||Self-introductions by the participants
|Abstract: No Abstract
|Douglas N. Arnold (University of Minnesota)
||Welcome and introduction
|Abstract: No Abstract
|Douglas N. Arnold (University of Minnesota)
||Seeing mathematics everywhere
|Abstract: From medicine to entertainment, mathematics is critical to meeting the needs of our future. In this Institute of Technology Public Lecture world-renowned mathematician and University of Minnesota mathematics professor Doug Arnold will highlight interesting examples in which the University’s Institute of Mathematics and its Applications (IMA) has solved problems in contemporary life.
Doug Arnold photoArnold will specifically highlight several examples where teams of mathematicians worked side-by-side with engineers and scientists to develop exciting solutions. IMA researchers have studied the movement of insects to build better robots, examined the body’s virus-fighting systems to develop enhanced computer security systems, and used imaging techniques to restore old films and improve surveillance video in a homicide investigation. Arnold will also discuss some of the IMA’s outreach work including a YouTube video on mathematics that has received nearly 1.5 million views within the last year.
Arnold is completing seven years as director of the IMA, the preeminent interdisciplinary mathematics research institute in the world. Arnold was recently awarded a 2008 Guggenheim Fellowship as one of only five mathematicians in the U.S. and Canada to receive the prestigious award. Arnold has been named a McKnight Presidential Endowed Professor of Mathematics, a position to which he was recently appointed by University President Robert Bruininks. Arnold also is the president-elect of the Society for Industrial and Applied Mathematics, the leading professional organization for applied mathematicians, computational scientists and engineers.
|José Bico (École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI))
|Abstract: When a liquid droplet is deposited on a flexible sheet, the
sheet may deform and spontaneously wrap the droplet. We propose to
address a problem in connection with this "capillary origami" experiment: does
a flexible rod put in contact with a liquid droplet spontaneously winds
itself around the droplet? In the positive situation, what is the maximum
length that can be packed inside the droplet? We will finally try to
connect this problem to damping issues in spider webs.
|Bjorn Birnir (University of California)
||Turbulent solutions of the stochastic
|Abstract: Starting with a swirling flow we prove the existence
of unique turbulent
solutions of the stochastically driven Navier-Stokes equation
in three dimensions.
These solutions are not smooth but Hölder continuous with index
1/3. The turbulent
solutions give the existence of an invariant measure that
the statistical theory of turbulence including Kolmogorov´s
scaling laws. We will
discuss how the invariant measure can be approximated leading
to a implicit formula
that can be used to compare with simulations and experiments.
|Philip Boyland (University of Florida)
||Topological kinematics of point vortex motions
|Abstract: Topological techniques are used to study the motions
of systems of point vortices. After symplectic
and finite reduction the systems become one-degree-of-freedom
Hamiltonian. The phase portrait of the reduced system is subdivided
into regimes using the separatrix motions, and
a braid representing the topology of all vortex motions
in each regime is computed. This braid also describes the isotopy
class of the advection homeomorphism induced by the vortex motion in the
surrounding fluid. The Nielsen-Thurston theory is then used to
analyze these isotopy classes, and in certain cases, lower bounds
for the complexity of the chaotic dynamics (eg. topological entropy)
of the advection are obtained. Similar analysis using the
Nielsen-Thurston theory applied to the stirring of two-dimensional
fluids will also be briefly described. The results illustrate
a mechanism by which the topological kinematics of large-scale,
two-dimensional fluid motions generate chaotic advection.
|Stephen Childress (New York University)
||Some remarks on vorticity growth in Euler flows
|Abstract: Motivated by some estimates of vorticity growth in axisymmetric flows without swirl, we re-examine
the paired vortex model of singularity formation proposed by Pumir and Siggia for Euler flows in three dimensions.
The problem is reformulated as a generalized system of differential equations. No supporting solutions of the system are known, and it is suggested that core deformation remains the most likely mechanism preventing the formation of a singularity.
|Paul Clavin (UMR CNRS-Universites d'Aix-Marseille I&II)
||Ablative Rayleigh-Taylor instability
|Abstract: Ablative Rayleigh-Taylor (R-T) instability is a special feature of the acceleration phase in inertial confinement fusion (ICF). Ablation stabilizes the disturbances with small wavelength, introducing a marginal wavelength. Due to a large temperature ratio, the conduction length-scale varies strongly across the wave, and the attention is limited to the intermediate acceleration regimes for which the length-scale of the marginal wavelength is in-between the smallest and the largest conduction length-scale.
The analysis is performed for a strong temperature dependence of thermal conductivity. At the leading order, the ablation front appears as a vortex sheet separating two potential flows 1, 2, and the free boundary problem takes the form of an extension of the pure R-T instability with unity Atwood number and zero surface tension. It shows also some analogies with the Kelvin-Helmholtz instability described by the Birkhoff-Rott equation. However, the hot flow of ablated matter introduces a damping at small wavelength which has a form different from the usual damping (as the surface tension for example). The nonlinear patterns are obtained by the same boundary integral method as used for revisiting the R-T instability 3. Unfortunately, a curvature singularity develops within a finite time, even though the short wavelengths are stabilised. Scaling laws are derived from numerical fitting and a self-similarity solution of the problem is exhibited close to the critical time 4.
