Events

Ethan Cotterill, University of Campinas

The irreducibility the Severi variety of plane curves V_{d,g} of fixed degree d and arithmetic genus g is a well-known and celebrated result of J. Harris from the 1980s. In higher (ambient) dimension, the situation changes drastically, and (the natural generalizations of) Severi varieties of curves with fixed numerical invariants in P^n are very often reducible and have components of larger-than-expected dimensions. In this talk, I will explain why this is the case, on the basis of a combinatorial analysis of unicuspidal rational curves with fixed ramification profiles.

MC 5417

Jiasheng Teh, McMaster University

Einstein metrics have long been considered as the canonical metrics in Riemannian geometry. The moduli space of Einstein metrics constitutes a diffeomorphism invariant of the underlying closed smooth manifold. In dimension four, they exhibit a balance between the rigidity of the constant sectional curvature metrics in low dimensions and the flexibility coming from higher dimensions. In this talk, we will outline the strategies to show that the moduli spaces of Einstein metrics for a certain family of closed 4-manifolds, the ones which admit a locally hyperKähler metric, are all path-connected. We will also present a Torelli theorem for semi-complex structures.

MC 5417

Ty Ghaswala, CEMC, University of Waterloo

Imagine you are handed a compact orientable surface with one boundary component. It is known that the Dehn twist about a curve isotopic to the boundary component is not quite like all the other Dehn twists. For one thing, it is central in the mapping class group of the surface. It turns out that such Dehn twists have a special property called co-finality in every left ordering of the mapping class group, making them even more amazing than originally thought! I will tell you what all of this means, and time permitting, we will discuss the implications for mapping class group actions on the real line.

This is joint with Adam Clay.

MC 5417

Sung Gi Park, Harvard University

In this talk, I will discuss the behavior of positivity, hyperbolicity, and Kodaira dimension under smooth morphisms of complex quasi-projective manifolds. This includes a vast generalization of a classical result: a fibration from a projective surface of non-negative Kodaira dimension to a projective line has at least three singular fibers. Furthermore, I will explain a proof of Popa's conjecture on the superadditivity of the log Kodaira dimension over bases of dimension at most three. These theorems are applications of the main technical result, namely the logarithmic base change theorem.

MC 5417

Daniel Alvarez, University of Toronto

The link between Poisson brackets and generalized Kähler structures goes back to the work of Lyakhovich and Zabzine in 2002. Later on, this relationship acquired a new depth thanks to the generalized geometry viewpoint of Hitchin and Gualtieri. In this talk I will discuss the remarkable way in which Poisson geometry and Lie theory can be applied to describe generalized Kähler structures from a global perspective. This talk is based on work in progress with Marco Gualtieri and Yucong Jiang. 

MC 5417

Eric Riedl, University of Notre Dame

In this talk, we describe two problems relating to plane curves, and describe how log tangent sheaves are key to solving both. First, we consider the natural question: when does the families of lines that intersect with a plane curve vary maximally in modulus? We show how the classical Grauert-Mulich theorem applied to the log tangent sheaf can solve this. Then we consider the question of the algebraic hyperbolicity of the complement of a very general quartic plane curve, and describe how we achieve an answer to this long-open problem, motivated by the Lang-Vojta Conjecture in number theory. This includes joint work with Xi Chen, Anand Patel, Dennis Tseng, and Wern Yeong.

MC 5417

Adrian Zahariuc, University of Windsor

The moduli space of stable nodal scaled marked lines constructed by Ziltener and Mau-Woodward is a compactification of the set of configurations of n distinct points on a line modulo translation. I will discuss a similar moduli space (in which the points are allowed to coincide) and some properties of it: it is equivariant, it represents a natural moduli functor, it is a degeneration of the Losev-Manin space, and it admits a small resolution by the augmented wonderful variety associated to the graphic matroid of a complete graph.

MC 5417

Hongyi Liu, University of California, Berkeley

A hyperkähler triple on a compact 4-manifold with boundary is a triple of symplectic 2-forms that are pointwise orthonormal with respect to the wedge product. It defines a Riemannian metric of holonomy contained in SU(2) and its restriction to the boundary defines a framing. In this talk, I will show that a sequence of hyperkähler triples converges smoothly up to diffeomorphims if their restrictions to the boundary converge smoothly up to diffeomorphisms, under certain topological assumptions and the “positive mean curvature” condition of the boundary framings.

MC 5417

Siyuan Lu, McMaster University

In this talk, I will first introduce the background and motivation for the study of Lagrangian phase equation. I will then discuss my recent work on the regularity of Lagrangian phase equation. In the second part, I will discuss some open problems relating to Lagrangian phase equation.

MC 5417

Michael Albanese, Department of Pure Mathematics, University of Waterloo

The question of which manifolds are spin or spin^c has a simple and complete answer. In this talk we address the same question for the lesser known spin^h manifolds which have appeared in geometry and physics in recent decades. We determine the first obstruction to being spin^h and use this to provide an example of an orientable manifold which is not spin^h. The existence of such an example leads us to consider an infinite sequence of generalised spin structures. In doing so, we determine an answer to the following question: is there an integer k such that every manifold embeds in a spin manifold with codimension at most k?

This is joint work with Aleksandar Milivojevic.

MC 5417