Logic ForAll: A Tour

From the blog post ‘Vanity Trip’. Translation: Valerria de Paiva: Brazilian Logician and Computer Scientist

While touring the math blogosphere I was very excited to find Logic ForAll’, a blog dedicated to making math accessible by mathematician and computer scientist Dr. Valeria de Paiva. She also writes in another great blog Women in Logic, which is used to organize and keep links to studies and graphs that show the extent of the problem and the tools other people have found to fight it. Women in Logic is also a Facebook group for women in Logic, philosophical, mathematical or computational or any other kind of formal logic that you care about. They have almost 500 members now and as described in the blog “so far, we have been finding it useful to discuss issues that affect us in our daily lives. There is also a Women in Logic spreadsheet with names of female logicians, organized by continent. This is an attempt at showing that there are plenty of female logicians around.”

Back in 2015, Dr. de Paiva was featured in the MAA article where she provides a brief description of her background and research interests.

“I am a Brazilian, from Rio de Janeiro, but I got my doctorate in Mathematics in Cambridge, UK, for work on “Dialectica Categories” written under Martin Hyland’s supervision. Working in Cambridge was a life-changing experience: I am now proud to say that my “academic great-grandfather” is none less than the founder of theoretical computer science, Alan Turing. I have, since my Cambridge days, worked on logical approaches to computation. My research interests include categorical proof theory, type theory, programming languages, logics for knowledge representation, logics of context, linear logic, intuitionistic modal logics and linguistic applications of logic. My work spans several different fields and I like all of my ‘hats’: mathematician, logician, computer scientist, and more recently computational linguist.”

One of the aspects I like most about the blog is the fact that it captures the life of a researcher quite well. Her writing combines a mix of styles that remind me of a mix between a classroom, research seminar, and talking with colleagues. I was curious to know more about the inspiration behind the ‘Logic ForAll’ blog so, in this tour, I hope to give you a glimpse of the blog’s style, content, and insights from Dr. Valeria de Paiva herself!

1. VRQ: Can you tell our readers a bit about yourself and your blog?

Valeria de Paiva: “Sure. I am a mathematician, an AI scientist and a computational linguist. I did my PhD in Cambridge, UK on Category Theory and I was a professor of Theoretical Computer Science at the University of Birmingham, UK, until I moved to the Bay Area, some twenty years ago. Here I have worked for some nine years at Xerox PARC and then in a series of other enterprises, like Nuance Communications and Samsung Research America. Along this way I have accumulated several different lines of research, so now I do work on several things, with different people. My blog started when I was teaching “baby logic” at Stanford and Santa Clara Universities and wanted to give students things to read. But nowadays its main function is to help me keep balancing these different projects. A sketchy description of the projects in the blog simply gives names to the buckets of things I do: Categorical Structures for (Linear) Logic, Constructive Modal Logics and IMLA, Lexical Portuguese Resources and OpenWordNet-PT, Lean Logic and Entailment and Contradiction Detection (ECD). This is in vaguely chronological order, but I actually work most when I have collaborators to play mental ping-pong with: I have an idea, you don’t like a bit of it, we try again, and the game goes on until we decide that we have a nice story of making a dent on our common ignorance.”

2.VRQ: What is the inspiration behind your blog?

Valeria de Paiva: “I’ve got inspiration from many mathematicians that I see trying to make mathematics more accessible to everyone. The name of the blog is “Logic ForAll”, now this is what I want, all people using logic formalized or not in the daily activities. But the name is also a pun, because in Brazilian Portuguese we have a dance and a style of music called “forro’ “. I only realized very late that the music (which is great and very danceable) comes from a mispronunciation of the English expression “for all”. So I wanted my blog to be like the music, fun and enjoyable and for all. Also, if possible full of little puzzles and games that it didn’t matter if you didn’t get them. It’s not about competition, it’s about fun!”

3.VRQ: What is the most interesting thing you’ve learned through blogging?

Valeria de Paiva: “I think I learned a while back that I only understand things when I am able to explain it to others — wherein others, I include myself. Once it’s written, it looks like another person did it, so I can debate it and discuss it all over again. I think one of the first posts in my blog, (I’m afraid I don’t remember where I copied it from) shows what I mean well 

From the webcomic Abstruse Goose.

You don’t just read mathematics, you fight it. An attitude that we should carry over to all kinds of things we read, right?”

4. VRQ:  Do you have advice for other mathematicians interested in creating their own blog?

Valeria de Paiva: “I do not think that I am good enough at this job, to be giving advice. My blog is a mess, I cannot keep the number of posts reasonable. I cannot find things I need that I know are there. I cannot write latex in it, I end up in a latex pidgin, where some things are their latex symbols, some others whatever name I prefer to give them, etc. But I’d suggest that any amount of demystification that we can do of mathematics is a good thing. It’s not rocket science, actually not even rocket science is rocket science, you just have to put the effort to understand it. And, as Barbara Fantechi was saying in Twitter the other day “most mathematicians aren’t like the gentlemen in this picture (Erdos and Tao). We’re not geniuses, just honest workers, motivated by a love of beauty, and patterns, and discovery. Most of us cover a variety of social roles, and not all of our time is for maths. We all count.” But we count more when we’re not impenetrable, when we have pictures and drawings, when we make our ideas more accessible, even if they do get a tiny bit less precise. It’s worth the trade-off, I say. Also, if English is not your first language (like it’s the case for me) using some grammar and spelling software does wonders for you (and your prepositions!).”

Some of her recent blog posts include:

“SICK (the data set) in these trying times” 

Here Dr. de Paiva describes her work with colleagues involving the Sentences Involving Compositional Knowledge (SICK) data set, provides a list of references, and shares her future research directions. In particular, she shares her article “Textual Inference: getting logic from humans, and her belief that their systems should learn from datasets that agree with human experience and how the single implication cases in SICK, they expected to find many problems. She mentions a few directions of the work towards addressing these problems.

