Thankfully for students in the Manresa Program, an integrated learning community in Rose Hill’s Loyola Hall, a class on everyday math takes place right downstairs. The first-year class is taught by Fordham College at Rose Hill Dean Maura Mast, Ph.D., and is based on the book Common Sense Mathematics (American Mathematical Society, 2021), which she co-authored.

“Our students come into this class knowing all sorts of amazing things about art, literature, science, math, chemistry, and yet the class is an eye-opener—even for the math students,” said Mast. “Not every family talks about practical finance or the context of interest rates for credit cards and loans.”

She said that most students come to the class with standard math knowledge from high school, but she asks them to approach the math in terms of important decisions they may have already started making, such as school loans. She also asks them to examine the motivations of institutions loaning that money to raise awareness of predatory practices such as undisclosed interest rates.

“These are the ways that I engage them and push them to think more deeply about things. [I tell them,] ‘There’s a formula that’s in there, don’t just let the bank do the math for you. Let’s go back and look up what that compounded interest is,’” said Mast, a recently named fellow at the Association for Women in Mathematics and specialist in the fields of differential geometry and quantitative reasoning. “Our consumer protections in this country are not all that great.”

The class also examines income distribution, such as the fair division of and access to financial resources, like mortgages.

“We look at why the median home price in Manhattan is a million dollars, and what that means in the Bronx,” she said.

The class also addresses how public policies such as redlining kept low-income Black and Latino families from acquiring property. It’s an aspect of the class that reflects the Manresa program’s focus on solidarity with the poor and marginalized.

The Manresa program is open to first-year Rose Hill and Gabelli School students and housed in Loyola Hall. Students take courses from a faculty member from Fordham College at Rose Hill or the Gabelli School who also acts as a core adviser. The program is supervised by First-Year Dean Robert Parmach, Ph.D.

“This is more than a random housing assignment—we are looking for students who are curious, creative, and open to new experiences,” said Mast, who also serves as a core adviser in the program.

One such new experience is creating a “mathematical autobiography,” which Mast uses in her class to ask students to examine their own math history.

“I ask them to tell me, ‘What role has math played in your life? How comfortable are you with numbers?’ Some have had a personal finance class in high school, but many are first-generation college students. I’d say they’re all over the map in terms of their experiences,” she said.

Mast encourages the use of technology in the class, from calculators to Excel spreadsheets that perform complicated formulas.

“Students are stressed in math class, especially when their previous experience hasn’t been good. I offer the technology as a way to say, ‘You don’t have to worry about doing arithmetic. Let’s focus on the answers and what the answers tell us,” said Mast.

One of Mast’s motivations for writing the book and designing the course was to further the goal of creating a mathematically literate public.

“I was determined that they have a positive experience of math and that they learn how important reading numbers is to them as citizens in our democracy,” she said.

“Being able to read a credit card statement and being able to read a New York Times article with numbers in a complicated context—we all need to be able to do that as responsible adults and members of our society. By the end of the course, they should understand that mathematics can tell us about how our society functions.”

“People don’t study number theory because it’s useful,” he said. “They study it because it’s beautiful.”

Raghuram’s research on number theory—a branch of mathematics devoted to the study of natural numbers and integers—has been recognized on an international scale. He has presented his work at the Max Planck Institute for Mathematics in Bonn, Germany, and served as a member of the Institute for Advanced Study in Princeton, New Jersey. His research has been funded by the prestigious Humboldt Research Fellowship in Germany and the National Science Foundation in the U.S. 

Raghuram is an accomplished mathematician in the U.S. and in his native India. He earned a bachelor’s degree in computer science and engineering from the Indian Institute of Technology at Kanpur (IITK) and a Ph.D. in mathematics from the Tata Institute of Fundamental Research. After holding postdoctoral positions at the University of Toronto, Purdue University, and the University of Iowa, he joined the faculty at Oklahoma State University, where he was granted tenure in 2011 as an associate professor. He went on to become the first chair of mathematics at the Indian Institute of Science Education and Research at Pune and a distinguished honorary professor at his alma mater, IITK. He is a fellow of the Indian Academy of Sciences and the Indian National Science Academy.

This fall, he joined Fordham’s faculty. In a Q&A with Fordham News, he spoke about his academic journey. 

