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IMA Newsletter #381

July 2008

News and Notes

Santosa begins his position as IMA director

Fadil Santosa starts his postion as the next director of the IMA on July 1, 2008. The staff and the directors of the IMA would like to give a warm welcome to Fadil and say thank you to Doug Arnold for his seven years of leadership and wish him well in his future endeavors.

Solicitation of Proposals for IMA Participating Institutions Conferences: All faculty members of the Participating Institutions of the Institute for Mathematics and its Applications are invited to submit proposals for the 2008-2009 IMA Participating Institution conferences. Deadline: November 15, 2008. Please see Solicitation of Proposals for more information.

IMA Events

PI Summer Graduate Program

Linear Algebra and Applications

June 30 - July 25, 2008

Organizers: Jason Grout (Iowa State University), Leslie Hogben (Iowa State University), Wolfgang Kliemann (Iowa State University), Yiu Tung Poon (Iowa State University)

Geometrical Singularities and Singular Geometries

July 14-25, 2008

Organizers: L. Mahadevan (Harvard University), Edward A. Spiegel (Columbia University), Thomas Witten (University of Chicago), Wendy Zhang (University of Chicago)
Schedule

Friday, July 4

All DayIndependence Day. The IMA is closed.

Monday, July 14

8:15a-9:00aRegistration and coffeeEE/CS 3-176 SP7.14-25.08
9:00a-9:15aWelcome to the IMAFadil Santosa (University of Minnesota)EE/CS 3-180 SP7.14-25.08
9:15a-10:05aIntroductory talkL. Mahadevan (Harvard University)EE/CS 3-180 SP7.14-25.08
10:05a-10:20aDiscussionEE/CS 3-180 SP7.14-25.08
10:20a-10:50aCoffeeEE/CS 3-176 SP7.14-25.08
10:50a-11:40aA catalogue of singularitiesJens Eggers (University of Bristol)EE/CS 3-180 SP7.14-25.08
11:40a-11:50aDiscussionEE/CS 3-180 SP7.14-25.08
11:50a-2:00pLunch SP7.14-25.08
2:00p-2:10pCalculation of complex singular solutions to the 3D incompressible Euler equationsMichael Siegel (New Jersey Institute of Technology)EE/CS 3-180 SP7.14-25.08
2:15p-2:25pTurbulent solutions of the stochastic Navier-Stokes equationBjorn Birnir (University of California)EE/CS 3-180 SP7.14-25.08
2:30p-2:40pSome open questions on similarity solutions for fluid film ruptureThomas Peter Witelski (University of Oxford)EE/CS 3-180 SP7.14-25.08
2:45p-2:55pCapillary pinch-off of a film on a cylinderJohn Lister (University of Cambridge)EE/CS 3-180 SP7.14-25.08
3:00p-3:30pSecond chancesEE/CS 3-180 SP7.14-25.08
3:45p-4:00pGroup Photo SP7.14-25.08
4:00p-6:00pReception and Poster SessionLind Hall 400 SP7.14-25.08

Tuesday, July 15

8:30a-9:00aCoffeeEE/CS 3-176 SP7.14-25.08
9:00a-9:50aAblative Rayleigh-Taylor instabilityPaul Clavin (UMR CNRS-Universites d'Aix-Marseille I&II)EE/CS 3-180 SP7.14-25.08
9:50a-10:00aDiscussionEE/CS 3-180 SP7.14-25.08
10:00a-10:30aBreakEE/CS 3-176 SP7.14-25.08
10:30a-11:20aHydraulic jump in a flowing soap filmWalter Goldburg (University of Pittsburgh)EE/CS 3-180 SP7.14-25.08
11:20a-11:30aDiscussionEE/CS 3-180 SP7.14-25.08
11:30a-2:00pLunch SP7.14-25.08
2:00p-2:10pSingularity theory and the inviscid pinch-off singularityPaul H. Steen (Cornell University)EE/CS 3-180 SP7.14-25.08
2:15p-2:25pBlowup dynamics of an unstable thin-film equationDejan Slepčev (Carnegie Mellon University)EE/CS 3-180 SP7.14-25.08
2:30p-2:40pAn open problem concerning breakup of fluid jets Michael Renardy (Virginia Polytechnic Institute and State University)EE/CS 3-180 SP7.14-25.08
2:45p-2:55pThe prospects for fission of self-gravitating massesNorman Lebovitz (University of Chicago)EE/CS 3-180 SP7.14-25.08
3:00p-3:30pSecond chancesEE/CS 3-180 SP7.14-25.08
6:30p-8:30pWorkshop DinnerTBA SP7.14-25.08

