Aftermath 2024

LETTER FROM THE CHAIR

DEAR ALUMNI AND MEMBERS OF THE DEPARTMENT,

I am pleased to extend a warm welcome to the latest edition of Aftermath. As the new chair of the Department of Mathematics, I am honored to present the achievements and developments that underscore the vitality and excellence of our community.

This year has seen significant strides in various facets of our department. Our faculty have demonstrated outstanding dedication and expertise in their roles, contributing to remarkable advancements in teaching methodologies, research endeavors, and dedication to the excellence of our students. Two exciting developments include the new VR section of MATH 1050 in which students practice class material by progressing through a virtual reality game with their classmates. A second development is the newly announced bioinformatics major. (Watch for

more details on these exciting launches.)

Equally noteworthy is the invaluable contribution of our dedicated staff; without their tireless efforts, our department couldn’t function as seamlessly as it does.

Concurrently, efforts to revamp the Math Center are well underway, with plans to expand its capacity and enhance the tutoring experience for our students. This resource serves as a haven for students from across campus, providing a welcoming environment where they can receive personalized assistance in virtually any level or area of undergraduate mathematics.

I am particularly proud to highlight an article within this issue featuring the announcement of Davar Khoshnevisan’s appointment as a Distinguished Professor. This title is reserved for faculty members who exemplify the highest goals

of scholarship and dedication to teaching. This recognition underscores not only Davar’s exceptional contributions but also the caliber of scholarship within our department.

As we eagerly anticipate the dawn of another academic year, I extend a warm welcome to incoming students, postdocs, and faculty. Your presence enriches our community, and we look forward to the contributions you will make to our shared pursuit of knowledge and discovery.

Finally, to our donors: Your generosity plays a pivotal role in advancing our mission of academic excellence and student success.

Thank you for your continued support and engagement.

Sincerely,

Chair

Tommaso

de Fernex

RECOGNITION

DISTINGUISHED PROFESSOR

DAVAR KHOSHNEVISAN

SIAM FELLOW

AARON L. FOGELSON

UNIVERSITY OF UTAH EARLY CAREER TEACHING AWARD

PRIYAM PATEL

GOLDWATER SCHOLAR

MUSKAN WALIA

FULBRIGHT SCHOLARS

CATHERINE WARNER

MOSES SAMUELSON-LYNN

RHODES SCHOLAR FINALIST

ELIZA DIGGINS

2024 PI MU EPSILON SPEAKER AWARD

NASH WARD

IN MEMORIAM

JOHN WARNOCK | Alumnus

1940 - 2023

Cover: Students actively engaged in the Math Center at the LeRoy Cowles Building

Aftermath is the official magazine of the Department of Mathematics, University of Utah, published in partnership with Marketing & Communications, College of Science.

Associate Director of Marketing & Communications: Bianca Lyon

Writer & Editor: David Pace

Designer/Photographer: Todd Anderson

Follow us on social media @uofumath

Prefer only a digital version of Aftermath? Send us an email. office@math.utah.edu

THE SOLACE OF NUMBERS

A REFUGE FOR MATH ANXIETY

PRIOR TO 2002, ANGIE GARDINER, FORMER DIRECTOR OF UNDERGRADUATE SERVICES IN THE DEPARTMENT OF MATHEMATICS, WAS WORKING WITH HER MATH TUTORS OUT OF TWO OLD WWII BARRACKS SITUATED BETWEEN THE COWLES AND WIDTSOE BUILDINGS ON PRESIDENTS CIRCLE.

Today, things are exponentially better, and not just because of the expanded space, with its beautifully rendered sandstone exterior accented with concrete and glass and illuminated naturally through skylights, rising into an elevated garden patio above. The $1.8 million T. Benny Rushing Mathematics Student Center, where the Math Center is housed, currently speaks to the collaborative, integrated space—both physical and mental—that offers what Gardiner calls a “continual review of mathematics” targeting

University of Utah undergraduates enrolled in math classes.