The occurrence of a curvature singularity indicates that the modifications to the inner structure of the vortex sheet can no longer be neglected. A non-local curvature effect is obtained by pushing the asymptotic analysis to the next order 5. The corresponding small pressure correction is showed to prevent the occurrence of the curvature singularity within a finite time.
|Itai Cohen (Cornell University)
||Investigating dislocation dynamics in degenerate crystals of dimer colloids
|Abstract: Colloidal suspensions consist of micron sized solid particles
suspended in a solvent. The particles are Brownian so that the
suspension as a whole behaves as a thermal system governed by
the laws of statistical mechanics. The thermodynamic nature of
these systems allows scientists to use colloidal suspensions as
models for investigating numerous processes that typically take
place on the atomic scale but are often very difficult to
investigate. In this talk I will describe how we use confocal
microscopy techniques to investigate the structure and dynamics
of these systems and gain an understanding of dislocation
nucleation and transport in colloidal crystals. Such
dislocations are examples of singular point defects in 2D
crystals and line defects in 3D crystals.
|Peter Constantin (University of Chicago)
||The zero temperature limit of interacting corpora
|Abstract: We consider examples of melts of corpora, that is collections of
compacts each having finitely many degrees of freedom, such as articulated
particles or n-gons. We associate to the melt the moduli spaces of
the corpora, compact metric or pseudometric spaces equipped with a Borel
probability measure representing the phase space measure. We consider
probability distributions on the moduli
spaces of such corpora, we associate a free energy to them, and show that
under general conditions, the zero temperature limit of free energy
minimizers are delta functions concentrated on a single corpus, the
ur-corpus. We give a selection principle for the ur-corpus.
This is a generalization of the isotropic to nematic transition but we
suggest that this language is appropriate for a larger class of n-body
interactions. This is work in progress with Andrej Zlatos.
|Mark Dennis (University of Bristol)
||Topological singularities in optical waves
|Abstract: Understanding of complicated spatial patterns emerging from wave interference, scattering and diffraction is frequently aided by insight from topology: the isolated places where some fundamental physical quantity -- such as optical phase in a complicated light field -- is undefined (or singular) organize the rest of the field. In scalar wave patterns, the optical phase is undefined at nodes at points in 2D, and lines in 3D, in general whenever 3 or more waves interfere. Similar singularities occur in optical polarization fields, and these quantized defects bear some morphological similarity to defects in other systems, such as crystal dislocations, diclinations and quantum vortices in condensed matter physics, etc.I will describe the features of these optical singularities, concentrating on three cases. The first will be three-dimensional optical speckle, familiar as the mottled pattern in reflected laser light. Natural speckle volume is filled with a dense tangle of nodal phase singularity lines. We have found in computer simulations that these lines have several fractal scaling properties. Secondly, by controlling the interference using diffractive holograms in propagating laser light, I will show how these nodal lines can be topologically shaped to give a range of loops, links and knots. Finally, I will describe the natural polarization pattern that occurs in skylight (due to Rayleigh scattering in the atmosphere), originally discovered in the 1800s by Arago, Babinet and Brewster. This pattern contains polarization singularities, whose global geometry has several physical interpretations and analogs.
|Efi Efrati (Hebrew University)
||Elastic theory of non-Euclidean plates
|Abstract: Thin elastic sheets are very common in both natural and
man-made structures. The configurations these structures assume in space are
often very complex and may contain many length scales, even in the case of
unconstrained thin sheets. We will show that in some cases, a simple intrinsic
geometry leads to complex three-dimensional configurations, and discuss the
mechanism shaping thin elastic sheets through the prescription of an intrinsic metric.
Current reduced (two-dimensional) elastic theories devised to describe thin
structures treat either plates (flat bodies having no structure along their
thin dimension) or shells (non-flat bodies having a non-trivial structure
along their thin dimension). We propose the concept of non-Euclidean plates,
which are neither plates nor shells, to approximate many naturally formed
thin elastic structures. We derive a thin plate theory which is a
generalization of existing linear plate theories for large displacements but
small strains, and arbitrary intrinsic geometry. We conclude by surveying
some experimental results for laboratory-engineered non-Euclidean plates.
|Jens Eggers (University of Bristol)
||A catalogue of singularities
|Abstract: We survey rigorous, formal, and numerical results on the formation of
point-like singularities (or blow-up) for a wide range of evolution
equations. We use a similarity transformation of the original equation with
respect to the blow-up point, such that self-similar behaviour is mapped to
the fixed point of an infinite dimensional dynamical system. We
point out that analysing the dynamics close to the fixed point is a useful
way of classifying the structure of the singularity. As far as we are aware,
examples from the literature either correspond to stable fixed points,
low-dimensional centre-manifold dynamics, limit cycles, or travelling waves.
We will point out unsolved problems, present perspectives, and try
to look at the role of geometry in singularity formation.
|G. Bard Ermentrout (University of Pittsburgh)
||Cables and dendrites
|Abstract: No Abstract
|G. Bard Ermentrout (University of Pittsburgh)
||Firing rates, filters, and introduction to the visual system
|Abstract: No Abstract
|G. Bard Ermentrout (University of Pittsburgh)
|Abstract: No Abstract
|Stephan Gekle (Universiteit Twente)
||High-speed jet formation after solid object impact
|Abstract: A circular disc impacting on a water surface creates a
remarkably vigorous jet. Upon impact an axisymmetric air cavity
forms and eventually pinches off in a single point halfway down
the cavity. Immediately after closure two fast sharp-pointed
jets are observed shooting up- and downwards from the closure
location, which by then has turned into a stagnation point
surrounded by a locally hyperbolic flow pattern.