Wednesday, July 16

8:30a-9:00aCoffeeEE/CS 3-176 SP7.14-25.08
9:00a-9:50aTBASidney Nagel (University of Chicago)EE/CS 3-176 SP7.14-25.08
9:50a-10:00aDiscussionEE/CS 3-180 SP7.14-25.08
10:00a-10:30aBreakEE/CS 3-176 SP7.14-25.08
10:30a-11:20aSome remarks on vorticity growth in Euler flowsStephen Childress (New York University)EE/CS 3-180 SP7.14-25.08
11:20a-11:30aDiscussionEE/CS 3-180 SP7.14-25.08
11:30a-2:00pLunch SP7.14-25.08
2:00p-2:10pViscous potential flow analysis of radial fingering in a Hele-Shaw cellDaniel D. Joseph (University of Minnesota)EE/CS 3-180 SP7.14-25.08
2:15p-2:25pLubrication theory in nearly singular geometries: when should one stop optimizing a reduced model?Jon Wilkening (University of California)EE/CS 3-180 SP7.14-25.08
2:30p-2:40pFragmentation under impactEmmanuel Villermaux (IRPHE - Institut de Recherche sur les Phénoménes Hors Équilibre)EE/CS 3-180 SP7.14-25.08
2:45p-2:55pDynamics of droplet breakup in a complex fluidJohn R. Savage (Cornell University)EE/CS 3-180 SP7.14-25.08
3:00p-3:30pSecond chancesEE/CS 3-176 SP7.14-25.08
7:00p-8:00pDiscussion SessionEE/CS 3-180 SP7.14-25.08

Thursday, July 17

8:30a-9:00aCoffeeEE/CS 3-176 SP7.14-25.08
9:00a-9:50aA new approach to regularity and singularity questions for a class of non-linear evolutionary PDEs such as 3-D Navier-Stokes equationSaleh A. Tanveer (Ohio State University)EE/CS 3-180 SP7.14-25.08
9:50a-10:00aDiscussionEE/CS 3-180 SP7.14-25.08
10:00a-10:30aCoffeeEE/CS 3-176 SP7.14-25.08
10:30a-11:20aTBADavid Quere (École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI))EE/CS 3-180 SP7.14-25.08
11:20a-11:30aDiscussionEE/CS 3-180 SP7.14-25.08
11:30a-2:00pLunch
2:00p-2:10pSingular jets in free-surface flowsSigurdur Thoroddsen (National University of Singapore)EE/CS 3-180 SP7.14-25.08
2:15p-2:25pHigh-speed jet formation after solid object impactStephan Gekle (Universiteit Twente)EE/CS 3-180 SP7.14-25.08
2:30p-2:40pCapillary windingJosé Bico (École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI)) SP7.14-25.08
2:45p-2:55pChaos in a one-dimensional cardiac modelDavid Schaeffer (Duke University)EE/CS 3-180 SP7.14-25.08
3:00p-3:30pSecond chancesEE/CS 3-180 SP7.14-25.08

