*How did you get interested in applying mathematics to ecology?*

When I was applying for college, I had to apply to a major before I went to college. I was looking through the majors, and I saw applied math, and I thought, “Not bad. Check.” Later my professor said, “Well, why don’t you think about applications of math to biology?” When I went to grad school, I tried out neurobiology, microbial biology, and molecular biology, and finally ecology again and loved it.

*Did you do any research as an undergraduate?*

Yes. When I was an undergrad, I did two REUs [Research Experiences for Undergraduates]. The first one was based on immunology. I took a class and did some research on a topic. The following summer I was in the Math and Theoretical Biology Institute, where I studied spatial spread of influenza. That was my first encounter with math and ecology. That’s what planted the seed in my head, but it took me a while to determine that I wanted to pursue it.

*What are you currently researching?*

I am working with an undergrad, Alex Meyer, on spatial synchrony of cicadas. Cicadas are insects that feed off of the roots of trees and live underground for sixteen years. On their seventeenth year, they emerge in huge numbers, reproduce and die, starting the cycle again. It’s amazing that cicadas emerge simultaneously. The other thing is that…[there is] usually one brood is in each location. There is no coexistence of the two broods [in the same place]. What we’re trying to do is to understand how that happened evolutionarily.

*What was your most difficult moment as a mathematician?*

My most difficult moment was making the decision of where I wanted to go. The places I had been to, prior to Williams, were big research universities where people tended to go once they found a permanent job. I assumed that was what I wanted, and I thought that I would follow this common trajectory. It became clear over time that I wanted to be in a small liberal arts college. Telling my mentors and the people I worked with that I was not going to follow the common path was challenging and nerve wracking.

*What was your favorite class in college?*

Differential equations. That was the subject that really got me most excited about math. On a very foundational level, it is the basis of all of the work I do, and I loved learning about it. The class gave me a clear direction to pursue math.

*Do you have any advice for other mathematicians?*

Use your resources. Talk to people. One thing I wished I did more was talk to people about the different options, because I assumed there was only one direction that everyone took. Even though I wasn’t completely satisfied with that one direction, I didn’t ask people about alternatives. It took me a long time to figure it out on my own.

*Julie Blackwood is Assistant Professor of Mathematics at Williams College. She hails from the great city of Buffalo, NY and enjoys spending time with her daughter. *

ANOTHER HAT PROBLEM

A hundred mathematicians have traveled to a hattery for the annual conference on colored hat problems. The following game is presented to them. Every mathematician receives a card with a distinct random real number. These cards are taped to the mathematicians’ foreheads, so that each mathematician can see the numbers of all ninety-nine of his or her colleagues, but cannot see his or her own assigned number. The hundred mathematicians then each individually select either a white hat or a black hat from the haberdashery’s infinite collection of white and black hats. The mathematicians are lined up by the proprietor of the hattery in increasing order of the numbers on their cards. If the hats on their heads are *alternating* in color (i.e., BWBWB … or WBWBW …), then they all win a lifetime collection of hats. Can the mathematicians agree on a strategy beforehand to guarantee a win?

Note that the mathematicians cannot talk to each other while selecting their hats, nor can they see what hat colors the other mathematicians are selecting. Their strategy, of course, can involve each mathematician choosing a color based on the numbers they see on their colleagues’ foreheads.

For example, if there were only two mathematicians, say Alice and Bob, then the game would be trivial – Alice would agree beforehand to choose a white hat and Bob a black hat (or vice-versa). What about for three mathematicians?

A HAND-SHAKING PROBLEM

You and your spouse are at a party of five couples (including you). As the party goes on, various introductions and hand-shakings occur. You observe that nobody shakes hands with their spouse, as presumably they already know them. At some point, you stop the party and ask each of the nine other people how many hands they have shaken. You are surprised to obtain nine distinct answers (i.e., one person shook no hands, one person shook hands with exactly one other person, one person shook hands with exactly two other people, and so on). How many people did your spouse shake hands with?