Counter-intuitively, however, this flow is not the mechanism
feeding the two jets. Using boundary-integral simulations we
show that only the inertial focussing of the liquid colliding
along the entire surface of the cavity provides enough energy
to eject the high-speed jets. With this in mind we show how the
natural description of a collapsing void (using a line of sinks
along the axis of symmetry) can be continued after pinch-off to
obtain a quantitative analytical model of jet formation.
|Walter Goldburg (University of Pittsburgh)
||Hydraulic jump in a flowing soap film
|Abstract: Joint work with S. Steers, J. Larkin, A. Prescott
(University of Pittsburgh), T. Tran, G. Gioia, P. Chakraborty,
G. Gioia, and N. Goldenfeld (University of Illinois, Urbana).
A soap film flows vertically downward under gravity and in a
steady state. At all lengths of the film, its thickness h(x) decreases
as the distance x from the top reservoir increases. But then h(x)
abruptly starts to increase and its downward flow velocity u(x)
correspondingly decreases to very small value. To explain this
nonmonotonic behavior in h(x) and u(x), it is necessary to
invoke the film's elasticity; one has a type of Marangoni
effect. The transition from subcritical flow speed to a
supercritical one at the thickening point, is akin to the
classical hydraulic jump. This transition will be explained,
but other findings, also to be described, are not yet understood.
|Dirk Hartmann (Ruprecht-Karls-Universität Heidelberg)
||Single cell mechanics: Mechanics of membranes and cytoskeletal networks
|Abstract: In this talk, I will first review existing theoretical approaches for mechanics of single cells. These are mainly determined by the membrane and the cytoskeleton. The most popular class of models among physicists are static models given in terms of energy functionals, e.g. the Canham-Helfrich energy modelling membrane mechanics or microscopic pseudo-spring / cable networks modelling cytoskeletal mechanics. On the other hand mathematicians often prefer to work in terms of continuous models based on conservation equations, e.g. the plate equation modelling mechanics of membranes on short length scales, a completely different class of models.
Considering mechanics of red blood cells, one of the most simplest cells, I will show how it is possible to link those different classes of models in a rigorous manner. Simulations of optical tweezers experiments based on finite element methods will be shown.
|Evan Hohlfeld (University of California)
||Point-instabilities, point-coercivity (meta-stability), and
|Abstract: For general non-linear elliptic PDEs, e.g. non-linear rubber
elasticity, linear stability analysis is false. This is because of the
possibility of point-instabilities. A point-instability is a non-linear
instability with zero amplitude threshold that occurs while linear
stability still holds. Examples include cavitation, fracture, and the formation
of a crease, a self-contacting fold in an otherwise free surface.
Each of which represents a kind of topological change. For any such
PDE, a point-instability occurs whenever a certain auxiliary
scale-invariant problem has a non-trivial solution. E.g. when sufficient strain
is applied at infinity in a rubber (half-)space to support a
single, isolated crease, crack, cavity, etc. Owing to scale-invariance,
when one such solution exists, an infinite number or geometrically
similar solutions also exist, so the appearance of one particular
solution is the spontaneous breaking of scale-invariance. We then identify
this (half-)space with a point in a general domain. The condition
that no such solutions exist is called point-coercivity, and can be
formulated as non-linear eigenvalue problem that predicts the critical
stress for fracture, etc. And when point-coercivity fails for a system,
the system is susceptible to the nucleation and self-similar growth of
some kind of topological defect. Viewing fracture, etc. as symmetry breaking
processes explains their macroscopic robustness.
Point-coercivity is similar to, but more general than,
quasi-convexity, as it can be formulated for any elliptic PDE, not just
Euler-Lagrange systems (i.e. for out-of-equilibrium systems, and so defining
meta-stability in a general sense). Indeed, these are just two
examples of a host of point-conditions, the study of which might be
called point-calculus. Time allowing, I will show that for almost any
elliptic PDE, linear- and point-instabilities exhaust the possible kinds
of instabilities. The lessons learned from elliptic systems will
be just as valid for parabolic and hyperbolic systems since the underlying
reason linear analysis breaks down – taking certain limits in the
wrong order holds for these systems as well.
|Mihaela D. Iftime (Boston University)
||On characteristic classes for the gravitational field and black holes
|Abstract: Many physical theories have mathematical singularities of some kind. A
spacetime singularity is "a place" where quantities that measure the
gravitational field ( e.g. spacetime curvature) "blows up".
The prediction of a singularity, such as the the big bang and the final
state of black holes is a signal that the classical gravitational theory has been pushed beyond the domain of its validity, and that we need a quantum theory to correctly describe what happens near the singularity. While no black hole can be visualized (in the literal meaning of that word) a meaningful picture of a black hole has been obtained by plotting curvature scalar polynomial invariants or Cartan scalars.