Friday, July 18

8:30a-9:00aCoffeeEE/CS 3-176 SP7.14-25.08
9:00a-9:50aTBAWendy Zhang (University of Chicago)EE/CS 3-180 SP7.14-25.08
9:50a-10:00aDiscussionEE/CS 3-180 SP7.14-25.08
10:00a-10:30aCoffeeEE/CS 3-176 SP7.14-25.08
10:30a-11:20aPatterns in dewetting liquid films: Intermediate and late phasesBarbara Wagner (Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS))EE/CS 3-180 SP7.14-25.08
11:20a-11:50aSecond ChancesEE/CS 3-180 SP7.14-25.08

Saturday, July 19

All DayNo scheduled activity
All DayNo scheduled activity SP7.14-25.08

Sunday, July 20

All DayNo scheduled activity SP7.14-25.08

Monday, July 21

8:30a-9:00aCoffeeEE/CS 3-176 SP7.14-25.08
9:00a-9:50aThe zero temperature limit of interacting corporaPeter Constantin (University of Chicago)EE/CS 3-180 SP7.14-25.08
9:50a-10:00aDiscussionEE/CS 3-180 SP7.14-25.08
10:00a-10:30aBreakEE/CS 3-176 SP7.14-25.08
10:30a-11:20aResolving dynamic singularities: from vortices to contact linesLeonid Pismen (Technion-Israel Institute of Technology)EE/CS 3-180 SP7.14-25.08
11:20a-11:30aDiscussionEE/CS 3-180 SP7.14-25.08
11:30a-2:00pLunch SP7.14-25.08
2:00p-2:10pGeometric flow approach to singularity formation and evolutionGuowei Wei (Michigan State University)EE/CS 3-180 SP7.14-25.08
2:15p-2:25pSingularities in Calabi-Yau varieties Mee Seong Im (University of Illinois at Urbana-Champaign)EE/CS 3-180 SP7.14-25.08
2:30p-2:40pWhat's new for microstructure David Kinderlehrer (Carnegie Mellon University)EE/CS 3-180 SP7.14-25.08
2:45p-2:55pElastic theory of non-Euclidean platesEfi Efrati (Hebrew University)EE/CS 3-180 SP7.14-25.08
3:00p-3:30pSecond chancesEE/CS 3-180 SP7.14-25.08
3:30p-4:20pGeneral relativity from A to A-Edward A. Spiegel (Columbia University)EE/CS 3-180 SP7.14-25.08
4:30p-6:30pPoster session and receptionLind Hall 400 SP7.14-25.08

Tuesday, July 22

8:30a-9:00aCoffeeEE/CS 3-176 SP7.14-25.08
9:00a-9:50aSingular minimisers in nonlinear elasticity and modelling fractureJey Sivaloganathan (University of Bath)EE/CS 3-176 SP7.14-25.08
9:50a-10:00aDiscussionEE/CS 3-180 SP7.14-25.08
10:00a-10:30aCoffeeEE/CS 3-176 SP7.14-25.08
10:30a-11:20aTopological kinematics of point vortex motionsPhilip Boyland (University of Florida)EE/CS 3-180 SP7.14-25.08
11:20a-11:30aDiscussionEE/CS 3-180 SP7.14-25.08
11:30a-2:00pLunch SP7.14-25.08
2:00p-2:10pSingularities associated with swelling of hyperelastic solidsThomas J. Pence (Michigan State University)EE/CS 3-180 SP7.14-25.08
2:15p-2:25pSome remarks on the symmetry of singular minimizers in elasticityScott J. Spector (Southern Illinois University)EE/CS 3-180 SP7.14-25.08
2:30p-2:40pInvestigating dislocation dynamics in degenerate crystals of dimer colloidsItai Cohen (Cornell University) SP7.14-25.08
2:45p-2:55pPoint-instabilities, point-coercivity (meta-stability), and point-calculusEvan Hohlfeld (University of California)EE/CS 3-180 SP7.14-25.08
3:00p-3:30pSecond chancesEE/CS 3-180 SP7.14-25.08
3:30p-3:45pGroup Photo SP7.14-25.08
6:30p-8:30aWorkshop dinner SP7.14-25.08

