SPECIAL BASIC NOTIONS SEMINAR: | Jean-Pierre SerreCollège de France |
Some simple facts on lattices and orthogonal group representations |

on Wednesday, May 03, 2017, at 3:00 pm in Science Center Hall D | ||

Afternoon tea will follow at 4:15 pm in the Math Department Common Room, 4th floor. |

JOINT DEPT OF MATHEMATICS AND CENTER OF MATHEMATICAL SCIENCES & APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Ilya SoloveychikHarvard School of Engineering & Applied Sciences |
Deterministic Random Matrices |

on Wednesday, May 03, 2017, at 3:00 pm in CMSA Building, 20 Garden St, G10 | ||

In many applications researchers and engineering need to simulate random symmetric sign (+/-1) matrices (Wigner's matrices). The most natural way to generate an instance of such a matrix is to toss a fair coin, fill the upper triangular part of the matrix with the outcomes and reflect it part into the lower triangular part. For large matrix sizes such approach would require a very powerful source of randomness due to the independence condition. In addition, when the data is generated by a truly random source, atypical non-random looking outcomes have non-zero probability of showing up. Yet another issue is that any experiment involving tossing a coin would be impossible to reproduce exactly, which may be crucial in computer scientific applications. In this talk we focus on the problem of generating n by n symmetric sign matrices based on the similarity of their spectra to Wigner's semicircular law. We develop a simple completely deterministic construction of symmetric sign matrices whose spectra converge to the semicircular law when n grows to infinity. The Kolmogorov complexity of the proposed algorithm is as low as 2 log (n) bits implying that the real amount of randomness conveyed by the semicircular property is quite small. |

GAUGE THEORY, TOPOLOGY AND SYMPLECTIC GEOMETRY SEMINAR: | Daniel Cristofaro-GardinerHarvard University |
Two or infinity |

on Friday, May 05, 2017, at 3:30 - 4:30 pm in Science Center 507 | ||

A central goal in symplectic and contact geometry is to better understand the dynamics of “Reeb” vector fields. About a decade ago, Taubes showed that any Reeb vector field on a closed three-manifold has at least one closed orbit. I will discuss recent joint work showing that, under some hypotheses, any Reeb vector field on a closed three-manifold has either two, or infinitely many, closed orbits. Key tools are an identity relating the lengths of certain sets of Reeb orbits to the volume of the three-manifold, and the theory of global surfaces of section as developed by Hofer, Wysocki, and Zehnder. |

SPECIAL LECTURE SERIES: | Jean-Pierre SerreCollège de France |
Cohomological invariants mod 2 of Weyl groups, Pt. 1 |

on Monday, May 08, 2017, at 3:00 - 4:00 PM in Science Center 507 | ||

The first lecture will mostly be a résumé of the first part of AMS ULECT 28; the second lecture will give an explicit description of the cohomological invariants of the Weyl groups. *Afternoon tea will follow the talks at 4:15 pm in the Math Department Common Room, 4th Floor. |

SPECIAL LECTURE SERIES: | Jean-Pierre SerreCollège de France |
Cohomological invariants mod 2 of Weyl groups, Pt. 2 |

on Tuesday, May 09, 2017, at 3:00 - 4:00 PM in Science Center 507 | ||

The first lecture will mostly be a résumé of the first part of AMS ULECT 28; the second lecture will give an explicit description of the cohomological invariants of the Weyl groups. *Afternoon tea will follow the talks at 4:15 pm in the Math Department Common Room, 4th Floor. |