These invariants have been primarly used in providing a local characterization
of the spacetime. In this talk I shall discuss the
equivalence problem more rigorously, and define a set of characteristic cohomology classes for the gravitational field.
|Mee Seong Im (University of Illinois at Urbana-Champaign)
||Singularities in Calabi-Yau varieties
|Abstract: Calabi-Yau manifolds are currently being
studied in theoretical physics to unify Einstein's general
relativity and quantum mechanics. Vibrating strings in string
theory live in 10-dimensional spacetime, with four of these
dimensions being 3-dimensional observable space plus time and
six additional dimensions being a Calabi-Yau manifold. In this
talk, I will discuss orbifold singularity on a Calabi-Yau
variety and the topology of crepant resolutions using the McKay
|Daniel D. Joseph (University of Minnesota)
||Viscous potential flow analysis of radial fingering in a Hele-Shaw cell
|Abstract: The problem of radial fingering in two phase gas/liquid flow in
a Hele-Shaw cell under injection or withdrawal is studied here.
The problem is analyzed as a viscous potential flow VPF in
which the potential flow analysis of Paterson 1981 and others
is augmented to account for the effects of viscosity on the
normal stress at the gas/liquid interface. The unstable cases
in which gas is injected into liquid or liquid is withdrawnfrom
gas lead to fingers. This stability problem was previously
considered by other authors with the viscous normal stress
neglected. Here we show that the viscous normal stress should
not be neglected; the normal stress changes the speed of
propagation of the undisturbed interface, it changes the growth
rate, and the numbers of fingers that grow the fastest and the cut-off number
above which fingers can not grow.
|Randall D. Kamien (University of Pennsylvania)
||The geometry of topological defects
|Abstract: The theory of smectic liquid crystals is notoriously difficult to study. Thermal fluctuations render them disordered through
the Landau-Peierls instability, lead to anomalous momentum dependent elasticity, and make the nematic to smectic-A transition
enigmatic, at best. I will discuss recent progress in studying large deformations of smectics which necessitate the use of nonlinear
elasticity in order to preserve the underlying rotational symmetry. By recasting the problem of smectic configurations geometrically
it is often possible to exploit toplogical information or, equivalently, boundary conditions, to confront these highly nonlinear problems.
Specifically, I will discuss edge dislocations, disclination networks in three-dimensionally modulated smectics, and large angle twist
grain boundary phases. Fortuitously, it is possible to make intimate comparison with experimental systems!
|David Kinderlehrer (Carnegie Mellon University)
||What's new for microstructure
|Abstract: Cellular structures coarsen according to a local evolution law,
a gradient flow or curvature driven growth, for example,
limited by space filling constraints, which give rise to random
changes in configuration. Composed of volumes, facets, their
boundaries, and so forth, they are ensembles of singlular
structures. Among the most challenging and ancient of such
systems are polycrystalline granular networks, especially those
which are anisotropic, ubiquitous among engineered materials.
It is the problem of microstructure. These are large scale
metastable, active across many scales. We discuss recent work
in this area, especially the discovery and the theory of the
GBCD, the grain boundary character distribution, which offers
promise as a predictive measure of texture related material
properties. There are many mathematical challenges and the
hint of universality.
|Peter R. Kramer (Rensselaer Polytechnic Institute)
||Stochastic mode reduction with metastability in biomolecular modeling
|Abstract: One approach to accelerating biomolecular simulations is to simulate explicitly only certain slow degrees of freedom of interest, incorporating the effects of the remaining "fast" variables through effective stochastic models. We illustrate a systematic multi-scale stochastic mode reduction procedure on a simple model problem with metastability — a high potential energy barrier separating different conformational states. Metastability is a prevalent feature in biomolecular systems. We show in particular how the metastability can lead to various effective stochastic equations for the slow degrees of freedom depending on the relations between the physical parameters and properties of the potential energy landscape. This work is in collaboration with Jessika Walter at Ecole Polytechnique Federale de Lausanne and Christof Schuette at the Free University of Berlin. Time permitting, I will also discuss some general observations concerning the application of multiple scale asymptotics to problems with three (or more) active time scales (joint work with Adnan Khan (Lahore) and Robert E. Lee DeVille (University of Illinois)).
©2006 Regents of the University of Minnesota. All rights reserved.
The University of Minnesota is an equal opportunity educator and employer.
|Arshad Kudrolli (Clark University)
||Experimental investigations of packing, folding, and crumpling
in two and three dimensions
|Abstract: We will discuss the packing and folding of a confined beaded
chain vibrated in a flat circular container as a function of chain
length, and compare with random walk models from polymer physics. Time
permitting, we will briefly discuss crumpling and folding structures
obtained with paper and elastic sheets obtained with a laser-aided
topography technique. We have shown that the ridge length distribution
is consistent with a hierarchical model for ridge breaking during crumpling.
