Past Algebraic and Symplectic Geometry Seminar

17 February 2009
15:45
Jacob Rasmussen
Abstract
Khovanov homology is an invariant of knots in $S^3$. In its original form, it is a "homological version of the Jones polynomial"; Khovanov and Rozansky have generalized it to other knot polynomials, including the HOMFLY polynomial. In the second talk, I'll discuss how Khovanov homology and its generalizations lead to a relation between the HOMFLY polynomial and the topology of flag varieties.
  • Algebraic and Symplectic Geometry Seminar
17 February 2009
14:15
Jacob Rasmussen
Abstract
Khovanov homology is an invariant of knots in $S^3$. In its original form, it is a "homological version of the Jones polynomial"; Khovanov and Rozansky have generalized it to other knot polynomials, including the HOMFLY polynomial. The first talk will be an introduction to Khovanov homology and its generalizations.
  • Algebraic and Symplectic Geometry Seminar
27 January 2009
15:45
Dominic Joyce
Abstract
Let $(M,\omega)$ be a symplectic manifold, and $g$ a Riemannian metric on $M$ compatible with $\omega$. If $L$ is a compact Lagrangian submanifold of $(M,\omega)$, we can compute the volume Vol$(L)$ of $L$ using $g$. A Lagrangian $L$ is called {\it Hamiltonian stationary} if it is a stationary point of the volume functional amongst Lagrangians Hamiltonian isotopic to $L$. </p> <p> Suppose $L'$ is a compact Lagrangian in ${\mathbb C}^n$ which is Hamiltonian stationary and {\it rigid}, that is, all infinitesimal Hamiltonian deformations of $L$ as a Hamiltonian stationary Lagrangian come from rigid motions of ${\mathbb C}^n$. An example of such $L'$ is the $n$-torus $ \bigl\{(z_1,\ldots,z_n)\in{\mathbb C}^n:\vert z_1\vert=a_1, \ldots,\vert z_n\vert=a_n\bigr\}$, for small $a_1,\ldots,a_n>0$. </p> <p> I will explain a construction of Hamiltonian stationary Lagrangians in any compact symplectic manifold $(M,\omega)$, which works by `gluing in' $tL'$ near a point $p$ in $M$ for small $t>0$. </p>
  • Algebraic and Symplectic Geometry Seminar

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