Benoit Charbonneau : Instantons and reduction of order via the Nahm transform
- Graduate/Faculty Seminar ( 141 Views )The instanton equations appear in gauge theory and generalize both the Maxwell equations and the harmonic equation. Their study has been and continues to be a very fertile ground for interactions between physicists and mathematicians. The object of this talk is a description of instanton solutions on S^1xR^3 due to Hurtubise and myself using the Nahm transform, a non-linear transformation that takes a system of PDE and produces a system of ODE or even a system of algebraic equations. This description allows us to answer existence questions for calorons.
Zhou Zhang : Volume Form and Scalar Curvature for K\ahler-Ricci Flow over General Type Manifold
- Geometry and Topology ( 151 Views )It is an interesting project guided by Tian's conjecture to use K\"ahler-Ricci flow with changing cohomology class in the study of general type manifold. The locally smooth convergence leaves quite some freedom for the global geometry. Meanwhile, volume form and scalar curvature have shown different behavior in infinite and finite time cases.
Hugo L. D. de S. Cavalcante : Digital Chaotic Circuits: part II - Characterization and Application
- Nonlinear and Complex Systems ( 153 Views )We discuss the characterization of chaos displayed by continuous time digital circuits, both numerically and experimentally. Continuous models for physical systems with switch-like behavior are used to simulate those circuits and their coupling. The effect of perturbations in the coupling and synchronization is also studied experimentally and numerically.
Lucy Zhang : Modeling and Simulations of Fluid and Deformable-Structure Interactions in Bio-Mechanical Systems
- Applied Math and Analysis ( 154 Views )Fluid-structure interactions exist in many aspects of our daily lives. Some biomedical engineering examples are blood flowing through a blood vessel and blood pumping in the heart. Fluid interacting with moving or deformable structures poses more numerical challenges for its complexity in dealing with transient and simultaneous interactions between the fluid and solid domains. To obtain stable, effective, and accurate solutions is not trivial. Traditional methods that are available in commercial software often generate numerical instabilities.
In this talk, a novel numerical solution technique, Immersed Finite Element Method (IFEM), is introduced for solving complex fluid-structure interaction problems in various engineering fields. The fluid and solid domains are fully coupled, thus yield accurate and stable solutions. The variables in the two domains are interpolated via a delta function that enables the use of non-uniform grids in the fluid domain, which allows the use of arbitrary geometry shapes and boundary conditions. This method extends the capabilities and flexibilities in solving various biomedical, traditional mechanical, and aerospace engineering problems with detailed and realistic mechanics analysis. Verification problems will be shown to validate the accuracy and effectiveness of this numerical approach. Several biomechanical problems will be presented: 1) blood flow in the left atrium and left atrial appendage which is the main source of blood in patients with atrial fibrillation. The function of the appendage is determined through fluid-structure interaction analysis, 2) examine blood cell and cell interactions under different flow shear rates. The formation of the cell aggregates can be predicted when given a physiologic shear rate.
Bill Allard : The Boundary Finder
- Graduate/Faculty Seminar ( 170 Views )(This abstract is in TeX source code. Sorry!) Fix a small positive number $h$. Let $$G=h\mathbb{Z}^2=\{(ih,jh):i,j\in\mathbb{Z}\};$$ thus $G$ is a rectangular grid of points in $\mathbb{R}^2$. Let $\Omega$ be an bounded open subset of $\mathbb{R}^2$ with $C^1$ boundary and let $E=\{x\in G:x\in\Omega\}$. {\bf Question One.} Given $E$ can one determine the length of $\partial\Omega$ to within $O(h)$? The answer to this question is ``yes'', provided $\Omega$ satisfies a certain natural ``thickness'' condition; without this additional assumption the answer may be ``no''. {\bf Question Two.} Is there a fast algorithm for determining the length of $\partial\Omega$. The answer to this question also ``yes''. In this talk I will describe the proof that the answer to Question One is ``yes'' and I will describe the fast algorithm whose existence is implied in the answer to Question Two. If time permits, I will describe some applications.
Davar Khoshnevisan : Nonlinear Stochastic Heat Equations: Existence, Growth, and Intermittency
- Probability ( 140 Views )We introduce some recent advances in the study of nonlinear stochastic heat equations, and related stochastic PDEs. Special attention will be paid to the local structure of the solution. In particular, we show that, frequently, the solution exhibits a form of intermittency. Time permitting, we discuss related connections to classical potential theory and mathematical physics as well.
