**Please note: all sessions are listed in Mountain Daylight Time (MDT = UTC-6:00)**

### Probability and the Geometry of the Laplacian and Other Operators.

*Lecture 1: Wednesday, August 4, 10:00 a.m. - 10:50 a.m.*

*Lecture 2: Friday, August 6, 10:00 a.m. - 10:50 a.m.*

**Rodrigo Bañuelos**, *Purdue University*

##### Abstract

The classical isoperimetric property (inequality) states that among all figures of equal area, the circle has the smallest perimeter. Equivalently, among all figures of equal perimeter, the circle encloses the largest area. In the first of these two talks the speaker will explore this property and its elegant connections (and generalizations) to Brownian motion and eigenvalues of the Laplacian. The notion of "stability" in these inequalities will be addressed and open problems will be mentioned. Departing from this, the second talk explores the question in the title of M. Kac’s famous 1966 paper "Can one hear the shape of a drum?" in the context of the geometry of the fractional Laplacian. Equivalently, by observing the trajectories of certain stochastic processes known as stable processes.

These talks are both expository, designed for general audiences. While interconnected, they are largely independent of each other. These talks intent to illustrate G. Pólya's statement that "the isoperimetric theorem, deeply rooted in our experience and intuition so easy to conjecture, but not so easy to prove, is an inexhaustible source of inspiration” from his book "Mathematics and Plausible Thinking.”

##### Biography

Rodrigo Bañuelos received his PhD from the University of California, Los Angeles in 1984. He was a Bantrell Research Fellow at Caltech and an NSF Postdoctoral Fellow at the University of Illinois. He moved to Purdue in 1987, where he served as Head of the Mathematics Department from 2007 to 2011. His research interests are in probability and its applications to harmonic analysis, partial differential equations and spectral theory. He has published over 100 research articles on these topics and has lectured on his research worldwide. He has served on many editorial boards and scientific committees, including the United States National Committee for Mathematics, MSRI’s Scientific Advisory Council, and IPAM’s Board of Trustees. He is a Fellow of the American Mathematical Society, a Fellow of the Institute of Mathematical Statistics and a Fellow of the Association for Women in Mathematics. He is a recipient of the Blackwell-Tapia Prize in Mathematics. Throughout his career, Bañuelos has been involved in many efforts to increase the representation of groups that have been historically excluded from the mathematical sciences.

Bañuelos was born in a farming community in the state of Zacatecas, Mexico. As a child, he had no formal schooling until his family moved to the US, two months shy of his 16th birthday. You may read about his journey to mathematics in his biographical profile, which is part of the “SACNAS Biography Project,” https://www.sacnas.org/team/rodrigo-banuelos-phd/

### Eigenvalues and Graphs

*Wednesday, August 4, 11:00 a.m. - 11:50 a.m.*

**Steven Butler**, *Iowa State University*

##### Abstract

One way to store information about a graph is by an array with entries indexed by pairs of vertices with each entry giving information about a relationship between the pair. The linear algebraist in us would say, ``let's change our names and instead of calling it an array, let us call it a matrix, which is an array with benefits''. Among these benefits are the eigenvalues and singular values of the matrix. The eigenvalues give information about the linear transformation to which the matrix corresponds, and this can capture some structural properties of the graph (often with just knowing a few of the extremal eigenvalues). This provides a way to obtain information about a graph with just a handful of parameters. We will explore several different possible matrices and look at some of the information that we can, and in some cases cannot, learn by studying the eigenvalues.

##### Biography

Steve Butler is the Barbara J Janson Professor of Mathematics at Iowa State University. He earned his PhD degree in 2011 from UC San Diego where he studied spectral graph theory under Fan Chung. He has worked extensively with Ron Graham, and is (currently) the last person to get an Erdos number of one. He has published over 70 papers in mathematics in topics ranging from circle packings and permutation enumeration to origami and card shuffling; has performed at the Iowa State Fair; and is co-author on the forthcoming book "Juggling Counts". More information about his research and teaching is at MathButler.org.

