Author Archives: diazlopez

Advice from Haimo Award Winners

What do award-winning mathematicians Federico Ardila-Mantilla, Mark Tomforde, and Suzanne Weeks have to say about teaching? Each of these mathematicians received the MAA Haimo Award at JMM 2020 and gave 20 minute talks. Here is a summary.

Federico Ardila-Mantilla used his time to ask us to reflect by thinking about three “open questions.” First, a story about Federico. A mathematician during the day, DJ at night, it is not surprising that Federico is interested in music. This interest led him to enroll in a timbales class at his local community center. His teacher? One of the top timbaleros of the 1970’s. After just two weeks of classes, this timbalero invited Federico to play with him at an event. It was a moving experience for Federico. What does a similar experience look like in a math environment? When do we see top mathematicians inviting newcomers into their work?

Undisturbed beech forest behave in a weird but spectacular way. “Apparently, the trees synchronize their performance so that they are all equally successful. And that is not what one would expect. Each beech tree grows in a unique location, and conditions can vary greatly in just a few yards. The soil can be stony or loose. It can retain a great deal of water or almost no water. It can be full of nutrients or extremely barren. Accordingly, each tree experiences different growing conditions; therefore, each tree grows more quickly or more slowly and produces more or less sugar or wood, and thus you would expect every tree to be photosynthesizing at a different rate… [However] The rate of photosynthesis is the same for all the trees. The trees, it seems, are equalizing differences between the strong and the weak.” – The Hidden Life of Trees. What does this synchrony look like in a mathematics classroom?

Born in 1952, bell hooks experiences the initial racial integration after segregation was deemed illegal. “Bussed to white schools, we soon learned that obedience, and not a zealous will to learn, was expected of us. That shift taught me the difference between education as the practice of freedom and education that merely strives to reinforce domination.” – bell hooks. What does a liberating math classroom look like?

Mark Tomforde had four original beliefs that have changed throughout his teaching experiences. They are:

– Old Belief 1: My main objective is to teach math content. The more content the better. Updated belief: My main objective is to teach my students to think. In 10 years what will students remember from this course? What do I want them to remember? I want to prepare my students for life, to develop good habits when engaging with facts, to work on improving their skills such as problem solving, attention to details, ability to communicate technical ideas, and to think deeply about a few topics.

– Old Belief 2: The more students I can reach the better. Updated belief: Quality is more important than quantity. I am a fixed quantity and the more projects I am involved with, the less of myself I can devote to them. For me, it is better to devote my time and energy to one or two projects and do it well rather than spending little time in a lot of projects.

– Old Belief 3: The best students are our future, teach to them. Updated belief: Direct teaching to all students paying particular attention to the ones that are struggling. Are our best students more important? Sometimes we think we are just training the future mathematicians and try and identify the “best students.” But, how many students are we not able to reach? Our goal should not be to find talent but rather to help students develop their talent.

– Old Belief 4: Good students are the ones who can prove theorems, solve problems and do well. Updated belief: There are many ways for students to be successful and contribute. When I teach math majors, I regard all of my students as the future of math. This is because math requires a much larger ecosystem than a group of researchers. In addition to researchers, we need teachers, advocates, PRs, writers, and many others are needed. In my classes, I ask students to find a need and contribute to fill that need. Some of the projects students conduct are: leading math circles, tutoring, creating guides for incoming students in the department, starting newsletters, and organizing department picnics.

Suzanne Weeks is a co-director of MSRI-UP and PIC Math, and has worked in many industrial projects. Her experiences have taught her it’s important to:

Show up – Take advantage of the opportunities that arise for you.

Sit in front – Be engaged, be involved, be fully in the room.

Raise your hand – Be an active participant, raise your hand, raise your voice, ask for stuff when you need it.

Make friends – You need to have a support group.

Be yourself – Your path is your own. Do what is best for you.

– Know your worth because you are worthy – Celebrate your successes.

Biomedical Data Sharing

AMS Invited Address speaker Bonnie Berger gave a captivating talk on Biomedical Data Sharing. There is a recurring issue in Computational Biology: genomic data is growing exponentially faster than computing power and data storage. If we want researchers to have access to the data, compressing data, sharing it, and decompressing it is not a viable solution due to the fact that decompressing is time consuming and requires a lot of storage. Bonnie’s solution: Compress the data and operate on the compress data. How?

Bonnie’s research team provided an algorithm that plots data points in a high-dimensional Euclidean space and covers the data points with spheres.

This is a representation of a core idea in Bonnie’s algorithm which is to group data points into spheres.

Then, by only using one point per sphere, the algorithm searches through the spheres, decides which are of interest to the particular problem of study and then goes back to these spheres and does a more thorough search in a region slightly larger than these spheres. The result? An algorithm that is much faster than the available methods and that recovers more than 99% of the results that  slower algorithms recover. This work has been cited thousands of times.

The blue line represents the running time of Bonnie’s full algorithm. The red line represents the time of running through the spheres in Bonnie’s algorithm. The black line represents the running time of one leading algorithm at the time Bonnie’s team presented theirs.

 

Bonnie then went to describe other problems for which they provided similar algorithms to reduce run time while providing great accuracy. One that seemed really interesting was to combine and cluster cell data that was obtained via multiple experiments by different research groups.

Bonnie’s algorithm clustered cell data from 26 different experiments.

