# Autumn Term 2010

Richard Thomas (Imperial). The Göttsche conjecture. Friday October 8th. Huxley 130, 1.30-2.30pm.

Fix a complex surface S with a sufficiently positive holomorphic line bundle L. The zeros of sections of L are complex curves in S. A general (d+1)-dimensional subspace of the sections of L gives a d-dimensional family of curves on S. They are generically smooth, but nodal curves appear in codimension-1, twice nodal curves in codimension-2, etc. (and more singularities besides). There should be a finite number of curves with d nodes. It is a classical question how many there are. I will outline a proof that the answer is topological, given by a universal degree d polynomial in the four numbers $c_1(L)^2, c_1(L).c_1(S), c_1(S)^2$ and $c_2(S)$.

Amin Gholampour (Imperial). Counting invariants for the ADE McKay quivers. Friday October 15th. Huxley 130, 1.30-2.30pm.

We study the moduli space of the McKay quiver representations Q associated to the finite subgroups $G < SU(3)$. Let $Y=Hilb_G(\mathbb{C}^3)$ be the natural Calabi-Yau resolution of $\mathbb{C}^3/G$. In the cases where the fibers of p are at most 1 dimensional, there is an equivalence of the abelian categories of such representations and the perverse sheaves on Y relative to p. By defining certain stability conditions on these abelian categories, the moduli spaces of Donaldson-Thomas and Pandharipande-Thomas invariants on Y , and of Szendroi invariants on Q are recovered. In the special case where $G < SU(2) < SU(3)$, we prove a relation between these invariants by means of the wall crossing. This gives explicit formulas for the invariants which allows verifying the conjectural Gromov-Witten/Donaldson-Thomas correspondence for Y, and the Donaldson-Thomas Crepant Resolution Conjecture for p.

Victor Pidstrygach (Gottingen). Nonlinear Dirac operator. Friday October 22nd. Huxley 130, 1.30-2.30pm.

One can define nonlinear analog of Dirac operator on a 4-manifold by replacing the clifford module with a hyperkahler manifold equipped with the suitable action of the spinor group.  We shall consider examples of suitable hyperkahler manifolds and discuss properties of harmonic spinors and some applications in gauge theory and geometry.

Song Sun (Imperial). Uniqueness of constant scalar curvature Kahler metrics. Friday October 29th. Huxley 130, 1.30-2.30pm.

We show that constant scalar curvature Kahler(cscK) metric “adjacent” to a given integral Kahler class is unique up to isomorphism. This generalizes the previous uniqueness heorems of  Donaldson and Chen-Tian, where the Kahler class itself is assumed to admit a cscK metric. This is also an infinite dimensional analogue of the Kempf-Ness theorem for semi-stable orbits. In this talk we shall emphasize this analogue. The techinal tools used in the proof are the Calabi flow and the metric geometry of the space of Kahler metrics. Joint with X-X Chen.

Vlad Lazic (Imperial). New outlook on Mori theory. Friday November 5th. Huxley 130, 1.30-2.30pm.

I will give a gentle introduction into Mori theory and then sketch how finite generation of certain algebras (proved by a self-contained argument in a paper by Paolo Cascini and me) implies in a straightforward way all the main results of Mori theory. This gives a new and much more efficient organisation of the theory. This is joint work with Alessio Corti.

Yunfeng Jiang (Imperial). Quantum orbifold cohomology under toric flops. Friday November 12th. Huxley 130, 1.30-2.30pm.

Let $\mathcal{X}\to \overline{X}$ and $\mathcal{X}^\prime\to\overline{X}$ be two contraction morphisms of smooth Deligne-Mumford stacks. Let $a=(a_0,\cdots,a_n)$ and $b=(b_0,\cdots,b_m)$ be two sequences of positive integers.  We assume that $\sum_{i=0}^{n}a_i=\sum_{j=0}^{m}b_{j}$. A rational morphism $f: \mathcal{X}\to\mathcal{X}^\prime$ is called a toric flop if the flopping locus $Z$ is isomorphic to a weighted projective bundle over a subscheme $S\subset \overline{X}$, with fibre the weighted projective space $\mathbb{P}(a_0,\cdots,a_n)$ such that the normal bundle $N_{Z/\mathcal{X}}|_{Z_{s}}\cong \oplus_{i=0}^{m}\mathcal{O}(-b_i)$ on the side of $\mathcal{X}$, and on the side of $\mathcal{X}^\prime$, the flopping locus $Z^\prime$ is isomorphic to a weighted projective bundle over a subscheme $S\subset \overline{X}$, with fibre the weighted projective $\mathbb{P}(b_0,\cdots,b_m)$ such that the normal bundle $N_{Z^\prime/\mathcal{X}}|_{Z^{\prime}_{s}}\cong \oplus_{i=1}^{n}\mathcal{O}(-a_i)$.  In the case of $m=n$ and $a_i=b_i$, the toric flop satisfies the so called  “Hard Lefschetz condition”.  In this talk I will talk about how  quantum orbifold cohomology changes under such toric flops.

Yaroslav Kurylev (UCL). Geometric convergence and inverse spectral problems. Friday November 19th. Huxley 130, 1.30-2.30pm.

We consider the inverse problems of the reconstruction of a Riemannian manifold from its spectral data (say, heat kernel) given on a part of the boundary or internal subdomain. In the first part of the talk we discuss the uniqueness in this problem while in the second part consider the question of stability and its relations to the issue of geometric convergence in proper classes of Riemannian manifolds.

Ken Millett (University of California, Santa Barbara). Knots, Ephemeral Knots and, Slipknots. Friday November 26th. Huxley 130, 1.30-2.30pm.

Knots, ephemeral knots and, slipknots occur with increasing probability and complexity as the length of random arcs increase. It is difficult, in practice, to locate them and to assess their size. Computer simulations give one a sense of what it true butthere are still many obscure aspects.  In addition, rigorous argument remains beyond present reach.  I will point out some examples of challenging open problems and a few conjectures.

Reto Mueller (Scuola Normale Superiore, Pisa). A compactness theorem for complete Ricci shrinkers. Friday December 3rd. Huxley 130, 1.30-2.30pm.

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. Contrary to previous work (for the compact case), we do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss-Bonnet with cutoff argument. This is joint work with Robert Haslhofer.

Andriy Haydys (Imperial). Fukaya-Seidel category and gauge theory. Friday December 10th. Huxley 130, 1.30-2.30pm.

In this talk I will first outline a new construction of the Fukaya-Seidel category, which is associated to a symplectic manifold  equipped with a compatible almost complex structure J and a J-holomorphic Morse function.  Then this construction will be applied in an in finite dimensional case of holomorphic Chern-Simons functional. The corresponding construction conjecturally associates a Fukaya-Seidel-type category to a smooth three-manifold.

Shing-Tung Yau (Harvard). The shape of Inner Space. Imperial College Colloquium, Wednesday December 8th. Room G16, Sir Alexander Fleming Building, Imperial College, 6.30-7.30pm.

Previous talks2009200820072006200520042003, 2002, 20012000, and earlier.