math-news
Moiré Magic
Highly tunable composite materials—with a twist.
The above animation shows the patterns created as two circles move across each other. Those patterns, created by two sets of lines offset from each other, are called moiré (pronounced mwar-AY) effects. As optical illusions, moiré patterns create neat simulations of movement. But at the atomic scale, when one sheet of atoms arranged in a lattice is slightly offset from another sheet, these moiré patterns can create some exciting and important physics with interesting and unusual electronic properties.
Mathematicians at the University of Utah have found that they can design a range of composite materials from moiré patterns created by rotating and stretching one lattice relative to another. Their electrical and other physical properties can change—sometimes quite abruptly, depending on whether the resulting moiré patterns are regularly repeating or non-repeating. Their findings are published in Communications Physics.
The mathematics and physics of these twisted lattices applies to a wide variety of material properties, says Kenneth Golden, distinguished professor of mathematics. “The underlying theory also holds for materials on a large range of length scales, from nanometers to kilometers, demonstrating just how broad the scope is for potential technological applications of our findings.”
Ken Golden
"We observe a geometry-driven localization transition that has nothing to do with wave scattering or interference effects, which is a surprising and unexpected discovery."
With a twist
Before we arrive at these new findings, we’ll need to chart the history of two important concepts: aperiodic geometry and twistronics.
Aperiodic geometry means patterns that don’t repeat. An example is the Penrose tiling pattern of rhombuses. If you draw a box around a part of the pattern and start sliding it in any direction, without rotating it, you’ll never find a part of the pattern that matches it.
Aperiodic patterns designed over 1000 years ago appeared in Girih tilings used in Islamic architecture. More recently, in the early 1980s, materials scientist Dan Shechtman discovered a crystal with an aperiodic atomic structure. This revolutionized crystallography, since the classic definition of a crystal includes only regularly repeating atomic patterns, and earned Shechtman the 2011 Nobel Prize in Chemistry.
Okay, now onto twistronics, a field that also has a Nobel in its lineage. In 2010, Andre Geim and Konstantin Novoselov won the Nobel Prize in Physics for discovering graphene, a material that’s made of a single layer of carbon atoms in a lattice that looks like chicken wire. Graphene itself has its own suite of interesting properties, but in recent years physicists have found that when you stack two layers of graphene and turn one slightly, the resulting material becomes a superconductor that also happens to be extraordinarily strong. This field of study of the electronic properties of twisted bilayer graphene is called “twistronics.”
Two-phase composites
In the new study, Golden and his colleagues imagined something different. It’s like twistronics, but instead of two layers of atoms, the moiré patterns formed from interfering lattices determine how two different material components, such as a good conductor and a bad one, are arranged geometrically into a composite material. They call the new material a “twisted bilayer composite,” since one of the lattices is twisted and/or stretched relative to the other. Exploring the mathematics of such a material, they found that moiré patterns produced some surprising properties.
“As the twist angle and scale parameters vary, these patterns yield myriad microgeometries, with very small changes in the parameters causing very large changes in the material properties,” says Ben Murphy, co-author of the paper and adjunct assistant professor of mathematics.
Twisting one lattice just two degrees, for example, can cause the moiré patterns to go from regularly repeating to non-repeating—and even appear to be randomly disordered, although all the patterns are non-random. If the pattern is ordered and periodic, the material can conduct electrical current very well or not at all, displaying on/off behavior similar to semiconductors used in computer chips. But for the aperiodic, disordered-looking patterns, the material can be a current-squashing insulator, “similar to the rubber on the handle of a tool that helps to eliminate electrical shock,” says David Morison, lead author of the study who recently finished his Ph.D. in Physics at the University of Utah under Golden’s supervision.
