For centuries, origami, the Japanese art of paper folding, has been a powerful technique for transforming two dimensional sheets into beautiful three dimensional sculptures. Recently, origami has made its foray into a new realm, that of physics and engineering, where it has been revolutionizing our concept of materials design. In this talk I will describe the new design principles we are uncovering for determining the shape, mechanics, and transformations of origami structures along with their usefulness in areas as diverse as solar sail design, architecture, and even fashion. Arguably however, the greatest strength of this new paradigm is the fact that origami is intrinsically scalable. Thus sculptures built at one size can be shrunk down smaller and smaller. This begs the question: what is the smallest fold one can make? Or in other words, how many folded angels can dance on the head of a pin? The rest of this talk will take a deep dive into how origami has been marching smaller and smaller in size. From folding by hand, to self-folding through shape memory alloys and even folding via polymer layers, I will argue that the ultimate limit for scaling down origami is set by folding a sheet of atomic dimensions. I will conclude by showing this vision: realized in the folds of a single sheet of graphene.

Self-organized behaviors are commonly observed in nature and human societies, such as bird flocks, fish swarms and human crowds. In this talk, I will present some celebrated mathematical models, with simple small-scale interactions which lead to the emergence of global behaviors: aggregation and flocking. I will discuss the models in different scales: from microscopic agent-based dynamics, through kinetic mean-field descriptions, to macroscopic fluid systems. In particular, the macroscopic models can be viewed as compressible Euler equations with nonlocal interactions. I will show some recent results on the global wellposedness theory of the systems, large time behaviors, and interesting connections to some classical equations in fluid mechanics.

Recently, the development of specialized techniques in mathematics known as exponential asymptotics has led to the successful resolution of long-standing problems in topics as varied as crystal growth, dislocations, pattern formation, turbulence, thin film flow, and hydrodynamics. These developments have emerged from the realization that in many such problems, exponentially small effects can significantly change the solutions of the underlying mathematical models.
In this talk, we will introduce the audience to the history, ideas, and basic techniques of exponential asymptotics, with particular emphasis on how to recognize when such approaches are necessary. We will discuss the 19th century struggles of the great Cambridge physicist G.G. Stokes to better understand what is now known as the Stokes Phenomenon. We will then show how this understanding would provide the key insight into resolving two famous problems: the problem of modelling dendritic crystal growth, and the Saffman-Taylor viscous fingering problem.
Our discussion will conclude with a glimpse of the present and future applications of exponential asymptotics, notably within the context of hydrodynamics and ship waves, and for the mathematical modelling of rupture and singularity formation in fluid flows.

Surface energy can be harnessed to introduce new features essential to a variety of engineering systems, such as large-area scalability for phase-change thermal diodes and non-clogging apertures for resistive Coulter counting. In the first case, surface energy drives dropwise condensate to spontaneously jump on a superhydrophobic surface, and the self-propelled motion is exploited to create a planar and orientation-independent thermal diode. In the second case, surface tension is manipulated electrohydrodynamically to produce a cone-jet liquid bridge, which serves as a tunable and deformable aperture for resistive pulse sensing. This presentation will cover both fundamental interfacial phenomena and practical engineering applications.

I will describe two simple models that incorporate only hard-sphere and geometrical constraints, yet provide quantitatively accurate predictions for the structural and mechanical properties of frictional packings of granular media and proteins. We first model static friction between grains by considering nominally spherical particles with periodically spaced asperities on the surface of the grains. This model captures the dependence of the average packing fraction and number of interparticle contacts on the static friction coefficient obtained from experiments, and has significant advantages over other models. Second, in the spirit of the Ramachandran map for the backbone dihedral angles of proteins, we develop a model for nonpolar amino acids that allows us to predict the allowed conformations of sidechain dihedral angles. Our predictions are quantitatively similar to the sidechain dihedral angle distributions obtained from known crystal structures. These two examples emphasize the power of simple physical models, which are able to predict important properties of soft and biological materials.

