In this talk I will discuss how physics concepts can be useful for understanding issues arising in the field of computational complexity, the study of the amount of computational resources needed to solve different problems. In particular, I will present a renormalization group construction similar to those used in studies of phase transitions that may be useful in distinguishing computational problems that can and cannot be solved efficiently.

Since the nineteenth century, conformal mapping has been used to solve Laplace's equation by exploiting the connection between harmonic and analytic functions. In this talk, we note that another special property of Laplace's equation -- its conformal invariance -- is not unique, but rather is shared by certain systems of nonlinear equations, whose solutions have nothing to do with analytic functions. This simple observation leads to some unexpected applications of conformal mapping in physics. For example, it generates a multitude of exact solutions to the Navier-Stokes equations of fluid mechanics and the Nernst-Planck equations of electrochemical transport. It also allows conformal-map dynamics for continuous and stochastic Laplacian growth (in models of viscous fingering and diffusion-limited aggregation, respectively) to be extended to a variety non-Laplacian growth phenomena on flat and curved surfaces. This formulation enables computer simulations of otherwise intractable problems, such as advection-diffusion-limited aggregation of up to 100,000 particles in a fluid flow (evolving with the aggregate). It also provides analytical insight into the average shape of fractal clusters grown by multiple, competing transport processes.

This talk will review the general theory of nonlinear elasticity including definitions of deformations, strains, and stress tensor and how symmetries are implemented in elastic energies. It will then consider two examples in which nonlinear rather than the more familiar linearized elasticity is essential: nonlinear response of biopolymer networks and liquid crystalline elastomers. The latter systems exhibit transitions from microscopically isotropic rubber phases to ones exhibiting varying degrees of liqiud crystalline order. In their idealized form, these transitions break continuous symmetries with associated Goldstone modes whose manifestation is the vanishing of certain shear elastic constants and soft non-linear stress-strain relations.

We consider the gasless solid fuel combustion model of the Self-Propagating High-Temperature Synthesis process in which combustion waves are employed to synthesize desired materials. Specifically, we consider the combustion of a solid sample in which combustion occurs on the surface of a cylinder of radius R. We consider solution behavior as R is increased. If R is sufficiently small, planar pulsating waves are observed. As R is increased transitions to more complex, nonplanar waves occur. The study of different wave types is important since the mode of propagation determines the structure of the product. We describe a variety of different waves, including (i) spin waves, (ii) counterpropagating (CP) waves of various types, (iii) alternating spin CP waves (ASCP), (iv) modulated spin waves, (v) bound states of asymmetric spin waves, (vi) modulated asymmetric spin waves, (vii) asymmetric ASCP waves, and others.

While materials are most commonly thought of as solids, liquids, or gasses, a tremendous variety of everyday materials (biological materials, consumer care products, foods, etc.) elude such easy classification. Rather, they fall somewhere in between -- e.g. solids on short time scales and fluids on long time scales. Over many decades, techniques in rheology have been developed to study how such materials deform and flow. Conventional rheology is 'macroscopic', in the sense that it requires milliliter quantities for analysis. Many materials, however, would be too difficult, too expensive, or impossible to procure in the amounts required for such (macro-) rheometry. In the past decade, "microrheology" has been developed to study such materials. Rather than externally forcing a macroscopic quantity of the material, small colloidal beads are introduced and driven into (Brownian) motion by thermal forces. Because the material remains in (or close to) equilibrium, the (frequency-dependent) linear-response properties of the material can be obtained from the fluctuating probe motion using the fluctuation- dissipation theorem. This, however, suggests another limit to microrheology -- nonlinear material properties (shear thickening or thinning, yield stresses, and so on) can not be obtained using conventional techniques. Here we will discuss recent experiments in which the colloidal probe is actively driven through the material in order to probe its nonlinear response. We will address various theoretical issues in such studies -- most crucially, what exactly is being measured, and how might these measurements be interpreted to give the material information one desires?

I would principally focus on (1) the non-rectilinear path of ascent of bubbles, proposing an explanation deduced from computational results, and (2) the loss of axial symmetry of a long (Taylor) bubble rising in a vertical tube.