Wednesday, July 23

8:30a-9:00aCoffeeEE/CS 3-176 SP7.14-25.08
9:00a-9:50aUltrasound as a probe of plasticity? The interaction between elastic waves and dislocationsFernando Lund (University of Chile)EE/CS 3-180 SP7.14-25.08
9:50a-10:00aDiscussionEE/CS 3-180 SP7.14-25.08
10:00a-10:30aCoffeeEE/CS 3-176 SP7.14-25.08
10:30a-11:20aTBAItamar Procaccia (Weizmann Institute of Science)EE/CS 3-180 SP7.14-25.08
11:20a-11:30aDiscussionEE/CS 3-180 SP7.14-25.08
11:30a-2:00pLunch SP7.14-25.08
2:00p-2:10pSelf similar rupture of thin films with slippage Andreas Münch (University of Nottingham)EE/CS 3-180 SP7.14-25.08
2:15p-2:25pStatic problems of the chiral smectic and bent core liquid crystals focusing on the role of the spontaneous polarizationJinhae Park (Purdue University)EE/CS 3-180 SP7.14-25.08
2:30p-2:40pOn characteristic classes for the gravitational field and black holesMihaela D. Iftime (Boston University)EE/CS 3-180 SP7.14-25.08
2:45p-2:55pSingularity formation in two-dimensional free surface dynamicsKonstantin Turitsyn (University of Chicago)EE/CS 3-180 SP7.14-25.08
3:00p-3:30pSecond chancesEE/CS 3-180 SP7.14-25.08
7:00p-8:00aDiscussion SessionEE/CS 3-180 SP7.14-25.08

Thursday, July 24

8:30a-9:00aCoffeeEE/CS 3-176 SP7.14-25.08
9:00a-9:50aTopological singularities in optical wavesMark Dennis (University of Bristol)EE/CS 3-180 SP7.14-25.08
9:50a-10:00aDiscussionEE/CS 3-180 SP7.14-25.08
10:00a-10:30aCoffeeEE/CS 3-176 SP7.14-25.08
10:30a-11:20aTBAMaria-Carme T. Calderer (University of Minnesota)EE/CS 3-180 SP7.14-25.08
11:20a-11:30aDiscussionEE/CS 3-180 SP7.14-25.08
11:30a-2:00pLunch SP7.14-25.08
2:00p-2:10pExperimental investigations of packing, folding, and crumpling in two and three dimensionsArshad Kudrolli (Clark University)EE/CS 3-180 SP7.14-25.08
2:15p-2:25pBuckled viruses, crumpled shells and folded pollen grains David R. Nelson (Harvard University)EE/CS 3-180 SP7.14-25.08
2:30p-2:40pTBALaurent Boué (École Normale Supérieure)EE/CS 3-180 SP7.14-25.08
2:45p-2:55pWeak turbulence of a vibrating elastic thin plateSergio Rica (Centre National de la Recherche Scientifique (CNRS))EE/CS 3-180 SP7.14-25.08
3:00p-3:10pSingular behaviors in drop impacts Christophe Josserand (Université de Paris VI (Pierre et Marie Curie))EE/CS 3-180 SP7.14-25.08
3:15p-3:45pSecond chancesEE/CS 3-180 SP7.14-25.08

Friday, July 25

8:30a-9:00aCoffeeEE/CS 3-176 SP7.14-25.08
9:00a-9:50aThe geometry of topological defectsRandall D. Kamien (University of Pennsylvania)EE/CS 3-180 SP7.14-25.08
9:50a-10:00aDiscussionEE/CS 3-180 SP7.14-25.08
10:00a-10:30aCoffeeEE/CS 3-176 SP7.14-25.08
10:30a-11:20aLengths and crossing numbers of tightly knotted ropes and bandsRobert B. Kusner (University of Massachusetts)EE/CS 3-180 SP7.14-25.08
11:20a-11:50aSecond chancesEE/CS 3-180 SP7.14-25.08
11:50a-12:00pConcluding remarksEE/CS 3-180 SP7.14-25.08