Over the past decade, much attention has focused on the behavior of hydrophobic particles at interfaces. These systems are of interest to scientists and engineers, for example, due to their potential for stabilizing drops and emulsions via jamming. This seminar will focus on the behavior of particulate 'rafts' that form when a monolayer of particles are placed at an air- liquid interface. The particles interact with the underlying fluid to form a quasi two-dimensional solid. Such particulate rafts can support both tension and compression, and they buckle under sufficiently large compressive loads. When a drop of surfactant is introduced into the system, fracture networks develop in the rafts. The fracture process exhibits features observed in other elastic systems, such as crack kinking, crack branching, and crack arrest. Moreover, there is a clear coupling between the praft fracture and the diffusion of the surfactant on the surface and through the 'porous' liquid-particle monolayer. As such, one can draw analogies between this system and others where crack growth interacts with fluid flow or mass transport. The seminar will present recent work in modeling the diffusion of surfactant into particle raft systems and the resulting formation of fracture networks. We will present both discrete models that track the motion of individual particles, as well as a new continuum model for poro-chemo-elasticity. Results that reproduce some of the quantitative and qualitative aspects of recent experimental studies of these systems will also be shown.

Cells exist in a complex mechanical environment that is both a source of applied forces and a means of mechanical support. An incomplete understanding of the mechanisms cells use to detect mechanical stimuli, a process termed mechanotransduction, is currently preventing advances in tissue engineering and hindering the understanding of several mechanosensitive disease states. Mechanical stimuli are sensed at focal adhesions (FAs), complex dynamic structures comprised of several hundred types of proteins that mediate physical connections between the extracellular matrix and the cytoskeleton. Detection of mechanical cues is thought to be mediated by mechanically-induced changes in protein structure, which, in elegant in vitro single molecule experiments, have been shown to induce new biochemical functions, such as changes in binding affinity as well as the formation of distinct protein-protein interactions. However, the existence and role of these mechanically-induced changes in protein function in living cells are not well understood. To enable the visualization of protein loading, we create Forster Resonance Energy Transfer (FRET)-based tension sensors that emit different colors of light in response to applied forces. The next step in the development of this technology is the use of these sensors to study the effects of mechanical loading on protein functions in living cells. To begin this process, we have refined two commonly used and powerful approaches, Fluorescence Recovery After Photobleaching (FRAP) and fluorescence co-localization to be compatible with FRET-based tension sensors. Initial efforts have focused on the mechanical linker protein vinculin due to its established role in regulating the response of FAs to mechanical loading. These novel techniques reveal that force affects both vinculin turnover as well as its ability to form distinct protein-protein interactions. Further use of these techniques should enable a wide variety of studies in mechanobiology involving different load-bearing proteins, subcellular structures, extracellular contexts, and cellular functions.

Many systems of interest can be represented by a network of nodes connected by edges. In many circumstances the existence of a giant component is necessary for the network to fulfill its function. Motivated by the need to understand optimal attack strategies, optimal spread of information or immunization policies, we study the network dismantling problem, i.e. the search of a minimal set of nodes whose removal leaves the network broken into components of sub-extensive size. Building on the statistical mechanics perspective we compute the size of the optimal dismantling set for random networks, propose an efficient dismantling algorithm for general networks that outperforms by a large margin existing strategies, and we provide various insights about the problem.

Granular materials are a part of a broad class of amorphous materials that display yield stress behavior. When the applied shear stress is below the yield stress, grains move temporarily, but only until finding a mechanically stable (MS) configuration that is able to resist the applied shear stress. Above the yield stress, the material is no longer able to find MS configurations. However, the geometrical reasons why MS states vanish at the yield stress is not well understood. In this talk, I will show evidence from molecular dynamics simulations that yielding in granular materials is akin to a second-order critical point, where the mechanical behavior is dominated by a correlation length that diverges at the yield stress. MS states exist above the yield stress for finite systems, but they vanish as the system size becomes large according to a critical scaling function. The packing fraction and coordination number for MS states are independent of the applied shear stress, implying that the critical behavior we observe is distinct from the well known jamming scenario. However, MS states at nonzero shear stress possess anisotropic force and contact networks, suggesting that the yield stress is set by the maximum anisotropy that can be realized in the large-system limit.

Diffusion over a network depends crucially on the pattern and timing of relations. This is particularly important for diseases carried over networks with relatively low volume and turnover. Here we explore both aspects using simulation tools. First, we ask how the shape of the distribution of number of partners affects multiple connectivity, and second we measure the exposure potential in dynamic networks across a wide array of structural patterns to identify the influence of "concurrency," the overlap in time of interactions among network nodes. We find that concurrency in low-volume settings has the same effect on epidemic spreading as a structural increase in the average degree. 

