## 10:15-11:00. "An Introduction to Category Theory" - Alistair Wallis (Heriot-Watt)

In around the 1930's and 1940's several mathematicians were interested in generalising ideas in various areas of geometry and topology (e.g. defining abstract manifolds, abstract varieties). Along the way was discovered / invented the concept of a category. Categories are a way of unifying concepts which come up all over the place in mathematics, and have found applications in theoretical physics and functional programming, as well of course as algebra, geometry and topology. In this talk, I will give the definition and describe some basic examples of a category, and describe functors and natural transformations.

## 11:05-11:50. Stiefel-Whitney classes and classification of Real Projective Spaces - Jesus Martinez Garcia (Edinburgh)

Stiefel-Whitney classes are an important invariant under continuous maps of vector bundles over topological real manifolds. Similar to Chern classes over complex manifolds, they have several applications to topology. I will recall the principal ideas of Singular Cohomology and introduce the theory of Stiefel-Whitney classes in an axiomatic fashion, giving some examples and computing basic cases as the tautological line bundle over the projective n-space. Time permitting I will apply it to cobordism theory.

## 11:55-12:40. "Introduction to String Theory" - Lisa Torlina (Edinburgh)

In this talk I will give a brief introduction to string theory as a potential candidate for quantum gravity.
Last century saw the emergence of two theories - quantum mechanics and general relativity - which fundamentally altered our physical understanding of the world and at the same time proved amazingly successful in explaining experimental observations. However, in situations where neither theory can be neglected (that is, at very high energies and small length scales), we find serious contradictions between the two.
In order to resolve this issue, string theory has emerged as a promising candidate for a single unified theory to describe all the forces and particles in the universe, which would apply at all energy scales and reduce to general relativity and quantum mechanics in the appropriate limits.
In this talk I will introduce the basic framework of string theory, sketch the way in which the graviton emerges, and discuss some important features, consequences and open questions in this area.

## 12:40-13:25. "An introduction to Mori Dream Spaces" - Dorothy Winn (Glasgow)

Mori Dream Spaces are a generalisation of toric varieties in the following sense: Cox's construction gives toric varieties as quotients of affine space under a torus action, whereas Mori Dream Spaces are quotients of irreducible subvarieties of affine space. They are just as fun to play with as toric varieties, are much more general (e.g. all Fanos are MDSs) and have lovely birational geometry (in that Mori's programme works).

## 14:15 - 15:00 Global dimension: A homological approach to measuring smoothness - Rollo Jenkins (Edinburgh)

In the dictionary between algebra and geometry, smoothness at a point on a variety is equivalent to regularity of corresponding the local ring. JP Serre showed that there is a homological description: that the 'global dimension' is finite. I'll briefly describe this and hopefully give some sort of justification for such a crazy idea, eg, it allows us to easily prove that a surface is smooth near a smooth embedded curve.

## 15:05 - 15:50. "Statistical Inference in Quantum Computation" - Alistair Wallis (Heriot-Watt)

The ideas of quantum computation revolve around the randomness that appears to be inherent to quantum systems. In this talk I would like to describe some theoretical results which say something about this inherent randomness. Much of the research into this randomness has been done in the past decade and so I will try to give a glimpse of what some people are currently studying.

## 16:05 - 16:50 An introduction to Lagrangian Fibrations - Daniele Sepe (Edinburgh)

Lagrangian fibrations are of interest in many different branches of mathematics, ranging from integrable systems to symplectic topology, from affine geometry to mirror symmetry. In this talk, I shall give a brief introduction to their invariants and how one may set out to attempt a classification of regular Lagrangian fibrations over a fixed manifold.