“Artifacts in NLP”

In this post, Dr. de Paiva shares the work that is being done to track the progress (i.e. frameworks, tasks, and datasets) in the area of Natural Language Processing (NLP). She explains that in the area of Natural Language Inference (NIL) which  “is the task of determining whether a “hypothesis” is true (entailment), false (contradiction), or undetermined (neutral) given a “premise””. As she remarks, many of the results reported on the NLI task seem to be the outcome of biases on the datasets constructed to detect inference, and these are called artifacts.  I really enjoyed reading this post and learning about the growing literature regarding NLP and modeling inference and its challenges.

Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)

Posted in Applied Math, Artificial Intelligence, Blogs, Category Theory, Math Communication, Mathematics and Computing, Theoretical Mathematics, women in math | Tagged , , , , , , | 2 Comments

Joyful Learning in the Early Years: A Tour

With schools shutting down for weeks or the rest of the semester in response to COVID-19, many guardians are concerned about how to support or even direct their children’s education from home. This seems particularly true when the children are young enough that online classes might not be feasible (or the school district has opted not to offer them).

Deanna Pecaski McLennan’s Joyful Learning in the Early Years blog offers educational resources for folks with young children.  McLennan “is an educator in Southern Ontario who is greatly influenced by the Reggio Emilia approach to Early Childhood Education. She believes in a play-based, exploratory, democratic learning environment,” according to her profile on the blog. (If you’re like me and don’t know much about the Reggio Emilia approach, the Compass School in Illinois has a blog post explaining a bit about that child-centered approach.) Here are a few interesting posts that could help parents and other caregivers keep their kids engaged with learning math during this stressful time.

“Spring Math”

“In times of uncertainty, helping one another is one of the best ways to get through the stress and worry of what awaits. I know that many educators and families right now are wondering how to help support children even when we can’t be together physically,” Pecaski McLennan wrote. She created the post “to continually provide ideas for how children can explore math in their natural world.”

“The CDC is asking us to engage in social distancing and being aware of what is recommended is important. Right now being outdoors in our yards, on trails, and in gardens is still safe and encouraged. I realize that some of us are limited by our personal circumstances and not everyone has access to a yard or natural trail. I will try my best to vary activities in order to meet as many circumstances as possible. I will also tweet ideas for math learning on a regular basis @McLennan1977,” she added.

Many of the activities discussed in the post could also be adapted to be done indoors if safe outdoor options aren’t available. At the end of the post, Pecaski McLennan shares a link to a free Kindle version of her Spring Math Walk book.

“Virtual Math Question”

“In our school hallway we have a dry erase board that asks rich, low floor high ceiling questions. Students and staff that walk by are encouraged to consider the question for a few days and then contribute their ideas using dry erase markers,” Pecaski McLennan wrote. For instance, she shares the question “If 24 is the answer, then what might the question have been?” By asking questions such as this, caregivers could give kids of multiple ages and levels of math background something to think about. After pondering the question for a few hours or days, the family could come together to discuss their varied answers to the question.

“Printable Pentominoes”

Pecaski McLennan shares a set of printable pentominoes. (She recommends printing them on cardstock and laminating them, but for easy, temporary, at-home use, they could also just be printed on ordinary printer paper and used without lamination.)

She describes these manipulatives as “an essential tool for any early childhood classroom,” because they encourage a positive attitude toward math, inspire children to cooperate and collaborate and “promote math thinking in a variety of areas including spatial reasoning (logic when solving puzzles, symmetry, reflection, rotation, design), measurement (considering the area and perimeter of designs), and number sense (counting the number of tiles or squares in a design, calculating the total number of squares using the anchor of 5).”

However, these tools can also be used in activities with older children. For instance, there are pentomino activities for middle schoolers on the

“Cereal Stringing”

While this activity isn’t inherently math-related, there are definitely ways to make it so. For instance, a caregiver and kids could each make a secret pattern using cereal on a pipe cleaner. Then the children could guess what pattern the adult created and if they can’t guess it, the adult could give them clues until they guess the pattern correctly. Each kid could then explain the pattern they created on their own bracelet. Alternatively, parents could play a game with kids in which everyone makes a bracelet without counting the number of cereal pieces they use. After the bracelets are made, everyone could make their own guess about how many pieces are strung on each bracelet, explain how they reached their guesses and then count the actual number of pieces together.

I also like this activity because it only requires a few materials and those can be easily swapped out. (Don’t have pipe cleaners at home? Use string or strips of fabric instead. Don’t have cereal? Use beads or help your kiddo thread stale popcorn onto string.)

The Joyful Learning in the Early Years blog abounds with other ideas that could be adapted to meet the needs of guardians educating their own children during the pandemic, even if outdoor access isn’t available or if the children they’re educating are older than the kindergarteners that Pecaski McLennan teaches.

Looking for additional ideas? The Bedtime Math website and app offer free activities. The new “Cabin Fever Math” section focuses on non-screen math activities that families can do together.

Thank you for reading! If you have ideas or feedback to share, please reach out in the comments or on Twitter (@writesRCrowell).

Posted in Book/App, Current Events, Interactive, K-12 Mathematics, Math Communication, Math Education, Mathematics and the Arts | Tagged , , , , , , , , , | Leave a comment

Old and New Math Celebrations

With all the news about the coronavirus, the uncertainty, and stress many are currently facing, I wanted to write a post with some levity ¹. What better day than this! Today is both the first International Day of Mathematics (IDM) and Pi Day. These two celebrations cause great joy in math enthusiasts and give space to learn new (and old) exciting facts.

International Day of Mathematics Logo.

Proclaimed by UNESCO back in November,  the goal of the IDM is to  “explain and celebrate the essential role that mathematics and mathematics education play in breakthroughs in science and technology, improving the quality of life, empowering women and girls, and contributing to the achievement of the Sustainable Development Goals of the 2030 Agenda (SDG1-17) of the United Nations.”