### Integer Programming for Kidney Exchange

*Wednesday, August 4, 1:00 p.m. - 1:50 p.m.*

**Sommer Gentry**, *United States Naval Academy*

*Photograph credit: Devon Rowland*

##### Abstract

People who volunteer as living kidney donors are often incompatible with their intended recipients. Kidney paired donation matches one patient and his or her incompatible donor with another pair in the same situation for an exchange. We represent patient-donor pairs be the vertices of a directed graph G, with edges connecting pairs if the donor of the source is compatible with the recipient of the sink. To find the best kidney exchanges, we maximize the sum of edge weights on disjoint cycles. I will first review various exponential-sized and polynomial-sized integer programming formulations proposed for this problem, and give an overview of integer programming solution methods to suggest why some formulations are more tractable than others.

Because a maximum edge-weight matching might not have the maximum cardinality; there is a risk of an unpredictable trade-off between quality and quantity of paired donations. The number of paired donations is within a multiplicative factor of the maximum possible donations, where the factor depends on the edge weighting. We design an edge weighting of G which guarantees that every matching with maximum weight also has maximum cardinality, and also maximizes the number of transplants for an exceptional subset of recipients, while favoring immunologic concordance.

##### Biography

Sommer Gentry is a Professor of Mathematics at the United States Naval Academy, and is also on the faculty of the Johns Hopkins University School of Medicine. She is a senior investigator with the Scientific Registry for Transplant Recipients. She has a B.S. in Mathematical and Computational Science and an M.S. in Operations Research, both from Stanford University, and a Ph.D. in Electrical Engineering and Computer Science from MIT. She designed matching optimization methods used for nationwide kidney paired donation registries in both the United States and Canada, and helped pass a law legalizing paired donation in the United States. Her redistricting work was also instrumental in pushing the Organ Procurement and Transplantation Network to make major policy changes that reduced geographic disparities in transplantation. Her work has attracted the attention of major media outlets including Time Magazine, Reader’s Digest, Science, the Discovery Channel, and National Public Radio. Gentry has received the MAA’s Henry L. Alder award for distinguished teaching by a beginning mathematics faculty member, was a finalist for the INFORMS Daniel H. Wagner prize for excellence in operations research practice, and was just named the US Naval Academy’s 2021 recipient of the Class of 1951 Civilian Faculty Excellence in Research award.

### Reflections in Teaching

*Saturday, August 7, 10:00 a.m. - 10:50 a.m.*

**Candice Price**, *Smith College*

##### Abstract

This year, I realized that I have been teaching for 19 years. “How is this possible when you are only 25 years old?” you ponder, perhaps out loud. Well, first... thank you, and second, but it is true! It has been 19 years since I started teaching. And 2020 has really shown me how far that journey has been. So take a short jaunt with me down memory lane where together we will reflect on lessons I have learned about teaching, and of course places where I am hoping to improve.

##### Biography

Candice Renee Price is an African-American mathematician and assistant professor at Smith College. Born and raised in California, Candice has a bachelor’s degree (2003) in Mathematics from California State University, Chico and a master's degree (2007) from San Francisco State University. She earned her doctoral degree (2012) in mathematics from the University of Iowa under the advisement of Isabel Darcy. Her main area of mathematical research is DNA topology, that is, knot theory applied to the structure of DNA but has interests in applications of mathematics to Biology and the Social Sciences. Candice is an advocate for greater representation of women and people of color in the STEM fields.

### A New Approach for Fighting Infectious Disease, Combining Game Theory and Graph Theory

*Thursday, August 5, 1:00 p.m. - 1:50 p.m.*

**Po-Shen Loh**, *Carnegie Mellon University*

##### Abstract

What happens when you've been thinking about graph theory and probability, and you're called to action to fight COVID?

The speaker will talk about his journey which uncovered a categorically new way to fight disease. It resulted in an app which is fundamentally different from every other app (and which resolves significant issues in "contact tracing apps").

Functionally, it gives you an anonymous radar that tells you how "far" away COVID has just struck. "Far" is measured by counting physical relationships separating you (https://novid.org, https://youtu.be/EIU-6FvwikQ).