This kind of abrupt transition from electrical conductor to insulator reminded the researchers of yet another Nobel-winning discovery: the Anderson localization transition for quantum conductors. That discovery, which won the 1977 Nobel Prize in Physics, explains how an electron can move freely through a material (a conductor) or get trapped or localized (an insulator), using the mathematics of wave scattering and interference. But Golden says that the quantum wave equations Anderson used don’t work on the scale of these twisted bilayer composites, so there must be something else going on to create this conductor/insulator effect. “We observe a geometry-driven localization transition that has nothing to do with wave scattering or interference effects, which is a surprising and unexpected discovery,” Golden says.
The electromagnetic properties of these new materials vary so much with just tiny changes in the twist angle that engineers may someday use that variation to precisely tune a material’s properties and select, for example, the visible frequencies of light (a.k.a. colors) that the material will allow to pass through and the frequencies it will block.
“Moreover, our mathematical framework applies to tuning other properties of these materials, such as magnetic, diffusive and thermal, as well as optical and electrical,” says professor of mathematics and study co-author Elena Cherkaev, “and points toward the possibility of similar behavior in acoustic and other mechanical analogues.”
Find the full study in Communications Physics.
Lindsey Henderson
Lindsey Henderson, Secondary Mathematics Specialist for the Utah State Board of Education.
Leading a revolution in how math is taught in Utah.
Lindsey Henderson (BS’2002 Mathematics Teaching, with a minor in geology) is poised to lead a revolution in how math is taught in Utah. As the Secondary Mathematics Specialist for the Utah State Board of Education, she and her team determine how Utah school kids are taught mathematics and which math courses are useful for their career paths following high school graduation.
In October 2021, Henderson and her team posted a survey asking Utah’s tech founders, executives, and chief technical officers to discuss the kinds of math typically used in tech careers. The results were interesting and surprising. According to the tech community, calculus is moderately useful because it teaches problem-solving skills, but it isn’t particularly useful in daily work. Other respondents thought calculus should be replaced immediately with more useful math-related subjects, such as AI and data science. Another study by Dr. David Bressoud, DeWitt Wallace Professor Emeritus at Macalester College says that studying calculus in high school is only marginally beneficial for the small number of students (18%) who study it.
“These responses reveal how much math skills in the workplace have changed and suggest how important it is for students to learn new and different skills for their future careers,” said Henderson. “Skills such as how to analyze and interpret data and spreadsheets and how to create other types of visual data.” Half of the respondents reported using algebra often, but the majority noted that they seldom, or never, use geometry, trigonometry, and especially calculus. The results of the survey mirror the results of a similar and larger survey created by professors at Stanford University and the University of Chicago in 2019, which can read about here. For now, the Utah State Board of Education is still collecting data and hasn’t yet decided on how best to update secondary mathematics courses and teaching for Utah students.
In 2021, Henderson received a Women Tech Award from the Women Tech Council for her educational leadership.
Lindsay Henderson
"My education at the University of Utah has made every difference in my life. I had a wonderful experience at the U, both socially and academically, and I feel it prepared me for success!"
At the U
Henderson grew up in Spanish Fork, Utah, and was raised by a mom who taught deaf students and a dad who taught kids who had been hospitalized due to mental illness. Her mom and grandfather have strong roots at the U since they both graduated from the university. Growing up in Utah County, Henderson’s family was considered an outlier because they always cheered for the University of Utah teams at sporting events.
“I saw the U as an opportunity to further my education and as a safe place to start learning how to become an adult,” said Henderson. She loved living on campus (in the Van Cott dorms and then eventually in Benchmark after the 2002 Winter Olympics). She worked at the front desk at Van Cott as well as at the Heritage Center before, during, and after the Olympics.
She was very good at math and enjoyed calculus and linear algebra. She decided that she wanted to earn a math degree because she loved learning, doing math, and she hadn’t heard of many women who had math degrees.
“My education at the University of Utah has made every difference in my life,” she said. “I feel so fortunate to have had access to such a high-quality institution of higher education. I had a wonderful experience at the U, both socially and academically, and I feel it prepared me for success!”
