Seminars, Colloquia, and Conferences
Colloquium
The colloquium meets on Fridays at 4:00pm in
BromfieldPearson 101, unless otherwise indicated.
Spring 2015
January 16, 2015
Live Experiments, Theorem
Curation and Natural Math Understanding: The Future of Mathematics by Computer
Stephen Wolfram
(CEO of Wolfram Research)
Abstract
Over the past quarter century, Mathematica and WolframAlpha have greatly
increased the level of automation possible in doing mathematics. What does
the future hold for doing math by computer? What new directions and qualitative
changes can we expect? I will discuss the implications of some of my and my
company's current practical and theoretical projects on the future of research
and education in mathematics. I'll probably try some live, unscripted,
mathematical experiments. And I'll show examples of perhaps unexpected
ways in which mathematical skills are used in the world of technology.
January 23, 2015
Lagrangian Particle Methods for
Vortex Dynamics
Robert Krasny
(University of Michigan)
Abstract
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 BiotSavart 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 KelvinHelmholtz
instability, the Moore singularity, spiral rollup, and chaotic
dynamics. Other examples include vortex rings in 3D flow, and
vortex dynamics on a rotating sphere.
January 30, 2015
Radon Transforms and Spherical Functions
Sigurdur Helgason
(Massachusetts Institute of Technology)
Abstract
The original problem is stated in the title of Radon's paper
from 1917: Determine a function on space X from its integrals over a
family of subsets of X. We shall discuss this problem form a general
viewpoint, leading up to integral transforms for homogeneous spaces in
duality, illustrated by several examples. We shall discuss several
instance of such problems in case X is a noncompact symmetric space.
February 13, 2015
Certified reduced basis methods and
reduced collocation methods
Yanlai Chen
(University of Massachusetts Dartmouth)
Abstract
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/realtime fashion. These include simulationbased design,
parameter optimization, optimal control, multimodel/scale
analysis, uncertainty quantification etc. Thanks to an
offlineonline procedure and the recognition that the
parameterinduced solution manifolds can be well approximated by
finitedimensional 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
(University of Wisconsin, Madison)
Abstract
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 TateShafarevich 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 TateShafarevich 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
Dynamical Chaos in Kepler Planetary Systems
Matthew Holman
(HarvardSmithsonian Center for Astrophysics)
Abstract
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 longterm 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 nearresonant systems,
it has also been applied to a broad range of systems. These include circumbinary
planets, as well as systems with additional nontransiting planets. Some of the
Kepler planetary systems exhibit evidence of dynamical chaos on remarkably short
time scales, yet these systems are likely to be longlived. I will highlight the
theoretical advances in our understanding of dynamical chaos and stability that
have been prompted by the Kepler planetary systems.
April 3, 2015
A differentialgeometric
construction of differential Ktheory
Scott Wilson
(Queen's College, City University of New York)
Abstract
Cohomology theories are among the fundamental objects of study in topology, having a
wealth of interesting structures and useful applications. Recent research has focused
on certain refinements of cohomology theories that produce nonhomotopy invariants.
These refinements are known as differential cohomology theories, since they often
incorporate the data of differential forms. In this talk I'll describe a new construction
of one such theory (differential Ktheory) that uses only familiar objects from topology
and geometry. As time permits, I will indicate some applications and connections with
other areas of mathematics.
April 10, 2015
How finite
are infinite groups?
Stefan Witzel
Abstract
There are various ways to measure how "finite" an infinite group is. For example
one might say that a group is "more finite" if it admits a finite set of generators
than if it doesn't. I will talk about a variety of such properties coming from
topology. I will then give an overview of what is known about about the finiteness
properties of several groups and how those can be proven.
April 17, 2015
Cascadic Multigrid
for Eigenvalue Problems and Its Application in Graph Problems
Xiaozhe Hu
(Tufts University)
Abstract
In this work, we develop a cascadic multigrid method for the elliptic
eigenvalue problems and show its optimality under certain assumptions.
We also develop an algebraic variant for the fast computation of the
eigenvalue problems of a graph Laplacian and explore the applicability
of such an eigensolver to the graph related problems. Numerical tests
for practical graphs are presented to show the efficiency of the proposed
cascadic multigrid method. This is a joint work with J. Urschel, J. Xu,
and L. Zikatanov.
April 24, 2015 at 3:00pm
Entropy for smooth systems
Todd Fisher
(Brigham Young University)
Abstract
Dynamical systems studies the longterm behavior of systems that evolve in time.
It is well known that given an initial state the future behavior of a system is
unpredictable, even impossible to describe in many cases. The entropy of a system
is a number that quantifies the complexity of the system. In studying entropy,
the nicest classes of smooth systems are ones that do not undergo bifurcations for
small perturbations.
In this case, the entropy remains constant under perturbation. Outside of the class of systems,
a perturbation of the original system may undergo bifurcations. However, this is a local phenomenon,
and it is unclear when and how the local changes in the system lead to global changes in the
complexity of the system. We will state recent results describing how the entropy (complexity)
of the system may change under perturbation for certain classes of systems.
Past Colloquia >
