finished (2022/04/25, finished 2022/11/12, listed until 2022/10/25)
*** this program has been closed and new applications are no longer accepted. ***| The Department of Mathematics at the University of Georgia invites application for undergraduate research in topology in Summer 2022. The program is open to all undergraduate students, regardless of major, who are US citizens, US permanent residents, or are based at a US university on a visa allowing them to legally work for the University of Georgia. The deadline for applications is May 6th, although applications received after the deadline may still be considered. Decisions will be announced as soon as possible. Should Covid-19 situation become unfavorable, the program may run remotely or have some remote components. The research project will last approximately 6 weeks, in June and July, although the exact timing is negotiable and can be worked out between students and mentors. There will be a $4125 total summer stipend. We will work with participants to find summer housing in Athens, either on or off campus. Dates: 6 weeks in June and July, 2022 (flexible) Title: 4-manifolds via surfaces 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. Activities: Students will spend an intensive 6 weeks engaged in collaborative activities such as the following:
Background / Prerequisites: Students are expected to have some experience with linear algebra, and to have an interest in studying unsolved problems in mathematics. While a formal course in proof-writing is not strictly required, the well-prepared participant should be willing to think carefully and deeply, and to be able to express their ideas clearly. We are committed to providing a positive and welcoming environment for all students, especially those from groups that have been historically underrepresented in academic mathematics. All undergraduate students are welcome to apply. Stipend: Students will receive a $4125 stipend. Application: Interested students should submit:
Deadline: May 6, 2022 |
)