The twisted K-theory of a space was introduced by Donovan-Karoubi (1970)
for torsion twistings and by Rosenberg (1989) in general. Motivated by
applications in mathematical physics, the theory has attracted a lot of
interest in recent years. In this talk I will review some of the
foundations of twisted K-homology, and outline a new proof of the
Freed-Hopkins-Teleman theorem describing the twisted equivariant
K-homology of simple, simply connected compact Lie groups.

A factor of a graph G is a spanning subgraph of G. A k-factor is a spanning k-regular subgraph. We describe some eigenvalue conditions that imply the existence of a 1-factor in a graph and discuss some open
problems.

In regression analysis the interest is in exploring the relationship between a response
variable and some explanatory variable(s). Without knowledge about an appropriate
parametric relationship between the variables, one often relies on nonparametric methods.
Local polynomial fitting leads to estimators of the regression function and its derivatives
up to a certain order. We very briefly discuss the basic properties of these estimators
when the regression function is smooth. In particular we pay attention to the behaviour
of the estimators in boundary regions. This local modelling technique is applicable in a
variety of applications.
When the regression function is non-smooth, e.g. discontinuous, the estimates are
inconsistent in the non-smooth points. We briefly discuss some available nonparametric
methods. We discuss, among others, a nonparametric estimation method that searches
for compromising between the properties of jump-preserving and smoothing. The method
chooses, in each point, among three estimates: a local linear estimate using only data to
the left of the point, a local linear estimate based on only data to the right, and a local
linear estimate using data in a two-sided neighborhood around the point. The data-driven
choice among the three estimates is made by comparing, in some appropriate way, the
weighted residual mean squares of the three fits. This results into a consistent estimate.
We establish asymptotic properties of the estimator, and illustrate its performance via
simulations and examples.

Regression surfaces can also exhibit non-smooth or irregular features. To recover e.g.
a non-smooth behaviour one should avoid using a nonparametric technique that smooths
away this particular behaviour. We discuss how to estimate via local linear fitting a non-
smooth regression surface, in case of fixed or random design. The proposed procedure can
also be used for image denoising. Applications to surface estimation and image denoising
are shown.
Sometimes the interest is in estimating the location of the points/curves at which e.g.
a function is non-smooth, say discontinuous. This relates to the problem of change-point
estimation or edge detection (or boundary estimation). We discuss how one can use the
previously discussed techniques to some other problems, such as estimating non-smooth
densities.
These lectures are partly based on joint work with Alexandre Lambert and Peihua Qiu.