Poster Sessions
Poster Blitz is on Tuesday, July 19, 5:00pm-5:40pm: Poster presenters will have 2 minutes to introduce their work with a one-two page beamer in the background. If you are presenting a poster please send the following to Nancy Watson by July 1st, 2016:
► Title and a brief abstract
► 1 to 2 pages of PDF slides for the Poster Blitz.
The best posters will receive prizes which will include certificates and financial awards totaling $1000.
Poster Blitz
Presenters will be in alphabetical order:
Davit Harutyunyan
Oliver Kanschat-Krebs
Arkadz Kirshtein
Yin Yang Lu
Matthew S. Mizuhara
Christopher Policastro
Mykhailo Potomkin
Matteo Rinaldi
Shawn Ryan
Xiaochuan Tian
Chong Wang
KiHyun Yun
Jia Zhao
Maxim Zyskin
Poster Prizes Ceremony
Wednesday, July 20, 12:00pm-12:10pm
Posters Abstracts
► Davit Harutyunyan (University of Utah)Gaussian Curvature as an Identifier of Shell Rigidity: Progress in the Theory of Shell Buckling► Oliver Kanschat-Krebs (University of Augsburg)
Abstract: In the present work we deal with shells with non-zero Gaussian curvature cut in the lines of the two orthogonal principal directions. We derive sharp Korn's first inequalities (linear geometric rigidity estimates) on that kind of shells for zero or periodic Dirichlet, Neumann, and Robin type boundary conditions. We prove, that if the Gaussian curvature is positive, then the optimal constant in the first Korn inequality scales like $h,$ and if the Gaussian curvature is negative, then the Korn constant scales like $h^{4/3},$ where $h$ is the thickness of the shell. This results can be viewed as classical in continuum mechanics, in particular shell theory, and and are the linear version of the famous geometric rigidity estimate by Frisecke, James and Müller for plates, 2002 (where they show that the Korn constant in the nonlinear Korn's first inequality scales like $h^2$), extended to shells with nonzero curvature. They can also be applied to find the scaling law for the critical buckling load of the shell under dead loads in one of the principal directions as well as to derive energy scaling laws in the pre-buckled regime. The exponents $1$ and $4/3$ appear in any sharp geometric rigidity estimate for the first time in the present work.
Periodic Homogenization of a Highly Heterogeneous Stefan Problem► Arkadz Kirshtein (Penn State University)
Abstract: Periodic homogenization of Stefan problems is an endeavour to understand a range of processes which include phase changes on small scales. As an example, it appears in models for harvesting maple syrup. Far-reaching results by A. Visintin from 2007 cover cases of certain media. However, highly heterogeneous media seem to play an important role, requiring an extension of the prior results. The current state of this progress is presented in the poster.
Energetic Variational Approach to Multi-Component Fluid Flows► Yin Yang Lu (Department of Mathematics and Statistics, McGill University)
Abstract: We use energetic variational approach and diffusive interface method to model multi-component flows. These variational approaches are motivated by the seminal works of Rayleigh and Onsager. The advantage of this approach is that we have to postulate only energy law and some kinematic relations based on fundamental physical principles. The method gives a clear, quick and consistent way to derive the PDE system. We explore different boundary phenomena in the case of two-component flows, as well as different approaches to modeling multi-component fluid flows. Techniques that are equivalent in the case of two phases, produce non-equivalent ternary systems. We are exploring advantages and disadvantages of each approach.
A Gradient Flow Approach to Coarsening in Polycrystalline Materials: The Grain Boundary Characteristic Distribution► Matthew S. Mizuhara (Pennsylvania State University)
Abstract: Polycrystalline materials are materials composed of several single crystals. They encompass a wide class of technologically useful materials. Predicting the coarsening in polycrystalline materials is desirable. One such theory is the theory of Grain Boundary Characteristic Distribution (GBCD), proposed by Kinderlehrer et al. The theory of GBCD employs a gradient flow approach, by describing the behavior or polycrystalline materials as a curve of steepest descent of the Gibbs free energy. Originally rigorously justified only in the 1D case, it has been recently extended to 2D case. Based on joint work with David Kinderlehrer.