GASOLINE ON A RACETRACK

You are driving a car at constant speed around a circular racetrack. Distributed around the racetrack are cans of gasoline. The net amount of gasoline on the racetrack is precisely enough to drive the car once around the track, but the gasoline has been divided into many small containers and scattered along the course of the ride. You start with no gasoline, but as you pass each can of gasoline you (instantaneously) pick it up and put it in the gas tank. You are allowed to start your ride at any point along the track. Is it always possible to complete a full lap?

For example, if the gasoline were divided into two containers, then you could start at whichever container had more gas and drive in the direction for which the distance to the other was shortest. (Check that this always makes it possible to complete a full lap!) In general, the gasoline may be divided into many containers of unequal size.

Special thanks this month to M. Miller and L. Alpoge for discussion and communication of the above riddles! If *you* have a riddle or puzzle that you think is cool, send it along (with a solution, if you like) and we may decide to post it.

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I had never questioned my decision to study math at a liberal arts institution. I loved the fact that my classmates and friends (and professors!) had such varied interests, such as the classics, philosophy, music, and art, and that these subjects were often woven together into interdisciplinary courses. I appreciated my small classes that fostered conversation with classmates and close contact with professors. I didn’t realize what I could have benefited from at a larger school: a deeper pool of math classes (at the undergraduate and graduate levels) from which to choose, as well as graduate students who I could look up to, learn from, and emulate.

When I started comparing myself to my peers, it was easy for me to focus on what I didn’t get out of my liberal arts degree. It was easy to believe that the “best” preparation for math graduate school would have been taking lots of high-level math courses as an undergraduate. It took effort for me to find the value that my liberal arts degree afforded me as a graduate student. But there is value there, and it’s worth it to remind myself of the ways in which my undergraduate experience did prepare me well for graduate school.

I’ve received a lot of advice from older graduate students and professors about making the transition to graduate school, and I learned quite a bit in my own first semester. Based on this advice and my own experiences I’ve compiled a few suggestions about how you can make a successful transition from a liberal arts college to graduate school. (This advice can apply more broadly to anyone starting graduate school, not just liberal arts students – everyone faces struggles in making this transition.)

**It’s okay to feel behind, but that doesn’t mean you’re less ready.** In my first semester of graduate school it seemed like so many of my classmates were already familiar with many of the topics in our graduate courses. It was almost all new to me. I worried from day one that I was already behind and I wouldn’t be able to catch up. Those fears weren’t completely unfounded: I did know less about certain subjects than my classmates, and I did have to spend more time than them puzzling over some homework assignments. But in my first semester of graduate school I took courses that were at the right level for me, and I was able to learn and understand the material despite starting from “behind.” As a liberal arts student, it can take some time to catch up to others in terms of sheer knowledge base, but after a few years everyone will be diving into their own research, and it won’t matter who started “ahead.”

**Don’t be afraid to ask questions.** Office hours were seen as an essential part of my undergraduate courses, and liberal arts colleges in general tend to have lots of close student-faculty interaction. I practically lived outside my professors’ offices as an undergrad, and I continued my frequent office hour attendance when I started graduate school. I think I go to my professors’ office hours more than any of my fellow graduate students, but I’ve gained so much understanding and cleared up so many misconceptions during those office hours. Whether you’re talking to a professor, a TA, or a classmate, take advantage of the wealth of knowledge around you. I promise you’re not the only one who has questions.

**Let your communication skills shine.** Almost all of my undergraduate math classes had one or more writing projects, as well as plentiful opportunities to present proofs and other work in front of the class. Each paper and each presentation honed my ability to express mathematical ideas in words, organize my thoughts in a logical way, and motivate the importance of the topic at hand. Liberal arts courses teach students to communicate effectively, a skill which is necessary in every discipline. Even the most brilliant mathematical insight would lose value if it couldn’t be explained to other mathematicians. The ability to communicate well is not only useful for writing papers, but also for writing clear homework and test solutions, TAing, giving talks, and writing applications for fellowships, conferences, and so on.

