GAUGE THEORY, TOPOLOGY & SYMPLECTIC GEOMETRY SEMINAR: | Sheng-Fu ChiuNORTHWESTERN UNIVERSITY |
Contact non-squeezability and Microlocal category |

on Friday, October 24, 2014, at 3:30 PM in Science Center 507 | ||

This talk focuses on the relation between microlocal categories and symplectic/contact topology. We will briefly describe how to assign a triangulated category to a given geometric object and how this assignment varies under Hamiltonian flows in the ambient manifolds. This allows us to retrieve Hamiltonian invariants from homological data in a systematic way. Finally, we will discuss its application to a contact non-squeezability problem posed by Eliashberg, Kim and Polterovich. This is a joint work with Dmitry Tamarkin. |

CENTER OF MATHEMATICAL SCIENCES & APPLICATIONS PHYSICAL MATHEMATICS SEMINAR: | Murad AlimHARVARD UNIVERSITY |
Lie algebra and differential rings of tt* geometries of Calabi-Yau sigma models |

on Monday, October 27, 2014, at 12:00 - 2:00 PM in Science Center 232 |

DIFFERENTIAL GEOMETRY SEMINAR: | Semyon DyatlovMIT |
Resonances in general relativity |

on Tuesday, October 28, 2014, at 4:15 PM in Science Center 507 | ||

We discuss long time behavior of linear scalar waves on Kerr and Kerr-de Sitter black hole backgrounds and their stationary perturbations. The physical motivation comes from the analysis of gravitational waves emitted during the ringdown stage of a large scale event (such as merging with another black hole). The properties of these waves, and their frequencies, called quasi-normal modes or resonances, depend on the structure of the set of all trapped light rays. In the considered Kerr(-de Sitter) case, the trapped set is r-normally hyperbolic and we can provide a detailed description of quasi-normal modes and long-time behavior of linear waves. |

NUMBER THEORY SEMINAR: | Jerry WangPRINCETON UNIVERSITY |
Pencils of quadrics and the arithmetic of hyperelliptic curves |

on Wednesday, October 29, 2014, at 3:00 PM in Science Center 507 | ||

In recent joint works with Manjul Bhargava and Benedict Gross, we showed that a positive proportion of hyperelliptic curves over Q of genus g have no points over any odd degree extension of Q. This is done by computing certain 2-Selmer averages and applying a result of Dokchitser-Dokchitser on the parity of the rank of the 2-Selmer groups in biquadratic twists. In this talk, we will see how arithmetic invariant theory and the geometric theory of pencils of quadrics are used to obtain the 2-Selmer averages. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: | C.F. Jeff WuGEORGIA INSTITUTE OF TECHNOLOGY |
A statistical view of uncertainty quantification: interface between statistics and applied mathematics |

on Wednesday, October 29, 2014, at 4:15 PM in Science Center 507 | ||

Because of the advances in complex mathematical models and fast computer codes, computer experiments have become popular in engineering and scientific investigations. Statisticians have worked on the design, modeling and computation aspects of computer experiments. Applied mathematicians have approached a closely related class of problem called UQ (uncertainty quantification). Interface between the two approached is made in the talk. Two problems on the statistical side are presented to illustrate this interface. 1. Consider deterministic computer experiments with tuning parameters which determine the accuracy of the numerical algorithm (e.g., mesh density in finite element analysis). To efficiently integrate computer outputs with different tuning parameters, a class of nonstationary Gaussian process models consistent with the knowledge in numerical analysis is proposed to model the integrated output. Estimation is performed by using Bayesian computation. Numerical studies show the advantages of the proposed method over existing methods. A related problem is given to illustrate the interplay between modeling and design. For this and a broader class of models with multi-levels of fidelity, the nested space-filling designs are most suitable. Some examples are given and the underlying mathematics discussed. 2. Calibration parameters in deterministic computer experiments are those attributes that cannot be measured or available in physical experiments or observations. Kennedy-O’Hagan (2001) suggested an approach to estimation by using data from physical experiments and computer simulations. We show that a simplified version of the original KO method leads to asymptotically inconsistent calibration. This calibration inconsistency can be remedied by modifying the original estimation procedure. A novel calibration method, called the L2 calibration, is proposed and proven to be consistent and enjoys optimal convergence rate. A numerical example and some mathematical analysis are used to illustrate the source of inconsistency. |

CENTER OF MATHEMATICAL SCIENCES & APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Remi Monasson *1st speaker*Laboratoire de Physique Theorique de l’ENS |
Crosstalk, transitions, and wormholes in an attractor network model of hippocampal place cells |

on Wednesday, October 29, 2014, at 12:00 - 2:30 PM in Science Center 232 |

CENTER OF MATHEMATICAL SCIENCES & APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Simona Cocco *2nd speaker*Laboratoire de Physique Statistique de l’ENS |
Inferring 'interactions' from correlations: application to memory consolidation and cell assemblies in the behaving rat |

on Wednesday, October 29, 2014, at 12:00 - 2:30 PM in Science Center 232 |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Ji Oon Lee - *1st speaker*KAIST |
Tracy-Widom Distribution for Real Sample Covariance Matrices with General Population |

on Friday, October 31, 2014, at 12:00 - 2:30 PM in Science Center 232 | ||

Consider a sample covariance matrix of the form XX^*. The sample X is an MxN real random matrix whose columns are independent multivariate Gaussian vectors with covariance Σ. We show that the fluctuation of the largest rescaled eigenvalue is given by Tracy-Widom distribution for a large class of general Σ. This is a joint work with Kevin Schnelli. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Mykhaylo Shkolnikov - *2nd speaker*PRINCETON UNIVERSITY |
Intertwinings, wave equations and beta ensembles |

on Friday, October 31, 2014, at 12:00 - 2:30 PM in Science Center 232 | ||

We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to wave equations and more general hyperbolic partial differential equations. The talk will be devoted to this recent development, as well as an algebraic perspective on intertwinings which, in particular, gives rise to a novel intertwining in beta random matrix theory. The talk is based on joint works with Soumik Pal and Vadim Gorin. |

DIFFERENTIAL GEOMETRY SEMINAR: | Larry GuthMIT |
Homotopical effects of k-dilation |

on Tuesday, November 04, 2014, at 4:15 PM in Science Center 507 | ||

The k-dilation of a map measures how the map stretches k-dimensional areas. If Dil_k f < L, then it means that for any k-dimensional submanifold S in the domain, Vol_k (f(S)) is at most L Vol_k(S). We discuss how the k-dilation restricts the homotopy type of a map. Our main theorem concerns maps between unit spheres, from S^{m} to S^{m-1}. If k > (m+1)/2, then there are homotopically non-trivial maps S^m to S^{m-1} with arbitrarily small k-dilation. I find this somewhat counterintuitive. The construction has a similar flavor to constructions in the h-principle literature with lots of wrinkling. On the other hand, if k is at most (m+1)/2, there every homotopically non-trivial map from S^m to S^{m-1} has k-dilation at least c(m) > 0. So there is a transition at k=(m+1)/2 between flexible behavior and rigid behavior. The first interesting case of the rigid behavior is k=3 and m=2. It was proven by Gromov in the 70's. The higher-dimensional cases are new. The main difficulty here is to connect the topology and the geometry. To detect that a map S^m to S^{m-1} is homotopically non-trivial requires tools from algebraic topology such as Steenrod squares. We have to connect the Steenrod squares with k-dimensional volumes of k-dimensional surfaces. (For the talk, I won't assume familiarity with Steenrod squares.) |