What is `Turbulence'? Is it the random interaction of `eddies'? Or the breakup of `big whorls into little whorls then little whorls into lesser whorls and so on to viscosity'? 40 years of observations of turbulent shear flows have revealed the presence and importance of coherent structures in the near wall regions. Those observations have led to a theory of a fundamental self-sustaining process in shear flows and the discovery of `exact coherent structures'. The latter are steady, traveling wave and periodic solutions of the Navier-Stokes equations. These solutions are unstable, yet they capture the statistics and structures of turbulent shear flows remarkably well, and force us to rethink our conventional view of turbulence.

Almost 100 years ago, it was postulated that the motor cortex should be viewed as a synthetic organ for complex motor actions such that elementary movements represented by individual motor cortical neurons could be combined in an almost infinite number of ways to generate the rich variety of complex motor actions that are ubiquitous in every day life. This view implies that motor cortex constitutes a sort of language of motor actions where individual motor cortical neurons encode the movement primitives of the language and the manner in which these neurons combine their activities to generate more complex motor actions constitute the grammar of action. Here we show that single motor cortical neurons encode time-dependent movement trajectories and not simply time-independent movement parameters. Moreover, we demonstrate that these movement trajectories can be combined using a simple addition rule when neurons fire simultaneously but independently. Finally, neurons that engage in significant synchronization combine their movement primitives through addition but with an additional gain factor which suggests that a functional role for synchronization may be to increase tuning sensitivity. Our findings are particularly topical given the recent excitement that has been generated by several studies demonstrating that electrical stimulation of motor cortex can elicit complex, time-evolving movements even at the single neuron level (Graziano et al., 2002; Brecht et al, 2004).

Ultracold quantum degenerate Fermi gases provide a remarkable opportunity to study strongly interacting fermions. In contrast to other Fermi systems, such as superconductors, neutron stars or the quark-gluon plasma of the early Universe, these gases have low densities and their interactions can be precisely controlled over an enormous range. A major goal has been the realization of superfluidity in a gas of fermions. Our observation of vortex lattices in a strongly interacting rotating Fermi gas provides definitive evidence for superfluidity. By varying the binding energy between fermion pairs, we have studied the crossover from a Bose-Einstein condensate of molecules to a Bardeen-Cooper-Schrieffer superfluid of loosely bound pairs. The crossover is associated with a new form of superfluidity. The observed transition temperatures normalized for the density of the gas by far exceed the highest transition temperatures achieved in high-T_c superconductors. We have extended those studies to interacting Fermi gases with imbalanced spin populations and observed a quantum phase transition at a critical imbalance, which is the Pauli limit of superfluidity.

Anderson's discovery of localization of waves in disordered media has profoundly affected our thinking about glasses. Glasses are by definition disordered solids, and they differ qualitatively from ordered solids. They have many more low-energy excitations and their relaxation after a disturbance extends over much longer times. The origin of these anomalies has naturally been thought to lie in their disorder. However, recent work at U of C and elsewhere suggests another origin. In addition to being disordered, glasses share another property: self-arrested mobility. The configuration is defined not by equilibrium Boltzmann probabilities or by an ad-hoc ensemble, but by succession of fluid states terminating in a jammed state where fluidity is lost. The randomness is thus defined kinetically rather than by an explicit assignment of probabilities. The extreme limit of kinetic jamming samples only a tiny subset of the jammed configurations. Only configurations immediately adjacent to fluid configurations are accessible.

This talk reviews recent progress in understanding marginally jammed solid as described in the work of Wyart, Silbert, Nagel and Witten. This work considers a system of hard spheres gradually compressed to a state of solidity. We argue that the marginally solid or jammed state is isostatic. The isostatic property is confirmed numerically. We account for several robust scaling behaviors discovered by Nagel and collaborators describing the effect of slight compression above the marginally jammed threshold. We characterize the low-frequency vibrational modes, account for the excess of near-contacts in the pair correlation function, and predict the growth of shear modulus with compression. We speculate on how the coupling of these modes might account for annealing and relaxation in glasses.

Highly correlated brain dynamics produces synchronized states with no behavioral value, while weakly correlated dynamics prevents information flow. In this talk we ask on why the brain should be critical, arguing in favor of the idea that the working brain stays at an intermediate (critical) regime characterized by power-law correlations. We discuss recent results describing the traffic between brain regions that continuously creates and reshapes complex functional networks of correlated dynamics (Phys. Rev. Lett. 94, 018102, 2005; Physica A 340, 756, 2004).

