Morse theory analyzes the topology of a smooth manifold by studying the behavior of its real-valued functions. From this, one obtains a well-behaved homology theory which provides further information about the manifold and places constraints on the smooth functions it admits. This idea has proven to be useful in approaching topological problems, playing an essential role in Smale's solution to the generalized Poincare conjecture in dimensions greater than 4.
This talk will include background information on contact structures and open book decompositions of 3-manifolds and the relationship between them. I will state the necessary definitions and include a number of concrete examples. I will also review some convex surface theory, which is the tool used in the main talk to investigate the contact structure – open book relationship.
Surface bundles lie in the intersection of many areas of math: algebraic topology, 2–4 dimensional topology, geometric group theory, algebraic geometry, and even number theory! However, we still know relatively little about surface bundles, especially compared to vector bundles. In this interactive talk, I will present the general (and beautiful) fiber bundle theory, including characteristic classes, as a starting point, and you the audience will get to specialize the general theory to surface bundles, with rewards!
Elliptic surfaces are some of the most well-behaved families of smooth, simply-connected four-manifolds from the geometric and analytic perspective. Many of their smooth invariants are easily computable and they carry a fibration structure which makes it possible to modify them by various surgical operations. However, elliptic surfaces have large Euler characteristics which means even their simplest handle-decompositions appear to be quite complicated.
You are probably familiar with the concept of a knot in 3 space: a tangled string that can't be pushed and pulled into an untangled one. We briefly show how to prove mathematical knots are in fact knotted, and discuss some conditions which guarantee unknotting. We then give explicit examples of knotted 2-spheres in 4 space, and discuss 2-sphere version of the familiar theorems. A large part of the talk is practice with visualizing objects in 4 dimensional space. We will also prove some elementary facts to give a sense of what working with these objects feels like.
A CAT(0) space is a geodesic metric space where triangles are thinner than comparison triangles in a Euclidean plane. Prime examples of CAT(0) spaces are Cartan-Hadamard manifolds: complete simply connected Riemannian spaces with nonpositive curvature, which include Euclidean and Hyperbolic space as special cases. The triangle condition ensures that every pair of points in a CAT(0) space can be connected by a unique geodesic. A subset of a CAT(0) space is convex if it contains the geodesic connecting every pair of its points.
This talk will give an elementary introduction to my joint work with Kyler Siegel that shows how cuspidal curves in a symplectic manifold X such as the complex projective plane determine when an ellipsoid can be symplectically embedded into X.
In her thesis, Maryam Mirzakhani counted the number of simple closed geodesics of bounded length on a (real) hyperbolic surface. This breakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, symplectic geometry, surface topology, and homogeneous dynamics. In this talk, I’ll discuss some of these connections and a qualitative strengthening of her theorem, describing what these curves, and their complements, actually (generically) look like. This is joint work with Francisco Arana-Herrera.
In the early 80's, Freedman discovered that the Whitney trick could be performed in 4-dimensions which quickly led to a complete classification of closed, simply connected topological 4-manifolds. With gauge theory, Donaldson showed that 4-manifolds differ greatly from their higher dimensional counterparts which uncovered the stark differences between topological and smooth results in dimension 4. In this introductory talk, we will give a brief overview this classification and why dimension 4 is so unique.
The Burau representation is a kind of homological representation of braid groups that has been around for around a century. It remains mysterious in many ways and is of particular interest because of its relation to quantum invariants of knots and links such as the Jones polynomial. In recent work, I came across a relationship between this representation and a moduli space of Euclidean cone metrics on spheres (think e.g. convex polyhedra) first examined by Thurston.
