On Navigating Collaborations

No matter what stage of your career you are in, collaborations can be crucial to build relationships in the math community and advance your research.

In this talk, Dr. Chelsea Walton (Rice University), “pulled back the curtains and showed us that the process of navigating collaborators is not magic, by sharing personal stories and proving some concrete tips and guidelines for working on collaborative research projects.

This talk is an amazing resource, in particular, for graduate students and early-career mathematicians.  I think this presentation is required listening/reading for the beginning of any new collaboration.

Figure 1. “Hello, Would you like to do math together?”

Hilariously infused with pictures of animals that honestly are so relatable. Walton gives us a flawless and straight to the point guide. Through her talk, she discussed the different stages of mathematical collaboration and associated keywords.

At the introduction stage of collaboration, the keyword was ‘access’, and it asks who is doing mathematical research? In the next stage, the starting stage, the keyword is dreaming. Ideally, you should pick a topic according to the diagram in Figure 2.

Figure 2: Considerations when picking a research topic.

Some people pick their team first then their project or vice versa (and everything in between). There is no wrong way of doing it, after all, it is your dream.

Here, also it’s the ideal time to settle logistics with collaborators. For example, what will be the main mode of the meetings? How often will you meet? What will be the main way of storing or sharing ideas? As a person involved in many group projects this is a crucial step in assuring that everyone is on the same page and that progress can be made in clear concrete ways.

The next few stages get to the math of the matter (keyword: priorities), the writing stage (keyword: momentum), and the concluding stage (keyword: polishing). There can be many roles (see Figure 3) in each of these stages, but the important thing is to share the load. A great tip for junior researchers from Dr. Walton is to “set the pace and finish the project, even if this means readjusting goals”.

Figure 3. Some of the many roles involved in a collaboration.

Figure 3. Some of the many roles involved in a collaboration.

But one of the most important (and easy to neglect) aspects of collaboration is reflection. Dr. Walton suggests asking yourself, are you happy with the product? Were your contributions valued? Because at the end of the day, any collaboration is building a relationship, and “when you don’t feel valued you can’t do good math”.

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On the “AMS Committee on Education Panel Current Issues: What can we do to support math majors and grad students in the time of pandemic?”

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!

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On “The Black Mathematician Chronicles: Our Quest to Update the MAD Pages”

On Friday, January 8, I attended this excellent talk by Edray Goins, a professor of mathematics at Pomona College in Claremont, CA. While I haven’t personally spoken with Goins before, I have read multiple media articles he has been quoted in, including the New York Times piece “For a Black Mathematician, What It’s Like to Be the ‘Only One’” and a profile of him published by Science News for Students. I had also visited the “Mathematicians of the African Diaspora” page but didn’t quite realize how much work had gone into creating the original site and recent efforts to update it.

The “MAD pages,” as they are known, were originally created by Scott Williams of SUNY Buffalo, but Goins said in the talk that after Williams retired, the pages languished. Since 2015, Goins has been working with collaborators to update the pages.

In the talk, Goins discussed the histories of some of the first African Americans to be awarded their PhDs in mathematics. Goins posted the slides for his talk on the JMM conference platform (to access those, just go to the page for his talk and look under “downloads”). On those slides, he describes a little bit more about those mathematicians. If you haven’t heard the talk, I recommend listening to it when the recording becomes available.

Goins also discussed how the National Association of Mathematicians was formed and shared more about the “Pomona Research in Mathematics Experience,” a 2020 NSF REU that had 8 participants. Those participants helped update the MAD pages database, wrote biographies of Black mathematicians and participated in virtual seminars (all of which except 2 are available on YouTube). Goins also spoke about efforts to create an oral history, which at least one audience member stated, would be a huge undertaking but also a tremendous asset to the mathematical and historical communities.

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Studying Contact Patterns for COVID-19

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.

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Anti-racism in mathematics: Who, what, when, where, why, and how?

