Balázs Szendrői (Oxford). Purity in Donaldson-Thomas theory. Friday May 3rd, Huxley 140, 1.30-2.30pm.
Donaldson-Thomas theory, which despite what you might think was invented in Oxford and not London, is the theory of associating numerical invariants to sheaves on Calabi-Yau threefolds, and more generally objects in 3-Calabi-Yau categories. Recenty, a cohomological version of this theory has emerged, which has a much richer structure, including Hodge-theoretic data. I will present a Hodge-theoretic purity result, and draw some conclusions for geometric engineering and quantum cluster theory.
Geometry day at KCL, Friday May 10th (no G&T seminar)
Alexander Getmanenko (University of Tokyo/IHES). Microlocal properties of sheaves and complex WKB. Friday May 17th, Huxley 140, 1.30-2.30pm.
Following a joint work with Dmitry Tamarkin, arXiv:1111.6325, I will discuss an application of Kashiwara-Schapira style microlocal theory of sheaves to the problem of existence of ramified analytic solutions of the Laplace transformed Schroedinger equation.
John Parker (Durham). Constructing Non-Arithmetic Lattices. Friday May 24th, Huxley 140, 1.30-2.30pm.
A lattice in a semisimple Lie group is a discrete group whose quotient has finite volume. A group of isometries in a semisimple Lie group is arithmetic if it is commensurable with the group of integral points in a linear algebraic group defined over the rationals. Arithmetic groups are lattices but not all lattices are arithmetic. In this talk I will survey the background material and then describe a joint project with Martin Deraux and Julien Paupert in which we construct new examples of non-arithmetic lattices in SU(2,1). These are the first new examples of such groups to be found since the work of Deligne and Mostow in the mid 1980s.
Danielle O’Donnol (Imperial). Legendrian Graphs. Friday May 31st, Huxley 140, 1.30-2.30pm.
A Legendrian graph is an embedding of a graph G in a contact 3-mainfold which is everywhere tangent to the contact structure. Central results in contact geometry use Legendrian graphs in their proofs. Together with Elena Pavelescu, I am studying Legendrian graphs in their own right. I will discuss some of our recent results in this area.
There is also a Geometric Analysis day at UCL, starting at 2pm on Friday May 31st.
Paul Johnson (Columbia, Colorado State). The topology of Hilbert schemes of points on orbifolds. Friday June 7th, Huxley 140, 1.30-2.30pm.
The Hilbert scheme of n points on C^2 is a smooth manifold of dimension 2n, with important connections to physics, geometry, representation theory, and combinatorics. One can also study the Hilbert scheme of points on C^2/G, for G a finite group. When G is a subgroup of SL_2, the resulting Hilbert schemes are huge active area of research, but much less is known if G is not. We describe our ongoing work studying the topology of these varieties when G is not a subgroup of SL_2.
No seminar on Friday June 14th
Claire Voisin (Centre de mathématiques Laurent Schwartz, Jussieu). Rational equivalence of zero cycles on K3 surfaces. Friday June 21st, Huxley 213 (Clore LT), 1.30-2.30pm.
We know since Mumford that the group of 0-cycles on a projective complex surface with a nonzero holomorphic 2-form is very big. We will show however that in the case of a K3 surface, there is some nice structure in this group, as discovered by successive work of Beauville-Voisin, Huybrechts, O’Grady and myself.
Mark Gross (University California, San Diego). Mirror symmetry and cluster algebras. Friday July 5, Huxley 140, 1.30-2.30pm.
Mirror symmetry is a subject which originated in string theory in 1990, and has since developed mathematically in many different directions. Roughly put, mirror symmetry gives a correspondence between different algebraic varieties, in which certain calculations on one variety (usually involving counting curves) coincides with a different type of calculation on the other (usually computing integrals of certain forms). Joint work with Bernd Siebert over the last 10 years has explored the underlying geometry of mirror symmetry, and the structures that have emerged in this work can now be applied to diverse situation.
In this talk I will discuss joint ongoing work with Keel, Kontsevich and Hacking. I will make a connection between structures called scattering diagrams, or wall-crossing structures, which emerged in the study of mirror symmetry, and cluster algebras of Fomin and Zelevinsky, which were developed to model aspects of canonical bases of Lusztig and Kashiwara. In particular, a construction of an analogue of theta functions, motivated by the homological mirror symmetry conjecture, allows the construction of canonical bases of cluster algebras in many cases, as well as the proof of a number of conjectures concerning cluster algebras.