|Robert B. Kusner (University of Massachusetts)
||Lengths and crossing numbers of tightly knotted ropes and bands
|Abstract: About a decade ago, biophysicists observed an
linear relationship between the combinatorial complexity
DNA and the distance traveled in gel electrophoresis
Modeling the DNA as tightly knotted rope of uniform
thickness, it was
suggested that lengths of such tight knots (rescaled to
thickness) would grow linearly with crossing numbers, a
of knot complexity. It turned out that this relationship
subtle: some families of knots have lengths growing as the
power of crossing numbers, others grow linearly, all
3/4 and 1 can be realized as growth rates, and it could be
that the power cannot exceed 2 [2-5]. It is still unknown
there are families of tight knots whose lengths grow
linearly with crossing numbers, but the largest power has
to 3/2 . We will survey these and more recent
developments in the
geometry of tightly packed or knotted ropes, as well as
physical models of knots as flattened ropes or bands which
similar length versus complexity power laws, some of which
we can now
prove are sharp . References:
 Stasiak A, Katritch V, Bednar J, Michoud D, Dubochet J
"Electrophoretic mobility of DNA knots" Nature 384 (1996)
 Cantarella J, Kusner R, Sullivan J "Tight knot values
linear relation" Nature 392 (1998) 237
 Buck G "Four-thirds power law for knots and links"
 Buck G, Jon Simon "Thickness and crossing number of
Topol. Appl. 91 (1999) 245
 Cantarella, J, Kusner R, Sullivan J "On the minimum
knots and links" Invent. Math. 150 (2002) 257
 Diao Y, Ernst C, Yu X "Hamiltonian knot projections and
thick knots" Topol. Appl. 136 (2004) 7
 Diao Y, Kusner R [work in progress]
|Norman Lebovitz (University of Chicago)
||The prospects for fission of self-gravitating masses
|Abstract: The idea that a single, rotating, self-gravitating
mass — like a star — can evolve into a pair of masses
orbiting one another — like a double-star — was suggested
over a century ago. The elaboration of the mathematical
details led to negative results and most astronomers
abandoned this idea in the 1920's. The negative results are
not decisive, however, and we discuss alternative mathematical
formulations of this problem and their prospects for positive
|John Lister (University of Cambridge)
||Capillary pinch-off of a film on a cylinder
|Abstract: Much of the work on capillary pinch-off, and on other
fluid-mechanical problems with changes in topology, has focused
on situations that lead to finite-time singularities in the
neighbourhood of which there is some kind of similarity
solution. Capillary instability in the absence of gravity of an
axisymmetric layer of fluid coating a circular cylinder is, by
contrast, an example of an infinite-time singularity. Even more
unusually, film rupture proceeds through an episodic series of
oscillations that form a diverging geometrical progression in
time, each of which reduces the remaining film thickness by a
factor of about 10.
|Fernando Lund (University of Chile)
||Ultrasound as a probe of plasticity?
The interaction between elastic waves and dislocations
|Abstract: Plasticity in metals and alloys is a mature discipline in the mechanics of materials. However, it appears that current theoretical modeling lacks predictive power. If a new form of steel, say, is fabricated, there appears to be no way of predicting its deformation and fracture behavior as a function of temperature, and/or cyclic loading. The root of this problem appears to be with the paucity of controlled experimental measurements, as opposed to visualizations, of the properties of dislocations, the defects that are responsible for plastic deformation of crystals. Indeed, the tool of choice in this area is transmission electron microscopy, which involves an intrusive measurement of specially prepared samples. Is it possible to develop non intrusive tools for the measurement of dislocation properties? Could ultrasound be used to this end? This talk will highlight recent developments in this line of thought.Specific results include a theory of the interaction of elastic, both longitudinal and transverse, bulk as well as surface, waves with dislocations, both in isolation and in arrays of large numbers, in two and three dimensions. Results for the isolated case can be checked with experimental results obtained using stroboscopic X-ray imaging. The theory for the many-dislocations case constitutes a generalization of the standard Granato-Lücke theory of ultrasound attenuation in metals, and it provides an explanation of otherwise puzzling results obtained with Resonant Ultrasound Spectroscopy (RUS). Application of the theoretical framework to low-angle grain boundaries, that can be modeled as arrays of dislocations, provides an understanding of recently obtained results concerning the power law behavior of acoustic attenuation in polycrystals. Current developments of instrumentation that may lead to a practical, non-intrusive probe of plastic behavior will be described.
|Ken Miller (Columbia University)
||Pattern formation in the visual system
|Abstract: No Abstract
|Andreas Münch (University of Nottingham)
||Self similar rupture of thin films with slippage
|Abstract: We recently developed a thin film model that
describes the rupture and dewetting of very thin
liquid polymer films where slip at the liquid/solid
interface is very large. In this talk, we investigate
the singularity formation at the moment of rupture
for this model, where we identify different similarity
|David R. Nelson (Harvard University)
||Buckled viruses, crumpled shells and folded pollen grains
|Abstract: The difficulty of constructing ordered states on
spheres was recognized by J. J. Thomson, who discovered the
electron and then attempted regular tilings of the sphere in an
ill-fated attempt to explain the periodic table. We first
discuss how protein packings in buckled virus shells solve a
related “Thomson problem”. We then describe the grain boundary
scars that appear on colloidosomes, drug delivery vehicles that
represent another class of solution to this problem. The
remarkable modifications in the theory necessary to account for
thermal fluctuations in crumpled amorphous shells of spider
silk proteins will be described as well. We then apply related
ideas to the folding strategies and shapes of pollen grains
during dehydration when they are released from the anther after
maturity. The grain can be modeled as a pressurized
high-Young-modulus sphere with a weak sector and a nonzero
spontaneous curvature. In the absence of such a weak sector,
these shells crumple irreversibly under pressure via a strong
first order phase transition. The weak sectors (both one and
three-sector pollen grains are found in nature) eliminate the
hystersis and allow easy rehydration at the pollination site,
somewhat like the collapse and subsequent reassembly of a folding chair.
|Theoden Netoff (University of Minnesota)
||Experimentalists' perspective–what can we measure?
|Abstract: No Abstract
|Timothy Newman (Arizona State University), Hans G. Othmer (University of Minnesota)
||Review and discussion of IMA workshop: Quantitative approaches
to cell motility and chemotaxis
|Abstract: This Wednesday, we are planning to discuss topics and questions arising from the recent IMA workshop on Quantitative Approaches to Cell Motility and Chemotaxis. Everyone is welcome to attend and take part. There will be no set agenda. Hans Othmer will lead the discussion. Hans has asked that everyone attending think over their experience at the workshop and distill important issues and open questions.