For more information, visit Bonnie Berger’s website: http://people.csail.mit.edu/bab/computing.html

Generating Ideas of Undergraduate Research Projects

How do Joe Gallian, Katie Johnson, Stephan Garcia and Colin Adams generate ideas for their many undergraduate research projects? Pamela Harris put together a panel and this is what they had to say.

Joe Gallian:

– Do original research, have high expectations, expect some frustration, and always ask students to write their results as they get them.

– Use computers to generate data. Some projects don’t need to involve proofs. They can analyze interesting data.

– Look at Math Magazine, Involve, and other journals that publish work done with undergraduates.

Katie Johnson:

– Any time you think of an idea that might work for an undergraduate research project, write it in a notebook. Bring the notebook with you to conferences and events.

– Keep a folder with interesting papers that undergraduates could read.

– Stay current, attend conferences, seminars, and colloquia (and bring your notebook).

– Once students are working with you, ask them to think of 3 to 5 questions that might lead to other research projects.

Stephan Garcia:

– When needed, stretch your research area a bit. Try new topics. Search for fertile ground. Trust yourself.

– Encourage your students to find variants of existing questions.

– Ask students to read a paper and come back with 20 questions about it. Many won’t be “good” questions but some might lead to interesting projects.

– Be flexible, if something is not working, be willing to pivot and adapt. Change assumptions; be creative.

Colin Adams:

– Have the antenna up. While you are at math events, always think about if there is a project that can be done by students.

– Find something students can compute (even if they don’t understand the math). Fill in the details later.

– Learn about related areas to your research. One way to do this is to teach courses related to these areas.

– At the end of courses, collect papers and give them to students to present on them. Often interesting projects arise in this way.

– Think about whether you can generalize results or change assumptions a bit.

“Have FUN!” – Joe Gallian

Didn’t do well in high school. Didn’t think he would go to college. After getting his PhD, he applied to 145 jobs and got one, just one… University of Minnesota Duluth. Then, he became the father of undergraduate research. I am talking about Joe Gallian.

During JMM 2020, Joe talked about his REU program. Then, Shah Roshan Zamir (a former student of Joe), premiered a documentary about Joe and the program, which has ran continuously since 1987. It was fun, informative, and it conveyed how much fun students have in the program. It will be shown again at 2020 MAA MathFest and will eventually be available in YouTube.

In the documentary, after singing a Beatles son, Joe says “When you go to a new place, make yourself valuable.” For him, it was believing in and conducting undergraduate research. In his program, arguably the most successful math REU, he has some guidelines. The first one: participants should have FUN. Joe organizes weekly field trips, students play Math Charades and other games, and throughout the year, the program has developed all sorts of funny traditions: Duluth mile, erase the whiteboard, halftime show, crossing the river, and many others. You have to ask him about them…

Crossing the river. Ask Joe…

His alumni, which include Manjul Bhargava and Melanie Wood, paint a picture of a humble, fun, caring, and supportive advisor. In the Documentary, he takes things lightly, has fun singing Beatles songs, and shares great pieces of advice. “Find something to do, something to love, and something to look forward to.” And “Have FUN!”

Finding solutions to x^3+y^3+z^3=n?

“Tony gave the best lecture I ever saw at Rice.” When Brendan Hassett introduces a speaker with such words, you know you are in for a treat. Tony Várilly-Alvarado gave an inviting AMS Invited Address Wednesday morning.

“Can you find three integers (you can use negative numbers) $x$,$y$, and $z$ such that $x^3+y^3+z^3=35$?” he playfully asked. What about $x^3+y^3+z^3=34$?

While $(3,2,0)$ and $(3,2,-1)$ provide solutions for these questions, if you consider $x^3+y^3+z^3=33$ the first solution appeared in 2019 and involves integers with 16 digits. And if you consider $x^3+y^3+z^3=32$, there are no solutions! What is going on? To explain this, Tony took us through the wonderful world of projective space and elliptic curves.

Let’s first look at an easier question by “only” considering tuples $(x,y,z)$ such that $x^2+y^2=z^2$, i.e., Pythagorean triples. Since once you have a solution, any multiple of it is also a solution, then we only need to consider $x,y,$ and $z$ to be co-primes. In other words, we can look at it with a projective lense.

To find solutions, rescale $z$ to be 1 and consider rational solutions to $(x/z)^2+(y/z)^2-1=0$.  Setting $X=x/z$ and $Y=y/z$, we reduce the problem to finding rational solutions to $X^2+Y^2=1$ in projective space aka where you are allowed to rescale $X$ and $Y$. Using a bit of projective geometry, it turns out that the map

$$(X:Y)\to (2XY:X^2-Y^2:X^2+Y^2)$$

describe all such solutions, hence all Pythagorean triples are of the form $(2XY,X^2-Y^2,X^2+Y^2)$ or multiples of it.

By converting algebraic equations to geometric objects in projective geometry, Tony explained that the equation $x^3+y^3=z^3$ has actually no non-zero solutions (this is the $n=3$ version of Fermat’s Last Theorem) and that, going back to the initial question, $x^3+y^3+z^3=n$ seems to only have solutions when $n\neq \pm4 \mod 9$. To this date, this remains an open problem.

To be fair to Tony, he also talked about several other interesting objects: perfect cuboids, magic squares and how using similar methods, one can provide evidence for the veracity of conjectures involving these objects.

I left the room impressed by the talk. It was engaging, interesting, and quite understandable despite it touching technical mathematical tools. Well done Tony!