{"id":149,"date":"2011-10-03T13:31:08","date_gmt":"2011-10-03T13:31:08","guid":{"rendered":"http:\/\/geometry.ma.ic.ac.uk\/seminar\/?page_id=149"},"modified":"2014-09-28T20:04:15","modified_gmt":"2014-09-28T20:04:15","slug":"autumn-term-2011","status":"publish","type":"page","link":"http:\/\/coates.ma.ic.ac.uk\/seminar\/?page_id=149","title":{"rendered":"Autumn Term 2011"},"content":{"rendered":"

 
\nMartijn Kool (Imperial). <\/em>Reduced classes and curve counting on surfaces. <\/strong>Friday October 7th, Huxley 140,\u00a01.30-2.30pm.<\/p>\n

Let S be a smooth projective surface with sufficiently ample line bundle L. The G\u00f6ttsche 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 G\u00f6ttsche’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.<\/p>\n

Matthew Rathbun (Imperial). <\/em>High Distance Knots in Closed 3-Manifolds. <\/strong>Friday October 14th, Huxley 140, 1.30-2.30pm.<\/p>\n

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the\u00a0stabilized splitting has arbitrarily\u00a0high\u00a0distance\u00a0\u00a0with respect to the\u00a0splitting surface. This generalizes a result of Minsky, Moriah, and\u00a0Schleimer for\u00a0knots\u00a0in S^3. We also show that in the complex of curves,\u00a0handlebody sets are either coarsely distinct or identical. We define\u00a0the coarse mapping class group of a Heegaard\u00a0splitting, and show that\u00a0for 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.<\/p>\n

Song Sun (Imperial). <\/em>Positively curved Sasaki manifolds. <\/strong>Friday October 21st, Huxley 140, 1.30-2.30pm.<\/p>\n

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. \u00a0I 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.<\/p>\n

Mauro Mauricio (Imperial). <\/em>Connect sum lens space surgeries<\/strong>. <\/strong>Friday October 28th, Huxley 140, 1.30-2.30pm.<\/span><\/em><\/p>\n

Motivated by particular DNA-protein interactions, we are interested in the following question:\u00a0 Can rational tangle replacement produce a connect sum of 4-plats from a single 4-plat?\u00a0 We study this problem via Dehn surgery on the double branch covers.\u00a0 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.\u00a0If both links share a common prime summand we conjecture that this is impossible when the complement is a prime tangle.\u00a0We describe a strategy for proving this conjecture using Heegaard Floer homology.\u00a0 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.\u00a0 These proofs builds on methods developed by Ozsvath and Szab\u00f3\u00a0for knots in\u00a0S3<\/sup>.<\/p>\n

Groups and Geometry in the Southeast I.<\/em> Friday October 28th, 2.30-5pm , UCL. Speakers: \u00a0Bergeron and Bartels. Further details.<\/a><\/p>\n

Danny Calegari (Cambridge & Caltech). <\/em>Ziggurats and rotation numbers<\/strong>. Friday November 4th, Huxley 140, 1.30-2.30pm.<\/p>\n

I \u00a0will \u00a0discuss \u00a0new \u00a0rigidity \u00a0and \u00a0rationality \u00a0phenomena \u00a0(related \u00a0to \u00a0the \u00a0phenomenon \u00a0of \u00a0Arnold \u00a0tongues) \u00a0in \u00a0the \u00a0 theory \u00a0of \u00a0nonabelian \u00a0group \u00a0actions \u00a0on \u00a0the \u00a0circle. I \u00a0will \u00a0introduce \u00a0tools that \u00a0can \u00a0translate \u00a0questions \u00a0about \u00a0 the \u00a0existence \u00a0of \u00a0actions \u00a0with \u00a0prescribed \u00a0dynamics, \u00a0into \u00a0finite \u00a0combinatorial \u00a0questions \u00a0that \u00a0can be\u00a0answered \u00a0 effectively. \u00a0There \u00a0are \u00a0connections with the \u00a0theory \u00a0of \u00a0diophantine \u00a0approximation, \u00a0and \u00a0with \u00a0the \u00a0bounded \u00a0 cohomology \u00a0of \u00a0free \u00a0groups. \u00a0A \u00a0special\u00a0case of \u00a0this \u00a0theory \u00a0gives \u00a0a \u00a0very \u00a0short \u00a0new \u00a0proof \u00a0of \u00a0Naimi’s \u00a0theorem \u00a0 (i.e. \u00a0the conjecture \u00a0of \u00a0Jankins-Neumann) \u00a0which \u00a0was \u00a0the \u00a0last \u00a0step \u00a0in \u00a0the \u00a0classification \u00a0of taut\u00a0foliations \u00a0of \u00a0 Seifert \u00a0fibered \u00a0spaces. \u00a0This \u00a0is \u00a0joint \u00a0work \u00a0with \u00a0Alden \u00a0Walker.<\/p>\n

