09:00 to 09:55 Registration and welcome 09:55 to 10:00 Opening remarks and welcome from Sir David Wallace (INI Director) INI 1 10:00 to 11:00 T Hausel ([Oxford])Arithmetic and physics of Higgs moduli spaces In this talk we discuss the connection between conjectures with Villegas on mixed Hodge polynomials of character varieties of Riemann surfaces achieved by arithmetic means and conjectures on the cohomology of Higgs moduli spaces derived by physicists Chuang-Diaconescu-Pan. INI 1 11:00 to 11:30 Morning coffee 11:30 to 12:30 Hyperkahler implosion Symplectic implosion is a construction in symplectic geometry due to Guillemin, Jeffrey and Sjamaar, which is related to geometric invariant theory for non-reductive group actions in algebraic geometry. This talk (based on joint work in progress with Andrew Dancer and Andrew Swann) is concerned with an analogous construction in hyperkahler geometry. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 14:30 Hyperpolygons and moduli spaces of parabolic Higgs bundles Given an $n$-tuple of positive real numbers $\alpha$ we consider the hyperpolygon space $X(\alpha)$, the hyperkähler quotient analogue to the Kähler moduli space of polygons in $\mathbb{R}^3$. There exists an isomorphism between hyperpolygon spaces and moduli spaces of stable, rank-$2$, holomorphically trivial parabolic Higgs bundles over $\mathbb{C} \mathbb{P}^1$ with fixed determinant and trace-free Higgs field. This allows us to prove that hyperpolygon spaces $X(\alpha)$ undergo an elementary transformation in the sense of Mukai as $\alpha$ crosses a wall in the space of its admissible values. We describe the resulting changes in the core of $X(\alpha)$ as well as the changes in the nilpotent cone of the corresponding moduli spaces of PHBs. If time permits, we will explain how to obtain explicit formulas for the computation of the intersection numbers of the core components of $X(\alpha)$ and of the nilpotent cone components of the corresponding moduli spaces of PHBs. As a final application, we describe the cohomology ring structure of these moduli spaces of PHBs and of the components of their nilpotent cone. This is joint work with Leonor Godinho. INI 1 14:40 to 15:10 R Rubio ([Oxford])Higgs bundles and Hermitian symmetric spaces We study the moduli space of polystable G-Higgs bundles for noncompact real Lie groups G of Hermitian type. First, we define the Toledo character and use it to define the Toledo invariant, for which a Milnor-Wood type inequality is proved. Then, for the maximal value of the Toledo invariant, we state a Cayley correspondence for groups of so-called tube type and point out a rigidity theorem for groups of so-called non-tube type. The proofs of these results are based on the Jordan algebra structure related to the tangent space of the Hermitian symmetric space given by G and are independent of the classification theorem of Lie groups. (Joint work with O. Biquard and Ó. García-Prada.) INI 1 15:20 to 15:50 J Martens ([Aarhus])Compactifications of reductive groups as moduli stacks of bundles Given a reductive group G, we introduce a class of moduli problems of framed principal G-bundles on chains of projective lines. Their moduli stacks provide equivariant toroidal compactifications of G. All toric orbifolds are examples of this construction, as are the wonderful compactifications of adjoint groups of De Concini-Procesi. As an additional benefit, we show that every semi-simple group has a canonical orbifold compactification. We further indicate the connection with non-abelian symplectic cutting and the Losev-Manin spaces. This is joint work with Michael Thaddeus (Columbia U). INI 1 16:00 to 16:30 Afternoon Tea 16:30 to 17:30 Refined curve counting on algebraic surfaces Let $L$ be ample line bundle on an an algebraic surface $X$. If $L$ is sufficiently ample wrt $d$, the number $t_d(L)$ of $d$-nodal curves in a general $d$-dimensional sub linear system of |L| will be finite. Kool-Shende-Thomas use the generating function of the Euler numbers of the relative Hilbert schemes of points of the universal curve over $|L|$ to define the numbers $t_d(L)$ as BPS invariants and prove a conjecture of mine about their generating function (proved by Tzeng using different methods). We use the generating function of the $\chi_y$-genera of these relative Hilbert schemes to define and study refined curve counting invariants, which instead of numbers are now polynomials in $y$, specializing to the numbers of curves for $y=1$. If $X$ is a K3 surface we relate these invariants to the Donaldson-Thomas invariants considered by Maulik-Pandharipande-Thomas. In the case of toric surfaces we find that the refined invariants interpolate between the Gromow-Witten invariants (at $y=1$) and the Welschinger invariants at $y=-1$. We also find that refined invariants of toric surfaces can be defined and computed by a Caporaso-Harris type recursion, which specializes (at $y=1,-1$) to the corresponding recursion for complex curves and the Welschinger invariants. This is in part joint work with Vivek Shende. INI 1 17:30 to 18:30 Welcome wine reception 18:45 to 19:30 Dinner at Wolfson Court (residents only)