The simple idea flips the incentives. Previous approaches focused on controlling you, preemptively removing you from society if you were suspected of being infected. This new tool lets you see incoming disease to defend yourself just in time. This uniquely aligns incentives so that even if everyone does what is best for themselves, they end up benefiting the whole. That solves the "tragedy of the commons", which has paralyzed much of the world.

This unique construction was made possible by many mathematical insights. During the talk, the speaker will highlight many places where it ended up being quite useful to have a history of thinking about research and competition math, and of interacting with the math enthusiast community.

##### Biography

Po-Shen Loh is a math professor at Carnegie Mellon University, and the national coach of the MAA's USA International Mathematical Olympiad team. He also dabbles in social entrepreneurship, founding the free math and science education platform expii.com which sees 500,000 visitors each month, supported by his online math courses that reinvent the middle school math curriculum with a focus on creative thinking (daily.poshenloh.com). He has featured in or co-created videos totaling over 10 million YouTube views, and runs a free weekly YouTube Live stream for students to ask him math questions on the spot. Upon the outbreak of COVID, he turned his mathematical attention to create NOVID, the first app to introduce the fundamentally different "network radar" paradigm for pandemic control.

Website: http://www.poshenloh.com/

### Lessons from 10+ Years of College Math Instructor Teaching Professional Development

*Friday, August 6, 11:00 a.m. - 11:50 a.m.*

**Stan Yoshinobu**, *University of Toronto*

##### Abstract

In this talk, I will highlight the intensive inquiry-based learning (IBL) workshop professional development model and findings from 10 years of data to identify key factors that influence uptake of IBL methods. IBL workshops can increase skills and knowledge, and ultimately influence instructor behavior in the classroom. Then using these insights, I’ll share thoughts on broader issues, including the general notion that intensive professional development workshops and follow-up support can be a key lever for change across a range of issues, such as inclusion and equity, mastery-based grading, course coordination, and more.

##### Biography

Stan Yoshinobu has been teaching courses in undergraduate mathematics and mathematics education for more than 20 years. His scholarly interests include active learning, inquiry-based learning, professional development in higher education, and diversity, equity, and inclusion in education. He also writes regularly about education related topics on The IBL Blog. Outside of work, Stan enjoys spending time with his wife and two children, hiking, photography, and rooting for the Los Angeles Dodgers.

### Complex Functions, Mesh Generation, and Hidden Figures in the NIST Digital Library of Mathematical Functions

*Friday, August 6, 2:00 p.m. - 2:50 p.m.*

**Bonita V. Saunders**, *National Institute of Standards and Technology*

##### Abstract

In 2010, the National Institute of Standards and Technology (NIST) launched the Digital Library of Mathematical Functions (DLMF), a free online compendium of definitions, recurrence relations, differential equations, and other crucial information about mathematical functions useful to researchers working in application areas in the mathematical and physical sciences. Although the DLMF replaces the widely cited National Bureau of Standards (NBS) Handbook of Mathematical Functions commonly known as Abramowitz and Stegun (A&S), it is far beyond a book on the web, incorporating web tools and technologies for accessing, rendering, and searching math and graphics content. I will discuss some interesting historical tidbits, but then focus on past and present technical research challenges being tackled to develop the DLMF’s graphics content. The DLMF currently contains more than 600 2D and 3D figures, and over 200 interactive 3D web visualizations of high level mathematical function surfaces that users can explore.

### Stories About How I Got Where I Am Today

*Thursday, August 5, 11:00 a.m. - 11:50 a.m.*

**Erica Flapan**, *Editor in Chief of the Notices of AMS*

##### Abstract

I will talk about my life, from elementary school to becoming the Editor in Chief of the Notices of the American Mathematical Society. While my history is quite different from that of most mathematicians, I hope that hearing stories about my trials and tribulations can inspire young mathematicians facing their own trials and tribulations to keep at it as I did and become mathematicians who can then tell their own stories to the next generation of young mathematicians. This talk will include a little bit of knot theory, a little bit of spatial graph theory, a little bit of chemistry, and a little bit of humor. But mostly, it will just be stories.