Aaron Naber : Orbifold Regularity of Collapsed Spaces and applications to Einstein Manifolds.
- Geometry and Topology ( 142 Views )Let (M_i,g_i) be a sequence of Riemannian n-manifolds with uniformly bounded curvature such that (M_i,g_i)->(X,d), a metric space, in the Gromov Hausdorff sense. Then we show that there is a closed subset S of X with codimension at least 3 and dimension at most n-5 such that X-S is a Riemannian Orbifold. We use this and an \epsilon-regularity theorem to show that metric spaces in the closure of the moduli space of Einstein 4-manifolds are Riemannian Orbifolds away from a finite number of points. This is joint with G. Tian.
Mark Huber : Conditions for Parallel and Simulated Tempering to be fast or slow
- Probability ( 142 Views )In Markov chain Monte Carlo, a Markov chain is constructed whose limiting distribution is equal to some target distribution. While it is easy to build such chains, for some distributions the standard constructions can take exponentially long to come near that limit, making the chain torpidly mixing. When the limit is reached in polynomial time, the chain is rapidly mixing. Tempering is a technique designed to speed up the convergence of Markov chains by adding an extra temperature parameter that acts to smooth out the target distribution. In this talk I will present joint work with Dawn Woodard (Cornell) and Scott Schmidler (Duke) that give sufficient conditions for a tempering chain to be torpidly mixing, and a related (but different) set of conditions for the chain to be rapidly mixing.
Elizabeth L. Bouzarth : Modeling Biologically Inspired Fluid Flow Using RegularizedSingularities and Spectral Deferred Correction Methods
- Applied Math and Analysis ( 146 Views )The motion of primary nodal cilia present in embryonic development resembles that of a precessing rod. Implementing regularized singularities to model this fluid flow numerically simulates a situation for which colleagues have exact mathematical solutions and experimentalists have corresponding laboratory studies on both the micro- and macro-scales. Stokeslets are fundamental solutions to the Stokes equations, which act as external point forces when placed in a fluid. By strategically distributing regularized Stokeslets in a fluid domain to mimic an immersed boundary (e.g., cilium), one can compute the velocity and trajectory of the fluid at any point of interest. The simulation can be adapted to a variety of situations including passive tracers, rigid bodies and numerous rod structures in a fluid flow generated by a rod, either rotating around its center or its tip, near a plane. The exact solution allows for careful error analysis and the experimental studies provide new applications for the numerical model. Spectral deferred correction methods are used to alleviate time stepping restrictions in trajectory calculations. Quantitative and qualitative comparisons to theory and experiment have shown that a numerical simulation of this nature can generate insight into fluid systems that are too complicated to fully understand via experiment or exact numerical solution independently.
Hubert Bray : An Overview of General Relativity
- Graduate/Faculty Seminar ( 154 Views )After brief introductions to special relativity and the foundations of differential geometry, we will discuss the big ideas behind Einstein's theory of general relativity. Einstein's theory replaces Newtonian physics not only as the best description of gravity according to experiments, but also as a philosophically pleasing and very geometric idea, which Einstein called his "happiest thought." We will also discuss the predictions made by general relativity, including the big bang and black holes, both of which are strongly supported by observations. We will discuss these ideas from a geometric perspective, and discuss some of the open problems and future directions that are currently being studied.
Ronnie Sircar : Games with Exhaustible Resources
- Probability ( 149 Views )We study N-player repeated Cournot competitions that model the determination of price in an oligopoly where firms choose quantities. These are nonzero-sum (ordinary and stochastic) differential games, whose value functions may be characterized by systems of nonlinear Hamilton-Jacobi-Bellman partial differential equations. When the quantity being produced is in finite supply, such as oil, exhaustibility enters as boundary conditions for the PDEs. We analyze the problem when there is an alternative, but expensive, resource (for example solar technology for energy production), and give an asymptotic approximation in the limit of small exhaustibility. We illustrate the two-player problem by numerical solutions, and discuss the impact of limited oil reserves on production and oil prices in the dupoly case. Joint work with Chris Harris (Cambridge University) and Sam Howison (Oxford University).
