and generalizations, as well as the physical systems they model, will be discussed.

I will discuss two simple models of contagion relevant to the desciption of social and biological spreading processes.

The first model aims to unify existing models of the spread of social influences and infectious diseases. This generalized model of contagion incorporates individual memory of exposure to a contagious entity (e.g., a rumor or disease), variable magnitudes of exposure (dose sizes), and heterogeneity in the susceptibility of individuals. Through analysis and simulation, we have examined in detail the mean-field case where individuals may recover from an infection and then immediately become susceptible again. We identify three basic classes of contagion models: epidemic threshold, vanishing critical mass, and critical mass respectively. The conditions for a particular contagion model to belong to one of the these three classes depend only on memory length and the probabilities of being infected by one and two exposures respectively. (For both models, a key quantity is the fraction of vulnerables, i.e., individuals who are typically infected by one exposure.) These parameters and their elaborations are in principle measurable for real contagious influences or entities, suggesting novel measures for assessing (as well as strategies for altering) the susceptibility of a population to large contagion events. We also study the case where individuals attain permanent immunity once recovered, finding that epidemics inevitably die out but may be surprisingly persistent when individuals possess memory. I will also discuss some related work by others.

The second model describes the spreading of social influences on networks, and is a natural extension of the threshold model due to Granovetter. For this model on various kinds of random networks, analytic results are known for when cascades (epidemics) are possible. In all cases, the density of the network must belong to an intermediate range referred to as the cascade window. When links are scarce, not enough individuals are connected for global spreading to occur, and when links are overly abundant, too few individuals are vulnerable. In our recent work for this model, we have examined the role of influentials (a.k.a. opinion leaders or, for a biological feel, super-spreaders). We examine cascades after they have occurred, as is invariably done for real cascades. Contrary to much ascribed to influentials, we find that highly connected nodes are not the chief determinants of whether or not a cascade will occur. While cascade initiators are typically more connected than the average individual, the discrepancy is not pronounced. We further observe that cascades arise through a multi-step process and that for dense networks, `early adopters' may in fact be less connected than on average. Also, for dense networks, cascades rapidly take off after a long and `quiet' build up period, making them difficult to identify until after they have been realized. In sum, influentials are limited in their effect since the condition for a cascade to occur is really a global one; there must be a sufficient population of vulnerables available, and it is the most influential of these vulnerables that dictate the spread of an influence.