Past Algebraic and Symplectic Geometry Seminar

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.

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.

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.

The space of smooth rational curves of degree d in projective space admits various moduli theoretic compactifications via GIT, stable maps, stable sheaves, Hilbert scheme and so on. I will discuss how these compactifications are related by explicit blow-ups and -downs for d

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 $L$.

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 $a_1,\ldots,a_n>0$.

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 $t>0$.

Bridgeland's notion of stability condition allows us to associate a complex manifold, the space of stability conditions, to a triangulated category $D$. Each stability condition has a heart - an abelian subcategory of $D$ - and we can decompose the space of stability conditions into subsets where the heart is fixed. I will explain how (under some quite strong assumpions on $D$) the tilting theory of $D$ governs the geometry and combinatorics of the way in which these subsets fit together. The results will be illustrated by two simple examples: coherent sheaves on the projective line and constructible sheaves on the projective line stratified by a point and its complement.

Let X be a Gorenstein orbifold and Y a crepant resolution of

X. Suppose that the quantum cohomology algebra of Y is semisimple. We describe joint work with Iritani which shows that in this situation the genus-zero crepant resolution conjecture implies a higher-genus version of the crepant resolution conjecture. We expect that the higher-genus version in fact holds without the semisimplicity hypothesis.

There is a non-degenerate 2-form on S^6, which is compatible with the almost complex structure that S^6 inherits from its inclusion in the imaginary octonions. Even though this 2-form is not closed, we may still define Lagrangian submanifolds. Surprisingly, they are automatically minimal and are related to calibrated geometry. The focus of this talk will be on the Lagrangian submanifolds of S^6 which are fibered by geodesic circles over a surface. I will describe an explicit classification of these submanifolds using a family of Weierstrass formulae.

We show that the leading terms of the number of absolutely indecomposable representations of a quiver over a finite field are related to counting graphs. This is joint work with Geir Helleloid.

Mean curvature vector is the negative gradient of the area functional. Thus if we deform a submanifold along its mean curvature vector, which is called mean curvature flow (MCF), the area will decrease most rapidly. When the ambient manifold is Kahler-Einstein, being Lagrangian is preserved under MCF. It is thus very natural trying to construct special Lagrangian/ Lagrangian minimal through MCF. In this talk, I will make a brief introduction and report some of my recent works with my coauthors in this direction, which mainly concern the singularities of the flow.