L3

We establish a correspondence between generalized quiver gauge theories in

four dimensions and congruence subgroups of the modular group, hinging upon

the trivalent graphs which arise in both. The gauge theories and the graphs

are enumerated and their numbers are compared. The correspondence is

particularly striking for genus zero torsion-free congruence subgroups as

exemplified by those which arise in Moonshine. We analyze in detail the

case of index 24, where modular elliptic K3 surfaces emerge: here, the

elliptic j-invariants can be recast as dessins d'enfant which dictate the

Seiberg-Witten curves.

I show how to construct some Fano 3-folds that have the same Hilbert series but different Betti numbers, and so lie on different components of the Hilbert scheme. I would like to show where these fit in to a speculative (indeed fantastical) geography of Fano 3-folds, and how the projection methods I use may apply to other questions in the geography.

We shall dedicate the first half of the talk to introduce

classical Higgs bundles and describe the fibres of the corresponding

Hitchin fibration in terms of spectral data. Then, we shall define

principal Higgs bundles and look at some examples. Finally, we

consider the particular case of $SL(2,R)$, $U(p,p)$ and $Sp(2p,2p)$ Higgs

bundles and study their spectral data. Time permitting, we shall look

at different applications of our new methods.

Expander graphs are sparse finite graphs with strong connectivity properties, on account of which they are much sought after in the construction of networks and in coding theory. Surprisingly, the first examples of large expander graphs came not from combinatorics, but from the representation theory of semisimple Lie groups. In this introductory talk, I will outline some of the history of the emergence of such examples from group theory, and give several applications of expander graphs to group theoretic problems.

I will discuss some recent progress connecting different quiver gauge theories and some applications of these results.

In April 2010 Eguchi--Ooguri--Tachikawa observed a fascinating connection between the elliptic genus of a K3 surface and the largest Mathieu group. We will report on joint work with Miranda Cheng and Jeff Harvey that identifies this connection as one component of a system of surprising relationships between a family of finite groups, their representation theory, and automorphic forms of various kinds Mock modular forms, and particularly their shadows, play a key role in the analysis, and we find several of Ramanujan's mock theta functions appearing as McKay--Thompson series arising from the umbral groups.

A cactus product is much like a wedge product of pointed spaces, but instead of being uniquely defined there is a moduli space of possible cactus products. I will discuss how this space can be interpreted geometrically and how its combinatorics calculates the homology of the automorphism group of a free product with no free group factors. Then I will reinterpret the moduli space with Outer space in mind: the lobes of the cacti now behave like boundaries and our free products can now include free group factors.

Twistor theory is a technology that can be used to translate analytical problems on Euclidean space $\mathbb R^4$ into problems in complex algebraic geometry, where one can use the powerful methods of complex analysis to solve them. In the first half of the talk we will explain the geometry of the Twistor correspondence, which realises $\mathbb R^4$ , or $S^4$, as the space of certain "real" lines in the (projective) Twistor space $\mathbb{CP}^3$. Our discussion will start from scratch and will assume very little background knowledge. As an application, we will discuss the Twistor description of instantons on $S^4$ as certain holomorphic vector bundles on $\mathbb{CP}^3$ due to Ward.

I will claim (and maybe show) that a lot of problems in differential geometry can be reformulated in terms of non-linear elliptic differential operators. After reviewing the theory of linear elliptic operators, I will show what can be said about the non-linear setting.

I will explain how moduli spaces of quadratic differentials on Riemann surfaces can be interpreted as spaces of stability conditions for certain 3-Calabi-Yau triangulated categories. These categories are defined via quivers with potentials, but can also be interpreted as Fukaya categories. This work (joint with Ivan Smith) was inspired by the papers of  Gaiotto, Moore and Neitzke, but connections with hyperkahler metrics, Fock-Goncharov coordinates etc. will not be covered in this talk.