1. Liquid and gas interactions often contain bubble interactions that often include liquid films. Simulation of those liquid films is challenging since liquid films quickly become thinner than the grid resolution, which leads to premature bursting or merging of the bubbles. We prevent this thinning process to make thin film last while bubbles are interacting, obtaining a realistic animation of bubble interactions. To prevent thinning, we apply a disjoining force designed to slow down the thinning process. However, since bubbles stay longer without bursting or merging, the volume loss of each bubble is noticeable. To solve this problem, we modify the pressure projection to produce a velocity field whose divergence is controlled by the proportional and integral feedback. This allows us to preserve the volume or, if desired, to inflate or deflate the bubbles. In addition to premature bursting and volume change, another difficulty is the complicated liquid surface, which increases memory and computational costs. To reduce storage requirement, we collocate the velocity and pressure to simplify the octree mesh. To reduce the computational complexity of the pressure projection, we use a multigrid method.

2. Back and Forth Error Compensation and Correction (BFECC) was recently developed for interface computation using a level set method. We show that BFECC can be applied to reduce dissipation and diffusion encountered in a variety of advection steps, such as velocity, smoke density, and image advections on uniform and adaptive grids and on a triangulated surface. BFECC can be implemented trivially as a small modification of the first-order upwind or semi-Lagrangian integration of advection equations. It provides second-order accuracy in both space and time. When applied to level set evolution, BFECC reduces volume loss significantly. We demonstrate the benefits of this approach on image advection and on the simulation of smoke, bubbles in water, and the highly dynamic interaction between water, a solid, and air. We also apply BFECC to dye advection to visualize vector fields.

Statistical mechanics models in two dimensions and their geometrical properties at a critical point can be represented by conformally invariant random scaling curves, which are examples of the Stochastic Loewner Evolution (SLE). The same models and curves can be studied on randomly fluctuating lattices, i.e., in presence of quantum gravity. I will describe the relation between the two approaches and its application to SLE. Transmutation properties of SLEs follow from it. Fine geometrical properties of systems of random scaling curves can be obtained in this way, like those of Brownian or self-avoiding paths.

A small concentration of polymers in a turbulent channel flow can reduce the drag significantly. A physical explanation of this phenomenon and of the revelant experimental results has been a significant challenge for more than 50 years. In this talk, after a short introduction to the problem, we describe a theoretical framework to understand the phenomenon and to predict qualitatively and quantitatively the experimental data.

We present the results of Direct Numerical Simulations (DNS) of turbulent flows seeded with millions of passive inertial particles. The maximum Reynolds number is $Re_{\lambda } \sim 200$. We study the case of particles much heavier than the carrier flows in the limit when the Stokes drag force dominates their dynamical evolution. We study both the transient and the stationary regimes. In the transient regime, we study the growth of inhomogeneities in the particle spatial distribution driven by the preferential concentration out of intense vortex filaments. In the stationary regime, we study the acceleration fluctuations as a function of the Stokes number in the range $St \in [0.16:3]$. We also compare our results with those of pure fluid tracers ($St=0$) and we show the almost singular behaviour of inertia for small Stokes values. Starting from the pure monodispersed statistics we also present a first attempt to characterized polydispersed suspensions with a given mean Stokes, $\overline{St}$.

Materials without crystalline order undergo plastic deformation in ways both similar and fundamentally different from their crystalline counterparts. In this talk I will review the theoretical models typically used to model the micromechanics of deformation in this class of solids. I will then discuss the ways in which molecular simulations do and do not agree with these theories.

In particular, molecular dynamics simulations of a number of amorphous systems analogous to metallic glasses reveal the structural changes that accompany plastic deformation and localization involve a decrease in the local short range ordering. We have simulated both two-dimensional and three-dimensional systems in nanoindentation [1], uniaxial tension [2] and compression [3] in plane strain. The degree of strain localization depends sensitively on the quench rate during sample preparation, with localization only arising in more gradually quenched samples. Careful analysis of the strain rate dependence of the localization allows us to extrapolate to the low strain rate limit. This analysis reveals a transition from localized flow to homogeneous flow at a critical value of the potential energy per atom prior to testing. This transition occurs in both two- and three- dimensional systems. The transition appears to be associated with the k-core percolation of short range order (SRO) in the two-dimensional system [2]. A generalization of the Frank-Kasper criterion permits the identification of SRO in the three-dimensional systems. Only in certain systems does this method predict a percolation transition corresponding to the transition in mechanical behavior [3]. I will discuss the non-uniqueness of this measure of SRO, and consider whether a more rigorous definition could be derived which applies to systems far from the hard-sphere limit. Recent results regarding the dynamics of shear localization will be discussed if time permits.

