Past Algebraic and Symplectic Geometry Seminar

I will talk about joint work with Dimca, respectively Behrend and Bryan, in which we refine the numerical DT-Behrend invariants of Hilbert schemes of threefolds by using vanishing cycle motives (a la Kontsevich-Soibelman) or mixed Hodge modules, leading to deformed MacMahon formulae.

I will talk about joint work with Dimca, respectively Behrend and Bryan, in which we refine the numerical DT-Behrend invariants of Hilbert schemes of threefolds by using vanishing cycle motives (a la Kontsevich-Soibelman) or mixed Hodge modules, leading to deformed MacMahon formulae.

I shall describe joint work with Gritsenko and Hulek in which we study the moduli spaces of polarised holomorphic symplectic manifolds via their periods. There are strong similarities with moduli spaces of K3 surfaces, but also some important differences, notably that global Torelli fails. I shall explain (conjecturally) why and show how the techniques used to obtain general type results for K3 moduli can be modified to give similar, and quite strong, results in this case. Mainly I shall concentrate on the case of deformations of Hilbert schemes of K3 surfaces.

We give a three dimensional generalization of the classical McKay correspondence construction by Gonzales-Sprinberg and Verdier. This boils down to computing for the Bridgeland-King-Reid derived category equivalence the images of twists of the point sheaf at the origin of C^3 by irreducible representations of G. For abelian G the answer turns out to be closely linked to a piece of toric combinatorics known as Reid's recipe.

Associated to a pencil of algebraic curves with singular fibres is a bundle of Jacobians (which are abelian varieties off the discriminant locus of the family and semiabelian varieties over it). Normal functions, which are holomorphic sections of such a Jacobian bundle, were introduced by Poincare and used by Lefschetz to prove the Hodge Conjecture (HC) on algebraic surfaces. By a recent result of Griffiths and Green, an appropriate generalization of these normal functions remains at the center of efforts to establish the HC more generally and understand its implications. (Furthermore, the nature of the zero-loci of these normal functions is related to the Bloch-Beilinson conjectures on filtrations on Chow groups.)

Abel-Jacobi maps give the connection between algebraic cycles and normal functions. In this talk, we shall discuss the limits and singularities of Abel-Jacobi maps for cycles on degenerating families of algebraic varieties. These two features are strongly connected with the issue of graphing admissible normal functions in a Neron model, properly generalizing Poincare's notion of normal functions. Some of these issues will be passed over rather lightly; our main intention is to give some simple examples of limits of AJ maps and stress their connection with higher algebraic K-theory.

A very new theme in homological mirror symmetry concerns what the mirror of a normal function should be; in work of Morrison and Walcher, the mirror is related to counting holomorphic disks in a CY 3-fold bounding on a Lagrangian. Along slightly different lines, we shall briefly describe a surprising application of "higher" normal functions to growth of enumerative (Gromov-Witten) invariants in the context of local mirror symmetry.

I'll explain (following Kontsevich and Soibelman) how cluster transformations intertwine non-commutative DT invariants for CY3 algebras related by a tilt.

I will describe some recent joint work with Davide Gaiotto and Greg Moore, in which we explain the origin of the wall-crossing formula of Kontsevich and Soibelman, in the context of N=2 supersymmetric field theories in four dimensions. The wall-crossing formula gives a recipe for constructing the smooth hyperkahler metric on the moduli space of the field theory reduced on a circle to 3 dimensions. In certain examples this moduli space is actually a moduli space of ramified Higgs bundles, so we obtain a new description of the hyperkahler structure on that space.

I will describe joint work with Mohammed Abouzaid, in which we complete the proof of homological mirror symmetry for the standard four-torus and consider various applications. A key tool is the recently-developed holomorphic quilt theory of Mau-Wehrheim-Woodward.

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.

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 $f$.

I'll define the category of B-branes in a LG model, and show that for affine models the Hochschild homology of this category is equal to the physically-predicted closed state space. I'll also explain why this is a step towards proving that LG B-models define TCFTs.

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.

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.

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

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.

There are two basic theories of curve counting on Calabi-Yau threefolds. Donaldson-Thomas theory arises by considering curves as subschemes; Gromov-Witten theory arises by considering curves as the image of maps. Both theories can also be formulated for orbifolds. Let X be a dimension three Calabi-Yau orbifold and let

Y --> X be a Calabi-Yau resolution. The Gromov-Witten theories of X and Y are related by the Crepant Resolution Conjecture. The Gromov-Witten and Donaldson-Thomas theories of Y are related by the famous MNOP conjecture. In this talk I will (with some provisos) formulate the remaining equivalences: the crepant resolution conjecture in Donaldson-Thomas theory and the MNOP conjecture for orbifolds. I will discuss examples to illustrate and provide evidence for the conjectures.