Looking forward to seeing you this Wednesday at 11:15 in the IMA seminar room.
|Timothy Newman (Arizona State University)
||Hans Othmer's group meeting: Recent measurements of the statistics of cell-sidedness in proliferating epithelial sheets
|Abstract: I will be speaking about recent measurements of the statistics of cell-sidedness in proliferating epithelial sheets. The reference (Gibson et al 2006).
I will be providing a critique of the model presented in this paper, and present some recent work in my own group which aims to improve the biological realism of the model.
|Jinhae Park (Purdue University)
||Static problems of the chiral smectic and bent core
liquid crystals focusing on the role of the spontaneous polarization
|Abstract: In this talk, I will present mathematical modeling of
crystals and discuss existence and partial regularity
results of minimum configurations in some special geometry. I
then speak about switching
problem between ferroelectric states and derive a formulae for
field. I will end my talk with the proof of hysteresis loop
spontaneous polarization and electric field which can be
applied to other materials including ferroeletric solids and
|Thomas J. Pence (Michigan State University)
||Singularities associated with swelling of hyperelastic solids
|Abstract: This talk will discuss certain singularities that
arise in the solution to boundary value problems involving the
swelling of otherwise hyperelastic solids. In this setting,
both non-uniform swelling and constrained swelling give rise to
nonhomogeneous deformation in the absence of externally applied
load. The standard singularities that are encountered in
nonlinear elasticity may occur, such as cavitation. Additional
singularities also arise, such as loss of smoothness associated
with the concentration of deformation on singular surfaces.
|Leonid Pismen (Technion-Israel Institute of Technology)
||Resolving dynamic singularities: from vortices to
|Abstract: When a physical object, which is perceived as a singularity on
a certain level of mathematical description, is set into
motion, a paradox may arise rendering dynamic description
impossible unless the singularity is resolved by introducing
new physics in the singular core. This situation, appearing in
diverse physical contexts, necessitates application of
multiscale matching methods, employing a simpler long-scale
model in the far field and a short-scale model with more
detailed physical contents in the core of the singularity. The
law of motion can be derived within this approach by applying a
modified Fredholm alternative in a region large compared to the
inner and small compared with the outer scale, and evaluating
the boundary terms which determine both the driving force and
dissipation. I give examples of applying this technique to both
topological (vortices) and non-topological (contact lines) singularities.
|Michael Renardy (Virginia Polytechnic Institute and State University)
||An open problem concerning breakup of fluid jets
|Abstract: We present a simple one-dimensional equation modeling slender
jets of a Newtonian fluid in Stokes flow. It would be desirable to have a proof
linking the asymptotics of surface tension driven breakup to the behavior of the
initial condition near the thinnest point of the jet. Despite the apparent
simplicity of the equations, the problem is open. I shall discuss some partial results.
|Sergio Rica (Centre National de la Recherche Scientifique (CNRS))
||Weak turbulence of a vibrating elastic thin plate
|Abstract: I will talk about a work in collaboration with G.
During and C. Josserand on the long-time evolution
of waves of a thin elastic plate in the limit of small
deformation so that modes of oscillations interact weakly. According to the
theory of weak turbulence (successfully applied in the past to plasma, optics,
and hydrodynamic waves), this nonlinear wave system evolves at
long times with a slow transfer of energy from one mode to
another. We derived a kinetic equation for the spectral transfer in terms of
the second order moment. We show that such a theory describes the approach
to an equilibrium wave spectrum and represents also an energy
cascade, often called the Kolmogorov-Zakharov spectrum. We perform
numerical simulations that confirm this scenario. Finally, I will discuss
recent experiments by A. Boudaoud and collaborators and N. Mordant.
|Jonathan E. Rubin (University of Pittsburgh)
||Introduction to the nervous system
|Abstract: No Abstract
|Jonathan E. Rubin (University of Pittsburgh)
||Simple models and networks
|Abstract: No Abstract
|Jonathan E. Rubin (University of Pittsburgh)
|Abstract: No Abstract
|John R. Savage (Cornell University)
||Dynamics of droplet breakup in a complex fluid
|Abstract: The dynamics of droplet breakup in Newtonian fluids are
described by the Navier-Stokes equation. Previous experiments
have shown that in many cases
the breakup dynamics follow a self-similar behavior where
successive drop profiles can be scaled onto one another. In
visco-elastic systems however,
the Navier-Stokes equation is not sufficient to describe
breakup. In this talk we will describe droplet breakup in a
visco-elastic surfactant system
which forms micellar, lamellar, and reverse-micellar phases at
various concentrations. We present results of the dynamics of
breakup in this system
and compare these to previously studied Newtonian systems.
|Michael Siegel (New Jersey Institute of Technology)
||Calculation of complex singular solutions to the 3D incompressible Euler equations
|Abstract: We describe an approach for the construction of singular
solutions to the 3D Euler equations for complex initial data.