That continuing review is robust as well—on-demand for students who at times just drop in with a quick query or for others who actually use the space as a study hall, raising differentcolored felt flags (depending on what area of math they need help with) at their work stations when they need one-on-one attention.

Some of the tutors are graduate students, required, as part of their teaching requirement to work a minimum of one hour per week. There is also a tutor cohort which is more of a fixture, logging many weekly hours. All of these skilled mentors are always at-the-ready,

willing and fluidly collaborative with their colleagues who may be more conversant with concepts than they are—whether it's quadratic equations, calculus, or trigonometry. There’s even a dedicated private space for Foundations of Analysis work where students are mentored through “proofs,” a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.

Discussion is a premium, says Lisa Penfold who took over the direction

Lisa Penfold sitting among the activity of the Math Center.

of the Center in 2018. "If a student discusses the problem with a tutor within 24 hours of a lecture,” she says, “understanding and retention increases” markedly. For the tutors, the benefits are mutual as they benefit in their preparation for the Graduate Record Exam, required for graduate school, by working as a tutor in the Math Center.

While designed first to be pragmatic (students need to pass math classes for their generals, but also for their STEM-related majors) the Center inspires students to move beyond the Steps (1,2,3…) to solving an assigned problem to really understand what the meaning is of what they’re looking at. It’s this undergirding mission that has made the Math Center at the U the powerhouse that it is. Tutors, who are skilled at deciphering the students' level of understanding, respond to students in real time.

Helping students to pass their math classes and to better understand math meanings can sometimes be complicated by the all too familiar “math anxiety.” Refrains like "I'll never be able to do math" or "math doesn't even walk in our family" are all too common for some who arrive at the U.

The Math Center, with its welcoming atmosphere, private group meeting rooms, computer lab and a break room/kitchen has become a sanctuary for students struggling with the

complexities of numbers, equations, and formulas. (It would seem the only thing they might be missing is a climbing wall.)

"Our objective is to make sure students feel welcome here and get the help they need," Penfold states. "We know there's a lot of baggage with mathematics."

Addressing that baggage is one of the Center's core missions. Through extraordinary patience and individualized teaching approaches, tutors work tirelessly to dismantle math anxiety. "We start by acknowledging it's valid," says tutor Caleb Albers, a PhD candidate in applied mathematics. "Then we can begin chipping away at those negative associations.”

“I find that helping students get more of a conceptual understanding … helps a lot,” says Stella Brower, a veteran tutor pursuing a master's

statistics. “I have experience tutoring many different types of learners. There are those that thrive working alongside you as you go through a problem, some that want a tutor to check their work, and then there are those that need a bit of guidance or a refresher on a concept.”

Albers echoes this approach. “The most important thing is to help them ‘learn how to learn’ math for themselves, instead of just showing them how to do one problem.”

The tutoring strategies are as diverse as the students themselves, and “meeting the student where they are,” is the standard operating procedure of the tutoring team. The objective of every interaction is to bring a deep commitment to unlocking each student's individual potential. Penfold encourages a culture of tutors commingling across disciplines, asking each other for support on esoteric concepts. Students from calculus and introductory courses are seen clustering together, facilitating an enervating—even fun—

"You'll see six to eight students sitting together,” she says, “some in calculus helping those in an entry-level course.”

The pandemic accelerated the Math Center’s evolution, prompting an online tutoring option that continues facilitating virtual support Monday through Saturday. As the U continues to grow its enrollment, Penfold insists on preserving the Center's uniquely personalized, student-centric approach. Penfold is often stepping onto the floor herself when every seat is taken and helping a student even as they are currently almost doubling

the size of the tutoring area and creating satellite centers like the one that recently opened in the Sutton Building to accommodate the number of students using the center.

“If a tutor has a question or encounters something shaky, they’ll ask another tutor. We have grad students, as well as undergrads— everyone talking about mathematics and working together.” Asked what she wants students to know about her facility, she immediately responds that she hopes they find out about the Center and that they then use it as

PUTTING FUN IN COMMUTATIVE ALGEBRA

frequently as needed.