Were you always good at math? 

I grew up in a middle-class home in Bengaluru, India. My father was a chemical engineer for India’s space research organization within the rocket propulsion unit. He and my mother emphasized the importance of doing well in school, where I was good at math. In eleventh grade, I won a state mathematics olympiad. I also did well on my entrance exam for competitive science and technology schools across India—I came second in the country. But I didn’t know you could have a career in mathematics.

How did you become a mathematician? 

At first, I majored in computer science. All the top-ranked students wanted to study it. Then I realized I enjoyed mathematics more than anything else. A professor mentored me and helped me read the right kinds of books. My parents were opposed to me studying mathematics because a career in computer science pays much more. There is some truth to that. Many of my former classmates are now millionaires who own companies in Silicon Valley. But for many mathematicians like myself, their career is a calling. 

In your new role at Fordham, you teach and advise undergraduates. How has that been going? 

This semester, I am teaching an intermediate linear algebra course to 23 students, ranging from first-year students to seniors who study mathematics, computer science, economics, and finance. The class size is just right. I can get to know the students on a one-on-one basis. At a public university, I once taught a course with 350 students, and it felt impossible to get to know them all. I also find Fordham an exceptionally friendly place. There’s this idea of New York City being a busy place where people are on the go, with no time for niceties. But Fordham is an antithesis to that.  

How would you describe your research? 

I try to understand all kinds of patterns within numbers. Sometimes they seem unclear and haphazard. But when you dig deeper, there are all these mysterious patterns. The first time I encountered this was with prime numbers. A number is prime if it has no divisors, like 5, 7, or 11. If you take the set of all prime numbers, you can see all the beautiful patterns that show up. 

You rely on prime numbers every time you use a credit card. The numbers in your 16-digit number and security code are encoded in a certain way. The process of taking a number and jumbling it up in a seemingly random way is actually systematic. Without the idea of a prime number, you can’t even get started on this subject. 

I don’t know if my research has immediate applications to things like Internet commerce. But who knows? There’s an essay called “The Usefulness of Useless Knowledge” by American educator Abraham Flexner, which says that knowledge drives and creates itself. Fifty years later, someone might realize that what we did is exactly the right tool that’s needed for some physical phenomenon. 

I once asked another Fordham math professor what his research process looks like. He said he brainstorms on his sofa. What about you? 

There is a little bit of lying on the couch and trying to meditate over the whole thing to see if I can assimilate all these various calculations. I think every mathematician has to meditate—it’s an intrinsic part of the process. But I also fiddle with numbers and equations on paper and on my iPad.

Over the past five years, I’ve been working on a 60-page manuscript for a research paper on number theory. In one section, I needed to prove a technical statement. It took me about six months of hard work and filling in hundreds of pages of calculations until I saw a pattern that explained how those things were behaving.

You helped to produce a documentary about Srinivasa Ramanujan, a pioneering mathematician, that has been screened across hundreds of schools and colleges in India. What was one of the highlights of working on this project? 

In the 1890s, Ramanujan grew up in a very poor place in India. Paper was expensive, so he wrote most of his mathematical equations on a small slate with chalk. He would write and erase until he found something worth recording on paper. Those sheets were bound together into three notebooks, now housed in a library. There is one scene in the documentary where another mathematician and I are sitting in the library, going through Ramanujan’s notebooks. Those notebooks are a national treasure. 

This interview has been edited and condensed for clarity. 

This semester, Farmakis, a native of Greece who called Brooklyn home, was teaching two sections of Math for Business: Calculus, a required course for first-year students in the Gabelli School of Business.

Melkana Brakalova-Trevithick, Ph.D., chair of the Mathematics Department, said Farmakis was an erudite, caring, hard-working, and devoted teacher. An accomplished mathematician and prolific writer, he split his time between Fordham and Brooklyn College, where he’d been teaching as an adjunct since 2007. He’d been an adjunct at Fordham since 2013.

“He had a unique style of putting different areas of mathematics together and making them accessible to a larger audience,” she said.

“His contributions and impact in the field were important to him. He loved what he did, and he did it well.”

Farmakis earned a B.S. in mathematics at the University of Athens in 1972. After completing further studies at the University of Paris, he worked in Paris, including a stint from 1980 to 1984 as technical director for the French newspaper Liberation.