[1] Y. Shi and M.L. Falk, "Structural transformation and localization during simulated nanoindentation of a non-crystalline metal film," Applied Physics Letters, Vol. 86, pp. 011914 (2005).

[2] Y. Shi and M.L. Falk, "Strain localization and percolation of stable structure in amorphous solids," Physical Review Letters, Vol. 95, pp. 095502 (2005).

[3] Y. Shi and M.L. Falk, "Atomic-scale simulations of strain localization in three-dimensional model amorphous solids," Physical Review B, in press.

Combinatorial games, which include chess, Go, checkers, Nim, Chomp, and dots-and-boxes, have both captivated and challenged mathematicians, computer scientists, and players alike. In this talk I will report on a new physics-inspired approach that reveals surprising connections between combinatorial games and key ideas from physics and nonlinear dynamics, most notably notions of scaling, renormalization, universality, and chaotic attractors. Using the game of Chomp as a prototype (which is one of the simplest of the "unsolved" combinatorial games), we find that there is an invariant geometric structure underlying the game that "grows" (reminiscent of crystal growth), and show how this growth may be analyzed using a renormalization technique. This in turn allows one to calculate detailed, probabilistic properties of the winning and losing positions of the game, answers longstanding questions that have appeared in the literature, and suggests a natural pathway toward a new class of algorithms for combinatorial games.

Particles suspended in a complex fluid can be subject to a variety of interactions including electromagnetic, elastic, entropic, and interfacial forces. The study of these forces has provided understanding of fundamental problems in fluid physics as well as exciting avenues for applications. For example, nematic liquid crystals are complex fluids possessing anisotropy that introduces both forces and torques on a suspended particle. Nematics, which are comprised of rod-like molecules, are characterized by an alignment of the long axes of the molecules. Anchoring of the alignment direction at a suspended particle's surface introduces boundary conditions on the alignment that obligate distortions with a corresponding cost in elastic energy. The tendency to minimize this energy leads to forces on the particles. This talk will focus on our studies to measure quantitatively such forces on highly anisotropic, wire-shaped particles and to control these forces in order to manipulate the particles. The talk will also describe recent experiments investigating the dynamical behavior of the wire-shaped colloids in strongly non-Newtonian surfactant solutions that display shear-induced nematic order.

Small particles floating on a turbulent fluid behave very differently than those that are neutrally buoyant. The floaters sample the velocity field at the surface but cannot follow local vertical velocity fluctuations in and out of the bulk. For that reason they form a compressible system and are seen to coagulate into string-like structures. The rate of change of the entropy of the floaters dS/dt is measured and compared with theoretical predictions and computer simulations. This rate is a random variable that takes on positive and negative values corresponding to coagulation and dispersion. Its measurement permits a test of the Fluctuation Relation of Gallavotti and Cohen.

It is well known that small neutral particles normally tend to aggregate due to the van der Waals forces. We discover a new universal long-range interaction between solid objects in polymer media that is directly opposite the van der Waals attraction. The new force could reverse the sign of the net interaction, leading to the net repulsion. This universal repulsion comes from the subtracted soft fluctuation modes, which are not present in the real polymer system, but rather are in its ideal counterpart. The predicted effect has a deep relation to the classical Casimir interactions, providing an unusual example of fluctuation-induced repulsion instead of normal attraction. That is why it is referred to as the Anti-Casimir effect. We also find that the correlation function of monomer units in a concentrated solution of long polymer chains follows a power-law rather than an exponential decay at large distances.

First I shall review spreading phenomena in statistical physics - percolation, directed percolation, sign percolation... - and discuss the types of transitions observed in these systems. Then I'll show that similar phenomena occur in knowledge networks and propagation algorithms.

Viruses present one of the most elegant examples of spontaneous self-assembly provided by nature. They can be produced both in the lab and in the living cells, with the highest degree of monodispersity. This aspect alone brings crucial amount of interest in virus self-assembly. In this talk we address electrostatic interactions inside RNA viruses, and, in particular, interactions between genome and protein tails. By mapping the problem onto complexation between charged polymers, we can predict properties such as spacial organization of the genome and constraints on the genome length. For example, it is a common belief that the genomic code uniquely determines protein sequence and the ultimate biological structures. We will argue that the reverse relation may also be true, and the virus genome is constrained in length to minimize electrostatic energy. This is found to be particularly true in wild type viruses that have undergone long periods of mutations.