It’s a tribute to the hard work and dedication of the Math Center team from the early days in old barracks to today’s open and accommodating facility with both online and in-person options, “This is probably the most difficult challenge faced as a tutor," says Brower. "The ability to change up your teaching methods on the fly is very important when helping a student that just isn’t quite getting it.”

The Math Center helps make that happen every day. <

THE WORD “FUN” IS A SUBJECTIVE ONE, BUT THAT IS HOW ANNE FAYOLLE DESCRIBES MATHEMATICS AS A DISCIPLINE.

A graduate student in mathematics at the U, Fayolle is a recent recipient of a multi-year scholarship from The Natural Sciences and Engineering Research Council (NSERC) which is Canada’s equivalent of the National Science Foundation’s Graduate Student Fellowship in the U.S.

MATH CREDENTIALS

As a PhD student working with Karl Schwede, Fayolle, who was born in France and grew up in Canada, clearly has her mathematics credentials. Before coming to the U, she studied first at McGill University, one of Canada's best-known institutions of higher learning and one of the leading universities in the world. There, she quickly developed an affinity for the independent learning model in which one is paired with a professor in a

one-on-one setting and in which both determine together what textbooks and papers they will study together. This conversational model of learning proved to be better than the lectureand-classroom-style model for Fayolle and helped solidify her desire to go to graduate school.

Following her bachelor’s degree, Fayolle returned to École Polytechnique Fédérale de Lausanne (EPFL), one of the most celebrated

public universities in Europe. As a master’s student, Fayolle studied with Carvajal-Rojas in commutative algebra. This involves working in modular arithmetic, in which addition works similar to time on a clock: 5 + 10 = 3. “The numbers working differently,” she says, “means one cannot use the traditional tools of calculus to study polynomial equations and the shapes they define. For instance, since the numbers work differently, we can't draw graphs exactly like we would over the real numbers. So our intuition derived from those graphs doesn't always work here. One has to rely on the underlying algebraic structures."

Those structures and the understanding they bring is one of Fayolle’s favorite parts of doing math, and it may speak to what Fayolle identifies as the “weird” aspect in commutative algebra, followed closely by the feeling of it being “fun.” The appeal is also philosophical. “You can get to the heart of why something works the way it does. I really enjoy the abstraction that comes with [commutative algebra]—trying to find the structure in abstract things.”

Part of her enjoyment in doing math is that singular moment when someone (or “some ones”) in the math

sector solves a persistent problem. “It’s [only] ‘hard’ until someone comes along and finds the right object or point of view of how things are working,” she says of breakthrough findings. “It’s suddenly less ‘weird’ because it makes more sense.” She explains that she’s been working in this positive characteristic realm for the past few years and is now used to it. Fayolle is especially interested in studying singularity theory, in positive and mixed characteristics and, fortunately, in Schwede has found a principal investigator/mentor at the U who “does cool math.”

EVERYONE CAN DO MATH

The multi-year NSERC fellowship will free up more of Fayolle’s time for research. Her ambition is to continue in academics as a post-doctoral researcher and then as faculty, if possible. “I like having stuff that has

more world impact independent from [just the study of] math. I think that pure math is intrinsically valuable, hard to justify by linking it to real world applications, but still necessary." This includes teaching.

“I think math is very scary to a lot of people," she says. "I personally think that everyone can do math. Everyone struggles, and I think that’s very important to emphasize when you’re teaching. I struggle in math. I don’t think struggling in math should be a barrier to doing math.”

In the meantime, Anne Fayolle continues in graduate school, sharing mathematics by organizing BIKES, the student commutative algebra seminar here at the U, as well as co-organizing an Association for Women in Mathematics conference later this year. She also skis on the weekends. “I was skeptical,” the Montreal native says, “when I first saw the [Utah “Greatest Snow on Earth”] license plates. But after I went skiing, I agreed.” It helps, she says, that it doesn’t get too cold and is not too icy.

