Past Algebraic and Symplectic Geometry Seminar

26 May 2009
15:45
Nicos Kapouleas
Abstract
I will survey the recent work of Haskins and myself constructing new special Lagrangian cones in ${\mathbb C}^n$ for all $n\ge3$ by gluing methods. The link (intersection with the unit sphere ${\cal S}^{2n-1}$) of a special Lagrangian cone is a special Legendrian $(n-1)$-submanifold. I will start by reviewing the geometry of the building blocks used. They are rotationally invariant under the action of $SO(p)\times SO(q)$ ($p+q=n$) special Legendrian $(n-1)$-submanifolds of ${\cal S}^{2n-1}$. These we fuse (when $p=1$, $p=q$) to obtain more complicated topologies. The submanifolds obtained are perturbed to satisfy the special Legendrian condition (and their cones therefore the special Lagrangian condition) by solving the relevant PDE. This involves understanding the linearized operator and its small eigenvalues, and also ensuring appropriate decay for the solutions.
  • Algebraic and Symplectic Geometry Seminar
19 May 2009
15:45
Kazushi Ueda
Abstract
A polynomial $f$ is said to be a Brieskorn-Pham polynomial if $ f = x_1^{p_1} + ... + x_n^{p_n}$ for positive integers $p_1,\ldots, p_n$. In the talk, I will discuss my joint work with Masahiro Futaki on the equivalence between triangulated category of matrix factorizations of $f$ graded with a certain abelian group $L$ and the Fukaya-Seidel category of an exact symplectic Lefschetz fibration obtained by Morsifying $f$.
  • Algebraic and Symplectic Geometry Seminar
7 May 2009
15:45
Eduard Looijenga
Abstract
This is an overview, mostly of work of others (Denef, Loeser, Merle, Heinloth-Bittner,..). In the first part of the talk we give a brief introduction to motivic integration emphasizing its application to vanishing cycles. In the second part we discuss a join construction and formulate the relevant Sebastiani-Thom theorem.
  • Algebraic and Symplectic Geometry Seminar
7 May 2009
14:00
Eduard Looijenga
Abstract
This is an overview, mostly of work of others (Denef, Loeser, Merle, Heinloth-Bittner,..). In the first part of the talk we give a brief introduction to motivic integration emphasizing its application to vanishing cycles. In the second part we discuss a join construction and formulate the relevant Sebastiani-Thom theorem.
  • Algebraic and Symplectic Geometry Seminar
28 April 2009
15:45
Geordie Williamson
Abstract
Triply graded link homology (introduced by Khovanov and Rozansky) is a categorification of the HOMFLYPT polynomial. In this talk I will discuss recent joint work with Ben Webster which gives a geometric construction of this invariant in terms of equivariant constructible sheaves. In this framework the Reidemeister moves have quite natural geometric proofs. A generalisation of this construction yields a categorification of the coloured HOMFLYPT polynomial, constructed (conjecturally) by Mackay, Stosic and Vaz. I will also describe how this approach leads to a natural formula for the Jones-Ocneanu trace in terms of the intersection cohomology of Schubert varieties in the special linear group.
  • Algebraic and Symplectic Geometry Seminar

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