James Pascaleff

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This webpage is out of date. I have moved to the University of Illinois at Urbana-Champaign.

The University of Texas at Austin
Department of Mathematics, RLM 8.100
2515 Speedway Stop C1200
Austin, TX 78712-1202

Office: RLM 11.166
Telephone: (512) 471-1135
Email: jpascaleff@math.utexas.edu (PGP public key)

I am an RTG postdoctoral fellow in the Department of Mathematics at the University of Texas at Austin. Previously, I was a student at MIT, where I received my PhD under the supervision of Denis Auroux. You can see my full CV here.

During the period 1 June–15 August 2014, I am in Bonn as a guest of the Max Planck Institute for Mathematics.

Spring 2014 teaching: Lagrangian Floer Homology (M 392C)

Research

I am interested in symplectic geometry, Floer theory and mirror symmetry. Currently I've been thinking about Floer theory for log Calabi-Yau surfaces, following work of Gross-Hacking-Keel. My thesis concerns special Lagrangian torus fibrations (the Strominger–Yau–Zaslow conjecture) and the Floer cohomology of Lagrangian sections of the torus fibration in certain cases where this fibration has singularities.

Articles

  • My arXiv and google scholar pages.
  • Floer cohomology of g-equivariant Lagrangian branes.
    (with Yankı Lekili, arXiv:1310.8609, submitted)
  • On the symplectic cohomology of log Calabi–Yau surfaces.
    (arXiv:1304.5298, submitted)

    Log Calabi-Yau manifolds (the complement Y-D of an effective anticanonical divisor D in a projective manifold Y) have a rich symplectic geometry. They are the subject of a mirror symmetry conjecture: there is a mirror variety M, and the Floer theory of Y-D is reflected in the algebraic geometry of M. For instance, following Gross–Hacking–Keel and Gross–Siebert, one expects to find the ring of regular functions on M sitting inside the symplectic cohomology of Y-D.

    This article contains some results towards this conjecture in complex dimension two. There is a basis for degree zero symplectic cohomology indexed by integral points in a certain SL(2,Z)-manifold (related to the Gross-Hacking-Keel theta functions). We also include a discussion of wrapped Floer cohomology of certain Lagrangian submanifolds, and a description of the product structure in a particular case. We also show that, after enhancing the coefficient ring, the degree–zero symplectic cohomology defines a family degenerating to a singular surface obtained by gluing together several affine planes.

  • Floer cohomology in the mirror of the projective plane and a binodal cubic curve.
    Duke Math. J., to appear. (arXiv:1109.3255, or thesis version)

    This paper explores an example of homological mirror symmetry, relating Lagrangian branes in a Landau-Ginzburg model (X, W) to coherent sheaves on the projective plane. The Landau-Ginzburg model is regarded as the mirror to the pair consisting of the projective plane along with a chosen anticanonical divisor, which we take to be the union of a conic and a line. This is the binodal cubic curve of the title. The topology of X and the superpotential W reflect this particular choice, whereas, for example, taking the anticanonical divisor to be the union of three transverse lines gives rise to the the "Hori-Vafa" mirror.

    This Landau-Ginzburg model is interesting from the point of view of the Strominger-Yau-Zaslow (SYZ) picture of mirror symmetry, because the total space X carries a special Lagrangian torus fibration with a singularity. The Lagrangian submanifolds we consider are sections of this torus fibration, meaning that they correspond to line bundles on the mirror projective plane, in line with general expectations from the SYZ picture.

    The algebra structure on the Floer cohomology of these Lagrangian submanifolds is computed, and matched with the algebra structure on the hom-spaces of the corresponding line bundles. The tools here are symplectic techniques developed by Seidel for counting pseudoholomorphic sections of Lefschetz fibrations. One of the later sections describes the "tropical" interpretation of the result: the Lagrangian Floer cohomology has a natural basis in bijection with integral points in an affine manifold, and the algebra structure counts particular tropical curves in this manifold. In the last section we show how our computation can be used to describe the wrapped Floer cohomology of the Lagrangian submanifolds as well.

Seminars & workshops

  • Graduate Geometry and Topology Current Literature Seminar: see departmental calendar (2012-2013 schedule)

As a graduate student, I co-organized two workshops:

Teaching