Seminars, Colloquia, and Conferences
Colloquium
The colloquium meets on Fridays at 4:00pm in
BromfieldPearson 101, unless otherwise indicated.
Fall 2015
September 18, 2015
Talks from the Directed Reading Program
Freddy Saia, Matt DiRe, Ryan HastingsEcho
(Tufts University)
Abstract:
Freddy Saia: "Continued Fractions and Transcendental Numbers" (mentor: Michael BenZvi)
We will begin by talking about continued fractions  what they are, some of their properties,
and the notation we will use to discuss them. We will then venture into seemingly unrelated territory,
presenting and proving Joseph Liouville's theorem on diophantine approximation. This theorem was used by
Liouville in the 1840s both to prove the existence of transcendental numbers (real numbers which are
not roots of any nonzero polynomials with rational coefficients) as well as to give specific examples
of such numbers, dubbed Liouville numbers (only a proper subset of the transcendentals, though still
uncountable). In this vein, we will use the properties of continued fractions we had previously
discussed to display a quick, neat construction of Liouville numbers.
Matt DiRe and Ryan HastingsEcho: "Gödel's Proof" (mentor: George Domat)
At the turn of the last century, mathematicians like Hilbert and Russell attempted to unify all
branches of mathematics within a single logical calculus. But, at the time, there were no tools
available to settle questions of inconsistency (contradictions derived within a system) or
incompleteness (existence of true, expressible statements with no formal proof) within a formal system.
In 1931, Kurt Gödel introduced a method, now know as Gödel numbering, which allowed him to translate
metamathematical statements into welldefined arithmetic relations expressible within the calculus.
Using Gödel numbering, he was able to explicitly express the statement “this statement is not expressible”
within the calculus, thus deriving a critical contradiction. The nuanced logic of this proof was
complex yet irrefutable, and it provided a definitive answer to the problem of inconsistency and
incompleteness.
September 25, 2015
An algorithm for phase retrieval with corrupted data
Paul Hand
(Rice University)
Abstract
Phase retrieval is the process of recovering a vector from phaseless
linear measurements. It is a challenging mathematical problem that
appears in Xray crystallography and other applications. The problem
is even more difficult in the presence of corrupted measurements, which
can occur because of occlusions, sensor failures, or sensor saturation.
In this talk we show that a wellknown semidefinite program can
successfully solve the phase retrieval problem under corrupted data.
Specifically, we show that under a Gaussian measurement model, any
signal can be recovered with high probability, provided there are
enough measurements and provided that at most a fixed fraction of
the measurements are arbitrarily corrupted. Of the standard classical
and recent methods to solve phase retrieval, this semidefinite program
is the only one that can succeed under arbitrarily corrupted data.
October 2, 2015
Understanding the Structure and Stability of Localized Patterns
Elizabeth Makrides
(Brown University)
Abstract:
Localized patterns, in which a spatially oscillatory pattern on a finite spatial range connects
to a homogeneous solution outside this range, have been observed in a wide variety of physical
contexts, from fluid flows to crime hot spots, and optical cavities to vegetative growth. The
bifurcation diagrams of such patterns often exhibit snaking behavior, in which a branch of
symmetric solutions winds back and forth between two limits of an appropriate parameter.
I will discuss analytical and numerical results on the existence, uniqueness, and stability
of both symmetric and asymmetric localized patterns. I will also discuss analytical results
on the appearance of localized solutions, including branch reorganization and drift speed,
upon perturbations breaking symmetries and/or variational structure, and provide numerical
illustrations of these results.
November 13, 2015
TBA
Matthew Douglass
(University of North Texas, National Science Foundation)
December 11, 2015
PathIndependent Integrals in Equilibrium ElectroChemoMechanics
Jianmin Qu
(Tufts University)
Abstract:
Using Noether's first theorem, we constructed two types of pathindependent
integrals in equilibrium electrochemoelastic systems and proved their uniqueness.
These pathindependent integrals are the extensions of the classical J and Lintegrals
in elasticity. Similar to their elastic counterparts, the electrochemoelastic Jintegrals
and Lintegrals represent energy release when a crack or a cavity undergoes a translation
and rotation, respectively. Results of this study established a theoretical foundation
for energy conservation laws in equilibrium electrochemoelasticity. Such conservation
laws are useful in modeling various phenomena in electrochemoelastic systems.
In addition, the pathindependent integrals obtained here provide a theoretical tool
for understanding and a practical tool for numerical evaluation of singular fields
in electrochemoelasticity.
Past Colloquia >
