Spring Term 2012

Simon Salamon (King’s College London). Twistor transforms of slice regular functions. Friday January 20th, Huxley 130, 1.30-2.30pm.

The class of slice regular quaternion valued functions of one quaternion variable includes power series of the form $\sum q^n a_n$ with $a_n \in H$, which have many interesting properties. Locally, they map a standard complex structure on H minus the real axis to another orthogonal complex structure, and lift to holomorphic maps between ruled surfaces of $CP^3$. I shall discuss one example in detail.

Alexei Skorobogatov (Imperial). Root systems, projective homogeneous spaces and del Pezzo surfaces. Friday January 27th, Huxley 130,  1.30-2.30pm.

I will give a survey of a series of joint papers with Vera Serganova that give a description of universal torsors over del Pezzo surfaces X in terms of G/P, where G is the simple Lie group such that X and G have the same root system. This leads to a presentation of the Cox ring of X by generators and explicit quadratic relations.

Kai Ishihara (Imperial). On nullification distance. Friday February 3rd, Huxley 130,  1.30-2.30pm.

The action of some site-specific recombinases can be modeled by a band surgery. Then we can consider the distance between two links in terms of coherent band surgeries. That is called nullification distance. In particular, the distance from an oriented link to an unlink is called the nullification number. Ernst-Montemayor-Stasiak enumerated the nullification number for all knots up to 9 crossings. The purpose of this talk is to give the table of the nullification distances between pairs of knots up to 7 crossings and two component links up to 6 crossings. We also discuss a relation between band surgery and rational tangle surgery.

Dietmar Salamon (ETH Zurich). Hyperkaehler Floer theory and the Fueter equation. Friday February 10th, Huxley 130,  1.30-2.30pm.

There is a natural analogue of the symplectic action functional on the space of maps from a 3-manifold  M, equipped with a volume preserving frame, into a hyperkaehler manifold  X. The critical points and gradient flow lines of this functional are solutions of the Fueter equation in dimensions 3 and 4, possibly with Hamiltonian type perturbations.
When  X  is flat, one can construct Floer homology groups and use them to prove existence theorems for solutions of the Fueter equation, in analogy to the Arnold conjecture for the torus. Conjecturally, one would expect some version of hyperkaehler Floer theory to extend to non-flat manifolds, such as the moduli space of ASD instantons over a K3 surface  S, and interact with the G2-type Donaldson-Thomas-Floer theory of  M x S.
(The talk is based on joint work with Sonja Hohloch and Gregor Noetzel.)

Andrew Lobb (Durham). 2-strand twisting and knot homology. Friday February 17th, Huxley 130,  1.30-2.30pm.

Introducing a long twist into a pair of strands in a knot diagram gives rise to a simple tensor factor of its (Khovanov or Khovanov-Rozansky) knot homology.  We explore how some straight-forward homological algebra can be brought to bear in this situation and deduce some consequences.

Karen Vogtman (Cornell). Automorphisms of free groups, hairy graphs and modular forms. Friday February 24th, Huxley 130,  1.30-2.30pm.

The group Out(F_n) of outer automorphisms of a free group shares many properties with both GL(n,Z) and mapping class groups of surfaces. These properties can often be proved by considering the action of Out(F_n) on a space of finite graphs known as Outer space, which I will describe.

The homology of the quotient space is an invariant of the group, and a theorem of M.Kontsevich relates this homology to the cohomology of a certain Lie algebra. Using this connection, S.Morita discovered a series of new homology classes for Out(F_n). I will explain how these classes can be reinterpreted in terms of “hairy graphs” and then show how this graphical picture leads to the construction of large numbers of new classes, including some based on classical modular forms for SL(2,Z).

Frances Kirwan (Oxford). Hyperkähler implosion. Friday March 2nd, Huxley 130, 1.30-2.30pm.

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 hyperkähler geometry.

Tom Bridgeland (Oxford). Quadratic differentials as stability conditions. Friday March 9th, Huxley 130, 1.30-2.30pm.

I’ll attempt to explain how spaces of meromorphic quadratic differential on Riemann surfaces can be viewed a spaces of stability conditions on a class of CY3 triangulated categories. It is possible to interpret these categories as Fukaya categories of non-compact threefolds fibering over the surface, but if you want to actually prove anything it’s probably better to define them using quivers. This work is joint with Ivan Smith, and borrows a fair amount from a physics paper of Gaiotto, Moore and Neitzke.

Bobby Acharya (King’s College London). G2 manifolds at the CERN Large Hadron Collider.  Friday March 16th, Huxley 130,  1.30-2.30pm.

I will describe how data from proton collisions at the LHC may provide evidence for the existence of extra dimensions with  G2-holonomy. I will emphasise some open geometric/analytical questions which I consider to be most relevant for the `physics applications’ of G2-manifolds.”

Yuri Tschinkel (CIMS New York University). Almost abelian Anabelian geometry and rational points. Friday March 23rd, Huxley 130, 1.30-2.30pm.

I will describe Galois-theoretic and geometric approaches to the construction of rational points on varieties over function fields (joint with F. Bogomolov).

Joseph Krasilshchik (Independent University of Moscow and Russian State University for the Humanities). Cohomological invariants of integrable systems. Friday March 23rd, Huxley 342, 3.30-4.30pm. (joint Geometry and Integrable Systems seminar).

The interpretation of a differential equation as a submanifold in the space of infinite jets allows one to construct two cohomological theories naturally related to an equation. The 1st one is the Vinogradov spectral sequence (or variational complex) and is responsible for the Lagrangian formalism constrained by the equation at hand. It contains invariants such as conservation laws, cosymmetries (characteristics), etc. The 2nd theory is constructed using the notion of the Nijenhuis bracket on vector-valued differential forms. The corresponding cohomology groups contain symmetries of the equation, its infinitesimal deformations and other important objects. Studying the relations between the two theories makes it possible to derive efficient formulas to compute recursion operators, Hamiltonian and symplectic structures.

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