On Thursday, January 7, I attended a significant portion of this panel with Viveka Brown (Spelman College), Tasha Inniss (Spelman College) and Pamela Estephania Harris (Williams College), which was moderated by Katherine Stevenson (California State University Northridge) and included a student panel facilitated by Harris. I had to leave the meeting early to get to another one, but I found everything that I heard in the part I was able to attend to be informative. If you didn’t catch the live event, I recommend watching the recording when it’s available. I’ll share some highlights here.

Inniss spoke about a multi-session workshop series sponsored by the National Academy of Sciences that focused on viewing the pandemic response within higher education through a lens focused on strength and resilience, as well as thinking about lessons that could be used to shape the post-pandemic world. She mentioned that students have expressed concerns about whether they’ll be able to complete their degrees on time in an online setting. She also discussed some of the ways that there are currently opportunities to do things differently in higher education, such as evaluating and dismantling systemic racism that affects students.

In these discussions “equity and access came up over and over again,” she said. There is a need for equitable access to things like wifi and computers/tablets, but also to tutoring and peer networks. Other themes present in these discussions included the importance of building positive community and classroom culture and some of the positive aspects of remote learning. In some cases, this is including a shift towards thinking about students and researchers more holistically and a shift towards a “kinder research culture.”

Brown spoke about building community in a virtual classroom. She provided definitions of community and community building. She discussed recommendations for community building in virtual instruction. These recommendations came out of multiple studies. Some of these recommendations included reaching out to students “early and often,” working with students to discuss and address problems, limiting lecture time and increasing discussion time and more.

Brown also spoke about the notion of a collective growth mindset and shared a little bit about some of the strategies that have worked in her virtual classroom for building community and creating a positive atmosphere. She said that when she started letting students into Zoom rooms early, in one particular class, a group of them started joining the room early to work together on problems, chat, eat, share screens, etc. She also finds breakout rooms and ice breaker activities to be helpful. Some ice breaker activities she likes include “Rose/Thorn” (students can share something positive (a rose) or something negative (a thorn)), “In Common” and The App (where students briefly describe the most interesting app on their phone or tablet). She also mentioned that VoiceThread has been a useful tool in her classroom.

The panel moderated by Harris featured Lucy Martinez (senior undergraduate student at Stockton University, Becky Tang, (PhD student at Duke University) and Giovanny Marquez (PhD student at UC Santa Cruz).

Martinez mentioned that when she moved back to her parents’ apartment during the pandemic, her siblings were trying to learn online at some of the same times she was and that was challenging. She said that learning while her siblings and parents were in the same apartment was manageable, but that some of her classmates were struggling to find a place to attend online classes that wouldn’t interfere with their focus.

One of Tang’s primary concerns as a third-year PhD student has been how first-year graduate students have been coping and the troubles they were facing at her school. She mentioned that adjusting to graduate school is often already difficult and trying to adjust to remote learning can augment those challenges. She said that while professors are sometimes aware of the struggles their students are facing, they “may not internalize” them to the extent that they should.

Tang also mentioned that when Duke’s fall break was cancelled, that left many students without much needed time to take a break and practice some self-care. She said that she wished more professors had adjusted the timeline of their assignments so students would have a bit of a break that week. She said that one professor she knew of did so and students really took notice.

At this point, Harris interjected to say “Faculty take note: even if spring break is cancelled, you can still adjust your timeline” for your class assignments.

Marquez noted that many students who have moved back home — including him — have faced some amount of increase in family responsibilities such as having to help with transportation or being expected to babysit siblings while they learn virtually, which may lead to the older students having to miss class. He said that while a lot of professors are sympathetic, these responsibilities can cause scheduling issues and that those who are “sticklers for being in class on time” should realize that isn’t always possible. He also mentioned the need to provide students with necessary equipment and services, such as tablets and access to counseling services.

Stevenson, who moderated the panel, commented (I think it was in the chat) “Too often we do not realize that grad students are also teachers and should be at the front of the line for equipment.”

Tang also said that some faculty are expecting student researchers to “maintain pre-pandemic productivity” for research, homework, etc. and that it would be helpful if mentors initiated more conversations about research expectations and progress.