Favorite Professors
After she received an associate’s degree from Utah Valley University (UVU), she transferred to the U. Her favorite math professors were Dr. János Kollár (linear algebra), Dr Alexander Balk (differential equations), and Dr. James Carlson (history of math). Although she had already taken linear algebra at UVU, she took the class again at the U just for the opportunity to study with Dr. Kollár. She was impressed with his teaching style, and today she still loves linear algebra.
She remembers Dr. Balk’s efforts to engage students in differential equations by using rainbow-colored chalk to make things exciting. She appreciated his interest in students, and she felt seen and valued by him, which contributed to her doing well in his class.
She still thinks most about a history of mathematics class taught by Dr. Carlson. She loved learning about how mathematical topics were discovered and about different cultures and the way they reasoned and made sense of the world mathematically. While she was taking his class, she suffered a life-threatening snowboarding injury and had to be hospitalized. Dr Carlson worked with her one-on-one to help her make up an incomplete grade. “He did it out of the goodness of his heart and because he truly cared,” she said. “To this day, I appreciate his willingness to work with me, and I will never forget how he adapted his teaching to support me during a difficult time.”
Dr. Mary Burbank in the College of Education was a non-mathematics professor who greatly influenced Henderson. She was Henderson’s student-teaching professor, and Henderson flourished in her classroom. “She was the kind of professor who really took the time to get to know you, and then she pushed you to grow in ways that you needed to grow,” she said. “I loved working with her and appreciated the trust and relationship building she invested in because it really helped me hone my teaching skills.”
Advice for Students
“Learn to be your biggest advocate—nobody else will do it for you. Make space for yourself and your ideas because you have wonderful thoughts to share and so much to learn from others,” she said. “Not everyone will recognize your potential, and that’s okay as long as you do. Your persistence and thirst for knowledge will benefit you in so many ways in the future!”
Teaching Math
Once Henderson graduated from the U, she accepted a position teaching integrated 8th grade science and pre-algebra at Bryant Middle School in the Salt Lake City School District. After two years teaching both mathematics and science, she was told that she wasn’t considered “highly qualified” to teach integrated science and that she would have to complete more coursework to continue. At that point, Henderson decided to become a full-time mathematics teacher. She worked hard to keep her middle school students interested in math, and she tried to harness their excitement for learning. She would integrate mathematical discovery and enrichment tasks into the standard curriculum, such as having students use Mobius Strips, build tetrahedral kites, and work on other mathematical projects.
After five years at Bryant, she began teaching mathematics at Highland High School. From there, she transferred to East High School, where she spent the next nine years. Throughout her 13-year teaching career, she taught all secondary grade levels and all math subjects offered in Utah, from pre-algebra to AP Calculus, except AP Statistics and continuing education courses. She enjoyed it when the state switched to the new integrated math standards in 2010. She loved teaching and integrating connections between algebra, geometry, and algebra II, instead of teaching each subject by itself. She started using task-based and inquiry-based mathematical learning experiences.
“I loved making my classroom a place where traditionally underrepresented groups of students felt safe, and I particularly enjoyed making math accessible to all students,” she said. “One of the ways I did this was by helping students see themselves differently—changing their thinking so that they began to see themselves as mathematical thinkers and doers.”
Towards the end of her teaching career, she decided to form her own business—Sugar House Instructional Design—so that she could consult on curriculum development projects. She taught and consulted with private STEM Education Technology (EdTech) startups. Her consulting practice went well, and she was quickly promoted from consultant to chief academic officer for a local startup called Zaniac. She spent two years there before moving to Because Learning! (formerly Ardusat) as the director of learning.
From Because Learning! she made the transition back to public education, but now she had her math and teaching skills, along with her newly honed people and project management skills. She landed a district-level leadership position in the Davis County School District, the largest public district in Utah, and served as their K-12 mathematics specialist for the next two years. She found that she had a talent for bringing together diverse stakeholders and achieving a consensus, along with building productive, positive communities of educators.