This lofty goal is hoped to be achieved by worldwide events for all in schools, museums, libraries, and other spaces. This year’s International Day of Mathematics theme is “Mathematics is Everywhere” and there is a great website dedicated to exploring many examples of this theme available in seven different languages. Among my favorite examples on the page is “Search for Alien Life”, “Predicting Weather”, and “Epidemic Analysis”. Fun fact, in “Search of Alien Life”, they talk about the famous Arecibo Message sent from Earth to space back in 1974 from Puerto Rico. As explained in Arecibo Message” by the SETI Institute,

“The message consists of 1679 bits, arranged into 73 lines of 23 characters per line (these are both prime numbers, and may help the aliens decode the message). The “ones” and “zeroes” were transmitted by frequency shifting at the rate of 10 bits per second. The total broadcast was less than three minutes. A graphic showing the message is reproduced here. It consists, among other things, of the Arecibo telescope, our solar system, DNA, a stick figure of a human, and some of the biochemicals of earthly life. Although it’s unlikely that this short inquiry will ever prompt a reply, the experiment was useful in getting us to think a bit about the difficulties of communicating across space, time, and a presumably wide culture gap.” – SETI Institute

Through their Twitter account, IDM also shared this really neat collective video featuring submissions from all over the world in many languages showcasing all the places you can find math around you.

The fact that this was the first celebration of the International Day of Math made me curious about the history of Pi Day. I was surprised to find that it started in 1988 and it was founded by physicist Larry Shaw. The first celebration was at the Exploratorium interactive science museum and consisted of a circular parade and eating fruit pies. You can still celebrate Pi Day at the Exploratorium by joining online and checking out some of their fun $\pi$ inspired activities.

The symbol for pi wasn’t used until the 1700s. As described in PiDay.org, before the symbol was used it was described as  “the quantity which, when the diameter is multiplied by it, yields the circumference” and other long, roundabout descriptions. In the 1700s, the Swiss mathematician and physicist named Euler formalized the use of the Greek lowercase letter, π, as the notation for pi. This is the first letter of the Greek word, perimetros, which loosely translates to “circumference.”

In “Getting Ready for Pi Day, and also the Playful Math Blog Carnival”, Joseph Nebus shares a few some of his Pi day content in his archive including “Six or Arguably Four Things for Pi Day” on different ways to compute $\pi$ and a great list of comic strips from previous years. In the Crooked Pencil blog, Priya Narayanan writes about Ramanujan: He who had the Pi & ate it too!

“While Ramanujan’s formulae were progressively more and more accurate, what is more important to us today is his approach to the calculations, which provided the foundation for the fastest- known algorithm that, in 1987, allowed mathematician and programmer Bill Gosper to use the computer to churn out the value of π to around 17 million decimal places. Later, mathematicians David and Gregory Chudnovsky used his formulae as the basis of their own variants that allowed them to calculate the value of π to an astounding 4 billion decimal places using their homemade parallel computer.”

The number $\pi$ has a really interesting history. In his book, Tales of Impossibility, David Richeson discusses how “compass and straightedge problems—squaring the circle, trisecting an angle, doubling the cube, and inscribing regular polygons in a circle—have served as ever-present muses for mathematicians for more than two millennia.” A review of the book in The Math Less Travel blog, describes the chapters as follows,

“Alternating with the “regular” chapters, Richeson includes a number of “tangents”, each one a short, fascinating glimpse into some topic which is related to the previous chapter but isn’t strictly necessary for driving the story forward (e.g. toothpick constructions, Crockett Johnson, origami, the Indiana pi bill, computing digits of pi, the tau vs pi debate, etc.). Even though none of them are strictly necessary, taken as a whole these “tangent” chapters do a lot to round out the story and give a fuller sense of the many explorations inspired by the problems of antiquity.”

You also find many cool facts in this short article, “Here’s how pi matters every day not just March 14“, in particular, what is the current Guinness World Record for computing $\pi$.

“The Guinness World Record for a calculation of pi was set in 2019 by Emma Haruka Iwao using Google cloud software. She calculated pi to 31,415,926,535,897 digits.”

Pretty amazing! You can hear from Haruka herself on how she achieved this here. Another really interesting find was that IBM has released a new tutorial as part of its open-source online textbook to estimate $\pi$ on a quantum computer.

“The thing we’re trying to do here is to stay away from computing a million digits of Pi and more to use the theme of Pi Day to educate people on what quantum algorithms look like.” – Abraham “Abe” Asfaw, Global lead of quantum education at IBM.

But what makes Pi so interesting? As explained by Tom Crawford in “Make your Own Pi” it turns up in many important theories like Heisenberg’s Uncertainty Principle, Einstein Field Equations, and Newton’s Gravitational Constant to name a few.

“You may know it in terms of circles, but it has the rather fantastic knack of cropping up in the most unexpected places… Quantum Theory? Check. Einstein’s Theory of Relativity? Check. Newton’s Law of Gravity? Check. Three of the most important theories we use to explain the universe, and each of them has a formula containing the number Pi.”

Whether you celebrate International Mathematics Day and/or Pi Day, stay safe! Have ideas or feedback to share with us? You can reach us in the comments below or on Twitter (@MissVRiveraQ)!

¹ Note: Though I will recommend watching this great video from 3Blue1bBrown’s on exponential/logistic growth and epidemics and this recent article “Coronavirus, by the Numbers”.

Posted in Applied Math, Current Events, History of Mathematics, Math Communication, Mathematics and Computing, Recreational Mathematics | Leave a comment

A Tour of Intersections: Poetry with Mathematics

I don’t know about you, but between coverage of the coronavirus outbreak and political discussions looking ahead to this year’s presidential elections, I have been encountering a lot of stress-inducing content lately. Reading poetry is a welcome break from that, so here is a roundup of a few of my favorite posts from the last few months on the Intersections: Poetry with Mathematics blog.

“A MATH WOMAN acrostic poem”

In this short post, JoAnne Growney challenges readers to “describe a MATH WOMAN in 9 words? and, what if those words’ first letters must spell MATH WOMAN?”

After all, March is still Women’s History Month.

I’m still thinking of ideas for my poem.

“Those trains in word problems — who rides them?”

This post is about the poem “A Problem in a Math Book” by Yehuda Amichai. The poem was originally written in Hebrew before being translated into more than 40 other languages, Growney’s post notes.

If I had to pick a favorite line from the poem, it would be this part (about two trains in a math problem): “And no one ever asked what happens when they meet.”

