Gauge Theory and Low-Dimensional Topology (Part II: Smooth Four-Manifolds)

Posted on September 2, 2017 by Irving Dai

In the last post, I attempted to give an overview of the state of affairs in four-manifold topology leading up to the introduction of gauge theory. In particular, we discussed the correspondence between (topological) four-manifolds and their intersection forms afforded by Freedman’s theorem, and briefly touched on the relevance of this relationship to the difference between continuous and smooth topology in dimension four. Today, we will elaborate on this a bit by describing what happens to Freedman’s theorem in the smooth category and will try to give a vague idea (to be expanded upon next time) of why gauge theory might have something to say about smooth manifolds.

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Posted in Math, Topology | Tagged , | 2 Comments

AMS Notices Spotlight September 2017

Posted on August 29, 2017 by Taryn Laird

Hello and welcome to the September AMS Notices Spotlight. Since the last spotlight, many of you have started a new school year and if you haven’t started yet you are getting ready to start very soon. With that in mind, take a moment before the busyness of the semester sets in and peruse the September AMS Notices. There are many great articles in this month’s notices, including a sampler from the three sectional meetings that are going to occur this fall and several articles about bikes and math. We mentioned in our last spotlight that every issue of the AMS Notices includes a dedicated graduate student section. In the graduate student section, there is usually an interview of someone notable, and at least one article written with graduate students in mind. This month we are highlighting one of these articles. Continue reading →

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A Pretty Lemma About Prime Ideals and Products of Ideals

Posted on August 23, 2017 by Tom Gannon

I was trying to prove a theorem in algebraic geometry which basically held if and only if this lemma held. Here’s the lemma:

Lemma: Given any ring $A$, a prime ideal $ \mathfrak{p} \subset A$, and a finite collection of ideals $I_j,$ where $j \in \{1, 2, … , n\}$, then if $I$ is the intersection of the ideals, then $I \subset \mathfrak{p}$ implies that $I_j \subset \mathfrak{p}$ for some $j \in \{1, 2, … , n\}$. Continue reading →

Posted in Algebra, Algebraic Geometry, Math | Tagged , , | 3 Comments

What to Do When a Group Gets Stuck Working on a Task

Posted on August 17, 2017 by James Sheldon

In my previous post, I discussed how to adapt a problem that you have found in order to make the problem groupworthy. One of the important things to consider when adapting real-world problems is to avoid giving step-by-step instructions and formulas to students.  Instead, a teacher should maximize the opportunities for groups to make their own decisions about problems. In other words, in order to have challenging and productive group discussions, there must be an element of uncertainty so that students engage with the problem and with each other.

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An Infinite Understanding

Posted on August 10, 2017 by Deborah Wilkerson

“Have you ever thought about how strange it is that we think about infinity every day, but most people think about it only on the rarest of occasions, if ever?” This is the text message I recently sent two of my close friends, who also happen to be mathematicians in my department. I was deep in the midst of studying for preliminary exams, trying to prove Riemann’s Theorem on removable singularities, when I started to think – really think – about infinity.

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Posted in Math, Math in Pop Culture, Mathematics in Society | 2 Comments