Gavin BROWN and Miles REID
Title: Tom and Jerry and Sarkisov links, I and II
Abstract: We construct Fano 3-folds anticanonically embedded in weighted projective space in codimension 4 using Kustin-Miller unprojection (Papadakis-Reid unprojection of Type I). This constructs those Fano 3-folds that we expect to admit Type I projections in at least two ways (Tom and Jerry), and extending the corresponding Sarkisov links distinguishes the cases according to the Mori fibre spaces they link to.

Alvaro Nolla de Celis
Title: Dihedral groups and $G$-Hilb
Abstract: The special McKay correspondence states that the minimal resolution $Y$ of the singular quotient $C^2/G$ by a finite subgroup $G$ in $GL(2,C)$ is the $G$-invariant Hilbert scheme $G\text{-Hilb}$. In this talk I will consider this correspondence when $G$ is a binary dihedral group in $GL(2,C)$, and explain how we can give an explicit description of $G\text{-Hilb}$ via $G$-graphs and the moduli space $M(Q,R)$ of stable representations of the McKay quiver.

Osamu FUJINO
Title: On log canonical flops
Abstract: We explain Koll\'ar's example of a log canonical flopping contraction. It does not have a flop.

Klaus HULEK
Title: Calculating the Mordell-Weil rank of elliptic threefolds
Abstract: In this talk I will discuss a method to calculate the Mordell-Weil rank of an elliptic threefold. The main idea is to reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective $4$-space. I then describe a method for calculating the cohomology of a certain class of singular hypersurface. As an application I discuss a class of elliptically fibered Calabi-Yau threefolds over del Pezzo surfaces which were first constructed by Hirzebruch. This is joint work with R. Kloosterman.

Akira ISHII
Title: Dimer models and the special McKay correspondence
Abstract: A dimer model is a bi-colored graph on a $2$-torus, which encodes the information of a quiver with relations. A typical example is given by a tessellation of regular hexagons, in which case the associated quiver is the McKay quiver for a finite abelian subgroup of $SL(3, C)$. When a dimer model is \lq non-degenerate', the moduli space of representations of the associated quiver with relations (with dimension vector $(1,1, ... ,1)$ and with respect to a generic stability parameter) is a crepant resolution of the $3$-dimensional Gorenstein affine toric variety associated with a lattice polygon, which is combinatorially determined by the dimer model. Fundamental questions on dimer models are:
1. Describe \lq consistency condition' on a dimer model, which ensures that the moduli space is derived equivalent to the path algebra of the associated quiver with relations.
2. For an arbitrary $3$-dimensional Gorenstein affine toric variety, construct dimer models which satisfies the condition in Question 1.
In this talk, I will explain the behavior of the dimer models under the removal of a vertex from the toric diagram (and taking the convex hull of the rest), and how this can be applied to the above two questions. The triangle case is crucial and depends on computations on the \lq special representations' of Wunram-Riemenschneider for two-dimensional cyclic quotient singularities. This is a joint work with Kazushi Ueda.

Yukari ITO
Title: Existence of crepant resolutions
Abstract: There are several kinds of generalizations of the McKay correspondence, but all of them need a crepant resolution. It is known that there exist crepant resolutions in dimension 2 or 3 for any Gorenstein quotient singularities, but it is not the same in higher dimension. I would like to talk on existence of higher dimensional crepant resolutions.

Anne-Sophie KALOGHIROS
Title: The defect of Fano $3$-folds
Abstract: Let $X$ be a quartic $3$-fold in $P^4$ with no worse than terminal singularities. The Grothendieck-Lefschetz theorem states that the Picard rank of $X$ is $1$, i.e. that every Cartier divisor on $X$ is a hyperplane section of $X$. However, no such result holds for the group of Weil divisors of $X$ if $X$ is not factorial. In this lecture, I will give a bound for the rank of the group of Weil divisors of $X$ when $X$ is a Fano $3$-fold with terminal Gorenstein singularities. This bound is optimal in the case of the quartic $3$-fold. This bound is obtained using birational geometry. I will also show how these methods yield a "classification" of non factorial terminal quartic $3$-folds.

Yongnam LEE
Title: Log minimal model program for the moduli space of stable curves of low genus
Abstract: In this talk, we completely describe the log minimal model program for the moduli space of curves of stable curves of genus $\le 3$, and the second small contraction of genus $4$. It is jointed work with Donghoon Hyeon. The log minimal model program for the moduli space of curves of stable curves of genus $g$ was introduced by Brendan Hassett. The first divisorial contraction, small contraction, and flip were worked out recently by Hassett and Hyeon.