“I think the license plates might be right.” <

HOW LONG CAN MENO PAUSE BE DELAYED?

by BRIAN MAFFLY

AT BIRTH, OVARIES IN GIRLS CAN CONTAIN ABOUT A MILLION TINY STRUCTURES CALLED PRIMORDIAL FOLLICLES, EACH OF WHICH CONTAINS AN EGG CELL.

As girls grow and experience adulthood, most of these follicles will die while only one follicle will survive each month to ovulate a mature egg. When the loss of primordial follicles is nearly complete, and only hundreds remain, women experience menopause, a time when menstrual cycles have ceased for 12 months.

New research that relies on a mathematical model developed by Sean Lawley, associate professor of mathematics at the U, indicates that it is possible to delay the onset of menopause, perhaps indefinitely, by implanting a woman’s own previously harvested ovarian tissue back into her body. This technique has already been successfully used to restore fertility in cancer patients.

This interdisciplinary work is a collaboration between Kutluk Oktay, a prominent physician, professor

of obstetrics, gynecology, and reproductive sciences and ovarian biologist at Yale University; Joshua Johnson, an ovarian biologist at the the University of Colorado School of Medicine; Jay Emerson, professor of statistics and data science at Yale; and Lawley, co-author of their study published in the American Journal of Obstetrics and Gynecology, or AJOG.

In the late 1990s, Oktay developed ways to harvest ovarian tissue from young cancer patients, freeze it, and then transplant it after she has undergone cancer treatments that would have left her menopausal and infertile. This technique is referred to as “ovarian tissue cryopreservation and transplantation.”

Oktay’s technique has enabled hundreds of cancer survivors to conceive and have children. It is substantially different from the common procedure of freezing eggs, which is effective in helping older women conceive through in-vitro fertilization but has no impact on menopause.

HOW MATHEMATICAL

MODELS ANSWER OTHERWISE UNANSWERABLE QUESTIONS

“A lot of the interest behind delaying menopause is fertility, but a lot of it also comes from the idea that functioning ovaries are better for a woman’s health,” Lawley says. “Menopause is associated with many health issues relating to cardiovascular disease, bone density, obesity, etc. Keeping ovaries functioning longer might delay or even prevent these health issues from starting.”

Lawley continues: “Math is being used to address the question of how long you can delay menopause and how that depends on different factors. We have developed a model of how ovaries age. The data come from a number of places, chiefly from primordial follicle counts inside ovaries.”

Oktay, the world’s leading expert in fertility preservation, wondered if the ovarian-tissue-cryopreservation-andtransplantation procedure could be applied to healthy women to forestall menopause and its negative effects.

"In the past few years, we’ve been developing mathematical models of how the ovaries age and what triggers menopause,” says Lawley. “It was extremely exciting when he [Oktay] contacted our group to see if our model could be used to help explore whether this procedure could be used to delay menopause."

It turned out Lawley’s model helped a lot. The new AJOG study concluded that the procedures Oktay pioneered for cancer patients would be likely to delay menopause in healthy women under certain conditions.

“We were faced with a number of important questions. The first is, will it work? Will it delay menopause and by how much?” Lawley says. “Next, how do you optimize the procedure? Are there age ranges that tissue should be removed? How does the number of follicles in a woman’s ovarian tissue influence how long the tissue will function?”

CALCULATOR

SHOWS HOW LONG MENOPAUSE CAN BE DELAYED

The team developed ways to address these questions using mathematical modeling during the AJOG study. This included the development of an online calculator that indicates how many years a woman’s menopause would be delayed by the procedure

according to modifications to four key data points: the woman’s age at time the tissue is preserved; the fraction of ovarian cortex tissue that is removed and preserved; the percentage of primordial follicles that survive; and the number of replacement procedures later in life.

Data in the paper and use of the online calculator show that all else being equal, and under certain assumptions, the younger the woman is when the tissue is preserved, the longer her menopause would be

recently published a separate study in Science Advances that showed that the timing of menopause in individual women is related to random gaps in the supply of growing ovarian follicles over time.