“Learning slowly . . savoring difficulty . . .”

Growney shares one of her own poems called “Reflection,” which is about her mathematics learning process.

I relate deeply to this line. ” My notes were three times as long as what I had read.”

“Poetically exploring the the invention of ‘i'”

Featuring an except from “The Mathematical i” by Punya Mishra.

“Dogs Know . . . Mathematics”

This piece explores the poem “Dogs Know” by Larry Lesser, which first appeared in the Journal of Humanistic Mathematics and was also featured on NPR. I really enjoy the section of the poem that talks about the dog solving a packing problem, but I think my favorite phrase from the poem is:

My dog knows trigonometry, tracking
periodic rhythms
of moon
and heart.

Finally, Growney’s blog also has two posts (this one and this one) about the three winning poems in the 2020 AMS Math Poetry Contest: “Outlier,” by Sabrina Little, “The Number Won,” by Austen Mazenko and “x² + y² = 1(ife),” by Chenyu Lin, Colorado Christian University.

Have ideas or feedback to share with us? You can reach us in the comments below or on Twitter (@writesRCrowell)!

Posted in Current Events, Math Communication, Math Education, Mathematics and the Arts, people in math, Recreational Mathematics, women in math | Tagged , , , , | 1 Comment

The Joy of x Podcast: A Tour

The Joy of x podcast logo.

The Joy of x podcast is a series of conversations with a wide range of scientists about their lives, work, and what fostered their passion. It is hosted by Steven Strogatz in collaboration with QuantaMagazine. The format of this podcast makes it seem like you are joining an intimate session where you are privy to the interviewee’s lived experiences and how it has influenced their journey so far. Steven Strogatz, an applied mathematician, and author, really sparks the curiosity of listeners by displaying his own joy for discovery and scientific quests. In each episode, you get a clear sense of the joy behind the search for answers for the big questions these mathematicians tackle. As said by Strogatz in Why I’m Hosting The Joy of x Podcast,

“Through this podcast, I’ve been learning about the inner lives of some of the most intriguing mathematicians and scientists working today. A few are old friends and colleagues, while others are people I’ve still never met in person: Until their voices came through my headset, I knew them only through their research. But in every case, I wanted to know what makes them tick. I wanted to know why they do what they do, what they’ve discovered, and why it matters to them and to the world.” – Steven Strogatz

In the press release, QuantaMagazine Launches new podcast ‘The Joy of x’, we get the full line-up of this series which will have 12 episodes (one per week) that run from 40 to 75 minutes each and features a fantastic group of scientists and mathematicians including “mathematical physicist Robbert Dijkgraaf, mathematical biologist Corina Tarnita, mathematician Alex Kontorovich, neurobiologist Leslie Vosshall, mathematician and retired NFL player John Urschel, theoretical cosmologist Janna Levin, mathematician Tadashi Tokieda, neurobiologist Cori Bargmann, astrophysicist Brian Keating, mathematician Moon Duchin, mathematician Rebecca Goldin, and psychologist Brian Nosek.” 

So far, the podcast has released six episodes featuring Priya Natarajan (Yale University), Alex Kontovorich  (Rutgers University), Leslie Vosshall (Rockefeller University), Robbert Dijkgraaf (Director of the Institute for Advanced Study in Princeton), Corina Tarnita (Princeton University), and John Urschel (Massachusetts Institute of Technology). There is a great variety in the fields represented by the interviewees. If you enjoy exploring all sorts of areas of science and knowing the minds behind interesting questions, this podcast is for you. You can listen to the trailer below!

For me, the power of this podcast is listening to people share their stories. This adds a new dimension to them that is missed in how we talk about science and math. In this post, I review and give a glimpse of the latest three episodes.

John Urschel: From NFL Player to Mathematician

In this episode, we listen to mathematician and former NFL football player John Urschel. Currently, he is a Ph.D. Candidate at MIT where he studies topics in Convex Geometry, Graph Theory, Machine Learning, and Numerical Analysis. He shares the pressures of “living two lives” as an NFL football player and graduate student. For example, how qualifying exams can feel like the only chance to prove that you belong in the fields and how strategic quitting can be a valuable skill. From a very young age, he discovered the joy in solving challenging puzzles which turned into a passion for math. While his work has many applications, what he really enjoys is “digging out the math that makes the machine work”. This translates into going from a real-world problem, moving it to a more abstract yet beautiful representation and towards a generalization. You can read more about his journey in this interview about his upcoming book “Mind and Matter: A Life in Math and Football”. 

Corina Tarnita and the Deep Mathematics of Social Insects

In this episode, we listen to Corina Tarnita a mathematical biologist with a passion for patterns. She nurtured her mathematical ability by tackling problem-solving from a young age through the encouragement of her mother and participation in math olympiads. Her love for math sparked a love for biology. As she elegantly put it, “there is something remarkable about the diversity of solutions that nature has found for this unbelievable complex problems.” Using mathematical models, she discovered that termite colonies and plant competition for resources drive the patterns that can be seen like “pepperoni slices on a pizza” in the grasslands of Namibia. What I enjoyed the most about this interview, what of course the connection between the math, the biology, and also a bit of chance. I won’t spoil the story but sometimes, even when math points in the right direction, it takes being in the field to connect the dots. You can read more about Tarnita’s story and work in A Mathematician Who Decodes the Patterns Stamped Out by Life.

Robbert Dijkgraaf on Exploring Quantum Reality

In the episode, we listen to Robbert Dijkgraff a mathematical physicist who rediscovered his passion for physics through art. Through his career, even as a young scientist, he valued the power of collaboration. He retells how collaborating with his childhood friend allowed them to use both their strengths to make discoveries. He spent two years at an art school in the Netherlands, where it brought him a new perspective, “it’s about how adventurous are you, are you willing to go to other fields?… You could just explore, explore, explore.” This allowed him to bring this sense of adventure to what it means to be a research scientist. He works in the field of string theory, a field that hopes to bring together the theory of general relativity (i.e. the theory of the very large) and quantum theory (i.e. the theory of the small). Matrix models, which have symmetry at its core, can be used to build strings, gravity, and space-time. What I loved about this interview is the great commentary on how we experience time and space differently as humans. You can read more about Dijkgraff’s views on mathematical conjectures in his recent post, “The Subtle Art of the Mathematical Conjecture”.

