Nathan Karst
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Equivalent unitary representations are unitarily equivalent

1/31/2013

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I found an elementary proof of a problem I saw on mathoverflow. Mateusz Wasilewski gave a nice proof, but I was hoping to find one that was solely linear algebraic. The problem states that if $\pi, \sigma: G \rightarrow U_n(\mathbb{C})$ are two equivalent unitary representations of a finite group, then $\pi$ and $\sigma$ are unitarily equivalent. 

The proof goes as follows. If $\pi$ and $\sigma$ are equivalent, then there exists $T \in GL_n(\mathbb{C})$ such that $T \pi(g) = \sigma(g) T$ for every $g \in G$. Let $T = W \Sigma V^*$ be the singular value decomposition of $T$, so that $T = (W V^*) (V \Sigma V^*)$ is the polar decomposition of $T$. We will show that $A = V \Sigma V^* \pi(g)$ is normal, that is $A^* A = A A^*$. $$\begin{align*}
A A^* &= (V \Sigma V^* \pi(g)) (\pi(g)^* V \Sigma^* V^*) \\
&= \sum |\lambda_i|^2 \\
A^* A &= (\pi(g)^* V \Sigma^* V^*)(V \Sigma V^* \pi(g)) \\
&= \sum |\lambda_i|^2,
\end{align*} $$ where $\lambda_i$ are the diagonal entries of $\Sigma$. 

The components of the polar decomposition of a normal matrix commute, so $(V \Sigma V^*) \pi(g) = \pi(g) (V \Sigma V^*)$. Then $$\begin{align*}
T \pi(g) &= \sigma(g) T \\
(W V^*) (V \Sigma V^*) \pi(g) &= \sigma(g) (W V^*) (V \Sigma V^*) \\
(W V^*)  \pi(g) (V \Sigma V^*) &= \sigma(g) (W V^*) (V \Sigma V^*) \\
(W V^*)  \pi(g) &= \sigma(g) (W V^*).
\end{align*}$$ Since $WV^*$ is unitary by construction, we have completed the proof. 

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The Atlantic on lecturing

1/30/2013

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The Atlantic ran an interesting article yesterday on lecturing in the digital age. I'd expect to see this type of article in the Chronicle of Higher Education or some of the other trade journals, but it's a little surprising to see such a discussion in a less specialized publication. 
To begin with, we lecturers must ask ourselves a basic question: why am I lecturing? What will I be able to get across to learners through a lecture that they could not get just as well and with less inconvenience by reading a book or working through an online learning module? 
Spoiler alert: the author of the article may be a little too pro-lecture for people who want to see lecturing really raked over the coals. 
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Algebraic topology and data analytics

1/26/2013

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The Guardian has a great article on a new startup that's using algebraic topology to analyze data sets. Just goes to show you that you never really know how theory will get brought into the real world. From the article: 
Modern business intelligence methods allow analysts to drill down into complex databases and find the answers to predetermined questions, but the emergent field of data science is concerned with finding the questions that should be asked of huge and often unstructured data in order to yield otherwise invisible results.

Using topology, data scientists can do exactly this, running algorithms that carry out what is effectively blind analysis of a database to reveal the inherent patterns therein, rather than showing correlations between preselected variables.
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Babson strategy video featuring new faculty members

1/24/2013

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I had the chance recently to talk with Len Schlesinger and Carolyn Hotckiss earlier this month with my colleagues Meghan MacLean and Gonzalo Chavez about being a new faculty member at Babson. It was interesting to hear so many different perspectives even within the same faculty demographic. With all the of the new hires happening and coming up, it's a really exciting time to be on campus.
This chat is part of a great series of monthly strategy videos. 
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Results from the marshmallow challenge!

1/24/2013

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I ended up trying out the marshmallow challenge yesterday in my linear algebra course. I had debated it because of the time it would take, but I figured in a 4 credit hour class, I could probably spare it. (I didn't run the demo in my calculus course which is 3 credit hours.) 

Of the 7 teams, 5 had their structures fall over or apart during before we could measure the height, and the maximum height achieved was 20 inches. So many failures actually made for a really productive discussion, because each group had an ideas about where they had gone wrong. 

