Seminars, Colloquia, and Conferences
The colloquium meets on Fridays at 4:00pm in
101, unless otherwise indicated.
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
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 7, 2014
Hydrokinetic approach to complex flows: the legacy of P.L. Bhatnagar
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
November 14, 2014
Nonlinear stability of coherent structures via pointwise estimates
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
December 5, 2014
Multiscale modeling using Lagrangian particle-based methods
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
January 23, 2015
Lagrangian Particle Methods for Vortex Dynamics
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
Certified reduced basis methods and reduced collocation methods
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.
In this talk, I will give a brief introduction of the RBM and
discuss recent and ongoing efforts to develop RCM, and the
accompanying parametric analytical preconditioning techniques
which are capable of improving the quality of the error
estimation uniformly on the parameter domain, and speeding up
the convergence of the reduced solution to the truth
approximation significantly. New ways of effectively bounding
the stability constants for the error estimation will also be
discussed. These results are critical in certifying the accuracy
of the reduced model and giving it a reliable predictive value.
February 27, 2015
Random groups in number theory and random integral matrices
Melanie Matchett Wood
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
Past Colloquia >