## Spring 2020

Organizers: Chuwen Wang and Hindy Drillick

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 hindy.drillick@columbia.edu or wang.chuwen@columbia.edu

The colloquium meets at **6:00pm, Wednesdays in Mathematics Room 507.** The talks will be followed by dinner.

Date |
Speaker |
Title of talk |

January 29 | Inbar Klang |
## Fixed Point TheoryGiven a map f from a topological space X to itself, one can ask whether it has fixed points, and whether it can be modified to remove the fixed points. I will discuss invariants in algebraic topology that answer these questions, the Lefschetz number and the Reidemeister trace. Time permitting, I will briefly discuss variations of this problem (equivariant, or in families), in which homotopy theory becomes very useful. |

February 5 | Konstantin Aleshkin |
## Computations in Special Kahler GeometrySpecial geometry is a structure which appears on deformation spaces of Calabi-Yau manifolds. Via mirror symmetry this structure is related to the rational curve counting on the mirror manifolds. Curve counting and special geometry play important role in string theory, so a lot of effort has been made to understand them. In the talk I will explain what special geometry is and how one can explicitly compute it in a class of examples. |

February 12 | Simon Brendle |
## The Isoperimetric Inequality |

February 19 | Mike Miller |
## Triangulations, homology cobordism, and Floer homologyA longstanding open question asked whether every high-dimensional topological manifold is triangulable. This was shown false by Ciprian Manolescu in 2013, using PDEs (unrelated to triangulations) on 3-dimensional manifolds. I will explain the history of the triangulation question, why this high-dimensional problem is related to a low-dimensional problem, why "Floer homology" seems to pop up, and time permitting, what people are still thinking about today. |

February 26 | Mohammed Abouzaid |
## Is there a hyperbolic Lagrangian in the quintic 3-fold?I will explain the state of our understanding about Lagrangian embeddings in projective varieties, focusing on the case of hypersurfaces. A particularly intriguing puzzle occurs for the degree 5 hypersurface in CP^4, which physicists predict should admit a Lagrangian manifold which admits a hyperbolic metric, but which no one has seen yet. |

March 4 | Michael Weinstein |
## Honeycomb lattices, Dirac points and GrapheneGraphene is a 2-dimensional material; a single atomic layer of carbon atoms arranged in a honeycomb pattern, extending to the macroscale. Many of the remarkable properties of graphene, e.g. quasi-particles whose dynamics are governed by the equations of massless Fermions and topologically protected energy-transport along edges, are related to the spectral properties of certain continuum (PDE) and discrete (tight-binding) model operators. In this talk I'll describe some of the mathematical theory. |