In 1991, he moved to New York City, and in 2006, he earned a master’s degree in mathematics at Hunter College in 2006. He earned a master’s degree in philosophy at the CUNY Graduate Center in 2007 and two years later, he earned a Ph.D. in mathematics there. His dissertation was titled Cohomnological Aspects of Complete Reducibility of Representations (Lambert, 2010).

He specialized in algebra, differential geometry, topology, and lie groups and their representations. He and Martin Moskowitz, Ph.D., his dissertation adviser at CUNY, co-authored Fixed Point Theorems and Their Applications, (World Scientific, 2013), and most recently, they co-wrote A Graduate Course in Algebra: Volume 1, and A Graduate Course in Algebra: Volume 2 (World Scientific, 2017).

His son Vadim Farmakis, who was 3 when the family moved to the United States, recalled his father as man who was close to his family, both figuratively and literally, as Vadim and his older brother Oleg live in the same Crown Heights apartment complex as their father.

“He was a great mathematician, a great father, and a great grandfather,” he said. “That’s the most important thing.”

In addition to Vadim and Oleg, Farmakis is survived by three grandchildren, Graciela, Brianna, and Osiris Farmakis. The family will be holding private funeral services, and a burial will take at a later date place in Greece.

Fordham’s Campus Ministry and Counseling Center are available for students and staff. The Counseling and Psychological Services can be contacted at 718-817-3275 or 212-636-6225. Campus Ministry can be reached at 718-817-4500.

“There is no conflict between science and religious belief,” he once told a television interviewer. “My faith is enriched by my science, and my science enriches my faith.”

Father Coyne rejected the view that science is the only way to true and certain knowledge, but he also challenged religious fundamentalists and proponents of the intelligent design movement, which he said reduces God to a “dictator God … who made the universe as a watch that ticks along regularly.”

In 2005, he published an essay, “God’s Chance Creation,” in the Catholic magazine The Tablet in response to a New York Times op-ed by Cardinal Christoph Schönborn, a prominent proponent of intelligent design.

“God in his infinite freedom continuously creates a world that reflects that freedom at all levels of the evolutionary process to greater and greater complexity,” Father Coyne countered. In an evolutionary universe, he wrote, God, like a loving parent, “is not continually intervening, but rather allows, participates, loves.”

Similarly, he challenged his friend Stephen Hawking, the Cambridge University physicist who wrote in A Brief History of Time that when scientists finally come to understand the formation of the universe, “it would be the ultimate triumph of human reason—for then we would know the mind of God.”

Father Coyne described Hawking’s concept of God as “something we need to explain parts of the universe we don’t understand. I tell him, ‘Stephen, I’m sorry, but God is a God of love. He’s not a being I haul in to explain things when I can’t explain them myself.”

George V. Coyne was born in Baltimore, Maryland, in 1933. After attending Loyola Blakefield, a Jesuit high school in Towson, Maryland, he entered the Society of Jesus in 1951. “They threw out the fishing net,” he once joked, “and I didn’t know any better.”

At the Jesuit novitiate in Wernersville, Pennsylvania, a professor of ancient Greek and Latin literature helped turn him on to astronomy, providing him with books on the subject and a flashlight so he could read them in bed after lights out. “It was forbidden fruit,” Father Coyne told The Catholic Sun in 2012, “but it was good fruit!”

He continued his studies at Fordham, where he earned a bachelor’s degree in mathematics and a licentiate in philosophy in 1957. Five years later, he earned a doctorate in astronomy at Georgetown, where he studied the chemical composition of the moon. He was ordained to the priesthood in 1965 and soon after that joined a team of researchers at the University of Arizona who were mapping the surface of the moon—something Jesuit scientists had been doing since the mid-17th century. Their research helped guide NASA as it planned the Ranger missions and Apollo crewed missions to the moon during the 1960s.

Father Coyne later focused his research on the life and death of stars, especially binary stars, and the evolution of protoplanetary discs, among other topics. He published more than 100 articles in peer-reviewed scientific journals. In 2009, he received the George Van Biesbroeck Prize from the American Astronomical Society, and an asteroid—14429 Coyne—was named after him.