Marquez said that his advisor has been having weekly meetings with him and while that’s been helpful, there are still instances where questions crop up outside of that time and it’s challenging because grad students can’t just pop in their advisors’ offices to discuss things they might be stuck on. He said “be a little more flexible with your time, if possible” and that even a 10-minute conversation can be quite helpful sometimes. Of course, the folks on the panel also acknowledged that professors are going through their own challenging situations as well during this difficult time.

I hope this post has given you a taste of what the panel discussed and that if you weren’t able to watch it live, you’ll consider watching the recording!

I dropped into today’s special session on mathematical biology to hear Yanyu Xiao of the University of Cincinnati speak. She talked about how her group modelled contact patterns in Ontario and used them to study the spread of COVID-19 this past year.

The standard model for predicting infectious disease spread is an agent-based model known as an SIR model. This acronym stands for Susceptible, Infected, and Recovered, referring to the three types of agents. Infected agents will remain so for a period of time until they recover (or die) and become immune, and in the meantime will spread their disease to susceptible agents. From this, you can derive a system of differential equations that describe how the number of susceptible, infected, and recovered people in a population evolves over time. To get the details right, you need data about how infectious the disease is and how often agents come in contact with one another.

Getting that data for a real-world situation like COVID-19 is easier said than done. People interact in tons of different ways, and establishing how likely one person is to infect another requires analyzing all of that complexity. How likely are those people to meet in the grocery store—and how likely is an infection to occur there? What if they meet at a backyard barbecue?

To answer these questions, Xiao explained, researchers distill the different settings in which people meet one another into four main groups: households, workplaces, schools, and community. They also split the population into age groups. For each setting, they can create a “contact matrix” of age groups. So, for example, in the workplace matrix, entry (i, j) represents the number of workplace contacts in age group j that a person in age group i has. I would imagine that this method comes in handy especially when studying a disease like COVID-19, whose behaviour seems to very drastically with age.

Reference contact matrices used in Xiao’s study.

Xiao and her colleagues used survey data from 2006 to create benchmark contact matrices, and used demographic adjustments to estimate the correct matrices for Ontario 2020. As Ontario went through its various shutdown and reopening phases, they modelled the overall contact matrix accordingly. For example: before any businesses or schools had been shut down or physical distancing measures recommended, the overall contact matrix was merely the sum of the matrices from each setting, C = C(Household) + C(Workplace) + C(Community) + C(School). But once schools shut down, C took a different form: C = (1 + p) C(Household) + C(Workplace) + (1 + q) C(Community). This formula reflects the fact that there were no longer any school contacts. But with children and teachers spending more time at home or out in their neighbourhoods, household and community contacts would increase.

Based on this data, Xiao could implement the SIR model to estimate the cumulative number of infections in Ontario, and find the parameters that best fit the data. The model could then be used to evaluate the reopening plan. And with the advent of vaccines, it may be used to analyze distribution strategies.

To conclude my first JMM, I stopped by the AMS Special Session on History of Mathematics to hear a talk on “A New Resource for the History of Mathematics: The Educational Times Online Database of Mathematical Questions and Answers.” 

A sampling of math questions posed in The Educational Times.

Published continuously from 1847 to 1923, The Educational Times journal served as a discussion forum for teachers throughout the United Kingdom. Readers played an active role in the math section. Each issue contained reader-submitted math questions and reader-submitted solutions to previous questions. From time to time, an issue would include a list of questions that had been submitted but not yet solved. In total, around 18,000 math problems graced the pages of The Educational Times. 

In their talk, Sloan Despeaux of Western Carolina University and Robert Manzo of the University of North Carolina-Chapel Hill described how they and collaborators created a searchable online database of the questions posed in The Educational Times and the related Mathematical Questions. Questions are indexed by the branch of math, and the database contains information on the gender, nationality, occupation, and educational background of the contributors. 

The first effort to catalog the questions in The Educational Times dates to the 1950s, when a Brown University professor and his students indexed many problems and solutions on paper notecards. In the 1970s, a Providence College professor unearthed those notecards and revived the project in digital form. Despeaux had the idea to create the full online catalog, an endeavor that her team began in 2016 and finished last year. 

Despeaux’s team designed the database to serve as a resource for historians of mathematics to gain insight into the development of math as a discipline. The questions and answers reveal which mathematicians were interested in which topics and hint at the networks between them. Well-known British mathematicians like James Joseph Sylvester, Arthur Cayley, and William Kingdon Clifford contributed prolifically to the questions and answers, but so did a multitude of math enthusiasts. 