The approach is based on a numerical simulation of complex traveling wave
solutions with imaginary wave speed, originally developed by Caflisch for
axisymmetric flow with swirl. Here, we simplify and generalize this
construction to calculate traveling wave solutions in a fully 3D
(nonaxisymmetric) geometry. Our new formulation avoids a numerical
instability that required the use of ultra-high precision arithemetic in the
axisymmetric flow calculations. This is joint work with Russ Caflisch.
|Jey Sivaloganathan (University of Bath)
||Singular minimisers in nonlinear elasticity and modelling fracture
|Abstract: We present an overview of a variational approach to modelling
fracture initiation in the framework of nonlinear elasticity.
The underlying principle is that energy minimizing deformations
of an elastic body may develop singularities when the body is
subjected to large boundary displacements or loads. These singularities often
bear a striking resemblance to fracture mechanisms observed in polymers.
Experiments indicate that voids may form in polymer samples
(that appear macroscopically perfect) when the samples are
subjected to large tensile stresses. This phenomenon of
cavitation can be viewed as the growth of infinitesimal
pre-existing holes in the material or as the spontaneous
creation of new holes in an initially perfect body. In this
talk we adopt both viewpoints simultaneously. Mathematically,
this is achieved by the use of deformations whose point
singularities are constrained to be at certain fixed points
(the "flaws" in the material). We show that, under suitable
hypotheses, the energetically optimal location for a single
flaw can be computed from a singular solution to a related
problem from linear elasticity.
One intriguing consequence of the above approach is that
cavitation may occur at a point which is not energetically
optimal. We show that such a disparity will produce
configurational forces (of a type previously identified in the
context of defects in crystals) and conjecture that this may
provide a mathematical explanation for crack initiation.
Much of the above work is joint with S.J. Spector (S. Illinois University).
|Dejan Slepčev (Carnegie Mellon University)
||Blowup dynamics of an unstable thin-film equation
|Abstract: Long-wave unstable thin-film equations exhibit rich
dynamical behavior: Solutions can spread indefinitely, converge to a
steady droplet configuration or blow up in finite time.
We will discuss the properties of scaling solutions that govern
the blowup dynamics. In particular, we will present how energy based
methods can be used to study the stability of selfsimilar blowup solutions
as well as other dynamical properties of the blowup solutions.
Strong connections to studies of blowup behavior in other equations will
|Scott J. Spector (Southern Illinois University)
||Some remarks on the symmetry of singular minimizers in elasticity
|Abstract: Experiments on elastomers have shown that triaxial tensions can
induce a material to exhibit holes that were not previously evident.
Analytic work in nonlinear elasticity has established that such
cavity formation may indeed be an elastic phenomenon: sufficiently
large prescribed boundary deformations yield a hole-creating
deformation as the energy minimizer whenever the elastic energy is of slow
In this lecture the speaker will discuss the use of
isoperimetric arguments to establish that a radial deformation, producing a
spherical cavity, is the energy minimizer in a general class of isochoric
deformations that are discontinuous at the center of a ball and produce a
(possibly non-symmetric) cavity in the deformed body. The key ingredient
is a new radial-symmetrization procedure that is appropriate for
problems where the symmetrized mapping must be one-to-one in order to prevent
interpenetration of matter.
|Paul H. Steen (Cornell University)
||Singularity theory and the inviscid pinch-off
|Abstract: Whitney's theorem tells us that folds and cusps are
generic in smooth mappings of a plane into a plane. Whitney's
work builds on Morse's and is extended by Thom's classification
of singularities of mappings (singularity theory). To the
extent that the pinch-off of an interface is a geometric
singularity, it is natural to ask what singularity theory says
about pinch-off. We explore this question for axisymmetric
surfaces. Curvature extrema, which coincide with either
curvature crossings or with profile extrema, are features whose
evolution can be tracked up to the instant of singularity. A
singularity theory classification is tested against
vortex-sheet simulations (theory) and against curvatures
extracted from images of evolving soap-films (experiment).
|Andrew M. Stein (University of Minnesota)
||The micromechanics of 3d collagen gels
|Abstract: Collagen is the most abundant animal protein and its mechanics have been studied in great detai. It takes on many morphologies, including skin, tendons, ligaments, individual fibers, and gels. Of particular interest is the mechanics of collagen-I gels. These gels provide a relatively simple structure that can be noninvasively observed by confocal microscopy and used as a scaffold for growing artificial tissues, and as a 3d environment for studying cell motility and tumor invasion. A critical first step in understanding these systems is to develop a model for the collagen gel alone. In this paper we give a successful theoretical model of the micromechanics of realistic networks.
|Sigurdur Thoroddsen (National University of Singapore)
||Singular jets in free-surface flows
|Abstract: Free-surface 'singular jetting' occurs in geometries where flow
focusing accelerates the free surface symmetrically towards a
line or a point. This is known to occur in a number of
configurations, such as during the collapse of free-surface
craters and of granular cavities as well as for capillary waves
converging at the apex of oscillating drops. Drops impacting
onto super-hydrophobic surfaces also generate such jets. We
will show recent work on characterizing such jetting, in
well-known and new jetting configurations. High-speed video
imaging, with frame-rates up to 1,000,000 fps, will be
presented and used for precise measurement of jet size and
velocity. The focus will be on three well-controlled
flow-configurations: During the crater collapse following the
impact of a drop onto a liquid pool and after the pinch-off of
a drop from a vertical nozzle. Finally, we will show a new
apex jet which is generated by the impact of a viscous drop
onto a lower-viscosity pool.