King\u2019s College Geometry Day 2. <\/em>Friday November 11th 11am-6.30pm. Speakers: Bourni, Haskins, Minicozzi, Tinaglia and Wickramasekera.\u00a0Further details.<\/a><\/p>\n

Seminar cancelled! <\/em> Friday November 18th, Huxley 140, 1.30-2.30pm.<\/span><\/strong><\/p>\n

De-Qi Zhang (Singapore and Max Planck).<\/em> Automorphisms of positive entropy on compact Kahler manifolds.<\/strong> <\/em>Friday November 25th, Huxley 140, 1.30-2.30pm.<\/span><\/p>\n

We show that a minimal compact Kahler manifold (or minimal normal projective variety) X of dimension n > \u00a02\u00a0has at most n-1 independent commuting (modulo elements of null entropy) automorphisms of positive entropy\u00a0and the maximality n-1 occurs only when X is a quotient of a compact torus.\u00a0We also propose an open question: if the representation G | NS_R(X) on the Neron Severi group is embedded as a Zariski-dense\u00a0subset in a real rank n-1 semi simple group, then is X birational to a quotient of an abelian variety?<\/p>\n

Stefan Bauer (Bielefeld). <\/em>Differential equations and stable homotopy<\/strong>. <\/strong>Friday December 2nd, Huxley 140, 1.30-2.30pm.<\/p>\n

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.<\/p>\n

Groups and Geometry in the Southeast II.<\/em> Friday December 2nd, 2.30-5pm , UCL.<\/a><\/p>\n

Michael Singer (Edinburgh). <\/em>Partial Bergman kernels and slope stability of toric varieties<\/strong>.<\/strong> <\/em>Friday December 9th, Huxley 140, 1.30-2.30pm.<\/p>\n

The Bergman kernel on a compact Kaehler manifold is the Schwartz\u00a0kernel of the L^2 projection onto the space of holomorphic sections of some\u00a0ample line bundle L.\u00a0 Much is known about the asymptotic behaviour of the\u00a0Bergman kernel for high powers of the bundle L. In this talk, we will\u00a0consider the `partial’ projection onto the subspace of sections vanishing to\u00a0order lk along a submanifold. Restricting to the toric case, we will prove\u00a0the existence of a distributional asymptotic expansion for the corresponding\u00a0partial density function (the value of the kernel on the diagonal), at least\u00a0to leading order, and indicate an application to the study of slope stability of\u00a0the polarized manifold (M,L).<\/p>\n

Ben Weinkove (UC San Diego). <\/em>Collapsing along the Kahler-Ricci flow<\/strong>. <\/strong>Friday December 16th. Huxley 140, 1.30-2.30pm.<\/p>\n

I will discuss the collapsing behavior of the Kahler-Ricci flow in two cases: \u00a0when the manifold is a projective bundle over an\u00a0algebraic 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.<\/p>\n","protected":false},"excerpt":{"rendered":"

  Martijn Kool (Imperial). Reduced classes and curve counting on surfaces. Friday October 7th, Huxley 140,\u00a01.30-2.30pm. Let S be a smooth projective surface with sufficiently ample line bundle L. The G\u00f6ttsche 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 …<\/p>\n

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