Gauge Theory and Low-Dimensional Topology (Part I: Historical Context)

Hi! This month, I thought I would start a brief series of articles describing the uses of gauge theory in mathematics. Rather than discuss current research directions in gauge theory (of which there are many), I hope to give an overview of the sorts of mathematical questions that gauge theory was first used to answer and a general idea of what it is all about. Our goal for today will be to contextualize the initial advances in low-dimensional topology due to gauge theory by giving a picture of the state of affairs before its introduction. We will thus spend this post establishing some basic terms and ideas for the uninitiated; we will wait until later posts to discuss gauge theory itself and what it can tell us about topology.

I have attempted to make this article as introductory and motivational as possible, especially to readers who are less familiar with the finer historical developments in low-dimensional topology. No background is needed except for (at best) a passing recollection of basic algebraic and differential topology. In several places I have been a bit cavalier with precise definitions for the sake of the exposition. All errors are (of course) mine!

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Some Funding Opportunities for Graduate Students in Mathematics

Most PhD programs in the United States fund their graduate students. Yet it is still beneficial for graduate students to obtain outside funding. A graduate student’s funding may be offered in the form of a teaching fellowship, but those wishing to relieve themselves of some time-consuming instructional duties may want to opt for outside funding. Obtaining external support is also an accomplishment and accomplishments look nice on a CV – success begets success. Also, many master’s students are not funded and many European PhDs are not funded either. Below, please find an (incomplete) list of some funding opportunities for master’s and PhD students in the mathematical sciences.

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The “Idea” of a Scheme

The mathematical concept of a “scheme” seems to pop up everywhere, but it’s hard to get a good grasp on what a scheme actually is. Any time you might ask someone what a scheme is in passing, there never seems to be enough time to explain it. On the other hand, if someone finds the time to internalize the full definition, it’s not immediately clear why a scheme is defined the way it is. The following interpretation helped me understand the idea of a scheme at a somewhat deeper level than a quick conversation — and hopefully it can help you too!

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Challenge by Choice: How to gain power in your own education

Hidden deep within the jungle of demands for graduate students is one unspoken, yet clearly important goal: become your own advisor. This painfully mysterious mission was once again apparent in a recent conversation that I had. As is typical when I feel powerless in the machinery of graduate school, I was standing in the library stacks, manically reading the table of contents of any book that looked interesting while repeating the mantra: “That’s fine, I will teach myself.” Searching for books never fails to remind me of why I love math and why I am in graduate school. But after settling on only checking out three books, I was quickly reminded that (for the most part) I am not quite capable of picking up a math book and reading it. Not to mention that checking out three math books is already unreasonable. So, logically, I found a professor I trusted and asked: “How do I find resources that are developmentally appropriate, but allow me to learn the things that I want?” The response: “That’s what an advisor is for.”

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How to Survive Grad School as a Woman in STEM

There are certain moments from grad school that will always stick with me: the conference in Boston where my usually quiet lab mate opened up to me; the nights I spent drinking cheap beer with my closest friends in the grungy, student-run bar; the time (okay, times) I cried in my advisor’s office. And you can be sure that I won’t ever forget my general exams, whose residual panic-inducing effects I can still feel, months later.

I value these memories because they add texture to my life as a graduate student, providing joy or throwing it into relief.

But I’ve also had experiences that extend beyond the normal ups and downs. I listened, trying not to cry, as a professor told me that I was too slow to do theoretical work. (A year later, I won an NSF grant to do just that.) I gritted my teeth as I, the only woman in the room, was asked to sort exams into piles, while my male colleagues graded them. These experiences didn’t make me stronger, happier, more resilient, or more confident. They just wore away at my well-being.

Learning to survive graduate school as a woman in STEM—or any minority, for that matter—means finding ways to manage the effects of constant, subtle antagonism, because that antagonism won’t make you a better scientist, mathematician, or engineer.

Here are seven things that will.

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