Event Legend:
SP7.14-25.08Geometrical Singularities and Singular Geometries
Abstracts
José Bico (École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI)) Capillary winding
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 Navier-Stokes equation
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 determines 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.
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.
Evan Hohlfeld (University of California) Point-instabilities, point-coercivity (meta-stability), and point-calculus
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 Correspondence.
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.
Christophe Josserand (Université de Paris VI (Pierre et Marie Curie)) Singular behaviors in drop impacts
Abstract: I will discuss different singular behaviors that arise when one consider the impact of drop on thin liquid films or solid surface. For instance, singularities can be observed for low velocity impacts on super-hydrophobic surface, related to classical surface singularities. I will then discuss in more details the condition of prompt splash when an impact is made on a thin liquid film. Self-similar behaviors are then exhibited which allow a simplified understanding of empirical scaling laws.
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.
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 approximately linear relationship between the combinatorial complexity of knotted DNA and the distance traveled in gel electrophoresis experiments [1]. Modeling the DNA as tightly knotted rope of uniform thickness, it was suggested that lengths of such tight knots (rescaled to have unit thickness) would grow linearly with crossing numbers, a simple measure of knot complexity. It turned out that this relationship is more subtle: some families of knots have lengths growing as the the 3/4 power of crossing numbers, others grow linearly, all powers between 3/4 and 1 can be realized as growth rates, and it could be proven that that the power cannot exceed 2 [2-5]. It is still unknown whether there are families of tight knots whose lengths grow faster than linearly with crossing numbers, but the largest power has been reduced to 3/2 [6]. We will survey these and more recent developments in the geometry of tightly packed or knotted ropes, as well as some other physical models of knots as flattened ropes or bands which exhibit similar length versus complexity power laws, some of which we can now prove are sharp [7].

References:

[1] Stasiak A, Katritch V, Bednar J, Michoud D, Dubochet J "Electrophoretic mobility of DNA knots" Nature 384 (1996) 122

[2] Cantarella J, Kusner R, Sullivan J "Tight knot values deviate from linear relation" Nature 392 (1998) 237

[3] Buck G "Four-thirds power law for knots and links" Nature 392 (1998) 238

[4] Buck G, Jon Simon "Thickness and crossing number of knots" Topol. Appl. 91 (1999) 245

[5] Cantarella, J, Kusner R, Sullivan J "On the minimum ropelength of knots and links" Invent. Math. 150 (2002) 257

[6] Diao Y, Ernst C, Yu X "Hamiltonian knot projections and lengths of thick knots" Topol. Appl. 136 (2004) 7