Timothy LOGVINENKO
Title: A derived approach to the geometric McKay correspondence
Abstract: We describe a three dimensional generalization of the geometric McKay correspondence constructed by Gonzales-Sprinberg and Verdier in dimension two. More precisely, we show that the Bridgeland-King-Reid derived category equivalence induces a natural geometric correspondence between irreducible representations of $G \subset SL_3(C)$ and closed subschemes of the exceptional set of $G\text{-Hilb}(C^3)$. Applied in dimension two this produces precisely the classical McKay. We further show that for an abelian G this correspondence can be calculated explicitly and that the result is intrinsically related to the toric combinatorics of the 'Reid's recipe'.

James McKernan
Title: Batyrev's conjecture
Abstract: I describe some work of Brian Lehmann who proves some results about the cone of nef curves, analagous to the cone theorem and contraction theorem, which concern the $K_X$-negative side of the cone of effective curves.

Yoshinori NAMIKAWA
Title: Induced nilpotent orbits and birational geometry
Abstract: Nilpotent orbit closures in a complex simple Lie algebra $g$ have symplectic singularities. In this lecture, I will pose a conjecture that any $Q$-factorial terminalization of them is obtained as a generalized Springer map, and will prove that the conjecture actually holds true when $g$ is a classical Lie algebra.

Keiji OGUISO
Title: Mordell-Weil groups of abelian fibered hyperkaehler manifolds and O'Grady's $10$-dimensional examples
Abstract: I would like to talk about Mordell-Weil rank of an abelian fibered hyperkaehler manifold. First, I notice that the maximal possible Mordell-Weil rank is $20$ among all currently known hyperkeahler manifolds. Then, I would like to explain how one can obtain examples having the maximal Mordell-Weil rank $20$, by using deformation and explicit birational geometry of O'Grady's $10$-dimensional examples.

Jongil PARK
Title: A new family of algebraic surfaces with $p_g = 0$ and $K^2 > 0$
Abstract: One of the fundamental problems in the classification of complex surfaces is to find a new family of simply connected surfaces with $p_g = 0$ and $K^2 > 0$. Although a large number of non-simply connected complex surfaces of general type with $p_g = 0$ and $K^2 > 0$ have been known, until recently the only previously known simply connected, minimal, surface of general type with $p_g = 0$ and $K^2 > 0$ was Barlow surface which has $K^2 = 1$. The natural question arises if there are other simply connected surfaces of general type with $p_g = 0$ and $K^2 > 0$ except Barlow surface. In 2004 I constructed a new family of simply connected symplectic $4$-manifolds with $b_2^+ = 1$ and $1 \le K^2 \le 2$ using a rational blow-down surgery. After this construction, it had been an intriguing question whether these symplectic $4$-manifolds admit a complex structure. In 2006 Yongnam Lee and myself constructed a new family of simply connected, minimal, complex surfaces of general type with $p_g = 0$ and $1 \le K^2 \le 2$ by modifying the symplectic $4$-manifolds constructed above. Our main techniques used here are a rational blow-down surgery and $Q$-Gorenstein smoothing theory. In 2007 Heesang Park , Dongsoo Shin and myself successfully found a right configuration to produce simply connected, minimal, complex surfaces of general type with $p_g = 0$ and $3 \le K^2 \le 4$ using the same technique as above. In this talk, I will sketch how to construct such a new family of simply connected surfaces of general type using a rational blow-down surgery and $Q$-Gorenstein smoothing theory. If time allows, I will also present a new family of surfaces of general type with $p_g = 0$ and small homology group. This talk is based on joint works with Yongnam Lee and with Heesang Park and Dongsoo Shin.

Thomas PETERNELL
Title: Varieties with generically nef tangent bundles
Abstract: I will discuss the notion of generically nef line bundles and vector bundles and basic properties. The main theme of the talk is to discuss which manifolds have generically nef tangent bundles and to indicate applications to classification theory.

Miles REID
Title: Big key varieties for surfaces with $p_g = 1$, $K^2 = 2$ (according to Stephen Coughlan)
Abstract:

Hiromichi TAKAGI
Title: $Q$-Fano $3$-folds and varieties of power sums
Abstract: We discuss some examples of $Q$-Fano $3$-folds which can be described by varieties of power sums.

Yukinobu TODA
Title: Stability conditions and Donaldson-Thomas type invariants
Abstract: Donaldson-Thomas invariant is a counting invariant of curves on Calabi-Yau 3-folds. Now there are several expected properties of generating functions of DT-invariants, say DT-PT correspondence, rationality conjecure, DT-NCDT correspondence, and flop invariance. In this talk, I will show that all the above properties can be deduced from the wall-crossing formula in a certain triangulated subcategory of the derived category of coherent sheaves.

Michael WEMYSS
Title: McKay Correspondence for two dimensional rational surface singularities
Abstract: I will describe a certain correspondence for all two dimensional quotient singularities, and will explain how some aspects generalize to all rational two dimensional surface singularities.