I want to leave you with one more thought. If you are in a graduate program and feeling like you aren’t ready, remember that you were not admitted by mistake. You were recommended by your undergraduate professors, and you were chosen by a graduate program to pursue a degree. Talk to an advisor or professor about finding the right courses for you, and trust that with a lot of hard work and help from classmates, TAs, and professors, you’ll make it through. It can be a jarring experience for anyone entering graduate school to suddenly find themselves surrounded by the best and brightest students. Take the time to identify your own personal strengths, and let those guide you toward a more confident outlook.

]]>*How did you get interested in mathematics? *

When I was in 7th grade, in Boston, I enrolled in an experimental program called The Math Circle, where students discover math on their own. We, the students, came up with all sorts of ideas about number theory, argued with each other, proposed conjectures, shot each other down, and eventually came up with all sorts of proofs. No one told us these things–we made them up. Turns out other people had discovered them earlier, but that wasn’t the point. The point was that we owned them–they were ours because we invented them! This experience showed me that math was creative and could be created by me. From then on, I was totally hooked.

*Do you take this kind of curiosity into your teaching at Williams College?*

Absolutely! I think it’s important to learn things before you know why they’re good for you. I also think it’s really important to have motivation from something other than an authority figure. That is what the Math Circle is all about. The teacher showed us this weird thing, and it was up to us to decide what we wanted to prove or discover. This greatly informed my ideas of teaching. My goal is not to transmit information; it’s to get people excited. Information is everywhere. There are textbooks and Wikipedia; we don’t need professors for information. The point is to excite people and get people to start exploring. I go in, I teach, I interact with students, I see a twinkle in an eye, and that’s it, I’ve made my day. In a university setting, you can’t exactly do a math circle model since there is a set amount of material that must be covered, but I try to make my classes student driven.

*I guess Math Circles really started you along the number theory route. Could you talk a little about your research in number theory?*

My research is in classical number theory. At a fundamental level, I am trying to understand structure and randomness within the primes. I think of primes as atoms: you take any whole number and break it down into parts, which are the primes. The question is if there is any sort of rhyme or reason to them. Is there a way to find out what the millionth prime is? Well, you could just go one by one and find the next prime and the next prime and eventually say, that’s the millionth one! This is obviously not a great way to do it. Is there a shortcut or a formula for figuring out the *n*th prime? My theorems are about subsets of the primes and trying to understand how noisy and how structured they are.

*How was your grad school experience?*

I was extremely lucky to have an amazing PhD advisor. His name is Kannan Soundararajan (he goes by Sound). He not only met individually with each of the students every week to talk about their progress, but also met with all of his students once a week all at the same time to talk about half-baked pre-ideas. They were along the lines of “What if I could do this?” or “Here’s a set-up of what I would like to work on.” We would shoot out all these pre-ideas. It was fascinating to see how people thought about and approached ideas. It was especially amazing to see how Sound thought and how his intuition worked.

*Was there a most difficult moment?*

I’d say all of it was difficult. There’s the course part of grad school, but at least the goals are well-defined. Then comes the research, which is frustrating. You have no idea what the goals are, and you don’t have a concrete, specific statement you are trying to prove. When I was out there, I really started getting demoralized by my lack of progress. I had been working for a while and I hadn’t really discovered much, if at all. It became clear to me that not only was I nothing compared to the faculty, but also I did not measure up to the students. In some ways it was inspiring, but also very demoralizing. I was getting pretty close to quitting.

*How did you overcome this impasse?*

I remember I was at lunch sitting next to Ravi Vakil at Stanford, and he asked me how things were going. And I said, “You know, not too well, I’m thinking of maybe quitting. I might be at the end.”

He said, “Why is that?” I said, “You know, it’s clear. I look at Sound, I look at myself. He’s supposed to be a mathematician, I’m not.” Then he said, “You know, most mathematicians aren’t as good as Sound.” I said, “Yea, but I look at the other students, and they’re on another level.” And he said, “You know, most mathematicians aren’t as good as the other students either.” His words had a huge impact on me. Later in the year, I ended up making my first progress and proved a theorem. Once I knew that it was possible for me to discover a result, there was a psychological shift.