Cell Motility Analytical and Numerical Study of the Motion of a Cell Membrane► Christopher Policastro (Berkeley Mathematics)
Abstract: The study of crawling eukaryotic cells has been of recent interest to biologists and mathematicians. Their motion is modeled by a 2D phase field consisting of an Allen-Cahn PDE coupled with a vectorial reaction-diffusion equation. The sharp interface limit (SIL) yields a non-linear and non-local evolution equation for the cell membrane. To prove short-time existence of curves propagating via this evolution we equivalently prove existence of solutions of a fully non-linear PDE. We prove short-time existence and uniqueness of solutions to the SIL and establish non-existence of traveling waves in a certain physical parameter regime. To compare this evolution to classical mean curvature motion, we develop an algorithm which resolves the non-linearity and non-locality of the SIL. We present simulations which show a departure from curvature motion dependent on the physical parameter and initial geometry. Part of this work was completed with my Ph.D. adviser L. Berlyand in collaboration with V. Rybalko and L. Zhang; the remainder is in progress with P. Zhang.
Estimates for volume preserving approximations► Mykhailo Potomkin (Mathematics Department, Pennsylvania State University)
Abstract: Analogues of the Korn inequality are formulated for $sl(n)$ and $sp(2n)$. Following a discussion of the matrix nearness problem for $SL(n)$, variants of the Friesecke-James-Müller inequality are formulated for $SL(n)$ and $Sp(2n)$. The approach requires bounding the approximation by a volume preserving map arising from the Brenier decomposition. For $n=3$ the estimates apply to nearly incompressible materials.
A Nonlinear PDE for 2-Point Correlation Function of a Many Particle System► Matteo Rinaldi (Department of Mathematical Sciences, Carnegie Mellon University)
Abstract: Systems of interacting particles described by a large number of coupled ODEs with random initial conditions appear in many problems of physics, cosmology, chemistry, biology, social science and economics. Many important phenomena in such systems may be characterized by 2-point correlation function which can be found by an appropriate truncation of the BBGKY hierarchy. In this poster I will present the derivation of such a truncation and the resulting PDE which is nonlinear and nonlocal, but instead is a closed form equation for 2-point correlation function. Moreover, it will be shown that key properties of 2-point correlation function are preserved by the truncation, and numerical comparison with direct simulations of the original coupled ODE system will be presented. This is joint work with L. Berlyand (PSU) and P.-E. Jabin (U. of Maryland).
Slow Motion for the One-Dimensional Swift-Hohenberg Equation► Shawn Ryan (Department of Mathematical Sciences, Kent State University)
Abstract: The behavior of solutions of the Swift-Hohenberg equation in a bounded interval $I \subset \mathbb{R}$ with periodic boundary conditions is studied. Combining results from $\Gamma$-convergence and ODE theory it is shown that solutions that start $L^1$-close to a jump function $v$, remain close to $v$. This can be achieved by regarding the equation as the $L^2$-gradient flow of a given energy functional and studying the asymptotic behavior of solutions of its Euler-Lagrange equation. The linearization of such equation provides almost sharp estimates on the tail of the associated energy.
Curvature Driven Foam Coarsening on the Sphere► Xiaochuan Tian (Columbia University)
Abstract: The von Neumann-Mullins law for the area evolution of a cell in the plane describes how a dry foam coarsens in time. Recent theory and experiment suggest that the dynamics are different on the surface of a three-dimensional object such as a sphere. This work considers the dynamics of dry foams on the surface of a sphere. Starting from first principles, we use computer simulation to show that curvature driven motion of the cell boundaries leads to exponential growth and decay of the areas of cells, in contrast to the planar case where the growth is linear. We describe the evolution and distribution of cells to the final stationary state.