Do you have suggestions of topics you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter! You can find me at @MissVRiveraQ.


Posted in Applied Math, Biomath, Current Events, Math Communication, Physics, Podcast | Leave a comment

ThatsMaths: A Tour

“The Great Wave off Kanagawa” by Katsushika Hokusai is discussed in a ThatsMaths post about rogue waves. Woodblock print by Katsushika Hokusai. Wikimedia Commons image credit: Meidosensei/CC

ThatsMaths is a blog by Peter Lynch, an emeritus professor of the University College Dublin’s School of Mathematics and Statistics. Many of the posts on the blog are articles that Lynch has written for the Irish Times. Please join me on a tour of some exciting ThatsMaths posts from the last several months.

“An English Lady with a Certain Taste”

In the early 1900’s, Dr. Muriel Bristol told two statisticians — Ronald Fisher and William Roach — that she could taste the order in which ingredients were added to tea.

Shortly after Fisher had moved to Rothamstead Research Station in 1919, he poured a cup of tea and offered it to Bristol. She declined, saying that she preferred the milk to be poured first. The arrogant young Fisher scoffed at this, insisting that it could not possibly make any difference, but Bristol maintained her stance, assuring him that she would always know the difference. Overhearing this exchange, another scientist, William Roach said, ‘Let’s test her,'” Lynch wrote.

He describes how statistics and combinatorics were used to determine if Bistol could actually taste the order in which ingredients were added to her cup of tea.

“Simple Curves that Perplex Mathematicians and Inspire Artists”

This post covers the Jordan Curve Theorem (including an extension of the theorem to higher dimensions), the traveling salesman problem and their intersections with art.

“The Jordan Curve Theorem states that every simple closed curve, no matter how complicated or convoluted, divides the plane into two regions, an inside and an outside. The theorem appears so trivial that it does not require a proof. But results like this can be much more profound than a first glance might suggest and, on occasions, things that appear obvious can turn out to be false,” Lynch wrote.

He then discusses Bernhard Bolzano’s work. (Lynch has also written a separate post focusing on Bolzano’s life and work.) “He claimed that, for a closed loop in a plane, a line connecting a point enclosed by the loop (inside) to a point distant from it (outside) must intersect the loop. This seems obvious enough, but Bolzano realized that it was a non-trivial problem,” Lynch wrote.

“For general curves it is quite difficult to prove since “simple” curves can have some bizarre properties, such as being jagged everywhere with no definite direction, or as being fractal in nature like the boundary of a snowflake. This makes it difficult to distinguish which points are inside and which are outside. The proof uses advanced ideas from the branch of mathematics known as topology,” he added.

“Hokusai’s Great Wave and Roguish Behaviour”

In this post, Lynch wrote about “The Great Wave off Kanagawa” woodcut by Katsushika Hokusai, rogue waves, non-linear modeling and the study of rogue waves in laboratory tanks.

“In recent decades, many enormous sea waves have been observed, removing all doubt about the existence of rogue waves. These waves have heights more than double the surrounding waves. In January 2014, the height of a wave off Killard Point in Co. Clare was measured at almost 30 meters. Although they are quite rare, rogue waves are part of the normal behaviour of the oceans,” Lynch wrote.

Lab studies on rogue waves utilize “mathematical theory, computer simulations, wave-tank experiments and observations” to “determine the critical factors for the formation of rogue waves. Mariners’ lives depend on their ability to avoid them, and new theoretical descriptions may enable us to anticipate their likely occurrence. Despite progress, many questions about rogue waves remain unanswered and research continues. The pay-offs include greater accuracy of wave predictions and saving of money and of lives,” Lynch noted.

“Chase and Escape: Pursuit Problems”

This piece describes the mathematical study of pursuit problems, beginning with the work of Pierre Bouguer, who, around the year 1730, produced “the first comprehensive treatment” of the subject, according to Lynch.

“From cheetahs chasing gazelles, through coastguards saving shipwrecked sailors, to missiles launched at enemy aircraft, strategies of pursuit and evasion play a role in many areas of life (and death). From pre-historic times we have been solving such pursuit problems. The survival of our early ancestors depended on their ability to acquire food. This involved chasing and killing animals, and success depended on an understanding of relative speeds and optimal pursuit paths,” he wrote.

The rest of the piece focuses on cyclic pursuit problems (more specifically, the N-bug problem).

Want to get in touch to share feedback or ideas for future posts? We welcome your comments below or on Twitter (@writesRCrowell).

Posted in Applied Math, Biomath, Blogs, History of Mathematics, Mathematics and the Arts, people in math, Statistics | Tagged , , , , , , | Leave a comment

In honor of Black History Month

February 1 marked the beginning of Black History Month. Its origin trace back to 1926, when the historian Carter G. Woodson pioneered “Negro History Week” in the second week of February because it coincided with the birthdays of former US president Abraham Lincoln (February 12) and abolitionist Frederick Douglass (February 14). Later, during the United States Bicentennial in 1976, President Gerald Ford recognized Black History Month, reminding Americans to “seize the opportunity to honor the too-often neglected accomplishments of black Americans in every area of endeavor throughout our history”.

In last year’s post, “On Mathematical Superpowers and Black History Month“, Rachel listed some of the great posts that have been published across many of the AMS blogs and highlights “some power problems that need to be addressed to make the mathematics community a more welcoming and opportunity-filled one for Black mathematicians and students.” Last year, SIAM News highlighted some of the African American heroes in mathematics in “Celebrating Black History Month” including Mary Jackson, Charles L. Reason, J. Ernest Wilkins Jr, Annie Easley, Katherine Johnson, Elbert Frank Cox, Dorothy Vaughan, and David Blackwell. Knowing their history, the power behind their pursuit of knowledge, and the trail the left for others to follow is a way to honor their place in our community. To preserve and share the stories of African American Elders, the National Visionary Leadership Project has recorded two video series featuring interviews with David Blackwell, the first African-American member of the National Academy of Sciences, and Evelyn Granville, one of the first African-American women to earn a Doctorate in mathematics.