##### Biography

Erica Flapan was a professor at Pomona College from 1986 to 2018. In addition to teaching at Pomona College, for most of the summers from 2000 until 2015, Flapan taught at the Summer Mathematics Program for freshmen and sophomore women at Carleton College. In 2011, Flapan won the Mathematical Association of America’s Haimo award for distinguished college or university teaching of mathematics. Then in 2012, she was selected as an inaugural fellow of the American Mathematical Society. From 2015-2017, she was a Polya Lecturer for the MAA. Since January 2019, she has been the Editor in Chief of the Notices of the American Mathematical Society.

Erica Flapan has published extensively in topology and its applications to chemistry and molecular biology. In addition to her many research papers, she has published an article in the College Mathematics Journal entitled “How to be a good teacher is an undecidable problem,” as well as three books. Her first book, entitled ``When Topology Meets Chemistry" was published jointly by the Mathematical Association of America and Cambridge University Press. Flapan also co-authored a textbook entitled ``Number Theory: A Lively Introduction with Proofs, Applications, and Stories" with James Pommersheim and Tim Marks, published by John Wiley and sons. Finally, in 2016, the AMS published her book entitled “Knots, Molecules, and the Universe: An Introduction to Topology”, which she wrote in collaboration with 18 mathematicians from all over the country.

### The Road to 2002 Sonic Boom Demonstrator

*Friday, August 6, 3:00 p.m. - 3:50 p.m.*

**Christine Darden**, *Retired from NASA Langley Research Center*

##### Abstract

I will open the lecture with some explanation of my childhood, my elementary school education in a segregated school that taught no higher mathematics classes than Algebra and Plane Geometry, and my experience in Plane Geometry during 11th grade at a boarding school that also taught no higher math class. During that 11th grade experience, I fell in love with the class and decided that I wanted to be a mathematician. After high school graduation, I enrolled in a college where all of the students who were planning to become mathematicians had taken Calculus and Trigonometry in high school. I will then share how 5 years after graduating with a B.S. Degree in Math and Physics Education and after having taught high school mathematics & physics for 2 years and having earned a master’s degree in Applied Mathematics, I was hired by NASA as a Data Analyst (Computer) where I worked for 5 years supporting Engineers in the Apollo Program.The year was now 1972 and the United States has just cancelled its Commercial Supersonic Transport Program because of the noise of the sonic boom. I was transferred to a section created to work on the softening of the sonic boom of a supersonic airplane. I will then explain the process of the sonic boom work that resulted in a demonstration of the softened sonic boom.

##### Biography

Christine Mann Darden is a native of Monroe, NC and a graduate of Allen High School in Asheville, NC. She has a BS Degree in Mathematics from Hampton Institute (now University) in Hampton, VA, the MS Degree in Applied Mathematics from Virginia State College (now University) in Petersburg, VA, and the D.Sc. Degree in Mechanical Engineering from George Washington University in Washington, DC. Darden also holds a Certificate of Advanced Study in Management from Simmons College Graduate School of Management in Boston, MA.

After nearly 40 years of service, Dr. Darden retired from NASA Langley Research Center in March 2007 as a member of Senior Executive Service. Her final assignment at Langley was as Director of the Office of Strategic Communications and Education (OSCE). In that position she was responsible for the Center’s external and internal communications, community outreach, governmental relations and educational outreach. Prior to the OSCE position, which Darden assumed in October 2004, Darden served as the Langley Assistant Director for Planning, responsible for the Langley strategic planning process, and oversight of the Center’s delivery on commitments. Darden also previously served as Director of the Aero Performing Center Program Management Office (APCPMO), as a Senior Program Manager in NASA's High Speed Research (HSR) Program Office, and for nearly 30 years as an internationally known researcher in high-speed aerodynamics and sonic boom research. Prior to her NASA career, Darden served as a Mathematics Instructor at Virginia State College and taught high school mathematics.

