Summer 2022: 4-manifolds via surfaces Leaders: Akram Alishahi and Peter Lambert-Cole Dates: June 6, 2022 and July 15, 2022 Description: A 4-manifold is a geometric object that locally looks like 4-dimensional Euclidean space, but may have interesting and surprising global structure. Spacetime in physics and general relativity is a basic example, but 4-manifolds also naturally appear elsewhere in mathematics, such as in algebraic geometry. Moreover, deep results over the past half-century have demonstrated that 4-manifold topology is enormously rich and fascinatingly complex. Unsurprisingly, it is quite hard to describe, let alone prove theorems, about 4-manifolds. There are many different schemes for describing a 4-manifold and these give different measures of the "complexity" of a 4-manifold. This project will focus on a new complexity measure on 4-manifolds, the multisection genus, and investigate the similarities and differences with well understood algebraic invariants of 4-manifolds, such as the Euler characteristic and (co)homology. The first question most people ask is "What is the fourth dimension? Time?". Meaning, they implicitly think that 4 = 3 +1 and 4-dimensional space consists of three spatial dimensions and one time dimension. But mathematicians have found that it's often more productive to think of 4 = 2 + 2 and that in order to get a handle on 4-dimensional objects, you should start with 2-dimensional surfaces. Therefore, the basic tools for this project will mainly involve curves on surfaces and related problems in linear algebra. Click here for more information! Summer 2021: Knots and Algebra Leaders: Akram Alishahi and Melissa Zhang Dates: June 1, 2021 -- July 9, 2021 Description: A knot is a tangled-up piece of string with the ends glued together. A useful way to depict a knot is to draw a knot diagram, which is a projection of the knot onto the plane, as if one had snapped a picture of it with a camera. While knot diagrams are very useful to look at, they bring up some interesting questions: How can we tell when two knots are actually the "same", i.e. we can wiggle one of them to look exactly like the other? For that matter, how can we tell when a knot is actually unknotted, i.e. just a regular circle without any kinks in it? How many times must a knot "pass through itself" for it to become unknotted? Even the simplest questions about knots can sometimes be surprisingly difficult to answer. Nevertheless, in the past century, we have developed a variety of different types of tools for studying knots, ranging from studying topological surfaces with knotted borders to constructing algebraic representations of knots. Moreover, much can be learned by simply drawing pictures of knots and mulling over them. The aim of this project is to introduce students to some of the basic tools and constructions in the field of knot theory with the goal of generating new ideas for approaching some surprisingly open questions. Along the way, students will learn how to interface with existing literature and how to communicate ideas to the scientific community. More information: https://akramalishahi.github.io/REU2021.html
