Autumn Term 2011

Martijn Kool (Imperial). Reduced classes and curve counting on surfaces. Friday October 7th, Huxley 140, 1.30-2.30pm.

Let S be a smooth projective surface with sufficiently ample line bundle L. The Göttsche conjecture (proved by several people) states that the number of \delta-nodal curves in a general \delta-dimensional linear subsystem of |L| is given by a universal polynomial in L^2, L.c_1(S), c_1(S)^2, c_2(S). We define a perfect obstruction theory for stable pairs on S by describing the moduli space as cut out by section of a vector bundle on a smooth space. The associated invariants are topological and generalise Göttsche’s invariants to the non-ample case. Next, we define quite general reduced Gromov-Witten and stable pair invariants of the canonical bundle K_S. In various cases, we relate (some of) these three invariants. In the sufficiently ample case, we prove a version of the MNOP conjecture. This is joint work with Richard Thomas.

Matthew Rathbun (Imperial). High Distance Knots in Closed 3-Manifolds. Friday October 14th, Huxley 140, 1.30-2.30pm.

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance  with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S^3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heegaard splitting, and show that for a splitting of genus at least 2, the coarse mapping class group is isomorphic to the ordinary mapping class group of the splitting. This is joint work with Marion Moore.

Song Sun (Imperial). Positively curved Sasaki manifolds. Friday October 21st, Huxley 140, 1.30-2.30pm.

The Frankel conjecture, as proved by Mori and Siu-Yau, states that the bi-holomorphism type of complex projective space could be characterized by a local curvature condition, namely the positivity of the bisectional curvature.  I will discuss an extension of this to Sasaki geometry (an odd dimensional companion of Kahler geometry). In particular, this yields certain orbifold version of the Frankel conjecture. This talk is based on a joint work with Weiyong He.

Mauro Mauricio (Imperial). Connect sum lens space surgeries. Friday October 28th, Huxley 140, 1.30-2.30pm.

Motivated by particular DNA-protein interactions, we are interested in the following question:  Can rational tangle replacement produce a connect sum of 4-plats from a single 4-plat?  We study this problem via Dehn surgery on the double branch covers.  By extending results of Boyer and Zhang on the maximal distance between exceptional Dehn fillings, we prove if there is a rational tangle replacement whose complement is a prime tangle that produces a connect sum of two 2-bridge links from a single 2-bridge link, then the distance of this surgery is at most one. If both links share a common prime summand we conjecture that this is impossible when the complement is a prime tangle. We describe a strategy for proving this conjecture using Heegaard Floer homology.  En route, we prove if positive integral m-Dehn surgery on a null-homologous knot in L(p,1) produces L(p,1)# L(m,1) then the genus of the knot is less than or equal to 1.  These proofs builds on methods developed by Ozsvath and Szabó for knots in S3.

Groups and Geometry in the Southeast I. Friday October 28th, 2.30-5pm , UCL. Speakers:  Bergeron and Bartels. Further details.

Danny Calegari (Cambridge & Caltech). Ziggurats and rotation numbers. Friday November 4th, Huxley 140, 1.30-2.30pm.

I  will  discuss  new  rigidity  and  rationality  phenomena  (related  to  the  phenomenon  of  Arnold  tongues)  in  the   theory  of  nonabelian  group  actions  on  the  circle. I  will  introduce  tools that  can  translate  questions  about   the  existence  of  actions  with  prescribed  dynamics,  into  finite  combinatorial  questions  that  can be answered   effectively.  There  are  connections with the  theory  of  diophantine  approximation,  and  with  the  bounded   cohomology  of  free  groups.  A  special case of  this  theory  gives  a  very  short  new  proof  of  Naimi’s  theorem   (i.e.  the conjecture  of  Jankins-Neumann)  which  was  the  last  step  in  the  classification  of taut foliations  of   Seifert  fibered  spaces.  This  is  joint  work  with  Alden  Walker.

King’s College Geometry Day 2. Friday November 11th 11am-6.30pm. Speakers: Bourni, Haskins, Minicozzi, Tinaglia and Wickramasekera. Further details.

Seminar cancelled! Friday November 18th, Huxley 140, 1.30-2.30pm.

De-Qi Zhang (Singapore and Max Planck). Automorphisms of positive entropy on compact Kahler manifolds. Friday November 25th, Huxley 140, 1.30-2.30pm.

We show that a minimal compact Kahler manifold (or minimal normal projective variety) X of dimension n >  2 has at most n-1 independent commuting (modulo elements of null entropy) automorphisms of positive entropy and the maximality n-1 occurs only when X is a quotient of a compact torus. We also propose an open question: if the representation G | NS_R(X) on the Neron Severi group is embedded as a Zariski-dense subset in a real rank n-1 semi simple group, then is X birational to a quotient of an abelian variety?

Stefan Bauer (Bielefeld). Differential equations and stable homotopy. Friday December 2nd, Huxley 140, 1.30-2.30pm.

The talk considers a space parametrizing certain differential equations. The topology of the space in question turns out to be related to spaces well known from stable homotopy theory: The classifying spaces of ko-theory and stable homotopy theory.

Groups and Geometry in the Southeast II. Friday December 2nd, 2.30-5pm , UCL.

Michael Singer (Edinburgh). Partial Bergman kernels and slope stability of toric varieties. Friday December 9th, Huxley 140, 1.30-2.30pm.

The Bergman kernel on a compact Kaehler manifold is the Schwartz kernel of the L^2 projection onto the space of holomorphic sections of some ample line bundle L.  Much is known about the asymptotic behaviour of the Bergman kernel for high powers of the bundle L. In this talk, we will consider the `partial’ projection onto the subspace of sections vanishing to order lk along a submanifold. Restricting to the toric case, we will prove the existence of a distributional asymptotic expansion for the corresponding partial density function (the value of the kernel on the diagonal), at least to leading order, and indicate an application to the study of slope stability of the polarized manifold (M,L).

Ben Weinkove (UC San Diego). Collapsing along the Kahler-Ricci flow. Friday December 16th. Huxley 140, 1.30-2.30pm.

I will discuss the collapsing behavior of the Kahler-Ricci flow in two cases:  when the manifold is a projective bundle over an algebraic variety and when the manifold is a product of an elliptic curve and a Riemann surface of genus larger than one. This is joint work with Song-Szekelyhidi and Song.

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