When a secondary mathematics specialist position opened at the Utah State Board of Education, Henderson saw an opportunity to work with a community of mathematicians at the state level. In the two years since she joined the board, she has built a community of more than 4,000 math or math-adjacent educators/leaders, significant growth from the 250 educators she inherited when she first began working for the state. Since Utah switched to an integrated secondary mathematics core in 2010, there have been remarkable student results. “I have been proud to unite the Utah Secondary Mathematics community around a shared common vision for mathematics education in Utah, as well as set the tone of the culture for what mathematics looks like for secondary math students in Utah,” she said. She also has been fortunate to work with higher education partners on several projects, most recently in updating the Secondary Mathematics endorsement requirements. Math endorsements are required in order to teach in a public school in Utah.
She has been doing a lot of collaboration with the local mathematics community and also with the Conference Board of Mathematical Sciences. She is preparing for the 2023-2024 school year when mathematics core standards are up for revision, with the goal of updating standards and determining the best path forward to ensuring that Utah high school graduates have the requisite math skills.
Henderson lives with her wife and kids in Salt Lake City. She loves to read, enjoys summer, and is drawn to water in nature—lakes, oceans, rivers. “I seem to stumble across parabolas a lot when I am out and about,” she said. “I’m also the crazy cat lady in the neighborhood—I have four Scottish kilt kittens that I adore. My wife and I really love living in Utah and all that it has to offer.”
Phi Beta Kappa
Muskan Walia Named Phi Beta Kappa Society Scholar.
Muskan Walia, a second-year student at the University of Utah Honors College, studying math
and philosophy, has been named a Key into Public Service Scholar by the Phi Beta Kappa Society. The Society is the nation’s most prestigious academic honor society, and the Key into Public Service award highlights specific pathways for arts and sciences graduates to launch public sector careers.
Chosen from nearly 900 applicants attending Phi Beta Kappa chapter institutions across the nation, the Key into Public Service Scholars hail from 17 states. These are high-achieving college sophomores and juniors, who display notable breadth and depth in their academic interests.
“I am extremely grateful and honored to be receiving this award from Phi Beta Kappa,” said Walia. “My community here at the University of Utah has provided me with a prodigious liberal arts and sciences education and has nurtured my interest in exploring the dynamics between science, society, and the public sector. I am excited for the incredible opportunity to further explore this interest this summer.”
Walia is an ACCESS Scholar and undergraduate researcher, working with Dr. Fred Adler, Professor of Biology and of Mathematics. In her research, Walia adapted an epidemiological SIR model for spread of disease to model the number of cells infected with SARS-CoV-2 in order to predict when different types of tests will produce false positives or false negatives.
“My summer in the ACCESS Scholars program sparked an interest and motivation to pursue a career in public service,” she said. “Being taught by faculty across the University of Utah in diverse disciplines, I learned about the intersections of science, communication, and policy and how scientists can practice the art of advocacy.
Muskan Walia
"My community here at the University of Utah has provided me with a prodigious liberal arts and sciences education and has nurtured my interest in exploring the dynamics between science, society, and the public sector."
“Working under the mentorship of Dr. Fred Adler has been invaluable. I wanted to be engaged in mathematics research that centered on justice and informed public policy. There was truly no better pairing than with Dr. Adler. He has wholeheartedly supported and encouraged my curiosity and passion to utilize mathematics principles to tackle the most pressing social justice related questions of our time.”
In addition to her studies, Walia currently serves as the ASUU student government Senate Chair and works as a youth environmental organizer in the Salt Lake City area. She founded a campaign to commit her local school district to a 100% clean electricity transition by 2030, and has assisted with the expansion of local clean energy campaigns in Utah school districts. She is also a leader and mentor at Utah Youth Environmental Solutions Network (UYES), where she supports the development of a new youth-based climate justice curriculum. Her experiences have cultivated a passion and commitment to community building, climate education, and environmental justice.
