日時 | 2019年5月6日（月）午後3時～6時30分 |

講演者（所属） | Anton Ayzenberg (Higher School of Economics) |

タイトル | 1） Quaternionic toric topology and complexity one torus actions 2） From toric topology to the application of Poincare conjecture in material science. (joint with Dmitry Gugnin) 3） How a mouse perceives the topology of the space? (joint with V.Chernyshev, R.Drynkin, and many others) |

場所 | 数学小講究室B（F４０５） |

概要 | 1）Historically, in toric topology two types of objects are studied: the
actions of a compact torus $T^n$ on manifolds and their real versions: the
actions of discrete torus $(Z/2)^n$ on manifolds. A lot is known about
the relation between moment angle manifolds and quasitoric manifolds, and
there is a real version of this relation, namely, the relation between
real moment angle manifolds and small covers. In 2012 Jeremy Hopkinson,
a graduate student of Nigel Ray, introduced a quaternionic version of these
stories. In quaternionic version a manifold is acted on by a noncommutative
group $(S^3)^n$
and a lot of interesting topology and combinatorics arise
that deserves further study. In my talk, I will describe briefly the main
points and difficulties which appear when you try to generalize from toric
topology to "quoric" topology (this is the short name for "quaternionic
toric" introduced by Hopkinson). I will concentrate on "quoric"
surfaces, which are 8-dimensional manifolds acted on by $(S^3)^2$. It happens
that there is an action of a compact 3-torus on each such manifold, which
gives a series of examples of complexity one torus actions. The $T^3$-orbit
spaces of these actions are homeomorphic to 5-spheres. 2）It is known that the right sided multiplication action of a compact torus $T^3$ on the Lie group U(3) of unitary matrices is free and the quotient is diffeomorphic to the complete flag variety Flag$(C^3)$. On the other hand, there exists the multiplication action of $T^3$ on U(3) from the left side: it reduces to the non-free action of $T^3$ on Flag$(C^3)$. Buchstaber and Terzic, using their theory of (2n,k)-manifolds, had proved that the orbit space Flag$(C^3)$/$T^3$ is homeomorphic to the 4-sphere. In other words, the double quotient $T^3$\U(3)/$T^3$ of a non-free two-sided multiplication action is a 4-sphere. There is a real version, which tells that the quotient of O(3) by the two-sided multiplication action of $(Z/2)^3$ is a 3-sphere. It happens, that the two-sided quotients of SO(3) by a pair of discrete groups acting from different sides appear in crystallography and material science under the name "misorientation spaces". We study misorientation spaces for different pairs of proper point crystallography groups. In many cases the misorientation space is a 3-sphere according to Poincare conjecture. In the remaining cases the topology of the misorientation space can also be completely described by Thurston's elliptization conjecture. In some cases it is possible to avoid the hardcore 3-dimensional topology: for several misorientation spaces we provide precise coordinates. As we hope, these can be used in applications. 3）In the last years it became quite popular in the world to combine topology with the brain study. Most of research is concentrated on the application of homology and persistent homology (I guess, because these can be calculated more or less efficiently). There is, however, a subfield in the brain study, which appears to be very geometrical and topological in its nature. The question is: how the location of a mammal is encoded in its brain? Nobel prize 2014 in physiology was given for discovery of place cells and grid cells in a brain of a mammal. A neural cell of this type fires when a mammal comes inside certain location of space. We may wonder: is it possible to recover the geometry and topology of the environment, given some data about neural activity? This task is very complicated and requires the cooperation of biologists, cognitive scientists, mathematicians, and specialists in big data analysis. I will give a general survey, and, if time allows, try to explain some fascinating homotopy theory beyond these problems. |

日時 | 2019年4月9日（火）午後5時～ |

講演者（所属） | Peter Crooks (Northeastern Univ.) |

タイトル | Kostant--Toda lattices and the universal centralizer |

場所 | 数学小講究室B（F４０５） |

概要 | Each finite-dimensional complex semisimple Lie algebra has a so-called universal centralizer, a hyperkähler variety of interest to geometric representation theorists. This variety carries a completely integrable system whose construction resembles that of the Kostant--Toda system. I will promote this resemblance to a precise relationship between the two aforementioned integrable systems. |

最終更新日: 2019年4月23日