Getting Better

One of my mathy friends recently said, “I thought that when I finished graduate school I would have learned pretty much all the math I’d ever know.” When I was getting ready to graduate, I felt the same way. Which was a bit scary, because I felt like I didn’t know very much. But I figured that my future research life would be applying the techniques I used in my thesis to similar problems. I imagined that I would pick up some new ideas and go deeper into the area, but that my baseline math knowledge was pretty much set. This has been entirely wrong.

In fact, I have found myself working more on problems that I have absolutely no idea how to solve, and have realized that my interests refuse to stay bounded by the math that I know best. And problems don’t care! They may seem like problems in number theory, but oh, suddenly here comes representation theory, and a whole bunch of group theory! So I find myself trying to learn whole new areas, or relearn things I saw in courses but couldn’t really absorb at the time. It can be really hard, with work and so many other things going on.

I have this sense that if I had been a better graduate student, everything would be so much easier now, and that I should probably just take a few years off and go back to school. It would be great to be in a structured environment, designed by experts to help me learn more math. In my post-thesis research life, I find that so many topics from my classes really did turn out to be important, and those that seemed irrelevant at the time (I mean, when was I going to need measure theory?) have turned up over and over again. And I constantly wish I knew stuff from classes that I couldn’t bring myself to take (classical groups, linear programming… the wish-I-knew-list grows every day). But I can’t go back to school. I have a job now! I wasted my chance! Nooooo!

Of course, I have to remind myself that I actually learned a ton in graduate school. And at the time I was stressed out, worried about the very immediate problems presented by my thesis and teaching, as well as eventually finding a job. I was the best graduate student I could be, given who I was and the time I had. So what do I do with this wistful sense of what I should have been? I just keep doing the best I can, trying to solve problems, learning things from books, going to talks, and asking people questions when I get the chance. And, slowly, I do keep learning new things and solving new problems

The regretful sense that I should have been a better graduate student is increasingly countered by a feeling of excitement in the realization that I am actually still growing as a mathematician. In fact, as I (through great struggle) learn more, it becomes a little easier to see bigger pictures, and I even learn faster. Graduate school was just a jumping off point, and I keep gaining new perspectives. I am so glad that I was wrong about my math trajectory! It is actually thrilling to realize that, far from being done learning, I can just keep getting better at this.

I don’t know why I didn’t realize it sooner. I mean, I guess I thought my professors just started out good at math; that they were of a different species of mathematician. I definitely didn’t think of them as still learning. Perhaps I had, without even knowing it, absorbed the idea of mathematics as a sort of inborn talent, that you could fulfill as a young person but which fades as time goes on. Thank G. H. Hardy, with his incredibly annoying statement in A Mathematician’s Apology, “No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man’s game.” Perhaps Hardy’s words applied to his life. After all, he was a Tripos star, at the top of his field from the very beginning. He may have just crammed in a huge amount of math early on, and never really felt that he was growing in the same way later in life. However, I definitely followed a different path, and many others have as well. Richard Guy, anyone? (Happy 100th birthday!) In fact, I am not the first to point out that this whole idea is suspect. I reject the notion that these early years are the peak of my career, and that people’s mathematical value or potential is determinable by their 40th birthday. Hooray for early achievers! But also, hooray for people who start later and move slower but keep getting better all the time.

Realizing that I can keep getting better as a mathematician has made me really empathize with my students, who often think that they are already basically good or bad at math, and are afraid that they have hit their limit and are not capable of understanding some difficult concepts. This takes me back to growth mindset, (which I just explained to my linear algebra class on the first day of class!) and how easy it is even for me to fall into the trap of thinking that I can’t grow in some way.

On the level of the profession, it is important to reach out to talented young people from all backgrounds, both to maximize mathematical progress and to make the mathematical world a more diverse place. However, I think that it is also important to keep the door open for those who are not early talents, who may not choose mathematics until much later but still have a lot to offer. This is vital to broadening the profession, since despite our best efforts, students from underrepresented groups in STEM or less affluent school districts may not come into contact with interesting mathematics until later in life, and may have far fewer opportunities to pursue mathematics young even if they are interested.

So that’s what I’m thinking about as I start the new semester. Good luck to everyone with the challenges of fall! I would love to hear your thoughts on all this in the comments.

Okay, this is a stretch... but I am totally getting better at math. Like, I used to know a flamingo's worth of math, and now I know many flamingo's worth of math. And someday...