*Leo Goldmakher is Assistant Professor of Mathematics at Williams College. Stephen Ai is a current sophomore at Williams studying mathematics and music; his favorite piece is JS Bach’s Mass in B Minor.*

- Refocus on what matters the most.

If you (like me) are still in the first few years of your program and are focused mainly on taking courses, recommit your attention to your study habits. Start out by scheduling more time than usual to work on homework. Find study partners to help make your work time more productive and more enjoyable; working with friends is often more fun and more useful than working alone.

If you have moved on to focusing on research, get back to treating it like a full-time job. Schedule hours during which you need to focus on your work every day, and hold yourself to the commitment, possibly by keeping track of your productive hours in a daily log, or by working around friends to keep each other focused.

- Get excited about teaching.

If you have a teaching assignment this semester, get excited about this opportunity to reach out to a new group of students and help them through their next math course. Although you will be devoting most of your time to your research and classes, the new semester poses an opportunity to improve your teaching, too. Come to your first few classes overly prepared with your lessons planned out and extra problems in mind in case your lesson runs shorter than expected. Review the material you’ll be covering so you are prepared for any questions that come your way, and remember to have fun! Teaching is a great opportunity to remember how much fun math can be.

- Remember to commit time to seminars.

If you began last semester attending your seminars regularly but gradually became busier and began skipping more frequently, take this new semester as a chance to start over. Look at the schedule of seminars offered this semester and add a few to your weekly schedule. Treating these as a course you must attend will help make participating in this social event a habit that you maintain throughout the semester. Remember that seminars are fun- these are a chance to spend time with the other people in the department whose interests mirror your own and to gain inspiration and encouragement from them, all while learning about a subject that interests you.

- Save time for yourself.

Just as important as refocusing on your work is remembering to prioritize yourself and your needs. If last semester began to overwhelm you and work started to eat up your free time, start fresh by scheduling in time for yourself this semester. Whether that means planning time to exercise, to make a good meal for yourself, or to spend some time away from math with your friends, remember that taking care of your mental health is an important part of being successful in graduate school. Taking some time off to unwind will help you to be more productive and focused when you should be working. Of course, it may take a while to find a good balance of work and relaxation this semester, but scheduling in time for what matters most to you in these first few weeks will make you more likely to continue to provide yourself with these opportunities as the semester goes on.

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In February 2013, the Wall Street Journal Reported, “Prison sentences of black men were nearly 20% longer than those of white men for similar crimes in recent years, an analysis by the U.S. Sentencing Commission found.” Is this evidence of racism, intentional or subconscious, on the part of judges? We might initially think so; after all, the statistics have already been adjusted for type of crime committed, so the outcome shouldn’t be skewed by one race committing more serious crimes. However, “The commission, which is part of the judicial branch, was careful to avoid the implication of racism among federal judges, acknowledging that they ‘make sentencing decisions based on many legitimate considerations that are not or cannot be measured.’” That may be, but some factors* can* be measured, and that is what we will try to suss out here using matrix multiplication.

Perhaps what we are really seeing is the effects of class. (Aside from the original WSJ quote, the following percentages are made-up). Suppose for a particular crime low-income defendants are handed down a long sentence 30% of the time, middle-income defendants 20% of the time, and high-income defendants 10%. It could be due to the high cost of good lawyers, biases by judges, harsher laws in states that have more poor people, etc. Now suppose 85% of black defendants are low-income, 10% are middle-income, and 5% are high-income. What percent of blacks should we expect to be receiving long sentences if the reason is class alone? Clearly the answer is:

Note that what we’ve just done is to take the dot product of two vectors:

Which we can also write like this, for reasons that will soon become clear:

More complicated than addition? Sure. But as this example illustrates the definition of dot product is not some arbitrary construction of interest only to mathematicians. There are times in life when multiplying the individual components of something separately and then adding them all up is extremely useful. Now suppose the percentages of low, middle, and high income *white *defendants are 60%, 25%, and 15% respectively. We can do the same thing we did for blacks by changing the values in the second vector:

But since the first vector is the same in both cases, let’s see if we can avoid having to write it twice:

And tada! You have matrix multiplication. It’s no more scary or complicated than vector dot products; we’re just doing those dot products side-by-side to save space. Those savings become even more apparent when we add more rows and columns. Perhaps we want to include a third column for Asian-Americans:

Or a separate row for death sentences:

And now that we have these percentages, we can start to make inferences…carefully. If the number in the top left corner of our final matrix is lower than the actual number of blacks serving long sentences, than we have good reason to suspect that racial disparities in sentencing cannot be fully explained by class differences. Or more broadly, if *any* of the numbers in our final matrix do not reflect the numbers we actually see, and we determine the differences are statistically significant, than we know that class cannot be the sole cause of the racial sentencing gap. Of course, there are many other things to control for including criminal record, age, whether the defendants live in heavily-policed urban areas, etc. And one can still ask the broader question of what societal and historical forces have led higher percentages of blacks to end up in poverty in the first place. I’m not claiming that you can fully explain racial sentencing disparities in one lesson. What I am claiming is three things:

- Matrix multiplication seems less arbitrary when it is motivated by a real-world problem
- Matrix multiplication appears less complicated and overwhelming when you start with the dot product of vectors and then expand it one row at a time
- There is plenty of room in the curricula of introductory college math courses to tackle race, class, and social justice

And indeed, there is no excuse to stand on the sidelines in an age of such inequality and injustice. Many math, science, and engineering students badly need to be exposed to the reality of these inequities, and what better a place than in a course that they value, in a context they find engaging? Students from other disciplines merely trying to fulfill their quantitative requirement might suddenly find that math is important to the world they live in and the values they hold. Moreover, both types of students will learn how math can serve as a valuable tool for fighting injustice. Even if they never use that tool themselves, they might someday be faced with a problem like this and realize that hiring someone with mathematical expertise would prove useful. And if you’re still not convinced, recall that much of our research is funded, however indirectly, by the American taxpayers. It helps to have voters to whom math matters.

Palazzolo, Joe. “Racial Gap in Men’s Sentencing.” *Wall Street Journal. *Feb 14, 2013. http://www.wsj.com/articles/SB10001424127887324432004578304463789858002 (Accessed January 21, 2016).

This will be the last installment in my job search series, though it will perhaps be the most important. In particular, I share research on negotiating a job offer.

If you receive a campus interview, congratulations! You are among 3-4 people who the search committee found suitable after a brief phone or conference discussion. Though the committee may have its favorite among those candidates, the campus visit can alter those perceptions drastically. The reality is that you are all equally qualified, so it’s best to be (a polished, groomed, suited up version of) yourself.

You will be peppered with questions on the day of the visit, so be sure to again review common interview questions. Practice the “elevator pitch” versions of your teaching philosophy and research directions. Keep in mind that, for better or worse, your best chance to make an impression on the department is the job talk. Practice your talk early and practice it often! This job talk advice from Prof. Lerman at UIUC very clearly spells out the goals for which you should strive, as does this article from the MAA’s FOCUS magazine.

In addition to reading advice articles, get out and talk to people in your department—especially postdocs and other recent hires who have fresh memories of the interview process. If your department is hiring, it would be a great idea to attend the job talks of candidates to get an up-close look. You may have other resources available at your campus such as mock interviews.

It’s a good time to dust off your nice clothes to ensure they still fit. A campus interview can involve a lot of walking as well as lunch and dinner with potential colleagues; you want to be well-dressed yet comfortable. This Academia.SE question provides useful opinions on dress etiquette.