More than 1,300 distinct authors contributed questions to The Educational Times. With this new database, historians of math surely have a host of untold stories to explore.

Blue whales, the largest creatures ever to roam the Earth, are highly migratory animals. Each summer, they travel northward along the California coast to forage krill. Then in the fall, they return to the southern breeding grounds where they pass the winter. Sadly, blue whales are endangered, facing the threats of fishing gear, ship strikes, and climate change. To mitigate these threats, researchers must understand the factors that influence the whales’ migration patterns.

Today at the AMS Special Session on Agent-Based Dynamics and Self-Organization in Biology, Stephanie Dodson of the University of California, Davis, gave a talk about her ongoing work to use agent-based models to study blue whale migrations. Agent-based models computationally simulate the actions and interactions of individuals in an attempt to uncover the large-scale dynamics of a community. 

Dodson considered a state-switching model where each whale chooses either a “transit” state (moving long distances with few turns) or a “forage” state (moving short distances with many turns) according to the current krill density and sea surface temperature at the whale’s location. The oceanic data used in the simulations came from the Regional Ocean Modeling System (ROMS) and the North Pacific Ecosystem Model for Understanding Regional Oceanography (NEMURO), covering the 2000-2010 migration seasons with a 3-kilometer spatial resolution and a daily temporal resolution.

Blue whales are more likely to forage in waters with low sea surface temperatures (SST) and high krill densities. The distributions of step lengths and turning angles used in Dodson’s model came from data from tagged whales.

Dodson initialized the simulations with the whales entering the region of interest from the south between May 1 and June 1. She showed the audience an animation of one simulated season, which clearly reproduced the broad-scale northward migration pattern. She explained that her model successfully captures differences from year to year but fails to show any southward migration in the fall. That’s because the southward migration is likely driven by additional factors beyond just sea surface temperature and krill density. (Her current project is investigating the role of social calls in the southward migration.)

Satellite data of the ocean often has gaps in time or space, forcing researchers to use lower-resolution data than they would prefer. To improve her model’s sophistication, Dodson compared its performance on the “gold standard” 3 km, 1 day data to its performance on 9 km, 1 day data and 3 km, 8 day data. She found that coarse spatial data caused the simulated whales to form clumps, which she fixed by lengthening the simulation’s time steps to make the average step distance comparable to the spatial resolution. On the other hand, coarse temporal data caused the simulated whales to stay in one state (transit or forage) for too long. The best way to address this issue, she explained, was to add in whatever higher-resolution temporal data was available, even if it had spatial gaps. 

If you want to read more, Dodson’s work, which she started as a graduate student with collaborators at NOAA, appeared last year in Ecological Modeling. It’s just one example of how math can illuminate animal migrations and inform conservation efforts.

Yesterday afternoon, I attended a discussion on the future of the National Science Foundation and how the mathematical sciences fit in. We heard from Karen Marrongelle, head of NSF’s Directorate for Education & Human Resources (EHR), and Tie Luo, acting deputy head of NSF’s Directorate for Mathematical & Physical Sciences (MPS). Karen Saxe, AMS Director of Government Relations, moderated the conversation.

“Envisioning the Future of NSF: A Guided Discussion with MPS and EHR Heads” touched on artificial intelligence, diversity and inclusion, COVID-19, and more.

To start, Luo and Marrongelle discussed the big picture of the NSF’s current work. Both of them spoke highly of the new NSF director, Sethuraman Panchanathan, who took the helm in June 2020 after unanimous Senate confirmation. “He’s strongly committed to inclusivity and innovative research,” Marrongelle said. “He’s an amazing thinker, a visionary.” Luo emphasized Panchanathan’s “energy and belief that there’s a talent in everyone.”

One exciting prospect for the NSF comes from this year’s National Defense Authorization Act, which became law after Congress overrode President Trump’s veto. The act authorizes (but does not appropriate) $4.8 billion over five years for NSF programs to support basic and applied research in artificial intelligence. The NSF hopes to fund AI institutes across the country (an early example is the AI Institute for Student-AI Teaming at the University of Colorado Boulder). Luo said that MPS is engaged in “AI for science”—developing AI tools that can solve scientific problems—as well as “science for AI”—digging into the math behind deep learning and related technologies.