Okay, this illustration is a stretch. But I’m going for it:  I am totally getting better at math. Like, I used to know a flamingo of math, and now I know many flamingos of math. And someday…

... someday I will know a giraffe of math! Okay sorry about that. I wanted to illustrate with a picture or a video of the Beatles singing "Getting Better," but do you know how totally copyrighted that stuff is?!? These are actually pictures from my summer trip to the San Diego zoo.

… someday I will know a giraffe of math! Okay, sorry about that. I wanted to illustrate with a picture or a video of the Beatles singing “Getting Better,” but do you know how copyrighted that stuff is?!? These are actually pictures from my summer trip to the San Diego zoo.

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Summer Research Wrap-up

We’re already back to school here at Hood, but I wanted to take a minute to reflect on my summer undergraduate research experience. Overall it was very positive. My students met all the goals I had for them, got to present at MathFest, and we all had a pretty great time.

I’d mentioned in a previous post that we used Slack extensively to coordinate our research. It ended up being a collaborative lab notebook of a sort, as well as an email replacement. I loved not having to search through email threads to find relevant bits of conversation or documented work. I loved it so much I’m trying to think of how I can force my classes and my department to use it with me.

When one of my students had to leave town for a couple of weeks, we also used an app called Baiboard in conjunction with Google Hangouts to keep in touch. Baiboard served as our collaborative whiteboard while we were chatting on Google, so that we could see what the others were writing without having to try to type things on the fly, or hold paper up to the camera. I will definitely use this app again for things like online office hours.

This summer was also my first experience teaching students to use LaTeX, and it went much better than I expected. They collaborated on Overleaf to write up their results and make their Beamer presentation for MathFest. They even got comfortable with the tikz package for their graphics. The only real help I gave them was a couple old documents of mine to modify, and they picked the rest up incredibly quickly (somehow). One student has a lot of CS experience so I’m not entirely surprised that he adapted well, but even my normally-computer-phobic student became a LaTeX whiz almost immediately. I think at that point in the suIMG_8086mmer they might have just been excited to have problems whose solutions were Googleable.

They wrote their presentation and practiced it for a couple of my department members, who had excellent advice. I haven’t even seen that many undergraduate talks, much less helped anyone write one, so I was very grateful for their experience. We ended up scaling down the scope of the talk a little and let them focus on a few main aspects of their results, which helped the cohesiveness a lot. The students rehearsed more than I ever have for anything in my life, and when the time came to present, they got over their nerves and just killed it. I was beaming.

Bringing students to a conference did present some interesting logistical challenges: spending three days – including about 12 hours in the car – with two rising sophomores meant I definitely overheard more juicy student gossip than I ever needed to. And I just assumed that they’d know how to get around a city on their own, even though they haven’t spent much time away from their parents. This proved false when I got a call asking me to rescue them from across town when they’d gotten on the wrong bus by accident.

But we all had a really good time at MathFest. I enjoyed getting to interact with these students outside of the classroom, and getting to know them outside of their mathematical interests. Turns out they’re pretty cool people! And they loved getting to see aspects of mathematics they’d never been exposed to, though I think they found a lot of the conference just plain overwhelming.

I probably won’t try to apply for this research grant next summer, as I’d rather take a break from managing a group and work exclusively on my own stuff for awhile. But I will definitely be doing this again in the future. I think this was about the best first experience I could have hoped for, and I think my students felt the same.




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National Service through Math

Original design for the "Be Patriotic" poster by Paul Stahr, 1917-18. Public Domain, from Wikipedia

Original design for the “Be Patriotic” poster by Paul Stahr, 1917-18. Public Domain, from Wikipedia

Math-ional Service? Two thoughts lead to one blog today. First: the recent political conventions (and non-stop political coverage for the last year before the conventions) have got me thinking about government and public service. The ideal of government is to serve the public. We all participate in government in a basic way when we vote, but surely there are other ways that citizens (like, say, mathematicians) can serve each other and the world through involvement in government. Second thought: once in a while, I want to do something else. I mean, do a job other than math professor. Occasionally, I want to do ANYTHING but be a math professor. Generally I am just looking for a temporary change, so I can use the skills that make me a good professor in a different way, or towards different goals… like maybe serving the public through working with the US government! Luckily, there are actually opportunities to do this as a professor. This blog is devoted to some government service opportunities for academic PhD mathematicians.