[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 outcomes.
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.
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 regimes.
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.
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 ferroelectric liquid crystals and discuss existence and partial regularity results of minimum configurations in some special geometry. I will then speak about switching problem between ferroelectric states and derive a formulae for critical field. I will end my talk with the proof of hysteresis loop between the spontaneous polarization and electric field which can be applied to other materials including ferroeletric solids and ferromagnetics.
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 contact lines
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.
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.
David Schaeffer (Duke University) Chaos in a one-dimensional cardiac model
Abstract: Under rapid periodic pacing, cardiac cells typically undergo a period-doubling bifurcation in which action potentials of short and long duration alternate with one another. If these action potentials propagate in a fiber, the short-long alternation may suffer abrupt reversals of phase at various points along the fiber, a phenomenon called (spatially) discordant alternans. Either stationary or moving patterns are possible. Echebarria and Karma proposed an approximate equation to describe the spatiotemporal dynamics of small-amplitude alternans in a class of simple cardiac models, and they showed that an instability in this equation predicts the spontaneous formation of discordant alternans. We show that for certain parameter values a degenerate steady-state/Hopf bifurcation occurs at a multiple eigenvalue. Generically, such a bifurcation leads one to expect chaotic solutions nearby, and we perform simulations that find such behavior. Chaotic solutions in a one-dimensional cardiac model are rather surprising--typically chaos in the cardiac system has occurred from the breakup of spiral waves in two dimensions.
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 be indicated.
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 growth. 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 singularity
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).
Saleh A. Tanveer (Ohio State University) A new approach to regularity and singularity questions for a class of non-linear evolutionary PDEs such as 3-D Navier-Stokes equation
Abstract: Joint work with Ovidiu Costin, G. Luo. We consider a new approach to a class of evolutionary PDEs where question of global existence or lack of it is tied to the asymptotics of solution to a non-linear integral equation in a dual variable whose solution has been shown to exist a priori. This integral equation approach is inspired by Borel summation of a formally divergent series for small time, but has general applicability and is not limited to analytic initial data. In this approach, there is no blow-up in the variable p, which is dual to 1/t or some power 1/tn; solutions are known to be smooth in p and exist globally for p in R+. Exponential growth in p, for different choice of n, signifies finite time singularity. On the other hand, sub-exponential growth implies global existence. Further, unlike PDE problems where global existence is uncertain, a discretized Galerkin approximation to the associated integral equation has controlled errors. Further, known integral solution for p in [0, p0], numerically or otherwise, gives sharper analytic bounds on the exponents in p and hence better estimate on the existence time for the associated PDE. We will also discuss particular results for 3-D Navier-Stokes and discuss ways in which this method may be relevant to numerical studies of finite time blow-up problems.
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 singularities. 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.

References:

