# 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 | ## Heegaard Floer homologyIn the early 2000s Peter Ozsváth and Zoltan Szabó constructed Heegaard Floer homology, a package of powerful invariants of smooth 3- and 4-manifolds and knots. Over the last decade, it has become a central tool in low-dimensional topology and used extensively to study important questions concerning unknotting number, slice genus, knot concordance, Dehn surgery, taut foliations, contact structures, and smooth 4-manifolds. I will give an overview of the construction of Heegaard Floer homology, provide concrete examples and discuss some of the topological applications. |

March 20 | No talk (Spring break) | |

March 26 (Tuesday) | Michael Woodbury | ## Some examples from the zoo of modular forms: From Maass forms and indefinite theta functions to automorphic forms on exceptional groupsI will give a survey of some rather disparate results on modular forms. I will begin discussing modular forms, in particular theta functions, from a very classical and concrete point of view, and will eventually move to the language of automorphic representations. The first portion of the talk will concentrate on joint work with Matt Krauel and Larry Rolen, and the latter part of the talk will be on joint work with Gordan Savin. |

April 3 | Julien Dubédat | ## Conformally invariant random structuresIn many natural situations, random objects arising as scaling limits are expected to be invariant under rotation and scaling. They can then be expected to be invariant under transformations that are locally to first order compositions of rotation and scaling, i.e. conformal maps. In two dimensions, there is an abundance of such conformal maps, which puts strong constraints on conformally invariant random structures. We'll discuss some examples and recent developments in that direction. |

April 10 | Toti Daskalopoulos | ## Nonlinear Geometric flowsWe will give an overview on the development of extrinsic Nonlinear Geometric Flows, emphasizing both the analytical and geometric point of views. Special emphasis will be given to the Mean curvature flow (an example of quasilinear diffusion), the Inverse Mean curvature flow (an example of ultra-fast diffusion) and to the Gauss curvature flow (an example of slow-diffusion). |

April 17 | Shotaro Makisumi | ## Diagrammatic Hecke categoryThe Hecke category has played an important role in geometric representation theory since its very beginning. More recently, a diagrammatic version of this category has emerged. I will describe this diagrammatic Hecke category and discuss some applications to representation theory. |

April 24 | Johan de Jong | ## Stratification of schemesIn this talk I will introduce the affine statification number on schemes. I will go over some examples and properties, and prove a bound on its size. The results are based on a joint paper by Mike Roth and Ravi Vakil. |

May 1 | Mohammed Abouzaid | ## Floer theory and the Arnol'd conjecture |