We are interested in the behaviour of Ricci-flat Kahler metrics on a compact Calabi-Yau manifold, with Kahler classes approaching the boundary of the Kahler cone. The case when the volume approaches zero is especially interesting since the corresponding complex Monge-Ampere equation degenerates in the limit. If the Calabi-Yau manifold is the total space of a holomorphic fibration, the Ricci-flat metrics collapse to a metric the base, which `remembers' the fibration structure.

We briefly survey line search algorithms for unconstrained
optimization.
Next, we consider the search direction and line search strategies used
in
several algorithms that implement a quasi-Newton method for simple
bounds,
including algorithm L-BFGS-B. In this context, we discuss two
currently-used line search algorithms and introduce a new method meant
to
combine the best properties of two different strategies. We present a
modified L-BFGS-B method using the new line search and demonstrate its
significant performance gains by numerical tests using the CUTEr test
set.

The asymptotic expansions to the Boltzmann equations provide a clue of the
connection from kinetic theory to fluid mechanics.
The Hilbert expansion turns out to be useful to verify compressible fluid
limits. As its applications, we rigorously establish the compressible Euler and
acoustic limits from the Boltzmann equation and the Euler-Poisson limit from
the Vlasov-Poisson-Boltzmann system. Moreover, we prove a global-in-time
convergence for a repulsive Euler-Poisson flow for irrotational monatomic gas.

The quaternions form an interesting and useful number system which is a (noncommutative!) extension of the complex numbers. We define the quaternions and give some of the famous history surrounding Hamilton's discovery of them. We describe some applications of quaternions to geometry and algebra.

As an example, consider the following problem. Given positive
integers $a_1,…,a_d$ that are relatively prime, let S be the set of
integers that can be written as a nonnegative integer combination of
these $a_i$. We can think of the $a_i$ as denominations of postage stamps
and S as the postal rates that can be paid exactly using these
denominations. What can we say about the structure of this set, S? What
is the largest integer not in S (called the Frobenius number)? How many
positive integers are not in S?

We attack these problems using the generating function $f_S(x)$, defined
to be the sum, over all elements s of S, of the monomials $x^s$. We will
build up the general theory of computing generating functions – for
this and other problems – and then use these generating functions to
answer questions we’re interested in. We will approach these problems
from an algorithmic perspective: what can we do in polynomial time?

In 1990, Jones received a Fields Medal, in part, for his work on knots and knot invariants. In particular, he developed what is now known as the Jones polynomial which can serve to distinguish two knots from one another. In this talk, we introduce the Jones polynomial and its basic properties and how it can be helpful to scouts who need to know their knots.

on-specific effects are ubiquitous in nature and have relevance in colloidal
science, electrochemistry, and geological and biological physics. The molecular
origin and the coarse-grained modeling of these effects are still widely unexplored.
In this talk we attempt to give more molecular insight into the individual
correlations in aqueous electrolyte systems which give rise to the ion-specific
behavior in bulk (e.g., the osmotic pressure) or in confinement (e.g., between
colloidal or biological surfaces). Particularly, we present a nonlocal
Poisson-Boltzmann theory, based on classical density functional theory,
which captures and rationalizes ion-specific excluded-volume correlations
(the 'size effect') in dense electrolytes and may help understanding the
restabilization of proteins, clays, and colloids at high salt concentrations.
The importance of electrostatic correlations at low dielectric constants is
briefly discussed.