The characterization of chaos as a random-like response from a deterministic dynamical system with an extreme sensitivity to initial conditions is well-established, and has provided a stimulus to research in nonlinear dynamical systems in general. In a formal sense, the computation of the Lyapunov Exponent (LE) spectrum establishes a quantitative measure, with at least one positive LE (and generally bounded motion) indicating a local exponential divergence of adjacent trajectories. Other measures are associated with certain geometric features of a chaotic attractor, e.g., the fractal dimension, and broadband frequency content. However, although the extraction of LE's can be accomplished with (necessarily noisy) experimental data, this is still a relatively data-intensive, sensitive (and frustrating) endeavor.
We present here an alternative, pragmatic approach to identifying chaos as a function of system parameters, based on frequency content and extending the concept of the spectrogram. This talk will describe this approach applied to systems of increasing complexity, ranging from direct numerical simulations of familiar archetypal systems like Lorenz and the pendulum to experimental data generated from mechanical systems. The accuracy and utility of the approach, including the effect of noise, is tested relative to the standard (LE) approach.

The glass problem is notoriously hard, but the recent exact solution of a microscopic model offers a novel perspective on the problem. In this seminar, I will discuss how contrasting entropic caging and isostaticity at the glass and the jamming transitions, respectively, reveals the presence of a Gardner transition. This onset of mechanical marginality then explains the presence of non-trivial critical exponents. I will also discuss how a family of finite-dimensional models reveals the clear role for caging geometry and hopping in the dynamical slowdown of colloid-like glass formers. Both advances greatly enrich the traditional mean-field description of glasses.

Many processes in nature occur in the form of rare but important events. Well known examples of such events include conformation changes of biomolecules, chemical reactions, and nucleation events during phase transformation. Rare events do not happen very often on the internal clock of the system (which makes their simulation very challenging), but this clock can be very fast and this leaves plenty of room for the appearance of rare events in our daily life. I will review classical theories for the description of rare events, recent theoretical developments such as Transition Path Theory, concept such as reaction coordinate or free energy of a reaction and I will discuss how to compute the pathway and rate of rare events efficiently using the String Method. As illustrations, I will discuss the hydrophobic collapse of a polymeric chain, phase transitions in the Ising model, and a genetic toggle switch.

We will discuss the application of multi-scale graph theory based methods to the detection of general structures in networks, lattices, and amorphous physical systems. These methods enable the detection of the "natural" system structures on all scales. We specifically analyze lattices and spin systems with defects and various glass formers (including an analysis based on experimental data) to ascertain dominant structures at different temperatures. We will discuss general features of the phase diagram related to this analysis.

The idea that glasses can become marginally stable at a Gardner transition has attracted significant interest among the glass community. Yet, the situation is confusing: even at the theoretical level, renormalization group approaches provide contradictory results on whether the transition can exist in three dimensions. The Gardner transition was searched in only two experimental studies and few specific numerical models. These works lead to different conclusions for the existence of the transition, resulting in a poor understanding of the conditions under which a marginally stable phase can be observed. The very relevance of the Gardner transition for experimental glasses is at stake.

We study analytically and numerically the Weeks-Chandler-Andersen model. By changing external parameters, we continuously explore the phase diagram and regimes relevant to granular, colloidal, and molecular glasses. We revisit previous numerical studies and confirm their conclusions. We reconcile previous results and rationalise under which conditions a Gardner phase can be observed. We find that systems in the vicinity of a jamming transition possess a Gardner phase. Our findings confirm the relevance of a Gardner transition for colloidal and granular glasses, and encourage future experimental work in this direction. For molecular glasses, we find that no Gardner phase is present, but our studies reveal instead the presence of localised excitations presumably relevant for mechanical and vibrational properties of glasses.

Gas-solids chemically reacting flows are omnipresent in many multiphase flow reactors in various industries like Chemical, Fossil and Nuclear. The challenging aspect of modeling these reacting flows are the wide range of both temporal and spatial scales encountered in these systems. The challenge is to accurately account and bridge (as seamlessly as possible) the length and time scales involved in the problem. First, the problem is introduced using biomass gasifier/pyrolyser and nuclear fuel coater with sample results as examples and provide an overview of the various models currently used at the different scales. In particular, the critical role of the granular dynamics in the overall performance of the reactors will be highlighted. The ongoing development of a multiphysics and multiscale mathematics framework for coupling various modeling methods over a range of scales will be presented. The development of a general wavelet-based multiscale methodology called compound wavelet matrix (CWM) for bridging spatial and temporal scales will be reported. Finally, the steps needed to generalize the current methodology for arbitrary heterogeneous chemically reacting flows or other applications involving multiscale/multiphysics coupling will be elucidated. The challenges and opportunities of employing these models for rapid deployment of clean energy solutions based on multiphase flow reactors to the market place will be discussed.