As is often mentioned, a campus invite also provides time for you to interview the department. Be prepared with a list of questions to ask the search committee. A full day interview usually incorporates a few short, one-on-one discussions with other faculty in the department. This is your chance to get into the nitty gritty about expectations for the following:

- teaching load, course assignments, areas of need;
- research and publishing;
- grant-writing and institutional support;
- main ingredients of the tenure process;
- summer activities and support;
- student advising.

I will close out this section with three selected articles from the Chronicle of Higher Education, which generally has good advice.

Having only negotiated a single software development job, I admit I was a bit naive about the negotiation process. Through my own reading and discussion about the academic job search, I have internalized the following rough model of the academic negotiation: The department, through the chair, entrusts the search committee to select the best person for the job. However, a dean or other administrator must be convinced of all decisions by the chair. So once an offer is put forth, the department chair acts as an intermediary and mostly wants to see the search wrapped up quickly and in your favor. Otherwise they must repeat the process, hoping that their second choice hasn’t already moved on. A corollary to this model, which I’ve seen echoed elsewhere, is that if you’re unsure about the feasibility of negotiating on a particular point, then simply ask the department chair.

Unless you end up as a department chair or administrator, you will likely never again have such direct influence over the actions of the chair and dean. You have the most power in the few days when an offer is on the table. After acceptance, you immediately take on the title of “Department Newbie“! (Which doesn’t confer much influence at all.)

I’m not suggesting you play hardball or ask for everything mentioned here, I’m merely trying to hammer home the idea that certain parameters of an offer are commonly negotiated. So much so, that a recent instance in which a job offer was rescinded spawned much weeping, wailing and gnashing of teeth across the internets. So do your research and negotiate!

Allow me to help you out with your research, which I feel is the meat of this post. The first thing everyone asks after hearing “job offer” is “how much?”, so you might as well begin with this primer from Inside Higher Ed. It’s often considered impolite to discuss salary, but I believe open information can help decrease pay inequality. If you happen to have an offer from a public university, then often times you can download *incredibly specific* salary information. Searching “state u faculty salary” often produces a detailed pay schedule or even an eerily convenient list of the salary of each employee by name. State law usually compels the dissemenation of this information, as well as surveys such as the Report on 2011 Faculty Recruitment and Retention Survey published by the California State University system. Though boring, such documents afford specific insight into institutional thinking. Another source of salary data is the The Annual Report on the Economic Status of the Profession published by the American Association of University Professors (salary data accessible here). The National Center for Education Statistics has fine-grained salary data which may be browsed through this helpful portal. You can break down information by institution, faculty rank and even view gender discrepancies. Really the only thing missing is the ability to see field-specific information. If you want a quick idea of how much room you have to negotiate salary, I’d look here first.

The are a number of potential negotiation points which are almost as good as increased salary. In general, you want to think in terms of your total *compensation*, not salary. One immediate concern is your relocation and most institutions offer at least partial support to offset these costs. Take a few hours to estimate moving costs and communicate these during negotiation. You may also have some ability to alter terms of retirement and non-cash benefits. For example, if you don’t get the extra salary you want, can you get an extra percentage point matched into your 401k? What about housing? Is it expensive? Then ask about subsidized faculty housing or mortgage assistance. Do you have family concerns such as parental or medical leave? Now is the time to investigate the possibilities!

I must confess ignorance about a very important benefit that is almost as important as salary: the start-up package. You may not need to build a lab like a chemist, but you should still think carefully about this aspect of the offer. For a mathematician, the essential purpose is to supplement your professional activities until you find external funding. Usually the start-up package will include a new computer and other office equipment. But as a mathematician you need to travel for research and professional development. Travel costs money and you really don’t want that coming from your (9 month!) salary. So ask for a travel fund. Figuring $1500 per trip (or twice that for international travel), one to two trips per year for three to five years and you’re looking at $10,000 before you gain tenure! I’m not suggesting you ask for that much in your start-up package, necessarily. But you will need something to form a backstop between external travel grants and your own salary.

