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## Seminars, Colloquia, and Conferences
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## Seminars, Colloquia, and Conferences## ColloquiumThe colloquium meets on ## Fall 2014## Spring 2015
September 19, 2014
September 26, 2014 The seminar starts by reviewing experimental data to illustrate the large deformation stress-strain response of nonlinear elastic materials. This is followed by a summary of the main ingredients of the nonlinear theory of elasticity and of suitable strain-energy functions to describe the isotropic and anisotropic responses of highly deformable materials. The second part of the seminar focuses on the coupling of mechanical and magnetic effects and on the development of constitutive equations for magnetoelastic materials. These smart materials typically consist of an elastomeric matrix and a distribution of nanoscale ferromagnetic particles and have the capability to change their mechanical properties by the application of a magnetic field. We summarize the relevant equations and propose a coupled free-energy formulation, which depends on the deformation gradient and on the magnetic induction. Finally, we discuss how constitutive equations are specialized to isotropic incompressible magneto-sensitive elastomers in either Lagrangian or Eulerian forms. October 3, 2014 A gas can be considered either as a large system of microscopic interacting particles, or as a continuous medium governed by fluid equations. A natural question is therefore to understand whether both kinds of models give consistent predictions of the dynamics. October 17, 2014 What can one say about maximal subgroups, or, more generally, subgroup structure of simple, finite or algebraic, groups? In this talk we will discuss how group representation theory helps us study this classical problem. We will also describe applications of these results to various problems, particularly in number theory and algebraic geometry. October 24, 2014 The Maxwell-Boltzmann ergodic hypothesis aimed to lay a foundation under statistical mechanics, which is at a microscopic scale a deterministic system. Similar complexity was discovered by PoincarĂ© in celestial mechanics and by Hadamard in the motion of a free particle in a negatively curved space. We start with a guided tour of the history of the subject from various perspectives and then discuss the central mechanism that produces pseudorandom behavior in these deterministic systems, the Hopf argument. It has been known to extend well beyond the scope of its initial application in 1939, and we show that it also leads to much stronger conclusions: Not only do time averages of observables coincide with space averages (which was the purpose for making the ergodic hypothesis), but any finite number of observables will become decorrelated with time. That is, the Hopf argument does not only yield ergodicity but mixing, and often mixing of all orders. October 31, 2014 The Central Limit Theorem (CLT) is one of the most fundamental theorems in probability theory. The CLT states that a sequence of appropriately scaled sums of i.i.d. random variables converges weakly to the standard normal distribution. Although mathematicians had worked on the CLT as early as the 1600s, William Feller gave a proof of the CLT in 1935 by employing L\'evy's continuity theorem. L\'evy's continuity theorem, a nontrivial result, establishes the equivalence of weak convergence for a sequence of random variables and the convergence of the characteristic functions for those random variables. In this talk, I will present the main ideas of our direct proof of the CLT which does not employ L\'evy's continuity theorem. In our proof, we transform a random variable into an i.i.d. sequence on [0,1] and then expand this sequence with respect to the Haar wavelet basis. November 7, 2014 Kinetic theory was originally devised by Ludwig Boltzmann to deal with rarefied gas dynamics. However, in the last decades it has become increasingly apparent that kinetic theory provides a very powerful and general mathematical framework for studying the dynamic evolution of a broad class of complex systems at large. Instrumental to this success, is the development of model Boltzmann equations capable of relinquishing most mathematical complexities without surrendering the essential physics. In this talk we shall discuss the special role played by lattice versions of the Bhatnagar-Gross-Krook model equation to model a variety of complex flows, ranging from fluid turbulence to quark-gluon plasmas. November 14, 2014 Coherent structures, for example traveling waves and periodic patterns, play a key role in determining the behavior of certain types of PDEs, such as reaction diffusion equations and viscous conservation laws. In particular, if they are stable, then they attract nearby initial data and thus qualitatively govern the local dynamics for large times. Determining the stability of coherent structures is often complicated by the presence of continuous spectrum for the associated linearized operator, as well as by details of the nonlinear terms in the PDE. In this talk, I will explain such difficulties in the context of patterns known as defects, and explain how they can be overcome using so-called pointwise estimates for the associated Green's function. December 5, 2014 Lagrangian particle-based methods have advantages for modeling physical phenomena involving multiphase flows, flows with free surfaces, advection-dominated flows, complex fluids, large-deformation of materials and soft matters. In this talk, I will discuss several particle-based methods including smoothed particle hydrodynamics, smoothed dissipative particle dynamics and dissipative particle dynamics. Their various applications in macroscale and mesoscale multiphysical modeling will be demonstrated. December 12, 2014 An abelian network is a collection of automata that live at the vertices of a graph and communicate via the edges. It produces the same output no matter in what order the automata process their inputs. Such a network can be viewed as an interacting particle system or as a model of asynchronous computation. This talk will survey the foundations of the subject, focusing on local-to-global principles, the halting problem, and groups and monoids associated to an abelian network. Based on joint work with Ben Bond (Stanford) and James Propp (Tufts). January 23, 2015 This talk will describe how Lagrangian particle methods are being used to study the dynamics of fluid vortices. These methods track the flow map using adaptive particle discretizations. The Biot-Savart integral is used to recover the velocity from the vorticity, and a tree code is used to reduce the computation time from $O(N^2)$ to $O(N\log N)$, where $N$ is the number of particles. I'll present computations of vortex sheet motion in 2D flow with reference to Kelvin-Helmholtz instability, the Moore singularity, spiral roll-up, and chaotic dynamics. Other examples include vortex rings in 3D flow, and vortex dynamics on a rotating sphere. February 6, 2015 Models of reduced computational complexity is indispensable in scenarios where a large number of numerical solutions to a parametrized partial differential equation are desired in a fast/real-time fashion. These include simulation-based design, parameter optimization, optimal control, multi-model/scale analysis, uncertainty quantification etc. Thanks to an offline-online procedure and the recognition that the parameter-induced solution manifolds can be well approximated by finite-dimensional spaces, reduced basis method (RBM) and reduced collocation method (RCM) can improve efficiency by several orders of magnitudes. The accuracy of the RB/RC solution is maintained through a rigorous a posteriori error estimator whose efficient development is critical. February 27, 2015 There are certain finite abelian groups that arise from objects in number theory that are quite mysterious and of great interest, for example the class group arising from a finite extension of the rational numbers, or the Tate-Shafarevich group of an elliptic curve y^2=x^3+ax+b (for some rational numbers a,b). We discuss the question of what a class group of a random extension, or the Tate-Shafarevich group of a random elliptic curve, looks like, and explain heuristics of several authors including Cohen and Lenstra, and Delaunay, for how these random groups behave. Finally we will relate the predictions of these heuristics to phenomena that can be seen and proven about random integral matrices. March 6, 2015 Of the Kepler planets that have been reported to date, a significant fraction are in systems with multiple transiting planets. In some cases, the signature of the gravitational interactions between planets in these systems can be seen in the variations of their times of transit. By carefully modeling the transit times, as well as investigating long-term stability, we are able to measure or constrain the masses and orbits of the transiting bodies in some of these systems, verifying that they are indeed planets. Although this approach is particularly effective for closely packed and near-resonant systems, it has also been applied to a broad range of systems. These include circumbinary planets, as well as systems with additional non-transiting planets. Some of the Kepler planetary systems exhibit evidence of dynamical chaos on remarkably short time scales, yet these systems are likely to be long-lived. I will highlight the theoretical advances in our understanding of dynamical chaos and stability that have been prompted by the Kepler planetary systems. |
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