Motivated by a recently identified severe discrepancy between a static and a dynamic theory of glasses, we numerically investigate the behavior of dense hard spheres in spatial dimensions 3 to 12. Our results are consistent with the static replica theory, but disagree with the dynamic mode- coupling theory, indicating that key ingredients of high-dimensional physics are missing from the latter. We also obtain numerical estimates of the random close packing density, which provides new insights into the mathematical problem of packing spheres in large dimension.

The physics of granular flow is of widespread practical and fundamental interest, and is also important in geology and astrophysics. One challenge to understanding and controlling behavior is that the mechanical response is nonlinear, with a forcing threshold below which the medium is static and above which it flows freely. Furthermore, just above threshold the response may be intermittent even though the forcing is steady. Two familiar examples are avalanches on a heap and clogging in a silo. Another example is dynamical heterogeneities for systems brought close to jamming, where intermediate-time motion is correlated in the form of intermitted string-like swirls. This will be reviewed in the context of glassy liquids and colloids, and more deeply illustrated with experiments on three different granular systems. This includes air-fluidized beads, where jamming is approached by density and airspeed; granular heap flow, where jamming is approached by depth from the free surface; and dense suspensions of NIPA beads, where jamming is approached by both density and shear rate. Emphasis will be given to measurement and analysis methods for quantifying heterogeneities, as well as the scaling of the size of heterogeneities with distance to jamming.

Numerical continuation is a tool for investigating the bifurcation structure of a nonlinear dynamical system with respect to the system parameters. It is most often used to "carve up" parameter space into regions of qualitatively different behaviour by finding and tracking bifurcations (e.g., Hopf bifurcations) as the system parameters change. This talk will give an introduction to the theory behind numerical continuation and go on to discuss recent developments in the field.

Particular attention will be paid to numerical continuation of systems with non-smoothness, motivated by the example of intermittent contacts in a model of orthogonal cutting (turning). Rich dynamical behaviour is present in this model due to the presence of a grazing bifurcation which denotes the transition point from constant contact of the cutting tool with the workpiece to intermittent contact. Using numerical continuation it is possible to elucidate the full bifurcation structure of the system, something that would be extremely difficult with other methods.

Finally, numerical continuation will be demonstrated as applied to a physical experiment (so-called control-based continuation): a nonlinear energy harvesting device. Numerical continuation in this context allows the investigation of a physical device without prior knowledge of a model. Both stable and unstable motions can be investigated and bifurcations found directly. As such these investigations may aid in establishing what an appropriate mathematical model could be.

My group studies colloidal suspensions, which are solid micron-sized particles in a liquid. We use an optical confocal microscope to view the motion of these colloidal particles in three dimensions. In some experiments, these particles arrange into a crystalline lattice, and thus the sample is analogous to a traditional solid. We study the interface between colloidal crystals and colloidal liquids, and find that this interface is quite sharply defined. In other experiments, the sample is analogous to a glass, with particles randomly packed together. The particles correspond to individual molecules in a traditional glass, and the sample exhibits glassy behavior when the particle concentration is high. This allows us to directly study the microscopic behavior responsible for the macroscopic viscosity divergence of glasses.

Granular materials exist all around us, from avalanches in nature to the mixing of pharmaceuticals, yet the behavior of these ``fluids'' is poorly understood. Their flow can be characterized by the continuous forming and breaking of a strong force network resisting flow. This jamming/unjamming behavior is typical of a variety of systems, including granular flows, and is influenced by factors such as grain packing fraction, applied shear stress, and the random kinetic energy of the particles. I'll present experiments on quasi-static shear and free-surface granular flows under the influence of external vibrations. By using photoelastic grains, we are able to measure both particle trajectories and the local force network in these 2D flows. We find through particle tracking that dense granular flow is composed of comparable contributions from the mean flow, affine, and non-affine deformations. During shear, sufficient external vibration weakens the strong force network and reduces the amount of flow driven by sidewalls. In a rotating drum geometry, large vibrations induce failure as might be expected, while small vibration leads to strengthening of the pile. The avalanching behavior is also strongly history dependent, as evident when the rotating drum is driven in an oscillatory motion, and we find that sufficient vibration erases the memory of the pile. These results point to the central role of the mobilization of friction in quasi-static granular flow.