Each Key into Public Service Scholar will receive a $5,000 undergraduate scholarship and take part in a conference in late June in Washington, D.C. to provide them with training, mentoring, and reflection on pathways into active citizenship.
Below are the names of the 2022 Key into Public Service Scholars and their chapter institutions:
Aylar Atadurdyyeva, University of Kansas
Miguel Coste, University of Notre Dame
Noelle Dana, University of Notre Dame
Grace Dowling, Clark University
Brandon Folson, Loyola University Chicago
Justin Fox, University of Maryland- College Park
Sora Heo, University of California - San Diego
Alec Hoffman, Clark University
Samiha Islam, State University of New York at Buffalo
Ruthie Kesri, Duke University
Katherine Marin, University of Florida
Sondos Moursy, University of Houston
Olivia Negro, Ursinus College
Emily Geigh Nichols, Stanford University
Paul Odu, University of Missouri
Vaidehi Persad, University of South Florida
Diba Seddighi, University of Tennessee
James Suleyman, Roanoke College
Jonah Tobin, Williams College
Muskan Walia, University of Utah
For more information about the scholarship and links to individual biographies of the recipients, please visit pbk.org/KeyintoPublicService.
Societal Impact Scholar
Ken Golden Named U Presidential Societal Impact Scholar
President Taylor R. Randall has named Ken Golden, Distinguished Professor of Mathematics, as an inaugural recipient of the University of Utah Presidential Societal Impact Scholar Award.
Dr. Golden and four other scholars are a select group of faculty. Recognized as experts in their respective fields and disciplines, they share and translate their scholarship, research, creative activities and ideas with opinion leaders, policy makers, the public and other audiences outside the university and in ways that can transform society.
Ken Golden
"Dr. Golden is among the rare group of top-level mathematical scientists who is able to reach to the broader public about one of the central issues of our time."
Golden is a brilliant expositor and a passionate advocate for public awareness of our changing climate and the critical role of mathematics in climate modeling. He has given over 40 invited public lectures since 2008, and over 500 invited lectures since 1984. His public lectures emphasize the rapid and significant loss of Arctic sea ice, and how mathematics is helping us predict the future of the Earth’s polar marine environment. Dr. Golden is among the rare group of top-level mathematical scientists who is able to reach to the broader public about one of the central issues of our time.
From tackling the social determinants of health and wellness, to addressing the underlying causes of crime and poverty, to designing interventions to curb poor air and water quality, to helping better inform public debate on society’s most pressing issues, these scholars’ works have a positive impact on people and institutions and help make our world a better, more equitable and enjoyable place in which to live.
The 2022 cohort of impact scholars are:
Kenneth Golden, Distinguished Professor, Department of Mathematics
RonNell Andersen Jones, Professor, College of Law
Michelle Litchman, Assistant Professor, College of Nursing
Susie Porter, Professor, College of Humanities and the School for Cultural and Social Transformation
Paisley Rekdal, Distinguished Professor, Department of English
The Presidential Societal Impact Scholar Award was conceived by and is supported by a gift from University of Utah Professor Randy Dryer.
Teaching Assistantship
Seungsu Lee Awarded Teaching Assistantship from the University of Utah
Graduate student Seungsu Lee has received a Teaching Assistantship Award from the University of Utah. The award is designed to bolster undergraduate education while providing graduate students with experience teaching in undergraduate environments. The opportunity is for full-time graduate teaching assistants.
“Receiving the award means a lot to me in different ways,” said Lee. “It tells me that my proposal is effective and will help many people who study math. Also, the award ensures support from the department and my mentor in implementing my proposal into an actual class. In terms of my career, the award confirms my teaching skills. I learned English as a second language, and I have a strong Korean accent, so receiving the award proves that one can develop communication and teaching skills to teach mathematics efficiently regardless one’s background.”