Jason Parsley : Helicity, Configuration Spaces, & Characteristic Classes
- Geometry and Topology ( 174 Views )The helicity of a vector field in R^3, an analog to linking number, measures the extent to which its flowlines coil and wrap around one another. Helicity turns out to be invariant under volume-preserving diffeomorphisms that are isotopic to the identity. Motivated by Bott-Taubes integration, we provide a new proof of this invariance using configuration spaces. We then present a new topological explanation for helicity, as a characteristic class. Among other results, this point of view allows us to completely characterize the diffeomorphisms under which helicity is invariant and give an explicit formula for the change in helicity under a diffeomorphism under which helicity is not invariant. (joint work with Jason Cantarella, U. of Georgia)
Jason Metcalf : Strichartz estimates on Schwarzschild black hole backgrounds
- Applied Math and Analysis ( 131 Views )In this talk, we will present some recent work on dispersive estimates for wave equations on Schwarzschild black hole backgrounds. We in particular will discuss Strichartz estimates and localized energy estimate. This is from a joint work with Jeremy Marzuola, Daniel Tataru, and Mihai Tohaneanu.
Michael Jenista : Dynamical Systems and the Conley Index
- Graduate/Faculty Seminar ( 164 Views )An introductory lecture to the Conley Index theory. We consider the flow case and introduce the key object of study: an index pair of an isolated invariant set. Index pairs are robust under perturbations and their homotopy type is invariant, making them an ideal tool for problems with error terms or even data-generated systems. The relevant tools are algebraic topology and some knowledge of continuous flows.
Krishna Athreya : Preferential attachment random graphs with general weight function
- Probability ( 146 Views )Consider a network of sites growing over time such that at step n a newcomer chooses a vertex from the existing vertices with probability proportional to a function of the degree of that vertex, i.e., the number of other vertices that this vertex is connected to. This is called a preferential attachment random graph. The objects of interest are the growth rates for the growth of the degree for each vertex with n and the behavior of the empirical distribution of the degrees. In this talk we will consider three cases: the weight function w(.) is superlinear, linear, and sublinear. Using recently obtained limit theorems for the growth rates of a pure birth continuous time Markov chains and an embedding of the discrete time graph sequence in a sequence of continuous time pure birth Markov chains, we establish a number of results for all the three cases. We show that the much discussed power law growth of the degrees and the power law decay of the limiting degree distribution hold only in the linear case, i.e., when w(.) is linear
Dan Rutherford : Generating families and invariants of Legendrian knots
- Geometry and Topology ( 137 Views )Legendrian knots in standard contact R3 have in addition to their topological knot type two classical invariants known as the Thurston-Bennequin and rotation numbers. Over the past decade several invariants have been developed which are capable of distinguishing between knots with identical classical invariants. The purpose of this talk is to describe interesting relationships between some of these new invariants. Major players in this talk are the Chekanov-Eliashberg DGA (Legendrian contact homology) and related objects, as well as combinatorial structures on front diagrams and homological invariants arising from the theory of generating families (due to Chekanov-Pushkar, Fuchs, and Traynor). The main new result (joint with Fuchs) is that, when a Legendrian knot is defined by a generating family, homology groups obtained by linearizing the Chekanov-Eliashberg DGA are isomorphic to the homology of a pair of spaces associated with the generating family.
Ralph Smith : Model Development and Control Design for High Performance Nonlinear Smart Material Systems
- Applied Math and Analysis ( 141 Views )High performance transducers utilizing piezoceramic, electrostrictive, magnetostrictive or shape memory elements offer novel control capabilities in applications ranging from flow control to precision placement for nanoconstruction. To achieve the full potential of these materials, however, models, numerical methods and control designs which accommodate the constitutive nonlinearities and hysteresis inherent to the compounds must be employed. Furthermore, it is advantageous to consider material characterization, model development, numerical approximation, and control design in concert to fully exploit the novel sensor and actuator capabilities of these materials in coupled systems.
In this presentation, the speaker will discuss recent advances in the development of model-based control strategies for high performance smart material systems. The presentation will focus on the development of unified nonlinear hysteresis models, inverse compensators, reduced-order approximation techniques, and nonlinear control strategies for high precision or high drive regimes. The range for which linear models and control methods are applicable will also be outlined. Examples will be drawn from problems arising in structural acoustics, high speed milling, deformable mirror design, artificial muscle development, tendon design to minimize earthquake damage, and atomic force microscopy.
