Spring Term 2013

Paolo Rossi (Zürich/Dijon). A 2D-topological field theory approach to integrability. Friday January 11th. Huxley 130, 1.30-2.30pm.

Inspired by some recent developments in Symplectic Field Theory, an holomorphic curve-counting theory by Eliashberg, Givental and Hofer that produces sophisticated invariants of contact manifolds, we propose 2D-TFT-like approach to (possibly infinite dimensional) integrable systems. This approach includes (but is not limited to) Dubrovin’s constrution of dispersionless integrable systems from Frobenius manifolds of topological type. In this case it sheds light on the Hamiltonian origin of topological recursion, as opposed to bi-hamiltonian recursion.

Dror Bar-Natan (Toronto). Meta-Groups, Meta-Bicrossed-Products, and the Alexander Polynomial. Friday January 18th. Huxley 130, 1.30-2.30pm.

I will define “meta-groups” and explain how one specific meta-group, which in itself is a “meta-bicrossed-product”, gives rise to an “ultimate Alexander invariant” of tangles, that contains the Alexander polynomial (multivariable, if you wish), has extremely good composition properties, is evaluated in a topologically meaningful way, and is least-wasteful in a computational sense. If you believe in categorification, that’s a wonderful playground.

See also http://www.math.toronto.edu/~drorbn/Talks/Imperial-130118/

No seminar.  Friday January 25th. Huxley 130, 1.30-2.30pm.

No seminar.  Friday February 1st. Huxley 130, 1.30-2.30pm.

Eugene Gorsky (Stony Brook). Torus knots and Cherednik algebras. Friday February 8th. Huxley 130, 1.30-2.30pm.

The relation between the quantum invariants of torus knots and characters of certain representations of the rational Cherednik algebras appears to be a source of new results both in knot theory and in representation theory. I will describe this relation on the level of polynomial invariants and give a conjectural description of the triply graded Khovanov-Rozanky homology of torus knots. The connections to the q,t-Catalan numbers of A. Garsia and M. Haiman will be also discussed.

Gabriel Paternain (Cambridge). Spectral rigidity and invariant distributions on Anosov surfaces. Friday February 15th. Huxley 130, 1.30-2.30pm.

I will discuss inverse problems on a closed Riemannian surface whose geodesic flow is Anosov. More specifically, I will try to establish spectral rigidity (a problem that has been open for some time) and surjectivity results for the adjoint of the geodesic ray transform. These surjectivity results imply the existence of many geometric distributions in H^{-1} invariant under the geodesic flow and play a key role to prove spectral rigidity. This is joint work with Mikko Salo and Gunther Uhlmann.

Dominic Joyce (Oxford). Categorification of Donaldson-Thomas invariants of Calabi-Yau 3-folds using perverse sheaves. Friday February 22nd. Huxley 130, 1.30-2.30pm.

Link to the abstract of the talk.

Ivan Smith (Cambridge). Symplectic Khovanov cohomology revisited. Friday March 1st. Huxley 130, 1.30-2.30pm.

Symplectic Khovanov cohomology is a Floer-theoretic invariant of oriented links in the 3-sphere, conjecturally equal to its combinatorial sibling. We will discuss joint work with Mohammed Abouzaid which leads to a partial proof of that conjecture. The new ingredient is a formality theorem for certain symplectic manifolds arising in Lie theory; the proof of formality uses ideas from homological mirror symmetry.

David Favero (Vienna). The Toric Mori Program and Homological Mirror Symmetry. Friday March 8th. Huxley 130, 1.30-2.30pm.

Homological Mirror Symmetry, conjectured by Kontsevich in 94, is an equivalence between categories appearing in algebraic and symplectic geometry. I will discuss some basic examples of this equivalence such as the projective plane and illustrate how symplectic deformations are “mirrored” by complex deformations. This viewpoint enhances Kontsevich’s conjecture with additional structure coming from a run of the Mori program.

Robert Berman (Chalmers). K-stability, singular Kähler-Einstein metrics and Perelman’s entropy functional. Friday March 15th. Huxley 130, 1.30-2.30pm.

The differential-geometric notion of a Kähler-Einstein metric on a Fano manifold X admits a natural extension to the case when X is a singular Fano variety; one simply demands that the metric be defined and Kähler-Einstein on the regular locus of X and that its total volume there be equal to the top intersection number of the first Chern class of X. In this talk (which is based on arxiv:1205.6214) I will discuss the proof of the fact that any Fano variety admitting such a metric is K-polystable. I will also explain how a similar argument shows that if the supremum of Perelman’s lambda-functional over the space of all Kähler metrics in the first Chern class of X coincides with the standard topological bound, then X is K-semistable. One technical ingredient in the proof is that any Kähler-Einstein metric as defined above extends to a current on X with bounded potentials, as recently shown in a joint work with Boucksom-Eyssidieux-Guedj-Zeriahi (see arXiv:1111.7158).

Richard Ward (Durham). Geometry of Spaces of Periodic Monopoles. Friday March 22nd. Huxley 130, 1.30-2.30pm.

This talk deals with BPS monopoles on R^3 which are either singly-periodic (monopole chains) or doubly-periodic (monopole walls). The moduli spaces of such monopole solutions are hyperkähler manifolds. My interest has been in seeing what the monopoles look like — in particular, what the moduli correspond to — and in finding novel examples of monopole dynamics corresponding to geodesics on the relevant moduli space (although in the doubly-periodic case, much less is known about the geometry). The generalized Nahm transform, and descriptions in terms of spectral data, both play an important part.

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