|Konstantin Turitsyn (University of Chicago)
||Singularity formation in two-dimensional free surface dynamics
|Abstract: Motivated by recent experiments on bubble pinch off
by Nathan Keim and Sid Nagel we study the nonlinear dynamics of
two-dimensional collapsing air bubble surrounded by ideal
fluid. We show that the dynamics can lead to several distinct
type of singularities: interface reconnections, cusps and
wedges. We analyze the critical dynamics of singularities
formation, and show that it is described by universal critical
exponents. Remarkably, there are strong similarities between
our system and the Hele-Shaw type systems. These similarities
support the conjecture that the critical dynamics of the free
interface is described by integrable equations.
|Emmanuel Villermaux (IRPHE - Institut de Recherche sur les Phénoménes Hors Équilibre)
||Fragmentation under impact
|Abstract: Fragmentation phenomena will be reviewed with a
particular emphasis on processes occurring with liquids, those
giving rise to drops (the case of solid fragmentation can
discussed also, depending on the audience requests). Examples
including impacts of different kinds, and raindrops will
specifically illustrate the construction mechanism of the drop
size distributions in the resulting spray.
|Barbara Wagner (Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS))
||Patterns in dewetting liquid films: Intermediate and late phases
|Abstract: We investigate the dynamics of a post-rupture thin liquid film
dewetting on a hydrophobised substrate driven by Van-der-Waals forces.
The stability of the three-phase contact line is discussed
numerically and asymptotically in the framework of lubrication models
by taking account of various degrees of slippage. The results are used to
explain some experimentally observed patterns.
Finally, we present some recent studies of the impact of slippage on
the late stages of the dynamics. Here, we present some novel coarsening
behaviour of arrays of interacting droplets.
|Guowei Wei (Michigan State University)
||Geometric flow approach to singularity formation and evolution
|Abstract: Geometric singularities are ubiquitous in nature, The
fascinating complexity of geometric singularities has attracted the
attention of mathematicians, engineers and physicists alike for
centuries. Geometric singularities commonly occur in multiphase systems at
the geometric boundaries. Their formation and evolution are often
accompanied with topological changes. In this talk, we argue
that the theory of differential geometry of curves and surfaces
provides a natural and unified description for the geometric
We show that geometric flows, particularly, the potential
driving geometric flows offer a powerful framework for the theoretical
analysis of singularity formation and evolution. Potential
driving geometric flows, derived from the Euler-Lagrange equation,
balance the intrinsic geometric forces, i.e., surface tension, with
potential forces. Geometric concepts, such as differentiable manifold,
tangent bundle, mean curvature and Gauss curvature, are utilized for
the construction of generalized geometric flows. The driving
potential can be the gravitation in describing the formation of droplets,
or have a double-wall structure in a phenomenological description
of phase separation, or be a collection of atomistic interactions
in a multiscale modeling of the solvation of biomolecules. Physical
properties, such as free energy minimization (area decreasing)
and incompressibility (volume preserving), are realized in our
paradigm of potential driving geometric flows. Finally, we discuss the
application of potential driving geometric flows to the multiscale
analysis of protein folding.
(1) P. Bates, G.W. Wei and S. Zhao, Minimal molecular surfaces and
their applications, J. Comput. Chem., 29, 380-391 (2008).
(2) S. N. Yu, W. H. Geng and G.W. Wei, Treatment of geometric
singularities in implicit solvent models, J. Chem. Phys., 126,
244108 (13 pages) (2007).
(3) P. W. Bates, Z. Chen, Y.H. Sun, G.W. Wei and S. Zhao,
Potential driving geometric flows, J. Math. Biology, in review (2008).
(4) G.W. Wei, Generalized Perona-Malik equation for image
restoration, IEEE Signal Processing Lett., 6, 165-167 (1999).
|Jon Wilkening (University of California)
||Lubrication theory in nearly singular geometries: when
should one stop optimizing a reduced model?
|Abstract: Shape optimization plays a central role in engineering and
biological design. However, numerical optimization of complex
systems that involve coupling of fluid mechanics to rigid or
flexible bodies can be prohibitively expensive (to implement
and/or run). A great deal of insight can often be gained by
optimizing a reduced model such as Reynolds' lubrication
approximation, but optimization within such a model can
sometimes lead to geometric singularities that drive the
solution out of its realm of validity. We present new rigorous
error estimates for Reynolds' approximation and its higher
order corrections that reveal how the validity of these reduced
models depend on the geometry. We use this insight to study
the problem of shape optimization of a sheet swimming over a
thin layer of viscous fluid.
|Thomas Peter Witelski (University of Oxford)
||Some open questions on similarity solutions for fluid film rupture
|Abstract: Finite-time topological rupture occurs in many models in fluid and
solid mechanics. We review and discuss some properties of the
self-similar solutions for such problems. Unresolved issues
regarding analytical forms of the solutions (stability and symmetry vs.
asymmetry) and numerical calculation methods (shooting vs. global
relaxation) will be highlighted. Further questions of interest arise in
post-rupture coarsening dynamics of dewetting thin films.