Trace Theorems for some Nonlocal Function Spaces► Chong Wang (George Washington University)
Abstract: It is a classical result on Sobolev spaces that any $H^1$ function has a well-defined $H^{1/2}$ trace on the boundary of a domain with enough smoothness. In this work, we present a nonlocal extension of such a trace theorem in a new function space which contains $H^1$ as a subspace. This new space is associated with a nonlocal energy norm which is characterized by a nonlocal interaction kernel defined heterogeneously. We show that the $H^{1/2}$ norm of the trace on the boundary is controlled by the nonlocal energy norm. This result is novel in the sense that the boundary trace is attained without imposing regularity of the function in the interior of the domain. The result has various applications. In particular, it provides a theoretical foundation to the coupling of nonlocal and local models.
A Stationary Core-Shell Assembly in a Ternary Inhibitory System► KiHyun Yun (Department of Mathematics, Hankuk University of Foreign Studies, Republic of Korea)
Abstract: A ternary inhibitory system motivated by the triblock copolymer theory is studied as a nonlocal geometric variational problem. The free energy of the system is the sum of two terms: the total size of the interfaces separating the three constituents, and a longer ranging interaction energy that inhibits micro-domains from unlimited growth. In a particular parameter range there is an assembly of many core-shells that exists as a stationary set of the free energy functional. The cores form regions occupied by the first constituent of the ternary system, the shells form regions occupied by the second constituent, and the background is taken by the third constituent. The constructive proof of the existence theorem reveals much information about the core-shell stationary assembly: asymptotically one can determine the sizes and locations of all the core-shells in the assembly. The proof also implies a kind of stability for the stationary assembly.
High Shear Stress Concentration in Stiff Fiber-Reinforced Composites► Jia Zhao (Department of Mathematics, University of North Carolina at Chapel Hill)
Abstract: When two stiff fibers are are closely located in a fiber-reinforced composite, the shear stress tensor, the gradient of the solution to a conductivity equation, can be arbitrarily large as the distance between two inclusions tends to zero. It is crucial to precisely characterize the blow-up of the gradient of such an equation. Here, we show that the blow-up of the gradient is characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue 1/2 of a Neumann-Poincare type operator, and we also review the recent progress on this subject.
A Multiphasic Complex Fluids Model for Cytokinesis of Eukaryotes► Maxim Zyskin (School of Mathematics, University of Nottingham)
Abstract: Cell Mitosis is a fundamental process in eukaryotic cell reproduction, during which parent cell's nucleus first dissembles leading to DNA and chromosome replication, then chromosomes migrate to new locations within the parent cell to form offspring nuclei which triggers cytokinesis leading to the formation of two offspring cells eventually. In this presentation, we develop a full 3D multiphase hydrodynamic model to study the fundamental mitotic mechanism in cytokinesis, the final stage of mitosis. The model describes the cortical layer, a cytoplasmic layer next to the cell membrane rich in F-actins and myosins, as an active liquid crystal system and integrate the extra cellular matrix material and the nucleus into a multiphase complex fluid mixture. With the novel active matter model built in the system, our 3D simulations show very good qualitative agreement with the experimental obtained images. The hydrodynamical model together with the GPU based numerical solver provides an effective tool for studying cell mitosis theoretically and computationally.
Transformation Groups and Discrete Structures in Continuum Description of Defective Crystals
Abstract: Davini's description of a defective crystal involves a continuum frame of 'lattice vector' fields (interpreted as averaged microscopic lattice), and a scalar dislocation density matrix. Those fields describe kinematics of a defective crystal, and may allow for elastic and plastic deformations. A truncation constitutive assumption on local energy density imply that 'lattice vector' fields form a finite dimensional Lie algebra, and may be classified In low spatial dimensions. We describe such classification for 2 dimensional crystals with dislocations, and variational problem which allows elastic and neutral plastic deformations.