From there, the conversation moved on to education, especially the NSF’s role in diversity, equity, and inclusion efforts in the mathematical community. Among the programs that Marrongelle and Luo mentioned was the Graduate Research Fellowship Program, which actively seeks to award funding to individuals that reflect the country’s diversity in gender, ethnicity, and type of educational institution. They both acknowledged that DEI efforts still have a long way to go in the mathematical sciences. Of the 935 US citizens who earned math or statistics PhDs in 2017-18, six were American Indian or Alaska Native, 81 were Asian, 27 were Black or African American, 34 were Hispanic or Latino, and two were Native Hawaiian or Other Pacific Islander.

Saxe also asked about the impact of the coronavirus pandemic on the NSF’s work. The CARES Act gave the NSF $75 million to distribute through its RAPID grant mechanism. These grants have funded research on COVID-19 treatment and vaccines, the pandemic’s effects on STEM faculty and students, risk communication on social media, and more. Still, making advances in other areas is an uphill struggle as the pandemic has thrown a wrench in the career trajectories of mathematicians at all levels. 

The discussion concluded with a look forward to the new Congress. Marrongelle emphasized that the NSF has enjoyed bipartisan support through many presidential administrations. Frequent meetings between congressional staff and NSF representatives, she said, help Congress understand the priorities of the NSF. Ultimately, the heart of the NSF is basic research and innovation, which seems poised to flourish in the coming years.

I always knew that math had come a long way in the last 200 years, but I never thought that natural history would ever come into it. Until I went to Kevin Lambert’s talk this afternoon on Thomas Robert Malthus’ population principle.

Lambert, of Cal State Fullerton, discussed Malthus’ 1798 essay which argued that populations would always grow faster than their food supply. His argument was based on a comparison between growth rates: the food supply would increase arithmetically, while populations grow geometrically.

This was based on little evidence, and dismissed outright by some. Malthus provided questionable support for the idea that populations would always grow geometrically, and no support at all for his claim that the food supply would grow arithmetically.

In preparing a second edition of his book, Malthus joined two colleagues on an expedition to Norway, Sweden and Russia. There, he collected historical and environmental data–recording notes on the cultures he saw and scientific data like temperatures. He investigated the way the natural environment informed populations and incorporated his observations and reasoning into his arguments.

Lambert argued that this method was typical for researchers of the era. Cambridge mathematician George Peacock also used historical research to formulate his principle of the permanence of equivalent forms. This led to a so-called analytical revolution at Cambridge, establishing the study of algebra and creating space for ideas like imaginary and negative numbers.

Malthus’ work may have helped pave the way for someone like Peacock. It’s stories like these that remind us how far math has come in just a short time. Perhaps in 200 years, mathematicians will be similarly baffled by 21st century research techniques.

During today’s special session on probabilistic methods in partial differential equations, I had the pleasure of hearing Zaher Hani of the University of Michigan speak on his recent work on the wave kinetic equation.

In the early twentieth century physicists began formulating the first principles of quantum mechanics. One of their key realizations is that waves are fundamental physical units in the same way particles are. This prompted scientists to start constructing a wave system analogy to Boltzmann’s theory of statistical mechanics for particles.

For those not familiar, Boltzmann statistics describes the average statistical distribution of non-interacting particles in a large system. Boltzmann’s equation describes how this distribution fluctuates with time. The question facing physicists was whether a similar framework could be derived for waves that satisfy the nonlinear Schrodinger equation. It is very difficult to understand the behaviour of these waves, says Hani, because there are so many possible solutions. But it turns out that there is a wave-analog of Boltzmann’s equation, the wave kinetic equation.

Until recently, though, it was not known how to derive this equation in a mathematically rigorous way. Hani and collaborators showed in 2019 that the wave kinetic equation held for short periods of time that depended on the details of the system, but it was suspected that a more universal bound on the time period should hold. Hani, together with Yu Deng of the University of Southern California, has recently improved the bound for certain types of waves and time domains.

It was definitely interesting to spend half an hour today learning about the comparison between wave statistics and Boltzmann statistics. The work involved not only differential equation techniques, but also Feynman diagrams and number theory—a fascinating illustration of how seemingly distinct areas of mathematics can intersect.