It is known since Bachelier 1900 that price changes are nearly uncorrelated, leading to a random-walk like behaviour of prices. However, compared to the simplest Brownian motion, price statistics reveal a large number of anomalies, such as fat tails and long memory in the volatility. The detailed study of trade by trade and order book data allows one to provide evidence for a subtle compensation mechanism that underlies the `random' nature of price changes. This compensation drives the market close to a critical point, which may explain the sensitivity of financial markets to small perturbations, and their propensity to enter bubbles and crashes. We argue that the resulting unpredictability of price changes is quite far from the neo-classical view that markets are informationally efficient.

Why don't glasses flow? An old explanation, yet unproven experimentally, is that the dynamics become sluggish in supercooled liquids, dense colloids, and granular assemblies because increasingly larger regions of the material have to move simultaneously to allow flow. We review recent physical arguments, theoretical models and experimental tools that suggest the existence of an underlying cooperative length that grows upon approaching the glass transition.

We will summarize recent progress on the distribution of eigenvalues (or singular values) of random correlation matrices, and their relevance to portfolio optimisation and econometric forecast. Some new results will be presented, in particular concerning the case of generic (rectangular) correlation matrices, and the statistics of the top eigenvalue in the presence of heavy tailed noise.

I got my Ph.D. for simulations of granular materials at U. Chicago, and yet now I am the Staff Director of the Research Subcommittee of the Committee on Science in the U.S. House of Representatives. Issues under the purview of the Research Subcommittee include oversight of the National Science Foundation, nanotechnology, information technology research, including cybersecurity, homeland security research, and math and science education at all levels. The talk will cover why I am no longer a practicing scientist, how I made the transition, and what it is like working on science policy in Washington, including what kinds of jobs ex-scientists have in DC. I will also touch on how Congress works and how it affects agencies of interest to materials researchers (such as NSF). I can also talk about the status of relevant bills and funding of science agencies and take questions on specific policy issues.

Biological regulatory networks are capable of sophisticated functions, such as integrating chemical signals, storing memories of previous molecular events, and keeping time. It is well-known that many regulatory proteins in these circuits form dimers and higher-order complexes. In my talk, I will discuss two important consequences of dimerization. First, ample experimental evidence suggests that protein subunits in vivo can degrade less rapidly when associated in complexes. This effect leads to a concentration dependence in the protein degradation rate, and our theoretical work demonstrates how this effect can enhance the function of bistable and oscillatory circuits. Second, active proteins can often be sequestered into inactive complexes. This "molecular titration" can lead to strong nonlinearities, and suggests a scenario for the rapid evolution of bistable or oscillatory circuits in nature.

There have been several attempts to construct statistical mechanical descriptions of athermal particulate systems, for example the Edward's entropy description for granular materials. Tapping experiments by Nowak, et al. (Phys. Rev. E, 57 (1998) 1971) and numerical simulations of granular shear flow by Makse, et al. (Nature 415 (2002) 614) have shown that the Edward's description may be promising for slow, dense granular flows. An essential assumption of the Edward's framework is that all jammed configurations are equally likely. However, this assumption has not been explicitly tested. Using numerical simulations, we create jammed hard particle packings using two experimentally relevant protocols: 1) a compression and decompression scheme and 2) a quasi-static shear flow. For both methods, we find that jammed packings are not equally likely, and thus we argue that the Edward's entropy description for granular materials should be reconsidered.

In traditional interpretations of quantum mechanics, the Born probability rule was introduced by fiat at an explicit or implicit 'collapse' process, outside the known dynamical equations. Attempts to formulate such collapse processes explicitly have not produced appealing theories. In recent years, many-worlds interpretations which avoid such processes have become popular. I present old arguments that the Born probability rule is unnatural within standard many-worlds accounts, which would appear to lead to very unfamiliar probabilities. I then present an idea by Hanson, which might possibly account for Born probabilities as actual ratios of numbers of experiencable worlds within standard quantum dynamics. In view of difficulties with that idea, I show that relatively palatable non-linear modifications of quantum dynamics could give rise to the proper ratios of numbers of outcomes without any fine-tuning.