Darden is a current or former member of several professional or honorary societies, including: Past National Secretary of the National Technical Association, Associate Fellow of the American Institute of Aeronautics and Astronautics (AIAA), Past Secretary of the AIAA Technical Committee on Aero-Acoustics, Beta Kappa Chi National Scientific Honor Society, Kappa Mu Epsilon Honorary Mathematics Society, Alpha Kappa Mu Honor Society, Sigma Pi Sigma Physics Society, and Kappa Delta Pi Educational Honor Society.

During her NASA career, Darden authored over 57 technical papers and articles, primarily in the areas of sonic boom prediction, sonic boom minimization, and supersonic wing design. She is recognized as a international expert in these areas. Darden has been recognized with dozens of awards and honors—including two NASA Medals, the Black Engineer of the Year Outstanding Achievement in Government Award and the Women in Science & Engineering Lifetime Achievement Award. One of Darden’s NASA Medals was for her active involvement in working with and encouraging students to pursue careers in math and science. Darden is also the recipient of 4 Honorary Doctorate Degrees---- Old Dominion University (’12), Muskingum University (’18), N.C. State University (’18), and George Washington University (‘2019). She was inducted into the Engineering Hall of Fame at George Washington University in 2017.

In 2016, Darden was included in the NY Times Best Seller, “Hidden Figures,” by Margot Lee Shetterly. She, and Walter, her husband of 56 years, have 3 daughters, 5 grandchildren, and 3 great-grand children.

### Surprising Discoveries by Three Amateur Mathematicians

*Saturday, Aug 7, 3:00 p.m. - 3:50 p.m.*

**Doris Schattschneider**, *Professor Emerita of Mathematics, Moravian College*

##### Abstract

It is amazing how intense curiosity and ingenuity can propel persons with little or no higher mathematical training to investigate mathematical problems and make surprising discoveries. Dutch graphic artist M.C. Escher (1898-1972), a failure at school mathematics, found answers to the question “Characterize shapes that will tile the plane in such a way that every tile is surrounded in the same manner.” American homemaker Marjorie Rice (1923-2017), not allowed any math beyond a high school general math course, found new answers to the question “Characterize convex pentagons that can tile the plane.” And Dutch sculptor Rinus Roelofs (b. 1954), with an undergraduate degree in applied mathematics and a degree from AKI School of Arts, discovered a new infinite family of uniform polyhedra through sculptural exploration. This lecture will give glimpses of how these three each asked and answered mathematical questions in their own unique way.

### Arithmetic and Digits

*Wednesday, August 4, 3:00 p.m. - 3:50 p.m.*

**Florian Luca**, *University of the Witwatersrand*

##### Abstract

In our recent paper in the Monthly (October, 2019) with Pante Stănică, we looked at perfect squares which arise when concatenating two consecutive positive integers like 183184 = 428^{2} with the smaller number to the left, or 98029801 = 9901^{2} with the larger number to the left. My talk will present variations on this topic with the aim of providing the audience with examples of numbers which are both arithmetically interesting (like perfect squares) while their digital representations obey some regular patterns. The examples will not be limited to perfect squares, but will also include other old friends like Fibonacci numbers and palindromes.

### 2020 Census, Lagrange's Identity, and Apportionment of the U.S. House of Representatives

*Thursday, August 5, 3:00 p.m. - 3:50 p.m.*

**Tommy Wright**, *U.S. Bureau of the Census*

##### Abstract

Given the impracticality of a pure democracy, the U.S. Constitution (1787) calls for a representative form of democracy where the people elect persons to represent them for governing. Each state gets a number of representatives in the U.S. House of Representatives "...according to their respective numbers..." as recorded in a census of the nation to be conducted every ten years starting in 1790. We make use of an elementary result known as Lagrange's Identity to provide a bridge between an insightful motivation and an elementary derivation of the method of equal proportions. The method of equal proportions is the current method for apportioning the 435 seats in the U.S. House of Representatives among the 50 states, following each decennial census. We highlight why the numbers from the census matter and affect our condition and behavior. We also present some historical comments about the first two methods of apportionment, as well as the method that preceded equal proportions.

