# Michael Zhao Memorial Student Colloquium

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

Date |
Speaker |
Title of talk |

September 18 | Mikhail Khovanov | ## What is categorification?Categorification lifts numbers to vector spaces and vector spaces to categories. In the talk we'll give a couple of examples and sketch more general setups where categorification works. |

September 25 | Alisa Knizel | ## Random tilingsRandom tilings of large planar domains by simple shapes (domino tiles or rhombuses obtained by gluing two unit triangles of the triangular lattice together) have many surprising features. They develop deterministic limit shapes, and the fluctuations around those shapes reveal universal laws of probability. The aim of the talk is to give an introduction to this topic. (slides) |

October 2 | Francesco Lin | ## Spectral geometry and the topology of hyperbolic 4-manifoldsThe Atiyah-Patodi-Singer index theorem describes, for a given Riemannian manifold with boundary, a relation between its geometry, topology and the spectral theory of the boundary. In this talk, we introduce the main protagonists of the theorem, and discuss some applications due to Long and Reid to the study of hyperbolic 4-manifolds. |

October 9 | Andrei Okounkov | ## Why is it interesting to think about elliptic cohomology ?Elliptic cohomology, especially equivariant elliptic cohomology, can be a very technical and not particularly intuitive subject. Yet I will try to argue that there is something one can gain from a better understand of elliptic cohomology, and its place in geometric representation theory and enumerative geometry. |

October 15 (Tuesday) | Chao Li | ## From quadratic forms to arithmetic intersection numbersWe begin with the classical theory of quadratic forms and illustrate its link to modular forms and intersection theory. We then motivate the notion of arithmetic intersection numbers, discuss their relation with the BSD conjecture and end with recent results concerning them. |

October 23 | Kyler Siegel | ## On the three-body problemThe three-body problem asks to describe the dynamics of three massive bodies (e.g. Earth, Moon, and Sun) subject to Newton's equations of gravity. The restricted three-body problem is a special case in which one of the bodies is considered to be nearly massless. Due to the chaotic nature of the dynamics, even simple questions about this problem have been open for centuries. Recently, it has been observed that the three-body problem can embedded into the world of symplectic geometry, opening up a whole range of new potential tools. I will give an elementary introduction and attempt to explain this new paradigm. |

October 30 | Gus Schrader | ## Quantum integrable combinatoricsIn addition to their applications to representation theory and geometry, quantum integrable systems can often shed light on problems in combinatorics and lead to exact and asymptotic enumeration of various objects. To illustrate this paradigm, I'll present Greg Kuperberg's proof of the Alternating Sign Matrix conjecture using the integrability of the six vertex model with domain wall boundary conditions, and discuss what might be some interesting directions for future work in the area. (slides) |

November 6 | Michael Harris | ## Introduction to the local Langlands correspondenceThe local Langlands correspondence relates two kinds of objects that seem to be quite distant from one another: Representations of reductive algebraic groups over a p-adic field F are conjectured to have parameters that are defined in terms of the Galois theory of F. The correspondence has been known completely for GL(n) for about 20 years, and for classical groups (orthogonal or symplectic groups) for about 10 years. I will talk about these results and say something about my recent work with Khare and Thorne on the exceptional group G2. |

November 13 | Carolyn Abbott | ## Hyperbolic groups and generalizationsIt is possible to learn a lot about a group by looking at how it acts on certain metric spaces. In this talk, we will introduce various metric spaces, with a focus on negatively curved spaces, also called delta-hyperbolic spaces. Our goal will to be to understand the properties of groups that admit particularly nice actions on hyperbolic metric spaces. Such groups are called hyperbolic groups, and have been well-studied by geometric group theorists. We will also define a recent generalization of hyperbolic groups, called acylindrically hyperbolic groups, which is the focus of much of my research. Such groups include mapping class groups, relatively hyperbolic groups, and most fundamental groups of 3--manifolds, among many others. |

November 20 | Eric Urban | ## Bernouilli numbers, Eisenstein series and cyclotomic unitsI will recall what the objects of the title are and explain how one can combine them in a new way to explain a deep Theorem of Mazur and Wiles (proving a conjecture of Iwasawa) that gives a formula for the cardinality of the p-part of the class groups of cyclotomic fields in terms of Bernouilli numbers. (slides) |

November 27 | No talk (Thanksgiving break) | |

December 4 | Andrew Blumberg | ## Algebra over the sphere spectrumIn this talk I will explain what abelian groups are in homotopy theory, and how they can be viewed as representing objects in the category of cohomology theories. |

December 11 | John Morgan | ## The Euler Characteristic: From combinatorics to homology to flows on manifolds.In the first instant he Euler characteristic is defined for any space that is divided in a reasonable way into finitely many cells, e.g., a finite simplicial complex. One lets n_k be the number of k-cells and defines the Euler characteristic as \sum_k(-1)^k n_k. One can compute the Euler characteristic by from the homology groups of the space by a similar formula: it is \sum_k(-1)^k rank(H_k), where H_k is the k^{th} homology group. When the space is a compact smooth manifold one can compute the Euler characteristic by counting (again with appropriate signs) the zeros of a vector field. This leads to a definition of the Euler characteristic in terms of the intersection of the diagonal in M x M with the graph of the diffeomorphism from M to M obtained by integrating the vector field. In this talk we will explore these various aspects of the Euler characteristic. |

Fall 2018