As I alluded to above, an academic salary traditionally only covers 9 months of the year. How will you support yourself during the summer? Here again, you may negotiate something into your start-up package. A salary for one summer gives you time to write research grants or develop other activities to supplement your income. A negotiation tip: You can’t simply ask for a free money (well, you can try). Instead you say, “I intend to host an REU in two years, would you support my time to develop a project via two months of summer support?” Then the following year, you can pay yourself one month salary courtesy of the NSF. Finally, your research may also require computing resources. If you don’t have grants to pay for this, you may need help to hit the ground running. Though focused on mathematics education, this article in the Notices provides context for the size of start-up packages, as does the Cal State report linked above.

I will conclude this section by mentioning a few common negotiating points for tenure track offers. In fact, I already heard some of these discussed during phone interviews.

- Most offers include a reduced teaching and/or administrative load for the first two years. You don’t want to be sitting on too many committees or preparing a bunch of new courses while you’re trying to lay the foundations of your research program.
- It may be possible to earn a pre-tenure sabbatical. One teaching-focused school I spoke with offered a standard half-year paid sabbatical during the fifth year.
- There may be flexibility in the tenure clock, in either direction. For instance, if you’ve completed a postdoctoral position and have a solid research program you can ask for a “no-penalty” review after five years instead of six, for instance.
- If you haven’t completed a postdoctoral position, but would like to, know that it is relatively common to be granted for a one-year deferment in your permanent position. Don’t withdrawal your tenure-track application at “Dream U” simply because you accepted a short-term research postdoc.
- Perhaps the most delicate to broach, many people have found success in solving the two-body problem! Some forward-looking institutions have even developed specific procedures for dealing with this scenario. Though for a look at some of the pitfalls (and, sadly, outright discrimination) of such a job search, I recommend this two-part series.

You never really stop applying for your job as an academic. When you’re not teaching or performing research, you have to continually write about your teaching and research to participate in professional development, win grants and gain tenure. And I’m sure such writing doesn’t stop after you gain tenure! If you’d like to take your mind off of your job applications, then start thinking about future programs in which you’d like to participate (I linked this helpful blog post previously). Have you applied to teaching-oriented jobs? Then you may want to check out programs like the Project NExT. As mentioned above, you may also want to begin looking into travel grants. There are many institutes which fund collaborative research such as the Simons Foundation, Mathematical Sciences Research Institute, the American Institute of Mathematics and the Banff International Research Station.

Happy job hunting!

]]>$$x_n=x_{n-1}-\frac{f(x_{n-1})}{f’(x_{n-1})},$$

and then, under certain assumptions, should converge to a root of .

When I’ve taught Newton’s method, I tried to stress that this method is not guaranteed to always work and can be fairly sensitive to the initial condition . Going into the precise details of when the method will in fact work is well beyond my first semester calculus students. That said, I am not sure they’ve ever really appreciated these points. In fact, to be completely honest, I am not sure I completely appreciated some of the aspects of these things before I started teaching.

That said, one way to visualize some of these complexities is via a cool program called FractalStream. For example, using FractalStream, we can run the following script:

iterate z – (z^3 – 1)/(3*z^2) until z stops.

and then turn on autocoloring – under the color settings – and we get the following interesting picture of the complex plane colored red, blue, and green:

Given the above script, FractalStream takes each point in the complex plane and iterates it under the given map until the sequence seems to stops. It then colors that initial point depending on where the sequence of iterates ended. That is to say, since we chose to iterate the function:

$$N(z)=z-\frac{z^3-1}{3z^2}$$

the above script actually performs Newton’s method using the polynomial for each initial value and labels each value by which root we end at (assuming we end at a root).

In the above example, we see there are three colors since has three roots. Specifically green corresponds to the root 1, blue to , and finally red to the root . Notice that while there is a large area round each root in which Newton’s method converges quickly to that root the areas sort of between each root show more complex behavior. In particular, in this region we see just how sensitive to the initial condition Newton’s method becomes. Additionally, using the orbit tool in FractalStream we get a sense that points in this region take longer to converge:

In this picture the white dot marks each forward iterate under the map of the point chosen.