This month’s Notices of the AMS features articles that showcase the research and contributions of Black mathematicians to the mathematical community. In “A word from…” by Robin Wilson, he summarizes the topics covered in this issue and emphasizes that the history of Black mathematicians is a part of the history of the American Mathematical Society, one not always centered around inclusion.

“Black history is American history, and the history of Black mathematicians in the United States is a part of the history of the American Mathematical Society. As with the history of the United States, the history of the AMS has not always been one of inclusion. With this special issue in honor of Black History Month, we shine light on some of that history, as well as uplift the efforts of mathematicians and institutions to redirect this tide of history and create equity in the field.” – Robin Wilson

A piece that caught my attention was Jesse Leo Kass, “James L. Solomon and the End of Segregation at the University of South Carolina”. In the article, he provides an overview of the impact segregation had on mathematics and how James L. Solomon, a former math graduate student, was one of the first three African American students to desegregate the university in 1963.

“The professional trajectories of African American mathematicians were profoundly shaped by legalized segregation and other exclusionary policies. Not only did such measures make it difficult for African Americans to obtain a college education, but those who persevered and wanted to work as professional mathematicians faced limited job opportunities. While HBCUs employed largely African American faculty, many other universities had formal or informal policies against hiring African Americans. Moreover, those who did secure academic positions still struggled to participate fully in academic culture. The career of William Claytor vividly illustrates these challenge.” – Jesse Leo Kass

In “Black and Excellent in Math”, Haydee Lindo writes for the MAA Math Values blog about the implicit and overt aggression that students and faculty of color face and how it is a key source of disparity in black mathematical achievement. She highlights the work of Ebony McGee, in search of navigating these challenges.

“How do we make ourselves bulletproof? We can’t. […] One of the key ideas seems to be this: when we are younger our attraction to Mathematics is often fueled by external encouragement from our teachers, high scores on tests, etc. As we grow more mature, black mathematicians and engineers remain successful by progressing, from being preoccupied with attempts to prove stereotypes wrong to adopting more self-defined reasons to achieve.”  – Haydee Lindo

Lindo also emphasizes the importance of cultivating affirming environments. For example, attending Historically Black Colleges and Universities (HBCU), taking courses with faculty of color, attending conferences with a focus on the success of minority STEM students, and moving towards culturally sustaining pedagogies. Giving back to the community through mentorship, service, and outreach plays a huge role in preparing future generations. In “Mathematics: The Key to Empowering Tomorrow’s Workforce”, Tanya Moore describes it elegantly.

“In the African-American tradition there is a phrase, Each One, Reach One, that reflects the value of bringing along others once you have acquired a certain level of knowledge or success. In the context of the mathematics community, this value is often reflected in the math-related activities and events that happen outside the classroom to prepare the next generation for their chosen educational and career paths. As technology promises to change the way we work by altering the landscape of the labor market, mathematics will take on a new level of importance. The role of service and outreach and the willingness for Each One to Reach One to increase mathematical engagement will matter even more.” – Tanya Moore

The workforce is also changing as data plays a bigger role in many career paths and in our lives. An amazing group that has data at its center is Data for Black Lives. This group of activists, organizers, and mathematicians are “committed to the mission of using data science to create concrete and measurable change in the lives of Black people”. During this month, “Mathematically Gifted & Black” highlights the contributions and lives of Black mathematicians. This website was founded in December 2016 by  Erica Graham, Raegan Higgins, Shelby Wilson, and Candice Price. Its name was inspired by the song “To Be Young, Gifted and Black” sung by Nina Simone and co-written by Weldon Irvine. I was so excited to read the profiles of the honorees of this year which so far include Asamoah Nkwanta, Felicity T. Enders, Kwame Okrah, Shea D. Burns, Kevin Corlette, Caprice Stanley, Abdul-Aziz Yakubu, Aissa Wade, Lorin Crawford, Omayra Ortega, Christopher C. Jett, and Loni Philip Tabb.  What I love about this website is summarized perfectly in “The Mathematically Gifted and Black Website“:

“The power of the personal story is helping people better understand one another and shred stereotypes. The mathematicians spotlighted were able to tell their stories in their own words, to discuss their proudest moments, in mathematics and in life, and to include personal stories of struggle along with inspirational anecdotes. All were allowed to be themselves, unapologetically.”

Do you have suggestions of topics you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter! You can find me at  @MissVRiveraQ.

Posted in Black History Month, Blogs, Current Events, History of Mathematics, Math Education, people in math, Publishing in Math, women in math | 2 Comments

Mathematical Enchantments: A Tour

Mathematical Enchantments, or “Jim Propp’s math blog” is about “adventures in fantastic realms you can build inside your head.” The blog has been discussed a few times on this blog in recent years. Welcome to my tour of a few interesting posts on the blog.

“Why this blog?”

I appreciate when the purpose and focus of a blog are well-defined, and this post really delivers on that front.

Propp wrote about math as “a consolation for living in a world without magic.” As someone who never believed in magic (even as a child) but wished for its existence, I relate to that sentiment.

“Lots of people (most notably Martin Gardner and more recently Arthur Benjamin, Persi Diaconis, Ron Graham, and Colm Mulcahy) have written and talked about the links between math and magic tricks, but hardly anyone talks about the way that math, for many people who do research in it, satisfies a craving for the fantastic that most of us haven’t outgrown (even if we’ve persuaded ourselves that we have). Indeed, I think that most children get glimpses, all too easily forgotten, of math as a wondrous ticket to other worlds,” Propp wrote.

“My goal in Mathematical Enchantments is to reawaken in my readers this childlike relationship to the subject, and to make this view of math enticing and even natural. And if you are an actual child, or an actual mathematician, and your sense of mathematical wonder is already awake and active, all the better! There’ll be lots of new games you can play. These things are fun, and fun is good,” he added.