Dr. Erica Graham is an assistant professor of mathematics at Bryn Mawr College. Her research is in the field of mathematical biology, with applications to endocrinology and physiology. As one of the co-creators of Mathematically Gifted and Black, Graham is also committed to efforts that address underrepresentation in the mathematical sciences.

In her talk, Anti-racism in mathematics: Who, what, where, why, and how?, Dr. Graham ’Five Ws and How’ for anti-racism as my vision for the mathematical community. This is particularly important a time where the implications (and harm) that are caused by white supremacist culture are showcased widely in the media due to recent events in Capitol Hill.  As she describes in her abstract,

 “The Black Lives Matter movement, and many like it has garnered widespread support for dismantling the racist structures woven into the fabric of our society at large. The academic discipline of mathematics–alongside many institutions of higher education–has also reached a point of reckoning in its history of institutionalizing racism. We must acknowledge the necessity, not choice, of persistent and active anti-racist work in realizing transformative, long-lasting change.” – Dr. Graham, from her abstract

This session “embraced humanity in the mathematical sciences explicitly”, as Carrie Diaz-Eaton, organizer of the MAA-SIAM-AMS Hrabowski-Gates-Tapia-McBay Session: Lecture remarked.  One of the main messages throughout was the idea that the nation is reckoning with what a Black life is worth in America, and we are being asked to reflect critically about what does that means and what is our role.

She began by asking, what do you see when you think about racism? What do you see in mathematics? Some examples mentioned were defensiveness, thinking that there is only one right way, paternalism, either/or thinking objectivity, and the right to comfort to name a few.


She made the distinction between white supremacy culture vs. the individuals. This distinction was very illuminating to me because, as she remarked, we have all acted in ways that promote white supremacy culture. The question becomes, how do we combat it? Parting from a definition of what antiracism is (and is not), we are reminded that antiracism is a set of actions not solely an idea or a policy.

We were asked to embrace the discomfort, be aware of our resistance, and examine our defenses. This talk was not aimed to convince us that anti-racist work is necessary or give us a one-hour solution. The style of the talk focused on discussing each of the five W’s (who, what, where, when, why), by providing a comfortable and uncomfortable answer.

And let me tell you, the comfortable answers are ones that I’ve constantly heard in my career in math. Contrasting them with uncomfortable answers felt like freeing the truth. Many times when we think of the five W’s of anti-racist work we are afraid to dive into the uncomfortable truths. But, as Dr. Graham remarked, imagine if we thought of math the same way, where would the field be? We must ask ourselves what are we prioritizing, reputations? What other upholders of white supremacy say? Are we exceptional racists or exceptional anti-racists? Truth is we are a long way from being exceptional anti-racists but we must name this behavior (or lack of behavior) so we may change it. It was a call to also reflect and evaluate myself and my actions because to be anti-racist, we must also combat racism within ourselves along with everywhere else.

“We need a bifurcation to move from racist to anti-racist.” – Dr. Graham

She concludes with a fantastic summary of the five W’s and how for anti-racism she discussed as her vision for the mathematical community.

Who: We should all, individually and collectively.

What: exert out privilege towards challenging the status quo and,

When: with immediate and persistent efforts,

Where: work within our respective institutions, organizations, and networks,

Why: to revolutionize mathematics as an anti-racist field,

How: by dismantling -thoroughly and permanently- the racist structures, policies, and practices on which the mathematical community was built.

In this summary of the talk, I can’t do it justice so I encourage everyone to give the talk a listen once its recording is shared because it is so worth-while.  Dr. Graham, thank you for your work and for sharing it with us at JMM.

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A 19th-century math discussion board

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 page from an old issue of The Educational Times, showing a column of math questions.

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.

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The math of blue whales’ migration patterns

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.

Graphs explaining the transit-forage state-switching model used by Dodson.

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.

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The National Science Foundation and the mathematical sciences

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.

Screenshot of Karen Saxe, Tie Luo, and Karen Marrongelle discussing the future of the NSF.

“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.

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Edward Thomas Malthus’ 19th century approach to mathematics

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.

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The Wave Kinetic Equation

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.

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