Despite an enormous range of applications and centuries of scientific study, understanding and predicting the flow of fluids remains a tremendous challenge, particularly when the flow is chaotic or turbulent. Turbulent flows tend to be characterized by violent fluctuations, enormous numbers of strongly coupled degrees of freedom, and significant variability in space and time. But despite all this complexity, turbulence is not random. Rather, it tends to self-organize into striking but transient patterns and features that arise from nonlinear interactions. Some of these "coherent structures," such as strong vortices, are readily apparent; others are more subtle. But how much can we learn or predict about the flow from studying coherent structures? The answer may lie in the energetics of the flow, since these same nonlinearities couple dynamics on different scales and, in turbulence, drive a net transfer of energy from the scales at which it is injected into the flow to the scales at which it is dissipated. To begin to make quantitative links between the nonlinear dynamics of the flow and the spontaneous generation of spatiotemporal order, I will discuss experimental results from a quasi-two-dimensional turbulent flow. Using a filtering technique, we extract the spatially localized scale-to-scale flux of energy, and show that it is linked to suitably defined coherent structures. I will also discuss the self-organization of the turbulent stress that drives this energy transfer.

Living organisms need to obtain and process information crucial for their survival. Information processing in living systems, ranging from signal transduction in a single cell to image processing in the human brain, are performed by biological circuits (networks), which are driven out of equilibrium. These biochemical and neural circuits are inherently noisy. However, certain accuracy is required to carry out proper biological functions. How do biological networks process information with noisy components? What is the free energy cost of accurate biological computing? Is there a fundamental limit for its performance of the biological functions? In this talk, we will describe our recent work in trying to address these general questions in the context of two basic cellular computing tasks: sensory adaptation for memory encoding [1,2]; biochemical oscillation for accurate timekeeping [3].

[1] The energy-speed-accuracy trade-off in sensory adaptation, G. Lan, P. Sartori, S. Neumann, V. Sourjik, and Yuhai Tu, Nature Physics 8, 422-428, 2012.
[2] Free energy cost of reducing noise while maintaining a high sensitivity, Pablo Sartori and Yuhai Tu, Phys. Rev. Lett. 2015. 115: 118102.
[3] The free-energy cost of accurate biochemical oscillations, Y. Cao, H. Wang, Q. Ouyang, and Yuhai Tu, Nature Physics 11, 772, 2015.

Motivated by the dewetting of viscous thin films on hydrophobic substrates, we study models for the coarsening dynamics of interacting localized structures in one dimension. For the thin films problem, lubrication theory yields a Cahn-Hilliard-type governing PDE which describes spinodal dewetting and the subsequent formation of arrays of metastable fluid droplets. The evolution for the masses and positions of the droplets can be reduced to a coarsening dynamical system (CDS) consisting of a set of coupled ODEs and deletion rules. Previous studies have established that the number of drops will follow a statistical scaling law, N(t)=O(t^{-2/5}). We derive a Lifshitz-Slyozov-Wagner-type (LSW) continuous model for the drop size distribution and compare it with discrete models derived from the CDS. Large deviations from self-similar LSW dynamics are examined on short- to moderate-times and are shown to conform to bounds given by Kohn and Otto. Insight can be applied to similar models in image processing and other problems in materials science. Joint work with M.B. Gratton (Northwestern Applied Math).

The cohesive strength of granular materials is a consequence of either cohesive bonding (capillary bridging, van der Waals forces) between the grains or the action of a binding solid or liquid material in the pore space. I first discuss the constitutive framework of the plastic behavior of granular materials with internal variables pertaining to the granular fabric. Then, I show how cohesive granular systems can be simulated by different methods accounting for capillary or solid bonding and in the presence of a binding solid or liquid. Finally, I focus on two issues: (1) How does local granular disorder affects the scale-up of cohesive interactions? (2) What are the respective roles of adhesion and volume fraction in the case of binding materials?

Both the 'reverse engineering' of biological networks (for example, by integrating sequence data and expression data) and the analysis of their underlying design (by revealing the evolutionary mechanisms responsible for the resulting topologies) can be re-cast as problems in machine learning: learning an accurate prediction function from high-dimensional data. In the case of inferring biological networks, predicting up- or down- regulation of genes allows us to learn ab intio the transcription factor binding sites (or `motifs') and to generate a predictive model of transcriptional regulation. In the case of inferring evolutionary designs, quantitative, unambiguous model validation can be performed, clarifying which of several possible theoretical models of how biological networks evolve might best (or worst) describe real-world networks. In either case, by taking a machine learning approach, we statistically validate the models both on held-out data and via randomizations of the original dataset to assess statistical significance. By allowing the data to reveal which features are the most important (based on predictive power rather than overabundance relative to an assumed null model) we learn models which are both statically validated and biologically interpretable.