Understanding the causes of growing follicles and ovulation even after menopause can help develop strategies to reduce symptoms of reproductive aging and support further development of interventions like Oktay’s so that the symptoms and consequences of menopause are delayed or never arise.

delayed, from a median 19.4 years for a 21-year-old woman to 3.4 years for a 40-year-old.

“If ovarian tissue can be frozen under the age of 30 years, in theory, menopause can even be eliminated in some cases,” the study says. “However, the feasibility and safety of delaying menopause beyond age 60 need to be clinically evaluated.”

THE MATHEMATICS OF OVARIAN AGING

Lawley and Johnson, along with University of Colorado colleagues Nanette Santoro and Mary Sammel,

Testing and validating Oktay’s procedures on healthy young women and then tracking their ovarian function for decades would not be feasible without mathematical modeling which allows predictions to be made about how ovaries will behave in different women under different circumstances and to predict “best-” and “worst-case” outcomes.

"There’s this huge interest and demand in developing clinical procedures like this that hold the promise of improving the health of women," says Lawley. "Mathematics is perhaps the only way to really get at some of these questions in the short term and help guide the first steps towards clinical interventions.” <

A longer version of this story first appeared in @TheU.

DISTINGUISHED PROFESSOR

by TOM ALBERTS

DAVAR KHOSHNEVISAN , DISTINGUISHED PROFESSOR OF MATHEMATICS

A member of the Department of Mathematics since 1993, Davar Khoshnevisan became a full Professor in 2001 and was chair of the department from 2017 to 2023. His appointment to the rank of Distinguished Professor beginning July 1 is a recognition of his outstanding contributions to the research, teaching, and service missions of the department.

Khoshnevisan’s research in probability theory spans three sub-concentrations: mathematical statistics, multi-parameter random processes, and stochastic partial differential equations. These topics are mathematically sophisticated yet have immediate applicability in engineering and applied sciences. His work in mathematical statistics is used in high dimensional data analysis, while his research in stochastic partial differential equations has been used to model the intermittency properties of sunspot phenomena over time. His work on multi-parameter random processes led to the resolution of several long-standing conjectures on fractal properties of random fields, some of which dated back to the 1930s.

A prolific author of over 135 research papers that have been cited over 1,800 times, fully one-quarter of Khoshnevisan’s articles have appeared in Annals of Probability or Probability Theory and Related Fields, the most well-regarded journals in the field.

Additionally, Khoshnevisan is known as an excellent expositor. His textbook Probability is used in graduate-level probability courses across the country, and his monograph Multiparameter Processes is the standard in the field. His upcoming book Gaussian Analysis is intended for beginning researchers. He has also served in an editorial role for several prestigious journals.

Related to his publications, Khoshnevisan has received numerous accolades in recognition of his prodigious research output and his professional service, including in 1998 the Rollo Davidson prize. In 2015 he was made a Fellow of the Institute of Mathematical Statistics, and in 2020 he was elected as a Fellow of the American Mathematical Society. Then, in 2018 he was recognized as a Medallion Lecturer at the Institute of Mathematical Statistics’ annual conference.

Of special note is Khoshnevisan’s mentorship of young mathematicians. At least 50 of his research articles are

jointly authored with his postdoctoral fellows and graduate students. At the U, Khoshnevisan has directly supervised 14 postdoctoral fellows, 10 PhD students, and nine master's of statistics students.

Khoshnevisan has been continuously funded by the National Science Foundation since 1997, with his most recent grant extending into 2026. This is a remarkable achievement given the exponential increase in probability research during that period and is an extraordinary validation of his research program. Equally impressive is that he was able to maintain his research funding while taking on the additional responsibilities of department chair. In 2020 alone, while chair and managing the pandemic, he published five remarkable articles on spatial ergodicity of solutions to stochastic partial differential equations. Already this work is being recognized as fundamentally important in the field.

