# Michael Zhao Memorial Student Colloquium

Spring 2019

Each week, the Michael Zhao Memorial Student Colloquium holds 45-minute talks by Columbia mathematics faculty about their own research.

The talks are intended for current PhD students in mathematics at Columbia. If you are an undergraduate student or external graduate student and would like to come, please email me at nguyen.dung@math.columbia.edu

The colloquium meets at **7:00pm, Wednesdays in Mathematics Room 307.**
The talks will be followed by dinner.

Date |
Speaker |
Title of talk |

January 23 | Giulia Saccà | ## Hyper-Kähler manifoldsI will give an overview of my perspective on the central role of hyper-Kähler manifolds in algebraic geometry both as the natural higher dimensional analogues of K3 surfaces and in terms of their relationship to cubic fourfolds. |

January 30 | Ivan Corwin | ## Random permutations, partitions and PDEsWe start with a seemingly innocuous question - what do large random permutations look like? Focusing on the structure of their "increasing subsequences" we encounter some remarkable mathematics related to symmetric functions (e.g. Schur and Macdonald), random matrices, and stochastic PDEs. No prior knowledge of any of this will be assumed. (slides) |

February 5 (Tuesday) | Dorian Goldfeld | ## Multiple Dirichlet SeriesThe Riemann zeta function and the Dirichlet L-functions are examples of Dirichlet series in one complex variable that were introduced to count primes.The Langlands L-function (associated to a certain infinite dimensional representation) can be thought of as a super generalization of the Riemann zeta function. But it is still a Dirichlet series in one complex variable. Multiple Dirichlet series are Dirichlet series in several complex variables that retain many of the properties of the Langlands L-function such as analytic continuation and functional equations. In this talk I will introduce a family of multiple Dirichlet series associated to root systems of classical Lie groups. |

February 13 | Oleg Lazarev | ## Symplectic flexibilitySymplectic manifolds often have interesting invariants coming from J-holomorphic curves that can be used to construct `exotic' symplectic manifolds, which are diffeomorphic but not symplectomorphic. I will describe recent developments in symplectic flexibilty - that for certain classes of symplectic manifolds diffeomorphic actually implies symplectomorphic- and explain how this phenomenon put constraints on certain J-holomorphic curve invariants. |

February 20 | Simon Brendle | ## Singularity formation in geometric flowsGeometric evolution equations like the Ricci flow and the mean curvature flow have played a central role in differential geometry. The main problem is to understand singularity formation. In other words, we want to know what happens in regions where the curvature becomes large. From one point of view, singularities are undesirable in that they can prevent us from continuing the solution. On the other hand, singularities tend to have a very deep special structure, which only reveals itself as the curvature becomes large. I will discuss some of the broad ideas underlying this theory, and indicate some recent developments. |

February 27 | Ovidiu Savin | ## The Optimal Transport Problem |

March 6 | Yihang Zhu | ## The spherical Hecke algebraThe spherical Hecke algebra is a wonderful confluence of representation theory, algebraic geometry, combinatorics, and number theory. In this talk I will introduce the three natural bases of the algebra, and discuss how the transition matrices between these bases encode rich and deep information. The talk will be based on concrete examples. |

March 13 | Linh Truong | |

March 20 | No talk (Spring break) | |

March 26 (Tuesday) | Michael Woodbury | |

April 3 | ||

April 10 | ||

April 17 | ||

April 24 | ||

May 1 | Alisa Knizel | |

May 8 | ||

May 15 |