We can create similar plots for other polynomials as well. For example, the plot for looks quite a bit different than the one for — in part because all the roots are real.

It’s pretty fun just playing around with what plots various polynomials produce!

Why these pictures look the way they do has to do with complex dynamics, which is a really awesome subject. If you want to learn more about these pictures and what is going on you might want to check out either *Dynamics in One Complex* *Variable* by Milnor or *Complex Dynamics* by Carleson and Gamelin. However, even for those not interested in complex dynamics these pictures might be something fun to show your students next time you teach Newton’s method!

In conclusion, if we apply real-life examples to our courses’ materials as I did in this example, then we can create an interactive-based math class.

To see more examples like this and other interactive methods, please see my course webpage. Best of luck and feel free to reach out if you have questions!

**References:**

Kaabar, M. K. A. (2012). __Mohammed Kaabar Website__, Available at http://www.mohammed-kaabar.net

The Joint Mathematics Meetings are being held this week in Seattle! In the most recent blog post, Derek gave advice on how to navigate the JMM when you are on the job market and/or presenting your own research. Here in this complementary post, you’ll find advice & resources for earlier-stage graduate students or first-time JMM attendees.

First, some **useful links** to get you ready to make the most of your JMM experience:

- @JointMath The official Twitter handle for all things JMM
- #JMM16 hashtag for anyone to share their favorite JMM moments
- 2016 JMM Mobile App: free app that compiles the most useful logistics info. See here for more information, or if you are on your mobile device, you can click here directly to download the app
- JointMeetingNews daily email updates and schedules will appear in your inbox if you’ve registered for the meeting, or you can find them online here
- The official JMM Newcomer’s Guide with the basic info and advice for the conference
- A piece I wrote for the Notices of the AMS about my experiences as a first-time JMM conference attendee

And some** perennial JMM favorites** to add to your schedule regardless of your area of interest:

**AMS Colloquium**talks (first talk is 1pm Wednesday): even if the talks are outside your main area of interest, attend the first and most accessible lecture in the series. It’s an unusual experience to listen to a math lecture amongst an audience of one thousand!**Graduate Student/First-Timer’s reception**(5:30pm-6:30pm Wednesday):*highly*recommended reception with delicious appetizers; it’s a great place for mixing and mingling after your first conference day and meeting other attendees and hearing about neat sessions, exhibits, talks, or events you might not have noticed on the extensive JMM program**Gibbs Lecture**(8:30pm-9:30pm Wednesday): you will have time to grab dinner with your new friends (from the above reception) before snagging a good seat for this accessible lecture**Who wants to be a mathematician**(Thursday morning): a national game-show style contest of the AMS**Joint Prize Session**(4:25pm Thursday): hear about the award-winning works of mathematicians from across the country as prestigious prizes are handed out to this year’s prize recipients**Mathematically Bent Theater**(6pm Friday): original math-themed theater performances are always an entertaining evening event; see a brief taste of what they do here**Exhibit hall**: set aside at least half an hour to browse the booksellers, math art displays, software demos, and freebie giveaways.

And I’ll leave you with a piece of personal advice:** Get off the Internet and embrace awkward encounters.**

In order to maximize your chances of having interesting and productive encounters at the JMM, try to leave your comfort zone and interact with lots of new people. If you let yourself always seek to minimize the awkwardness that you feel, you’ll probably end up spending a large portion of the JMM break time seated at one of the round tables and browsing the Internet. But you can browse the Internet any day! So unless you have a specific conference tidbit to look up, try to use your breaks putting yourself in awkwardness’s way by mixing and mingling with attendees you don’t know; the rewards are priceless and you will feel more comfortable the more you try it. You traveled long and far to attend the JMM, and to really make the most of the conference experience, it’s best to minimize your browser and maximize the in-person aspect of the meetings.

Do you have JMM conference advice you’d like to share? What’s your favorite part of the JMM? Leave a comment below!

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