“I am thrilled that Davar Khoshnevisan has been selected as a Distinguished Professor for the 2023-24 academic year,” says Department Chair Tommaso de Fernex. “This is a great accomplishment and well deserved recognition.” <

SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS

LAST NOVEMBER, THE U’S STUDENT CHAPTER OF THE SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS (SIAM) HELD A MODELING COMPETITION FOR UNDERGRADUATES.

Teams chose one of three modeling problems that were based on realworld situations and worked together to analyze the problem, generate figures and results, and write a report about their findings.

The reward for the winning team was their registration fee, paid-in-full, for the COMAP Mathematical Contest in Modeling (MCM) in February 2024. The U’s SIAM student chapter hosted this competition to help participants prepare for the MCM and give

them confidence in their abilities as scientific modelers.

Two teams participated in the contest, one with two members, Maxwell Archibald and Jack Perry, and another team with three members, Joshua Villarreal, Garrett Iverson and Dominic Cinaglia. Archibald and Perry studied the population dynamics of honeybees and won first place with their thorough and well-organized report while Villarreal, Iverson and Cinaglia looked at CO2 levels and their correlation with climate change which won second place for their strong statistical analysis and informative report. The judges were so impressed with the quality of both teams' submissions they decided to fund both for the MCM in February.

The U SIAM student chapter is extremely grateful for the financial support of the Department of Mathematics, without which the chapter would not have been able to fund these five deserving students.

SIAM FELLOW

In late March SIAM announced the 2024 Class of SIAM Fellows. The group includes Aaron L. Fogelson, professor of mathematics, recognized for pioneering work on mathematical modeling and numerical methods for platelet aggregation and blood clotting. Through their various contributions, SIAM Fellows help advance the fields of applied mathematics, computational science, and data science. <

THE BEAUTY OF MATHEMATICS

by FRED ADLER

REASONS TO DO MATHEMATICS: IT IS IMPORTANT (SOLVES AN OPEN PROBLEM), IT IS USEFUL (CURES CANCER), AND IT IS BEAUTIFUL.

These good reasons are not mutually exclusive, and my own ideal, rarely achieved, is to combine all three. In case you are curious, the dull talk exemplified one of the bad reasons (it is hard), that I'll say no more about.

So what is this vaunted mathematical beauty? Is mathematical beauty the same as beauty in the arts and nature, or does it just happen to go by the same name?

Faced with a problem of this magnitude, poet and Distinguished Professor Katharine Coles and I decided to do what we do best. Talk about it. This year's Symposium on Science and Literature took on the idea of beauty, bringing together poet Claudia Rankine, physicist Brian

Greene, and neuroscientist/artist Bevil Conway for three days of discussion, April 10-12. As part of the preparation, we are jointly teaching a course this semester on the theme of beauty to a small class of remarkable students, half from math, half from English. The English students are facing the trauma of making sense of math and physics and attempting to see the beauty therein. The math students are facing the terror of making sense of complex poetry and attempting to see its beauty. And we are all taking on the collective challenge of reading philosophy to peek behind the curtain to ask what beauty is.

Before revealing the answer, I'll share some of the mathematical ideas we have discussed, largely following the charming The Joy of x by Steven Strogatz, inspired by his popular series for the New York Times online called "The Elements of Math.” Given the mixed group, the mathematics, in the spirit of Strogatz's book, is fundamental and not technical.

We began with an age-old question: What does the golden ratio have to do with rabbits? The golden ratio appears in geometry, describing the shape of a

sum of the previous two numbers, and the sequence can be generated by counting the population of immortal and fecund rabbits who produce babies every month and take just two months to mature. The beauty, we decided, lies in the unexpected connection of geometry and arithmetic.