(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.
Visitors in Residence
Hillel Aharoni Hebrew University 7/13/2008 - 7/26/2008
Donald G. Aronson University of Minnesota 9/1/2002 - 8/31/2009
Daniel J. Bates University of Minnesota 9/1/2006 - 8/31/2008
Yermal Sujeet Bhat University of Minnesota 9/1/2006 - 8/31/2008
José Bico École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI) 7/13/2008 - 7/25/2008
Bjorn Birnir University of California 7/13/2008 - 7/19/2008
Laurent Boué École Normale Supérieure 7/12/2008 - 7/26/2008
Philip Boyland University of Florida 7/13/2008 - 7/25/2008
Justin C. Burton University of California 7/13/2008 - 7/25/2008
Maria-Carme T. Calderer University of Minnesota 7/14/2008 - 7/25/2008
Hannah Callender University of Minnesota 9/1/2007 - 8/31/2009
Enrique Cerda Villablanca University of Santiago de Chile 7/13/2008 - 7/25/2008
Stephen Childress New York University 7/13/2008 - 7/25/2008
Christophe Clanet École Polytechnique 7/13/2008 - 7/25/2008
Paul Clavin UMR CNRS-Universites d'Aix-Marseille I&II 7/13/2008 - 7/25/2008
Itai Cohen Cornell University 7/13/2008 - 7/25/2008
Peter Constantin University of Chicago 7/20/2008 - 7/25/2008
Ludovica Cecilia Cotta-Ramusino University of Minnesota 10/1/2007 - 8/30/2009
Robert Deegan University of Michigan 7/13/2008 - 7/20/2008
Mark Dennis University of Bristol 7/13/2008 - 7/25/2008
Olivier Dubois University of Minnesota 9/3/2007 - 8/31/2009
Efi Efrati Hebrew University 7/13/2008 - 7/26/2008
Jens Eggers University of Bristol 7/13/2008 - 7/25/2008
Marco Antonio Fontelos Consejo Superior de Investigaciones Científicas (CSIC) 7/13/2008 - 7/25/2008
Gilles Andre Francfort Université de Paris XIII (Paris-Nord) 7/13/2008 - 7/23/2008
Luiz Carlos Garcia de Andrade Universidade do Estado do Rio de Janeiro 7/13/2008 - 7/25/2008
Stephan Gekle Universiteit Twente 7/12/2008 - 7/19/2008
Walter Goldburg University of Pittsburgh 7/13/2008 - 7/18/2008
Jason E. Gower University of Minnesota 9/1/2006 - 8/31/2008
Milena Hering University of Minnesota 9/1/2006 - 8/31/2008
Peter Hinow University of Minnesota 9/1/2007 - 8/31/2009
Evan Hohlfeld University of California 7/13/2008 - 7/25/2008
Anette (Peko) Hosoi Massachusetts Institute of Technology 7/13/2008 - 7/25/2008
Mihaela D. Iftime Boston University 7/13/2008 - 7/24/2008
Mee Seong Im University of Illinois at Urbana-Champaign 7/12/2008 - 7/25/2008
Richard D. James University of Minnesota 7/13/2008 - 7/25/2008
Sookyung Joo University of California 7/13/2008 - 7/26/2008
Daniel D. Joseph University of Minnesota 7/13/2008 - 7/25/2008
Christophe Josserand Université de Paris VI (Pierre et Marie Curie) 7/20/2008 - 7/27/2008
Leo Kadanoff University of Chicago 7/13/2008 - 7/25/2008
Randall D. Kamien University of Pennsylvania 7/20/2008 - 7/25/2008
Lami Kim Seoul National University 7/13/2008 - 7/26/2008
David Kinderlehrer Carnegie Mellon University 7/13/2008 - 7/25/2008
Maciek Dominik Korzec Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS) 7/12/2008 - 7/25/2008
Arshad Kudrolli Clark University 7/13/2008 - 7/25/2008
Robert B. Kusner University of Massachusetts 7/13/2008 - 7/25/2008
Norman Lebovitz University of Chicago 7/13/2008 - 7/25/2008
Eun Joo Lee Seoul National University 7/13/2008 - 7/26/2008
Anton Leykin University of Minnesota 8/16/2006 - 8/15/2008
Chun-Chi Lin National Taiwan Normal University 7/13/2008 - 7/26/2008
John Lister University of Cambridge 7/13/2008 - 7/18/2008
Yang Liu University of Georgia 7/13/2008 - 7/25/2008
Kaspar Andreas Loeffel Massachusetts Institute of Technology 7/19/2008 - 7/25/2008
John Lowengrub University of California 7/13/2008 - 7/25/2008
Fernando Lund University of Chile 7/20/2008 - 7/25/2008
L. Mahadevan Harvard University 7/13/2008 - 7/25/2008
Shreyas Mandre Harvard University 7/13/2008 - 7/26/2008
Carl Modes University of Pennsylvania 7/13/2008 - 7/26/2008
Andreas Münch University of Nottingham 7/14/2008 - 7/25/2008
Sidney Nagel University of Chicago 7/13/2008 - 7/25/2008
David R. Nelson Harvard University 7/20/2008 - 7/26/2008
Jinhae Park Purdue University 7/13/2008 - 7/26/2008