"When I teach, I love to communicate with students, tell them what they’re doing correctly, and teaching them how to do mathematical reasoning. In particular, I like the moment when students understand what I’m teaching about a mathematical concept, and I can see the “aha” moment in their faces."
Lee will be teaching an asynchronous online class for Math 2270—Linear Algebra--and will have responsibility for creating lecture videos for the department website. Asynchronous learning allows an instructor flexibility in creating a learning environment that will allow for different kinds of learners and learning styles. Lee’s academic advisor is Professor Karl Schwede, and his mentor for the project is Assistant Professor (Lecturer) Matt Cecil.
“I like to chat about mathematics with other people,” said Lee. “When I teach, I love to communicate with students, tell them what they’re doing correctly, and teaching them how to do mathematical reasoning. In particular, I like the moment when students understand what I’m teaching about a mathematical concept, and I can see the “aha” moment in their faces.”
When Lee was a child, his father showed him the magic square. The magic square is a square array of numbers in which all the rows, columns, and diagonals add up to the same sum, which is called the magic constant. This is the fun part in working through the square—you get the same number when you add numbers for each row, column, or even diagonals. “As far as I can remember, the magic square marked the first time that I ever saw a mathematical puzzle,” said Lee. He was very interested in the algorithm to solve the magic square. As he got older, he started to do more and more math. When he was in high school, he had a great math teacher, who showed him rigorous ways to think about calculus by using epsilon and delta. This was a turning point for Lee that made him decide to forge a career in math.
He completed his undergraduate degree at Yonsei University in South Korea. “I got interested in algebraic geometry when I was an undergraduate,” he said. “Unfortunately, my university’s graduate school didn’t focus on this area of math, so I searched online and was excited to see that the U’s Math Department has a huge research group in algebraic geometry. I was so happy to be accepted to the department’s program.” After he earns a Ph.D., he plans to seek a research position.
Graduate Research Fellowship
Sanghoon Kwak has been awarded a Graduate Research Fellowship (GRF) from the University of Utah.
Sanghoon Kwak, a third-year Ph.D. candidate in the Department of Mathematics, has been awarded a Graduate Research Fellowship (GRF) from the University of Utah. The purpose of the GRF is to provide graduate students with an opportunity to do full-time research during an academic year. Recipients are selected and evaluated on the quality and impact of their research or creative project, their achievements, and their potential for success.
“I am tremendously honored and humbled to receive a GRF,” said Kwak. “It’s a huge affirmation of the work I’ve done for last three years and an encouraging nod to my future work. The fellowship will allow me to have more solid blocks of time to dedicate to my research. I also want to recognize the support, trust, and patience I’ve received from my advisors, Distinguished Professor Mladen Bestvina, and Assistant Professor Priyam Patel.”
"I also want to recognize the support, trust, and patience I’ve received from my advisors, Distinguished Professor Mladen Bestvina, and Assistant Professor Priyam Patel."
Kwak studies geometric group theory, which is an area of mathematics devoted to studying groups, endowing them with a metric, and treating them as geometric objects. Geometric group theory is a relatively new area of mathematics, providing a variety of applications to geometry, topology, group theory, number theory and graph theory.Many junior researchers have been drawn to this field, and the Math Department at the university has one of the leading groups.
In his research, Kwak works on the group of symmetries of infinite graphs that correspond to infinite-type surfaces. In the fall of 2021, Dr. Bestvina and Dr. Yael Algom-Kfir, a lecturer at the University of Haifa in Israel who received her Ph.D. in mathematics from the U in 2010, conducted a pioneering study on the symmetry group of infinite-type graphs. Based on this study, Kwak and other colleagues in the Math Department were able to develop a complete classification, of which infinite-type graphs have symmetry groups with “interesting” geometry. The GRF will allow him to continue his work in this area.