The most elegant and venerable link between geometry and numbers is the Pythagorean theorem, that the sum of the squares of the sides of right triangle is equal to the square of the hypotenuse. Where do those squares come from anyway? I know three broad classes of proof. The first is rather pretty, involving drawing squares on the sides and hypotenuse and cleverly chopping them to get them to match. The second, which I came up with when I couldn't figure out how to do the first, is rather ugly, involving drawing lines, taking ratios, and doing a bunch of nasty algebra.

The best proof, which I had not seen before, was attributed to the teenage Einstein in one of the books we read for the class, A Beautiful Question by Nobel-prize winning physicist Frank Wilczek. It is based on what we mean by area. If you take any shape and make it twice as big by stretching equally in all directions, the area gets bigger by a factor of 4. That's where the squares come from— if you made the shape 3 times as big, the area would be 3^2 = 9 times bigger. Rather than building on tricky drawing or algebra, this proof requires adding just one line to the picture, and then thinking. In mathematics, beauty lies in deep simplicity. And, as in music and the arts, that kind of simplicity has to be earned.

I became interested in mathematics because of the magic of numbers. And large numbers have an allure all their own. The Fibonacci series, like rabbit populations, grows rather fast. But what if you want to write down really huge numbers? We can use the way that mathematical ideas build on themselves, recalling the progression of arithmetic in elementary school. Addition is repeated counting (6+7 = 13 means counting to six

(6*7 = 42 means adding up seven 6's). Exponentiation is repeated multiplication (6^7 = 279,936 means 6*6*6*6*6*6*6, multiplying together seven 6's). The numbers are starting to get pretty big. But

to really turbocharge, let's try repeated exponentiation. Donald Knuth invented "arrow notation" to handle this question. 6 7 is 6 raised to the 6th power seven times, or 6^6^6^6^6^6^6. There's really no way to say how big this number is. Even 6 3 has 36,305 digits written in decimal notation. But no matter how absurdly large these numbers become, they are still nothing compared with infinity. The beautiful has the sense of the inexhaustible, the beauty of a poem, the face of one you love.

During the course, we have touched on many other mathematical questions. Is the quadratic formula ugly, or does it have "inner beauty"? Is there a beautiful poetry behind the existential angst of probabilities? Will I

ever get over my prejudice against fractals?

Along the way, we've learned a few things. Good things happen when geometry and algebra get together. Beauty has an element of surprise, evoked by connections between apparently different things. Beauty arises when complexity meets simplicity and when simplicity meets complexity. Einstein was a beautiful and deep thinker. Keats was a great poet who evoked deep thoughts with beautiful words.

There is a toast attributed variously to G.H. Hardy and other famous mathematicians: “Here’s to pure mathematics. May it never be useful for anything!” The Enlightenment philosopher Immanuel Kant argues that beauty indeed must lie outside anything useful, attractive, or even morally good. But mathematics has the remarkable power to surprise us with beauty when it seeks to be useful, and with usefulness when it seeks beauty. <

Fred Adler is Professor of Mathematics and

REMEMBERING JOHN WARNOCK

AS A HIGH SCHOOL STUDENT AT OLYMPUS HIGH IN SALT LAKE CITY, CO-FOUNDER AND FORMER CEO OF ADOBE JOHN WARNOCK , WHO PASSED AWAY AUGUST 19TH AT AGE 82, FOUND A MENTOR IN MATH TEACHER GEORGE BARTON. “HIS APPROACH WAS REALLY QUITE SIMPLE,” WARNOCK ONCE RECALLED.

“He instructed us to pick up a collegelevel textbook for algebra, solve every problem in the book, then move on to the next subject, trigonometry, and do the same. And after that, go on to analytic geometry. By following his advice and solving a lot of problems,

The auspicious career of Warnock and other brilliant University of Utah alumni who changed the world through computer science was in high relief last year when a sampling of the scrappy and now legendary bunch assembled on campus to commemorate their roles as 3-D graphics pioneers. The occasion was a celebration of 50 years of the U’s Kahlert School of Computing, and Warnock was presented with an IEEE Milestone award.

