JHU-UMD Complex Geometry Seminar

Departments of Mathematics

Johns Hopkins University & University of Maryland

Fall 2012 - Spring 2013

Date: Tuesdays at 4:30pm.
Room: Krieger Hall 308 (JHU), Mathematics Building 3206 (UMD).

Organized by: Y.A. Rubinstein , B. Shiffman , R. Wentworth, S. Wolpert, Y. Yuan .

The seminar is a combination of a learning and a research seminar. The first 30 minutes of each talk are a "trivial notions" talk defining all basic notions, giving examples and intuition to the subject, and should be accessible to a beginning graduate student. The next 50 minutes are a regular seminar talk.

  • September 18, UMD
    Vamsi Pingali (Stony Brook),
    Some analytic and computational aspects of Chern-Weil forms
  • A generalised Monge-Ampere equation arising out a natural question from Chern-Weil theory shall be presented. The difficulties involved with the analysis shall be discussed along with a connection to Kahler geometry. On the computational side of Chern-Weil theory, a computation of Chern forms for certain trivial bundles with non-diagonal metrics shall be shown along with an application or two.

  • October 2, JHU
    Ioana Suivaina (Vanderbilt)
    ALE Ricci-flat Kahler surfaces and weighted projective spaces
  • I will give an explicit classification of the ALE Ricci flat Kahler surfaces, generalizing previous classification results of Kronheimer. These manifolds are related to a special class of deformations of quotient singularities of type C^2/G, with G a finite subgroup of U(2). I finish the talk by explaining the relations with the Tian-Yau construction of complete Ricci flat Kahler manifolds.

  • October 23, UMD, 3:30pm, room 2300 (note special time AND room)
    Robert Penner (Aarhaus and Caltech)
    Counting chord diagrams
    A linear chord diagram on some number b of backbones is a collection of n chords with distinct endpoints attached to the interiors of b intervals. Taking the intervals to lie in the real axis and the chords to lie in the upper half-plane associates a fat graph to a chord diagram, which thus has its associated genus g. The numbers of connected genus g chord diagrams on b backbones with n chords are of significance in mathematics, physics and biology as we shall explain. Recent work using the topological recursion of Eynard-Orantin has computed them perturbatively via a closed form expression for the free energies of an Hermitian matrix model with potential  V(x)=x^2/2-stx/(1-tx). Very recent work has moreover shown that the partition function satisfies a second order non-linear PDE which gives a generalization of the Harer-Zagier equation that arises for one backbone.

    Tamas Darvas (Purdue) at 4:40pm, room 3206 (usual room)
    Morse theory and geodesics in the space of Kahler metrics
  • Given a compact Kahler manifold let H be the set of Kahler forms in a fixed cohomology class. As observed by Mabuchi, this space has the structure of an infinite dimensional Riemannian manifold, if one identifies it with a totally geodesic subspace of H, the set of Kahler potentials. Following Donaldson's program, existence and regularity of geodesics in this space is of fundamental interest. Supposing enough regularity of a geodesic u : [0; 1]--> H, we establish a Morse theoretic result relating the endpoints with the initial tangent vector. As an application, we prove that on all Kahler manifolds, connecting Kahler potentials with smooth geodesics is not possible in general.

  • November 6 (rescheduled due to weather), JHU
    Gang Liu (Minnesota)
    Some comparison theorems for Kahler manifolds with Ricci curvature bounded from below
  • Comparison theorems are a fundamental tool in Riemannian geometry. When the Ricci curvature is bounded from below, one has Bishop-Gromov volume comparison, Bonnet-Myers theorem on the diameter, comparison theorems on the spectrum of the Laplacian, and more. In the Kahler setting, Li and Wang established analogous comparisons when the bisectional curvature has a lower bound. In this talk, I will discuss some comparison theorems on Kahler manifolds when the Ricci curvature has a lower bound.

  • November 13, UMD
    Boris Hanin (Northwestern)
    Correlations and Pairing of Zeros and Critical Points of Random Polynomials
  • The goal of this talk is to explain how the zeros and holomorphic critical points of random polynomials are correlated. The motivation for studying this question comes from the Gauss-Lucas theorem, which states that the critical points of a polynomial in one complex variable lie inside the convex hull of its zeros. I will explain that, in fact, zeros and critical points appear in rigid pairs. I will present some results about the geometry of these pairs, and I will try to give some physical intuition for why they should appear in the first place.