“The Paintball Party Problem and the Habit of Symmetry”

Propp wrote about time he was “the showrunner” of his son’s ninth birthday party. He had to decide how to configure seven games of two “four on four” paintball teams so that each boy attending the party would be on his son’s team the same number of times, and ideally, would also be teammates with each of the other party attendees the same number of times.

He describes how the notions of randomness and quasirandomness, the geometry of cubes, finite fields and other mathematical ideas informed his solution. The team schedule “took me under five minutes, if you leave out the time I spent learning abstract algebra and coding theory thirty-plus years earlier,” Propp wrote.

“Knots and Narnias”

This post starts with the idea of portals similar to those in The Chronicles of Narnia books by C.S. Lewis and the His Dark Materials trilogy by Philip Pullman. Propp then shows how to use “mathematical scissors and glue” to construct different types of wonky, complex portals. His post includes a link to an older video of Bill Thurston (1946-2012) discussing similar ideas.

“Time and Tesseracts”

I love A Wrinkle in Time (the classic book by Madeleine L’Engle and the 2018 film), so this post especially appeals to me. Propp begins with a discussion of forth dimensional space, which he defines as “a space that at every points admits four mutually perpendicular lines, in no particular order, but not five.” He then delves into a discussion of tesseracts, hypercubes and music as a tool for thinking about higher dimensions.

Want to get in touch with feedback or to share ideas for future blog posts? Reach us in the comments or on Twitter (@writesRCrowell)!

Posted in Blogs, Interactive, Math Communication, Mathematics and the Arts, people in math, Recreational Mathematics | Tagged , , , , , , , , | 1 Comment

Traffic and Other Jams

Photo by Nabeel Syed on Unsplash.

Most people can relate to (or feel) the frustration caused by being stuck in traffic, waiting in a queue to board a plane, or circling the parking lot to find a space. Routes that could take 30 minutes can turn into hours, congested aisles of passengers cause bottlenecks, or while on your third-round around the parking lot you see someone behind you take the only space. In this post, I share some of the interesting math behind common jams you might find yourself in.

In her article, Can a city ever be traffic jam-free?, Katia Moskvitch highlights the environmental, health, and economic implications of traffic jams.

“Jams are not only frustrating, they are also a major contributor to air pollution, and that’s bad not just for our climate, but everybody’s health too. According to researchers at the Harvard Center for Risk Analysis, congestion in the 83 largest urban areas in the United States caused more than 2,200 premature deaths in 2010 and added $18bn to public health costs. Then there is the economic cost of lost hours (both work and leisure) and delayed shipments. Drivers in the 10 most-congested cities in the United States sit around 42 hours in traffic jams every year, wasting more than $121bn in time and fuel while doing so.”- Katia Moskvitch

With such high implications, you can see why traffic modeling has become a big part of applied mathematics research. From the same article, I loved this quote by Gabor Orosz (University of Michigan) which illustrates how traffic flows can be understood through analogies ( such  as fluid and gas flow, to the movement of birds and skiers) but still, “although such analogies may help scientists to gain some understanding, it is becoming more and more obvious that traffic flows like no other flow in the Newtonian universe”. I became more curious about the math behind traffic modeling after reading Mathematicians have solved traffic jams, and they’re begging cities to listen by Arianne Cohen. This article summarizes some of the key points of the work by Alexander Krylatov  and Victor Zakharov (St. Petersburg University) whose research tackles traffic modeling from an optimization perspective. Along with Tero Tuovinen, they are also authors of the book Optimization Models and Methods for Equilibrium Traffic Assignment which gives new approaches,  algorithms, methods, prospective implementations developed by the authors on the problem of traffic assignment.  Cohen highlights that four ideas that could reduce traffic jams are the following,

  1. All drivers need to be on the same navigation system. Cars can only be efficiently rerouted if instructions come from one center hub. One navigation system rerouting some drivers does not solve traffic jams.
  2. Parking bans. Many urban roads are too narrow and cannot be physically widened. Traffic-flow models can indicate where parking spots should be turned into lanes.
  3. Green lanes. For cities that want to increase electric car use, special lanes should be created for electric cars, providing an incentive for their use.
  4. Digital twins. Traffic demands and available infrastructure can only be balanced with digital modeling that creates an entire “twin” of existing roadways. The software will be “an extremely useful thought tool in the hands of transport engineers.”

After reading the article, I was curious to see if other perspectives on these matters were out there. In response to Cohen’s article, Daniel Herriges writes that human behavior is a strong factor in traffic congestion that is difficult (if not impossible) to account for with models.

“As long as we build a growing city around roads for cars, it’s a pretty sure bet that people in their cars are going to find ways to fill up those roads. We can’t build or network-engineer our way out of congestion, but we can bankrupt ourselves trying. There’s a better way to deal with traffic—and “deal with” does not mean “solve.” It is to make our places resilient to congestion, so that if and when it happens, it doesn’t destroy our quality of life. This means 15-minute neighborhoods: more destinations within walking distance of home. It means a range of ways to get around so nobody is forced into just one option, and a well-connected street network so there are many paths from A to B.”  – Daniel Herriges

The two perspectives are fascinating! This is not the first time that traffic models have appeared around the internet as the solution to traffic jams. Many researchers have tackled versions of these questions using different areas of math. For example, back in 2007  “Traffic jam mystery solved by mathematicians”.

In Traffic Modelling: Is Beating Traffic a Zero-Sum Game? Paul Sobocinski asks if self-driving cars that stick to one lane lead to less time on the road than humans switching lanes? He finds through simulations that opportunistic lane changing (i.e. weaving through lanes of traffic to shorten a commute) is not a zero-sum game. In fact,

“Opportunistic lane changing can benefit all drivers on the road if exercised judiciously. This means not changing lanes too frequently (i.e. adhering to a reasonable minimum time in lane), and only changing lanes if it saves a significant amount of time (i.e. the time saved in the new lane is 90% or higher). What do the results tell us about how to be a better driver? To state it simply: Be patient. Change lanes, but not frivolously. Everybody wins. Experienced drivers will likely not find this conclusion surprising.” – Paul Sobocinski

Following the same spirit, Jenna Marshall explains in Where to park your car, according to math the research of physicists Paul Krapivsky (Boston University) and Sidney Redner (Santa Fe Institute) which ordinary differential equations and simulations to find the best parking space (i.e. the one that lets you spend the least amount of time in the lot). As conveniently shown in the video below, they consider three strategies: meek (i.e. grabs the first space available), opportunistic (i.e. gambles on finding a space right next to the entrance), and prudent (i.e. drives past the first available space, betting finding another other space further in).