Thomas J. Pence Michigan State University 7/13/2008 - 7/25/2008
Leonid Pismen Technion-Israel Institute of Technology 7/13/2008 - 7/26/2008
Trevor Potter University of California 7/12/2008 - 7/26/2008
Itamar Procaccia Weizmann Institute of Science 7/13/2008 - 7/25/2008
David Quere École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI) 7/15/2008 - 7/20/2008
Michael Renardy Virginia Polytechnic Institute and State University 7/13/2008 - 7/25/2008
Sergio Rica Centre National de la Recherche Scientifique (CNRS) 7/20/2008 - 7/26/2008
Leif Ristroph Cornell University 7/13/2008 - 7/18/2008
Fadil Santosa University of Minnesota 7/1/2008 - 6/30/2010
John R. Savage Cornell University 7/13/2008 - 7/25/2008
David Schaeffer Duke University 7/13/2008 - 7/20/2008
Deena Schmidt University of Minnesota 9/1/2007 - 8/31/2009
Laura Schmidt University of Chicago 7/13/2008 - 7/25/2008
Shaun Sellers Washington University 7/13/2008 - 7/26/2008
Chehrzad Shakiban University of Minnesota 9/1/2006 - 8/31/2008
Michael Shearer North Carolina State University 7/14/2008 - 7/19/2008
Michael Siegel New Jersey Institute of Technology 7/13/2008 - 7/25/2008
Jey Sivaloganathan University of Bath 7/13/2008 - 7/25/2008
Dejan Slepčev Carnegie Mellon University 7/12/2008 - 7/25/2008
Scott J. Spector Southern Illinois University 7/13/2008 - 7/26/2008
Edward A. Spiegel Columbia University 7/13/2008 - 7/25/2008
Paul H. Steen Cornell University 7/13/2008 - 7/17/2008
Andrew M. Stein University of Minnesota 9/1/2007 - 8/31/2009
Peter Taborek University of California 7/13/2008 - 7/25/2008
Saleh A. Tanveer Ohio State University 7/15/2008 - 7/20/2008
Sigurdur Thoroddsen National University of Singapore 7/13/2008 - 7/25/2008
Carl Toews Duquesne University 7/27/2008 - 7/29/2008
Konstantin Turitsyn University of Chicago 7/13/2008 - 7/25/2008
Erkan Tüzel University of Minnesota 9/1/2007 - 8/31/2009
Henrik Bernhard van Lengerich Cornell University 7/13/2008 - 7/25/2008
Emmanuel Villermaux IRPHE - Institut de Recherche sur les Phénoménes Hors Équilibre 7/15/2008 - 7/20/2008
Vincenzo Vitelli University of Pennsylvania 7/13/2008 - 7/26/2008
Barbara Wagner Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS) 7/13/2008 - 7/25/2008
Zhian Wang University of Minnesota 9/1/2007 - 8/31/2009
Guowei Wei Michigan State University 7/13/2008 - 7/25/2008
Jon Wilkening University of California 7/13/2008 - 7/25/2008
Thomas Peter Witelski University of Oxford 7/13/2008 - 7/19/2008
Thomas Witten University of Chicago 7/13/2008 - 7/25/2008
Giovanni Zanzotto Università di Padova 7/13/2008 - 7/25/2008
Victor Zernov South Bank University 7/13/2008 - 7/25/2008
Hongchao Zhang University of Minnesota 9/1/2006 - 8/31/2008
Wendy Zhang University of Chicago 7/13/2008 - 7/25/2008
Legend: Postdoc or Industrial Postdoc Long-term Visitor
IMA Affiliates:
3M, Arizona State University, Boeing, Carnegie Mellon University, Corning, ExxonMobil, Ford, General Electric, General Motors, Georgia Institute of Technology, Honeywell, IBM, Indiana University, Iowa State University, Johnson & Johnson, Kent State University, Lawrence Livermore National Laboratory, Lockheed Martin, Los Alamos National Laboratory, Medtronic, Michigan State University, Michigan Technological University, Microsoft Research, Mississippi State University, Motorola, Northern Illinois University, Ohio State University, Pennsylvania State University, Purdue University, Rice University, Rutgers University, Sandia National Laboratories, Schlumberger-Doll, Schlumberger-Doll Research, Seoul National University, Siemens, Telcordia, Texas A & M University, University of Central Florida, University of Chicago, University of Cincinnati, University of Delaware, University of Houston, University of Illinois at Urbana-Champaign, University of Iowa, University of Kentucky, University of Maryland, University of Michigan, University of Minnesota, University of Notre Dame, University of Pittsburgh, University of Tennessee, University of Texas, University of Wisconsin, University of Wyoming, US Air Force Research Laboratory, Wayne State University, Worcester Polytechnic Institute