I had a small class, so we had mostly teams of three. It was interesting to see each team's process, but there were definitely common themes: 
  1. Most teams started building almost immediately. 
  2. Most teams hadn't load tested their structure with just 2 minutes left or so. 
  3. Most teams had at least one person who was significantly less engaged than the others. 
  4. Most teams had significantly underestimated the weight of the marshmallow. 

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Most of these are the common mistakes, and the challenge's website does a good job of preparing the instructor to talk about how these impacted the team's performance in this challenge, and how these general ideas affect projects all over the place. For instance,  assuming that marshmallows are light and fluffy really does impact your prediction of how difficult it will be to support one, and the students can draw a pretty clear (and slightly corny) metaphor from the experience. 

Another interesting part was bring in "fail fast, fail cheap" in the discussion of why so few groups prototyped. I pointed out that this isn't just an entrepreneurial princple; we'll be writing some code in this course, and the fastest way to give yourself a headache is try to write a big chunk of code without testing the components. FFF a productive idea all around, and I think the students appreciated seeing it in a different context.

Overall, I thought this was a great first day exercise. The students seemed to really enjoy it. We had a good discussion about a wide range of (non-mathematical) ideas that will be important in the course, including teamwork, prototyping, validation of assumptions, creativity in unstructured problems, etc. And everyone got to eat some marshmallows. What's not to love?


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The marshmallow challenge as a first day activity

1/23/2013

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There was a discussion on the Silver 2012 Project NExT email list about first day activities in class, and someone mentioned the Marshmallow Challenge. If you have time to watch a quick TED talk, check it out. It seems like a great way to build some camaraderie and start a discussion about some important ideas. 

Part of Babson's entrepreneurial training is "fail fast, fail cheap". Students are taught to go forward iteratively with their ventures rather than rolling all their hopes into a single, giant, untested idea. And it seems that the Marshmallow Challenge is trying to accomplish exactly the same thing. 

I'd be interested to see how graduating Babson seniors do in this challenge, especially given that graduating business students tend to do really poorly. 

Let me know if you give this a try! I'd love to know how it goes!

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Linear arboricity

1/22/2013

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I spent a lot of time this intersession trying to prove the following statement:
The edge set of a graph $G$ with maximum degree $\Delta$ can be decomposed into $\lceil {\Delta \over 2} \rceil$ disjoint classes such that each vertex in $G$ is incident to at most 2 edges in any class. 
I had mainly been thinking about constructing an algorithm that would construct one class per iteration, but I couldn't prove any algorithm I could come up with would terminate in $\lceil {\Delta \over 2} \rceil$ steps. 

I was also reviewing a paper over the break, and one of the references looked interesting with respect to this problem I had been working on. I followed the breadcrumbs are found the linear arboricity conjecture, first raised by Akiyama, Exoo, and Harary in 1981:
A linear forest is a forest in which every tree is a path. The linear arboricity $la(G)$ of a graph $G$ is the minimum number of linear forests whose union is the edge set of $G$. The linear arboricity conjecture asserts that for every simple graph $G$ with maximum degree $\Delta$, we have $la(G) \leq \left[ {\Delta + 1 \over 2} \right]$. 
(I'm using Alon's definition here.) This is really similar to the problem I'd been working on, the key difference being that my version allowed for cycles while linear arboricity does not. 

The conjecture holds for small $\Delta$, namely $\Delta \leq 6$ and $\Delta = 8, 10$. There are also some probabilistic results. In 1988, Alon showed that one needs at most ${\Delta \over 2} + c \Delta^{3/4} \log(\Delta)^{1/2}$ linear forests in the edge decomposition. And in 1994, McDiarmid and Reed verified the conjecture for almost all graphs. 

It's amazing how much time I spent working on a problem that's probably  too tough for me to take down in a couple of weeks just because I didn't know what to call the thing I was looking for.  But the process, like always, was fun. I just think we'll have to settle for some more piecemeal results in our paper. 
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    Nathan Karst

    Doing and teaching mathematics. 

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