##### Biography

Since joining the U.S. Census Bureau in January 1996 as a research mathematical statistician, Tommy Wright has provided the overall technical leadership for the Center for Statistical Research & Methodology (CSRM) (formerly Statistical Research Division) which is the Census Bureau's statistical and methodological research and collaborative/consulting facility. CSRM researchers are engaged in collaborative work applying known statistical methods and in research for new and better statistical methods motivated by practical problems encountered in measuring and releasing data on the behavior and condition of the nation's people, places, and businesses.

Between 1979 and 1996, he was a research staff member of the Mathematical Sciences Section at Oak Ridge National Laboratory (ORNL) where his collaborative research focused on probability sampling and the design of sample surveys for large energy related national studies sponsored by many different government agencies.

Tommy has over 35 years of undergraduate/graduate teaching experience in statistics and mathematics at Knoxville College; University of Tennessee-Oak Ridge Graduate Program; University of Tennessee, Knoxville; and most recently Georgetown University as adjunct faculty since 2009. He was an ASA/NSF/Census Research Fellow (1993-1996) pursuing research into using probability sampling methods to improve the constitutionally required decennial census count.

At the Census Bureau, he is currently engaged in the consideration of several problems, including: expressing uncertainty in overall rankings based on sample surveys; assessing the variability in census counts treated by a disclosure avoidance algorithm; and thinking about a role for big data with official government statistics.

Some recent results bring together his interests in optimal sample allocation, apportionment of the U.S. House of Representatives, and Lagrange's Identity.

Tommy was born and grew up in Birmingham, Alabama. He received the M.S. and Ph.D. in statistics from The Ohio State University, the M.S. in mathematics from the University of Tennessee, and the B.S. in mathematics from Knoxville College. His contributions in collaborative research, teaching, and service have led to professional recognition: (1) Elected Member, International Statistical Institute (1989) and (2) Fellow, American Statistical Association (1995).

### We Begin with a Deck of Cards …

*Saturday, August 7, 11:00 a.m. - 11:50 a.m.*

**Robert W. Vallin**, *Lamar University*

##### Abstract

We all know there are lots of fun games and activities that come from a standard deck of cards. As they say during 3 a.m. infomercials, “But wait, there’s more!!” A deck is also the gateway to a myriad of different ideas in mathematics. In this event we start with some of the more straightforward ideas like counting and then move on to some other fun things that we can play with. If you have a deck of cards, keep them handy.

##### Biography

Robert Vallin earned his PhD from North Carolina State University in 1991, studying classical real analysis. Since then he has gone on to publish in analysis, topology, number theory (accidentally), and several other topics. Several years ago he took a minicourse in mathematical card magic and became hooked on recreational mathematics. He is founder and chair of the SIGMAA on Recreational Mathematics and involved in both the Gathering for Gardner and the MOVES (Mathematics of Various Entertaining Subjects) conferences. He is currently a professor at Lamar University in Beaumont, TX, where he has learned to really dislike hurricanes.

### Who Are the Frodos and Celies of Mathematics?

*Thursday, August 5, 10:00 a.m. - 10:50 a.m.*

**Michael Dorff**, *MAA Past President, Brigham Young University*

##### Abstract

Who should be in the modern-day Hall of Fame of Mathematicians? For me, it is not a collection of people whom you would find in a traditional list of mathematicians. Instead, they are the mathematicians who have impacted people's lives whether in small quiet ways or through breakthrough actions. Find out who some of these people are and why I consider them to be the modern Superheroes of Mathematics.

##### Biography

Michael Dorff is the past President of the Mathematical Association of America (MAA) and a professor of mathematics at Brigham Young University. He earned his Ph.D from the University of Kentucky. He is interested in promoting mathematics to the general public, in math careers in industry, and in undergraduate research. He co-directs the MAA PIC Math program (Preparation for Industrial Careers in the Mathematical Sciences) and was the founder of CURM (Center for Undergraduate Research in Mathematics). He is a Fellow of the AMS, a CUR Fellow (Council on Undergraduate Research), and a Fulbright Scholar in Poland. He is married with 5 daughters. In any free time he has, he enjoys reading, writing, running, and traveling (he has traveled to 49 U.S. states and 48 countries).