But before he was known as the co-founder with the late Charles Geschke of Adobe, Warnock was propelled by his high school teacher into the U’s math department. There Warnock earned a BS and MS in mathematics before decamping to the College of Engineering where he earned a PhD in electrical engineering/ computer science. It was an exciting time. The U was one of 15 renowned universities that had a contract with the Advanced Research Projects Agency, prompted by the worrisome launch of the Russian Sputnik satellite during the Eisenhower era. A node on the original internet known

as ARAPNET, the U was the first university to offer online registration to its students, and Warnock, as part of his dissertation research was busy at work, just days (and long nights), ahead of when the portal dropped, having developed the recursive subdivision algorithm for hidden surface elimination that made computer graphics possible and that would eventually carry his name.

Twenty-five years post Sputnik, Adobe launched which, inarguably, lofted desktop publishing into the stratosphere with its soon-to-launch PostScript language. The information technology sector has never been the same since, epitomized by Warnock’s appeal to the U students during the 2020 commencement: "The rest of your life is not a spectator sport. Your job in life is to be an active player, to make the world a better place.”

Warnock is survived by his wife and three children. <

The Warnock Chair, endowed in 2001 by John and Marva Warnock, supports early career faculty in mathematics, attracting outstanding talent and enhancing departmental excellence. It is hard to overstate the impact that this has had, and will continue to have, on the growth and excellence of the Department.

FACULTY RETIREE JIM KEENER

by FRED ADLER

THE RETIREMENT OF DISTINGUISHED PROFESSOR JIM KEENER LEAVES AN UNFILLABLE HOLE IN THE WORLD-RENOWNED MATHEMATICAL BIOLOGY PROGRAM THAT HE DID SO MUCH TO BUILD.

Or at least would if he was not planning to continue to provide wisdom and insight to students and faculty alike from his office where the door (and minds) is (are) always open.

Jim joined the University of Utah faculty in the Department of Mathematics in 1978 and started a research program that set the standard for mathematical biology by combining creative modeling built on the devilish details of complex biological phenomena with development of the new mathematics needed to understand those models and answer biological questions.

cardiac arrythmias, and his perhaps even more influential work on ranking (which inspired the first version of Google's original PageRank algorithm) emerged from his interest in college football.

His three books, Principles of Applied Mathematics (1988), Mathematical Physiology (with James Sneyd, and winner of the 1998 Association of American Publishers' "Best New Title in Mathematics"), and the new Biology in Time and Space: A PDE Modelling Approach (2021) are paragons of exposition and laid the foundation for the rigorous and deep applied mathematics that underlie any serious quantitative study of biology.

Jim is the visionary leader for the U’s graduate program in mathematical biology. The success of this program reflects his own strengths:

collaboration, ability to communicate with both mathematicians and biologists, and the taste to identify and solve important problems. The reach of this program is extraordinary, with graduates holding leadership roles in research universities, liberal arts colleges, industry, and in the mathematical biology community. Students in the program today have a built-in international network of friends, colleagues and mentors.

Like most people who get this much done, Jim is fun to be around. His "cabin parties" are legendary, although what actually happens there can never be revealed. Dinner with Jim is a delightful mix of challenging debate and convivial laughs. His joy in inquiry and in people builds on a deep spirituality and love of nature that underpin the wisdom that we look forward to sharing for many

Thank you for making Math students a priority! Generous and thoughtful alumni and donors like you are helping students succeed in Mathematics! To meet the demand for STEM graduates in Utah, our top priorities include Math Department Scholarships and other Areas of Greatest Need.

Scholarships and hands-on experiences are more important than ever for our students. Your support can help them stay enrolled, focus on their studies, graduate on time, and avoid burdening debt!

Our goal is to provide direct scholarship support to mathematics students and to increase the graduation rate of math majors.

Every donation is important— even $25, $50, or $100 will make a big difference. Will you help Math students today?

We appreciate all that you do for students here in the College of Science. Together, we can reach our goal to create a brighter future for science and math education in Utah.

For more information about giving to Mathematics, contact:

DeGooyer