So, what is the answer? Being more prudent. However, the authors also acknowledge the limitations of their work.

“If you really want to be an engineer you have to take into account how fast people are driving, the actual designs of the parking lot and spaces — all these things,” he remarks. “Once you start being completely realistic, [every parking situation is different] and you lose the possibility of explaining anything.”- Sidney Redner

Finally, if you are a frequent flier you may have wondered about the best way to board an airplane. In Mathematician crunches the numbers to find most efficient way to board a plane, CBC radio interviews Eitan Bachmat whose work with Rami Pugatch (Ben-Gurion University), Sveinung Erland, Vidar Frette (Western Norway University), and Jevgenijs Kaupužs (University of Liepaja) tackles the airplane boarding policies using a Lorentzian-geometry-based analysis. As explained by Bachman,

“In our latest studies, we’ve been looking at random boarding versus if you have two groups of people — some which are slower and some which are faster. For example, people without luggage — they’re supposed to be the fast group. And people with luggage, the slow group. A lot of the times it happens they board first people who have children and need assistance. That would be a slow group.So, if you have a fast and slow group, what we found is that you should board the slow passengers first, which is kind of counterintuitive and surprising, I think.”- Eitan Bachmat

When asked about the mathematics behind this project his layman explanation what very insightful!

“OK. So I’ll try my best to keep it really simple. So the same math can describe very different things. I can say three plus three equals six, and three apples plus three apples equals six apples. Or, it could be three houses plus three houses equals six houses. Apples and houses have nothing in common. But, sort of, the math that describes the situation is the same. And what turned out, and that was very surprising, is that when you and 200, or 300, other people board the airplane, in terms of the mathematics, you’re doing a quite complicated computation in relativity theory about the aging of some free-falling particle and some model of the universe.” –  – Eitan Bachmat

Next time you are in a jam, you can rest easy knowing that a lot of cool mathematics is happening behind the scene. Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)

Posted in Applied Math, Game Theory, Mathematics and Computing, Physics, Traffic Modeling | Tagged , , , , , , | Leave a comment

Flygskam, Textbooks in Braille and More

This year’s JMM have come and gone, but many related posts are still available. Here are just a few that I recommend.

“To be or not to be there: Conferencing in the age of flygskam”

In this piece for the Graduate Student Blog, Aram Bingham wrote about why he didn’t attend this year’s JMM. He discusses the impacts of air travel to conferences on the environment and encourages mathematicians to consider alternative options, such as video conferences or “campaign[ing] for the construction of a carbon-neutral/negative conference center at the geographic/population center of the US (near the Nebraska-Kansas border, or somewhere in central Missouri, or somewhere else depending on how you measure), with connecting high-speed rail, to be used for all national scientific conferences.”

However, Bingham’s post isn’t advocating that mathematicians completely nix conference air travel, but, instead, consider carefully their conference travel decisions. “I’m not calling for any sort of a heroic abandonment of all air travel by the scientific community or advocating the use of sanctimonious hashtags (see #istayontheground),” he wrote. “I’m sure I will fly again for a conference, and probably even use a paper cup or two for coffee when I have forgotten my reusable mug. I just want to point out that the path of minimizing the consequences of our own actions is too tempting for a community that should be taking leadership, and that this path is made even easier by the fact that individualistic resource consumption and accumulation is still de rigeur in this country in general. Non-conformity might initially require a little bit of courage, but I think it’ll be a bit easier for the rest of society, and result in less political strife, if scientists act first,” he added.

World Premiere of ‘Secrets of the Surface'”

The film is about the life and achievements of Maryam Mirzakhani. “Following the screening was a Q&A moderated by Hélène Barcelo of MSRI, with panelists Ingrid Daubechies, Amie Wilkinson, Jayadev Athreya, Tatiana Toro, all mathematicians who knew Mirzakhani; also on the panel were Erica Klarreich, a math journalist who narrated the film, and George Csicsery, the director and producer,” wrote Leila Sloman, who attended the screening and discussion. She presents “an incomplete and slightly edited transcript of the panel.”

“Creating a math textbook accessible to the blind”

For this post on the JMM 2020 blog, Leila Sloman interviewed several mathematicians about their work on automating the process of converting math textbooks into Braille formats. Samantha Faria also interviewed the team behind the Math That Feels Good project.

“Bank of REU/Grad Fair Questions”

For the e-Mentoring Network in the Mathematical Sciences blog, undergraduate students Lucy Martinez and Eduardo Torres Davila compiled lists of recommended questions for students to ask while they evaluate whether a certain REU or graduate school might be the right fit for them.

“We attended the Joint Math Meetings (JMM) conference in Denver to present our research from our work at the Mathematical Sciences Research Institute Undergraduate Program,” Martinez and Torres Davila wrote. “At JMM, there was a fair of graduate schools and research experiences for undergraduate programs, which was attended by universities from all parts of the nation. At each booth, were university professors and current graduate students who could talk about their PhD program in the mathematical sciences…Although, we attended the fair to find out more about the programs offered, there were points in our conversations in which we were unsure what to ask. We quickly realized that if we had a list of questions prior to attending the conference, we could have been more prepared. We talked to Dr. Pamela E. Harris about our situation and she recommended that we go on a scavenger hunt for questions we could have asked!” Martinez and Torres Davila added.

As always, thanks for reading! If you want to reach me with ideas or suggestions, please leave a comment below or find me on Twitter (@writesRCrowell).

Posted in Blogs, Current Events, Events, Issues in Higher Education, K-12 Mathematics, Math Communication, Math Education, people in math, women in math | Tagged , , , , , , , | Leave a comment