Nowdays computers solve so many difficult tasks for us that we take them for granted. They can automatically land an airplane, detect a problem in your car, create special effects for movies, write music, and understand your questions over the telephone. Ask yourself, why is it that computers can predict global weather patterns yet, if you plug in a camera, they cannot tell a dog from a cat? The list of seemingly trivial visual tasks that computers cannot do goes on and on. For example, humans easily recognize their friends at a distance of 5-10 meters. Most people have no problems locating objects around them. We effortlessly percieve 3D shapes from just looking at photos or paintings. We can do all of this because our brain analyzes images projected onto our eyes' retinas in real-time. "Computer vision" is an area of computer science devoted to making computers understand images, much as humans do. Images come from a digital photo/video cameras connected to a desktop PC, laptop, or cellphone. They also come from MRI/CT scanners, electronic microscopes, ultrasound sensors, and many other sources. This talk will present some of the challenges facing computer vision and some of its recent advances. We will also discuss some mathematical models widely used in computer vision. In particular, discrete models based on Markov Random Fields and related models from based on deferential and integral geometry. We will also discuss some related optimization issues.

In the first lecture we will introduce some basic notions to study the behaviour of polynomial mappings, we will see what can happened at infinity and how the bifurcation set is related to the set of asymptotic critical values. Then, in the next two lectures we will focus on the asymptotic variety of polynomial mappings, i.e. on the set of points at which such a map fails to be proper. The geometry of this set and how we can get explicit description of it will be discussed.

Topological K-theory was the first "extraordinary" cohomology theory introduced into algebraic topology. It is extraordinary not only in that it generalized ordinary cohomology theory but also because of its immediate success in solving important problems and in contributing to the development of new mathematical theory. Topological K-theory was developed in the late 1950's by Atiya and Hirzebruch, building on the work of Grothendieck. During the 1960's in particular it found a number of important applications by Atiyah and collaborators. The goal of my talk is to explain the basics of this classical theory and to outline some of its important applications, namely those concerning how it can be used to study Lie groups and their classifying spaces. The usefulness of equivariant K-theory is a particular theme of these applications.