  • November 27, JHU
    Jian Song (Rutgers)
    Analytic minimal model program with Ricci flow
    I will introduce the analytic minimal model program proposed by Tian and me to study formation of singularities of the Kahler-Ricci flow. We also construct geometric and analytic surgeries of codimension one and higher codimensions equivalent to birational transformations in algebraic geometry by Ricci flow.
  • December 11, UMD
    Norm Levenberg (Indiana)
    Characterization of meromorphic functions and projective hulls
  • Reese Harvey and Blaine Lawson introduced the notion of the projective hull of a closed subset in a complex projective space with the hope of generalizing a result of John Wermer on the polynomial hull of a real-analytic curve in a complex affine space. Both notions of "hull" can be understood in terms of an extremal (quasi-)plurisubharmonic function associated to the underlying set. We begin by giving background motivation, definitions and examples of these hulls in the setting of pluripotential theory; and we include a complex geometric interpretation of the projective hull. Then we utilize these ideas to give conditions characterizing holomorphic and meromorphic functions in the unit disk in the complex plane in terms of certain weak forms of the maximum modulus principle. These characterizations are joint work with John Anderson, Joe Cima and Tom Ransford.

  • February 12, UMD
    Slawomir Dinew (Rutgers)
    Interior (ir)regularity for the complex Monge-Ampere equation
  • The complex Monge-Ampere operator arises in many geometric problems. When studying its local properties it is natural to ask for its interior regularity theory. This is crucial if analysis is performed in coordinate charts. Quite contrary to linear differential operators there is however no general purely interior result. In the talk we shall present several additional conditions under which such results can be obtained. We shall give several examples suggesting what is the expected behavior under different regularity assumptions.

  • February 26, JHU
    Xiaochun Rong (Rutgers)
    Continuity of extremal transitions and flops for Calabi-Yau manifolds
  • We will discuss metric behavior of Ricci-flat Kahler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We will prove a version of Candelas and de la Ossa's conjecture: Ricci-flat Calabi-Yau manifolds related via extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi-Yau manifolds and a compact metric space in the Gromov-Hausdorff topology. This is joint work with Yuguang Zhang.

  • March 12, JHU
    Dan Coman (Syracuse)
    Convergence of the Fubini-Study currents for singular metrics on line bundles and applications
  • April 2, UMD
    Guangbo Xu (Princeton)
    Gauged linear sigma model and adiabatic limits
  • The physics theory of gauged linear sigma model combines the theory of maps (the sigma model) and gauge theory. In dimension 2, it is naturally related to holomorphic vector bundles over Riemann surfaces and Gromov-Witten invariants of projective spaces (or more general varieties). In this talk, I will discuss, from a mathematical perspective, of some simple examples in gauged linear sigma model. I will also discuss about how to use the adiabatic limits of such theory to solve a natural equation (the vortex equation) in gauged linear sigma model over the complex plane.

  • April 16, UMD
    Simon Donaldson (Imperial)
    Informal talk on Kahler-Einstein geometry
  • April 30, JHU
    Ben Weinkove (Northwestern)
    Geometric flows on complex surfaces
  • I will discuss the behavior of the Kahler-Ricci flow and a new flow generalizing it, called the Chern-Ricci flow, recently introduced by M. Gill. The Chern-Ricci flow can be defined on any complex manifold. I will describe what is known about these flows in the case of complex surfaces, with an emphasis on examples.

  • May 14, UMD
    David Witt-Nystrom (Chalmers)
    Local circle actions on Kahler manifolds and the Hele-Shaw flow


Driving directions to JHU. Park in South Garage (see map) on any level (except the reserved spaces). Take a ticket when entering. The Department will provide a visitor parking pass to use when exiting.
Driving and parking directions to UMD. Park in Paint Branch Drive Visitor Lot (highlighted in yellow in the lower right corner of the second map in the previous link), or in Regents Drive Garage (highlighted in the upper right corner). If you arrive after 4pm you do not need to pay: see the instructions in the previous link.