You may have heard of the American Association for the Advancement of Science (AAAS) Science and Technology Policy Fellowships. These are year-long fellowships open to professional researchers and educators with PhDs from any STEM field (master’s degrees in Engineering are also ok). The program accepts applicants to work in seven different areas to help with government policy related to science in technology. Fellows are based in Washington DC and receive a $75,000-$100,000 stipend plus benefits—this is a viable job for a full year. There are both congressional and executive branch fellowships available. The AMS sponsors one congressional fellowship each year specifically for a mathematician.

People do this program at many different stages of math life—early career, sabbatical, or transitioning out of academic mathematics. This work is right in line with my interests, so I applied for the Science and Technology Policy Fellowship two years ago. The application process is outlined on the website above. If you are chosen as a semi-finalist (I was), you will do a video-conference interview. Before my interview I was asked to write a briefing memo, explaining an area in which my expertise would be helpful to a policy maker. The people on the interview panel were very engaging. We talked about the memo I’d written and my interests in math and public policy. I have to say that the interview was a positive experience in itself. Though I was eventually became a finalist, I received an offer from Villanova at the same time, so I didn’t actually get to participate in the program. However, I think this fellowship is fantastic and I hope I get the chance to do this at some point in my career.

Another opportunity that gets less press is the Institute for Defense Analyses (IDA) Centers for Communication and Computing summer program for faculty, known as SCAMP. The Center for Communications Research has two offices, one in Princeton, NJ, the other in La Jolla, CA. Each summer these centers run 10-week summer workshops in which visiting and permanent researchers work on problems of interest to the government, including “cryptography, cryptanalysis, algorithms, high-performance computing, information processing, signal processing, and network security, as well as related areas of pure and applied mathematics,” according the CCR website.  I talked to several people who have gone to SCAMPs, and they reported really enjoying the problems and collaborative atmosphere. One thing to consider is that attending these workshops requires obtaining a security clearance, which is not a quick (or necessarily easy) process. It probably makes sense to start working on this early in the fall if you are interested in attending the following summer. If you are interested in joining a SCAMP, email CCR La Jolla or CCR Princeton for more information.

Finally, an opportunity that gets incredibly little press: the Defense Science Study Group (DSSG). I recently heard about it from an alumnus who had a great experience in the program. An internet search made the DSSG seem very mysterious—I found a few references from alumni (including this and this), but very little official information. (Apparently there is a real website, which was down but will hopefully be up again soon.) Here’s the general idea: the DSSG is a group of science and engineering professors that meet/go on adventures for approximately 20 days per year for two years. A new class is chosen every two years from a field of faculty nominated by their home universities. The participants learn about national security issues, from active US government security personnel. As their brochure explains, “Group members interact with top-level officials from the Department of Defense (DoD), and other Government organizations, various Intelligence agencies, The White House, and Congress. Visits to military bases throughout the United States provide members with a unique perspective of operating forces and allow DSSG members to meet with senior commanders responsible for our nation’s defense. Tours of defense laboratories and industrial facilities provide further insight into the technical dimensions of national security.”

The goal of this program is to forge connections between scientists and engineers and the national security community, and to encourage academic scientists and engineers to contribute their knowledge to addressing national security issues. It is administered by IDA and sponsored by the Defense Advanced Projects Research Agency (DARPA). For a math professor with strong national security interests, who is interested in doing things like flying in military planes and visiting aircraft carriers, this program would be a perfect fit. However, individuals can’t directly apply for this program. To get into the program, you need to be nominated by someone, usually an official at your university (like a provost). While some institutions have internal application processes, many do not. Some provosts may not know about this opportunity or may not have considered nominating a mathematician for it. With that in mind, the alum I talked to suggested that if you are interested you should definitely contact your provost about a nomination. Recruitment for the next two-year class begins this September/October.

Of course, programs like these are not the only math-related ways to serve/effect change in government. You could always just go to Congress, Mr. Smith style. The Association for Women in Mathematics is doing this (okay, visiting congressional offices, not addressing Congress). In 2015, the AWM leadership and student chapter members visited Capitol Hill offices to meet with congressional staffers. Georgia College students, from a chapter led by Dr. Marcela Chiorescu, describe their group’s experience in DC in this press release. Villanova’s AWM student chapter is hoping to visit congressional offices this December—so excited about this!

Are there more programs out there? Other ideas on how to serve or get involved? Let me know in the comments!

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