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Seminars, Colloquia, and Conferences

Colloquium

The colloquium meets on Fridays at 4:00pm in Bromfield-Pearson 101, unless otherwise indicated.

Fall 2014

Date Speaker Title
September 15 Esther Kann
University of Linz, Industrial Mathematics Institute
A Weighted Wavelet Method for Region of Interest Tomography
September 19 James Propp
University of Massachusetts, Lowell
How well can you see the slope of a digital line (and other applications of the "tapering trick")?
September 26 Luis Dorfmann
Tufts University
Modeling of highly deformable materials
October 3 Laure Saint-Raymond
École Normale Supérieure, Paris
From molecular dynamics to kinetic theory and hydrodynamics
October 17 Pham Tiep
University of Arizona
Representations and subgroup structure of simple groups
October 24 Boris Hasselblatt
Tufts University
Statistical properties of deterministic systems by elementary means
October 31 Patricia Garmirian
Tufts University
A New and Direct Proof of the Central Limit Theorem
November 7 Stephen Wolfram
CEO, Wolfram Research
TBA
November 14 Margaret Beck Nonlinear stability of coherent structures via pointwise estimates
November 21 Andreas Arvanitoyeorgos
Tufts University
The dynamics of the normalized Ricci flow on some homogeneous spaces
December 5 Wenxiao Pan
Northwestern National Laboratory
Multiscale modeling using Lagrangian particle-based methods

Abstracts

September 19, 2014
How well can you see the slope of a digital line (and other applications of the "tapering trick")?
James Propp, University of Massachusetts, Lowell

In many situations, we want to estimate some asymptotic quantity using data taken from a finite window. How can we best compensate for the error introduced by the finiteness of the window?

There is a simple general method for doing this that often gives markedly improved estimates, even though the method requires no information about the relationship between window-size and error. The method is called "tapering" in the signal-processing literature and "smoothing" in the number-theory literature, but outside of these two disciplines it does not seem to have gotten the attention it deserves.

I will discuss applications of tapering related to such topics as computational geometry, almost-periodic functions, lattice-point enumeration, number theory, quasicrystals, and derandomized Markov chains. No special background in any of these fields (or indeed much of anything beyond calculus) is required, though a rudimentary knowledge of Fourier analysis will be helpful for those who want to understand when and why tapering works.

September 26, 2014
Modeling of highly deformable materials
Luis Dorfmann, Tufts University

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
From molecular dynamics to kinetic theory and hydrodynamics
Laure Saint-Raymond, École Normale Supérieure, Paris

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.

In this lecture, we will focus on the validity of the fluid approximation in the particular case of rarefied gases, using kinetic theory as an intermediate level of description as suggested by Hilbert in his sixth problem. We will present landmark partial results, both on the low density limit and on the Navier-Stokes limit of the Boltzmann equation, giving an hint of the mathematical tools used to establish these convergences, and discussing the challenging open questions.

October 17, 2014
Representations and subgroup structure of simple groups
Pham Tiep, University of Arizona

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
Statistical properties of deterministic systems by elementary means
Boris Hasselblatt, Tufts University

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 Poincaré 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
A New and Direct Proof of the Central Limit Theorem
Patricia Garmirian, Tufts University

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.

The theorem of Skorokhod can be applied to the CLT to establish the existence of a sequence of random variables converging almost surely to the standard normal distribution. In this talk, I will also discuss our present research in which we apply Skorokhod's theorem to a sequence of appropriately scaled sums of i.i.d. random variables.

November 14, 2014
Nonlinear stability of coherent structures via pointwise estimates
Margaret Beck

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
Multiscale modeling using Lagrangian particle-based methods
Wenxiao Pan

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.

Spring 2015

Date Speaker Title
January 23 Robert Krasny
University of Michigan
Title: Lagrangian Particle Methods for Vortex Dynamics
February 27 Melanie Matchett Wood
University of Wisconsin, Madison
TBA
April 10 Stefan Witzel TBA

Abstracts

January 23, 2015
Lagrangian Particle Methods for Vortex Dynamics
Robert Krasny

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.

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