Kwak has always enjoyed the beauty, simplicity, and universality of math. “One of the things I like about mathematics, compared to the other sciences, is that mathematical knowledge has no expiration date,” he said. “An established fact in mathematics, as long as it is rigorously proved, rests forever. For me, publishing a paper is like putting a small stone out there that will last. The stone could be a part of a cornerstone of a castle to build on; it could be placed on top of a pyramid of stones; or it could serve as a kind of Rosetta stone that unlocks understanding between different fields; or it could contribute to a mosaic of stones that helps us understand a larger piece of a picture.”
He received a bachelor’s degree in mathematics from the Korea Advanced Institute of Science and Technology (KAIST) in South Korea. During his undergraduate education, Dr. Bestvina visited KAIST and gave a lecture on geometric group theory. Kwak attended his presentation and wanted to learn more about the correspondence between surfaces and graphs. Following graduation, Kwak was accepted to the U for graduate school. After he receives his Ph.D., he hopes to continue his research and teach at a university.
Below is an example of Kwak’s work.
NSF Fellowship
Eamon Quinlan-Gallego, receives a Mathematical Sciences Postdoctoral Research Fellowship from the National Science Foundation.
The three-year fellowship is awarded to support future leaders in mathematics and statistics by helping them participate in postdoctoral research that will enhance their development. “Receiving this fellowship is an incredible honor, and it will allow me to dedicate myself to research full-time for four semesters and extend my stay in Utah for an extra year. It will also give me funds to travel to conferences and visit collaborators,” he said.
Quinlan-Gallego studies solutions to polynomial equations and their singularities. For example, in pre-calculus, the equation y = x^2 defines a parabola in the plane. This parabola is smooth—it doesn’t have any sharp corners; however, occasionally, polynomial equations can fail to be smooth. These non-smooth points, called singularities, are ubiquitous across mathematics, and their study is a fundamental problem.
Eamon Quinlan-Gallego
"Receiving this fellowship is an incredible honor, and it will allow me to dedicate myself to research full-time for four semesters and extend my stay in Utah for an extra year."
Typically, Quinlan-Gallego uses two different techniques to study these singularities. First, he can associate certain differential equations to them whose behavior allows them to be classified in different ways. Second, he can study singularities using “modulo-p.” He fixes a prime number (usually denoted by p, but in this case, for example, we could use p = 5). Working “modulo-5” means that when he looks at a polynomial equation, like y^3 = x^2, instead of thinking about it in the real-number system (as you would in pre-calculus), he thinks of it in clock arithmetic. This means that he does all of the algebra using a clock with 5 hours. For example, if our clocks had 5 hours, and it was 4 o’clock and 2 hours pass, it is 1 o’clock. In clock arithmetic, we would say that 4 + 2 = 1. Similarly, 4 x 2 is usually 8 but in our clock, we have 4 x 2 = 3. “By working in this clock arithmetic, we lose all of the “geometry,” but we gain a host of other tools we can use, and the hope is that as the prime p selected gets larger and larger, the behavior of the singularity modulo-p approaches the real behavior,” he said. He also likes to combine these two techniques and think about differential equations modulo-p.
He was good at math as a kid but until he was a senior in high school, he thought he would become a biologist. Then two things happened: he realized he only wanted to study biology because the idea of going to remote islands to look at creatures no one had seen before sounded cool, but learning about all the chemical reactions going on in the mitochondria wasn’t so exciting--and he read Stephen Hawking’s book A Brief History of Time and became fascinated by how mathematics is used to learn about things that are far away in space and time. At that point, he switched from studying biology to physics. The jump from physics to mathematics was much more straightforward when he realized he was enjoying his math classes more than experimental physics.