The local structure of biological networks can be studied by "counting" network motifs. Although the number of possible motifs with 8 or more nodes becomes overwhelming, it is possible to study large-size network patterns if we focus on specific motifs such as cycles. Cycles are interesting because in directed graphs they can form feedback and feed-forward loops, which usually have important dynamic functions in biological pathways. We studied large-size cycles and characterized their statistical properties in different networks. Cycles can be studied as a one-dimensional (slightly generalized) Ising model, with edges representing spins. It turns out that the statistical ensembe of this Ising model accounts for many of the statistical properties observed in the cycles for all networks studied. We found that for each network, two parameters in the Ising model, the nearest-neighbor coupling constant J and a chemical potential for undirected links m, are enough to account for the ensemble characteristics of cycles of any size between 3 and 20. In most biological networks studied, the fitted-coupling parameter J of the Ising model is sufficiently negative that there is a clear "anti-ferromagnetic order" which in terms of network motifs implies that there is a considerable depletion of feedback loops and selection for bifan-like cycles. This may have important signal processing implications, as will be exemplified in a particular cellular signal transduction network.

This work done in collaboration with Avi Ma'ayan, Guillermo Cecchi, John Wagner, Ravi Rao and Ravi Iyengar.

Systems of heterogeneous agents often display collective properties that cannot be deduced in a simple way from the behaviour of the individual. There are many examples of these systems in biology, such as bacterial colonies, fish schools, insects swarms, starling flocks and mammals herds. In particular, the coordinated patterns displayed by starlings before roosting time are a fascinating phenomenon whose intrinsic microscopic mechanisms are still unknown. In this talk I will present the first results of a highly interdisciplinary research project, whose aim is understanding collective motion in organic systems, from the case study of starling flocks to, eventually, herding behaviour in socio-economic contexts.

I will focus on the most innovative part of our research, namely the experimental reconstruction of the tridimensional positions and trajectories of individual birds during flocking. To this aim, we performed a non-trivial stereoscopic analysis of sets of digital images taken at high resolution and frequency with an appropriately calibrated apparatus. The results of this analysis can be compared with the predictions of existing models and represent the starting point for further modeling of collective motion and its biological interpretation.

Granular media consist of macroscopic, athermal particles that ``jam'',

into a solid-like state when subjected to a confining pressure. Recent studies of this jamming transition in systems of frictionless particles have shown, quite remarkably, that the jamming point has many features of a critical point, exhibiting power law scalings of various quantities nearby. We study the origin of this scaling behavior by analyzing the linear response of these packings to mechanical perturbations. The response to local forcing fluctuates over a length scale that diverges at the jamming transition. The response to global shear or compression becomes increasingly non-affine near the jamming transition. This is due to the proximity of floppy modes, the influence of which we characterize by the relative displacements of neighboring particles. We show that the local response also governs the anomalous scaling of elastic constants and contact number. If time permits I will shortly discuss the consequences of adding friction to the interaction between the particles.

The locomotion of most fish and birds is realized by flapping wings or fins transverse to the direction of travel. Here, we study experimentally the dynamics of a wing that is "flapped" up and down but is free to move in the horizontal direction. In this table-top prototype experiment, we show that flapping flight occurs abruptly at a critical flapping frequency as a symmetry-breaking bifurcation. We then investigate the separate effects of the flapping frequency, the flapping amplitude, the wing geometry and the influence from the solid boundaries nearby. Through dimensional analysis, we found that there are two dimensionless parameters well describe this intriguing problem that deals with fluid-solid interaction. The first one is the dynamical aspect ratio that combines four length scales, which includes the wing geometry and the flapping amplitude. The second parameter, the Strouhal number, relates the flapping efforts to it resultant forward flight speed. Overall, we emphasize the robustness of the thrust-generating mechanisms determining the forward flight speed of a flapping wing, as observed in our experiments.

The modeling of transcriptional regulation of genes in cells relies on representations of dynamical processes that occur in complex networks of interacting elements. A starting point for understanding such processes is the analysis of large random networks of Boolean gates. A theory of the "order-chaos" phase transition in such networks reveals surprising dynamical structures, which I will explain. Though Boolean logic is an intuitively appealing framework for describing transcriptional interactions, care must be taken to separate generic network behavior from artefacts of the Boolean model. I will discuss the relation between attractors of synchronous and asynchronous dynamics in Boolean networks and the continuous dynamics that arise in more realistic models of transcriptional networks.