Organized by Robert Berman, Semyon Klevtsov, Paul Wiegmann, and Steve Zelditch

April 20 – June 19, 2015

This program centers on the use of holomorphic sections of high powers of positive Hermitian holomorphic line bundles over a Kahler manifold to construct projective embeddings, Bergman Kahler metrics, and Gaussian random fields. The main tool is the Bergman kernel and its large N asymptotics, on and off the diagonal. Its asymptotics has many applications in geometry, probability theory and mathematical physics.

In geometry one often uses the Kodaira-type embedding of the manifold to the projective space of the holomorphic sections, in order to pull back the Fubini-Study metric to define a space of algebro-geometric metrics, known as Bergman metrics. For large N, they approx- imate all Kahler metrics, and in fact the finite dimensional symmetric space geometry of the space of Bergman metrics approximates the full infinite dimensional symmetric space geometry of the space of all Kahler metrics. In particular, one may study the geodesics on the latter, approximating them by the geodesics on the space of Bergman metrics. One can also study integral geometry on the space of metrics and use the finite dimensional approximations to define random Kahler metrics, with potential applications in physics.

In another directions Gaussian random holomorphic sections of the line bundle induce random zero sets and point processes. The two point function of the process equals to the Bergman kernel and, therefore, its asymptotics control the geometry of random zero sets. One may also define a canonical point process from the Bergman kernel and expect that as the number of points tends to infinity it approximates important metrics. The Bergman kernel has further uses in dimension one to study the geometric properties of random normal matrix ensembles.

All these applications are partly inspired by the Yau-Tian-Donaldson program of relating GIT stability and Kahler-Einstein metrics, but the focus of the program is on holomorphic stochastic geometry and mathematical physics.

**Speaker and Seminar Schedule:**

The weekly seminars will take place in** room 313 **(unless otherwise specified below).

Date and Time | Title | Presenters | |

4/27 at 10:00am – Room 313 | Hele-Shaw flow and holomorphic discs | Julius Ross (Cambridge) | |

4/27 at 11:00am – Room 313 | On the Agmon estimate on noncompact manifold | YZhiqin Lu (University of California, Irvine) | |

4/28 at 2:15pm – Room 313 | Hele-Shaw flows and holomorphic discs (continued) | David Witt Nystrom | |

4/29 at 10:00am – Room 313 | Hele-Shaw flow and holomorphic discs (continued) | Julius Ross (Cambridge) | |

4/29 at 11:00am – Room 313 | Geometry of the quantum Hall effect | Semyon Klevtsov (University of Cologne) | |

5/11 at 11:00am – Room 313 | Bergman kernel and Feynman diagrams | Hao Xu (Harvard University) | |

5/12 at 2:00om – Roon 313 | The second fundamental form of Kodaira embeddings, and quantization | Joel Fine (Universite Libre de Bruxelles) | |

5/14 at 1:00om – Roon 313 | Two-dimensional random geometry: continuous and discrete approaches | Ivan Kostov (IPhT, CEA-Saclay) | |

5/14 at 2:00om – Roon 313 | Geometrical responses of quantum Hall states | Alexander Abanov (SCGP) | |

5/26 at 10:00am – Room 313 | Two-dimensional random geometry: continuous and discrete approaches, Part 2 | Ivan Kostov (IPhT, CEA-Saclay) | |

5/27 at 10:00am – Room 313 | Geometrical responses of quantum Hall states, Part 2 | Alexander Abanov (SCGP) | |

5/27 at 11:15am – Room 313 | Geometrical responses of quantum Hall states, Part 3 | Andrey Gromov (Stony Brook University) | |

5/27 at 4:00pm – Room 313 | Singular metrics and equidistribution | George Marinescu (University of Cologne) | |

5/28 at 10:00 am – Room 313 | Quantum Hall effect on Riemann surfaces | Semyon Klevtsov (University of Cologne) | |

5/28 at 1:30pm – Room 313 | Ward Identities for Fractional Quantum Hall states | Tankut Can (SCGP) | |

6/2 at 10:00am – Room 313 | Pairing between zeros and critical points of random polynomials | Boris Hanin (MIT) | |

6/4 at 10:00am – Room 313 | The metric completion of the space of Kahler potentials and applications, part 1 | Tamas Darvas (Purdue University) | |

6/4 at 1:00pm – Room 313 | Large N expansion of beta-ensembles on arbitrary contours | Anton Zabrodin (ITEP and HSE) | |

6/4 at 2:30pm – Room 313 | Singular metrics and equidistribution, part 2 | George Marinescu (University of Cologne) | |

6/5 at 10:00am – Room 313 | The metric completion of the space of Kahler potentials and applications, part 2 | Tamas Darvas (Purdue University) | |

6/5 at 1:00pm – Room 313 | Equidistribution for sequences of line bundles on normal Kahler spaces | Dan Coman | |

6/8 at 1:30pm – Room 313 | Quantitative equidistribution of beta-ensemble | Joaquim Ortega-Cerda (Universitat de Barcelona) | |

6/8 at 2:30pm – Room 313 | What is the topology of a random real algebraic set? | Damien Gayet (Universite Joseph Fourier Grenoble 1) | |

6/9 at 10:00am – Room 313 | Some aspects of infinite dimensional complex geometry | Laszlo Lempert (Purdue University) | |

6/10 at 11:00am – Room 313 | Kahler geometry in string compactification | Michael Douglas | |

6/11 at 10:00am – Room 313 | Geodesics flows and the nonlinear sigma model | Peng Gao (Harvard University) | |

6/11 at 1:00pm – Room 313 | Constraints on canonical Poincar type Kahler metrics | Hugues Auvray (University Paris-Sud) | |

6/12 at 10:00am – Room 313 | Introduction to interpolation and sampling, part 1 | Dror Varolin (Stony Brook University) | |

6/12 at 11:10am – Room 313 | Introduction to interpolation and sampling, part 2 | Joachim Ortega-Cerda (Universitat de Barcelona) | |

6/12 at 2:00pm – Room 313 | Holography, Probe Branes and Isoperimetric Inequalities | Frank Ferrari (Universite Libre de Bruxelles) |