Jer-Chin Chuang : Subdivisions and Transgressive Chains
- Geometry and Topology ( 171 Views )Combinatorial transgressions are secondary invariants of a space admitting triangulations. They arise from subdivisions and are analogous to transgressive forms such as those in Chern-Weil theory. In this talk, I characterize transgressions that are path-independent of subdivision sequence. The result is obtained by using a cohomology on posets that is shown to be equivalent to higher derived functors of the inverse (or projective) limit over the opposite poset.
Mark Stern : Geometry of stable Yang-Mills connections
- Geometry and Topology ( 155 Views )On a compact 4-manifold, every self-dual connection and every anti-self-dual connection minimizes the Yang-Mills energy. In this talk, I will answer the converse question for compact homogeneous 4-manifolds. I will also survey related stability results in other dimensions.
Simon Brendle : Minimal Lagrangian diffeomorphisms between domains in the hyperbolic plane
- Geometry and Topology ( 148 Views )Let $\Omega$ and $\tilde{\Omega}$ be domains in the hyperbolic plane with smooth boundary. Assume that both domains are uniformly convex, and have the same area. We show that there exists an area-preserving, orientation-preserving diffeomorphism $f: \Omega \to \tilde{\Omega}$ such that the graph of $f$ is a minimal surface in $\mathbb{H}^2 \times \mathbb{H}^2$. Moreover, we show that the set of all such diffeomorphisms is parametrized by the circle.
Yunliang Yu : FDS
- Presentations ( 148 Views )FDS (fds.duke.edu) is a content management system (CMS) widely used across Duke for schools and departments to effectively maintain their faculty research and teaching related web pages and reports. In this talk we'll cover some fundamentals of FDS and give a short tutorial on the FDS templates. We hope this talk will help everyone (either webmasters, web developers and designers, and FDS group managers, or interested faculty/staff members) to use FDS better.
John Voight : On Moduli of Nondegenerate Curves
- String Theory ( 224 Views )We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. Such nondegenerate polynomials have become popular objects in explicit algebraic geometry, owing to their connection with toric geometry; however, despite their ubiquity, the intrinsic property of nondegeneracy has not seen much detailed study. We prove that every curve of genus $g \geq 4$ over an algebraically closed field is nondegenerate in the above sense. More generally, let $\mathcal{M}_g^{\textup{nd}}$ be the locus of nondegenerate curves inside the moduli space of curves of genus $g \geq 2$. Then we show that $\dim \mathcal{M}_g^{\textup{nd}} = \min(2g+1,3g-3)$, except for $g=7$ where $\dim \mathcal{M}_7^{\textup{nd}} = 16$; thus, a generic curve of genus $g$ is nondegenerate if and only if $g \geq 4$
Seung-Yeal Ha : Uniform L^p-stability problem for the Boltzmann equation
- Applied Math and Analysis ( 146 Views )The Boltzmann equation governs the dynamics of a dilute gas. In this talk, I will address the L^p-stability problem of the Boltzmann equation near vacuum and a global Maxwellian. In a close-to-vacuum regime, I will explain the nonlinear functional approach motivated by Glimm's theory in hyperbolic conservation laws. This functional approach yields the uniform L^1-stability estimate. In contrast, in a close-to-global maxwellian regime, I will present the L^2-stability theory which establishes the uniform L^2-stability of several classical solutions.
Eckehard Schoell : Time-delayed feedback control - from nano to neuro
- Nonlinear and Complex Systems ( 186 Views )We review recent developments in the control of deterministic and stochastic nonlinear dynamics by time-delayed feedback methods [1]. We point out how to overcome the alleged odd number limitation for unstable periodic orbits, and discuss the control of complex chaotic or noise-induced space-time patterns. Our findings are applied to a selection of models ranging from semiconductor nanostructures, like resonant-tunneling diodes [2], to neural systems. [1] E. Sch{\"o}ll and H.G. Schuster (Eds.): Handbook of Chaos Control (Wiley-VCH, Weinheim, 2008), second completely revised and enlarged edition. [2] E. Sch{\"o}ll, Nonlinear spatio-temporal dynamics and chaos in semiconductors (Cambridge University Press, Cambridge, 2001).