Quinlan-Gallego was raised in Spain—his mother is Spanish and his father is American. After high school, he left Spain to study in Scotland at the University of Glasgow. “There was this great program for citizens of the European Union that allowed me to study in Scotland for free,” he said. “I had a wonderful time in Glasgow, and I was given so many amazing opportunities.” During his undergraduate years, he also participated in an exchange program at the National University of Singapore for a year.
Once he completed his bachelor’s degree, he knew he wanted to come to the U.S. for graduate school. He was accepted to the University of Michigan and began working under Professor Karen Smith, who serves as the William Fulton Distinguished University Professor of Mathematics. He also spent more than a year in Tokyo, again as an exchange graduate student, at the University of Tokyo.
He’s looking forward to continuing his work at the U. “The department has many experts in modulo-p methods and a host of other very interesting topics, so I’m looking forward to learning as much as possible from them and moving forward in my research.”
NSF Fellowship
Alex Rasmussen receives a Mathematical Sciences Postdoctoral Research Fellowship from the National Science Foundation.
The three-year fellowship is awarded to support future leaders in mathematics and statistics by helping them participate in postdoctoral research that will enhance their development.
Rasmussen is a Research Assistant Professor in the department. “I’m very grateful to be recognized for my research and to the people who helped me along the way, including my advisors, collaborators, mentors, and teachers,” he said. “The award will allow me to devote more time to my research program. In addition, it will enable me to take on more activities to serve the math community, such as mentoring undergraduates and organizing conferences.”
Rasmussen’s work focuses on symmetries of geometric objects. Specifically, he’s interested in symmetries of spaces that are “negatively curved.” “The geometry of negatively curved spaces is quite unlike that of our own space, and it makes them exotic and also very beautiful,” he said.
A bunch of symmetries form a group, and a group can be thought of as symmetries of many different negatively curved spaces at the same time. A large part of Rasmussen’s research is spent on classifying the different spaces associated to one group. He finds the subject interesting because it allows him to draw pictures, engage his creative and aesthetic senses, and use tools from other fields, such as commutative algebra.
Alex Rasmussen
"I’m very grateful to be recognized for my research and to the people who helped me along the way, including my advisors, collaborators, mentors, and teachers."
In high school, he wasn’t especially interested in math. He did well at it but found it somewhat dry and mechanical. His first math class at Colby College was a multivariable calculus class taught by Scott Taylor, Associate Professor and Department Chair. Taylor used pictures of curves and surfaces in his teaching. This was a revelation to Rasmussen, who began to discover the beauty, depth, and creativity of math. From that point on, he took more math classes.
He received a bachelor’s degree in mathematics and began a graduate program at the University of California Santa Barbara, where he received a master’s degree. He obtained a Ph.D. in mathematics from Yale University in 2020.
He has a few research goals he’d like to work on over the next few years. These include classifying hyperbolic actions of metabelian groups and classifying geodesic laminations on infinite type surfaces. Metabelian groups are a wide class of relatively simple groups that can still have complicated hyperbolic actions. Geodesic laminations are 1-dimensional objects on surfaces consisting of long straight lines interacting in complicated ways. “These are pretty hard problems that will keep me busy for a while. Along the way, many other related problems will pop up naturally,” he said.
MAA Teaching Award
Kevin Wortman, an Associate Professor and Director of Undergraduate Studies in the University of Utah Department of Mathematics, has been honored with the 2022 Mathematical Association of America (MAA) Distinguished Teaching Award for the Intermountain Region.
Kevin Wortman
The award honors professors of mathematics whose efforts have been recognized as influential beyond their own institutions. Since 2004, Wortman is the fifth U mathematics faculty member to receive this MAA award. Previous U math faculty recipients include Don Tucker, Nicholas Korevaar, Peter Alfeld, and Anne Roberts. Wortman joined the U's Math Department in 2007.
The Mathematical Association of America, with more than 25,000 members, is the primary professional organization for teachers of undergraduate mathematics. The MAA Intermountain Region includes all colleges and universities in Utah and southern Idaho.