Brother Guy Consolmagno, S.J., the current director of the Vatican Observatory and a former Loyola Chair in Physics and Astronomy at Fordham University, was a graduate student at the University of Arizona’s Lunar and Planetary Lab in the 1970s when he first met Father Coyne. They later worked together at the Vatican Observatory.

“His instructions to each of us upon arriving at the observatory [were] simply: ‘Do good science,’” Consolmagno said at the funeral Mass for Father Coyne, held on Feb. 17 at LeMoyne College. “The science itself was the goal. And he gave us the space to make it happen.”

Under Father Coyne’s leadership, the Vatican Observatory expanded from its traditional base at Castel Gandolfo south of Rome to a second site on Mount Graham in Arizona, where in the early 1990s he presided over the installation of the Vatican Advanced Technology Telescope, or VATT. He founded the Vatican Observatory Summer School, a research academy for graduate students, creating opportunities for women to become affiliated with the observatory research staff for the first time.

From 2006 to 2011, he served as president of the Vatican Observatory Foundation, and for the past eight years he held the McDevitt chairs in religious philosophy and physics at LeMoyne College, where he taught courses in astronomy, cosmology, and the relationship between science and religion.

Father Coyne often described parallels between science and religion while maintaining that they are independent pursuits. The Bible, he often noted, predates modern science, which he saw as neutral, a discipline that doesn’t have theistic of atheistic implications in itself.

In a 2010, Father Coyne and Consolmagno were interviewed by Krista Tippett, host of the NPR show On Being. Father Coyne spoke about the temptation to imagine a “God of the gaps.”

“We tend to want to bring in God as a God of explanation, a God of the gaps. … Newton did it,” he said. “If we’re religious believers, we’re constantly tempted to do that. And every time we do it, we’re diminishing God and we’re diminishing science. Every time we do it.”

He said that his belief in God and his scientific knowledge led him to consider, “What kind of God would make a universe like this?”

“I marvel at this magnificent God,” Father Coyne said. “He made a universe that I know as a scientist that has a dynamism to it, it has a future that’s not completely determined. … This is a magnificent feature of the universe.”

“My knowledge of the life and death of stars,” he added, “leads me also to know that there’s a unity, a whole universe with respect to life and death. If stars were not being born and dying, we would not be here. The sun is a third-generation star. It was only after three generations of stars that we had the chemical abundance to make an amoeba, to make primitive life forms, and through that to come to ourselves.”

He said his religious faith, like science, is something that he worked at constantly.

“Every morning I wake up, I have my doubts, I have my uncertainties, I have to struggle to help my faith grow, because faith is love,” he said, and “love in marriage, love with friends and brothers and sisters is not something that’s there once and for all, and always kind a rock that gives us support. And so what I want to say is ignorance in doing science creates the excitement in doing science, and anyone who does it knows that discoveries lead to a further ignorance.”

Ultimately, he said, “Doing science to me is a search for God, and I’ll never have the final answers, because the universe participates in the mystery of God. If we knew it all, I’d sit under a palm tree with my gin and tonic and just let the world go by.”

“It’s an honor to be recognized for the work I’ve done advocating for women to be successful in mathematics—and, more broadly, for diversity,” Mast said.

Mast was among 18 U.S. scholars recognized for increasing the visibility and success of women in the mathematics field. 

As a mathematician and a woman, Mast said she knows firsthand the challenges that women in STEM face. She was one of the few female graduate students in her mathematics Ph.D. program at the University of North Carolina in the 1990s—and the only one in her original class to finish the program, she said. She also recalled people who told her, “You’re not going to have a problem getting a job because you’re a woman. Universities are supposed to be hiring women now.” 

But thanks to the work of Mast and her colleagues, women are getting closer to achieving parity with their male counterparts. 

Over the past two decades, Mast has promoted the participation of women in math through her leadership in several organizations. As co-chair of the Joint Committee on Women in the Mathematical Sciences, she designed panels for women in math, including a panel on balancing professional and family life. As a member of the AWM executive committee, she helped mitigate implicit bias in the honors and awards processes in the mathematics community. With three colleagues, she co-edited the book Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America (Springer International Publishing, 2017), which celebrates the contributions of women in mathematics. 

As the first female dean of Fordham College at Rose Hill, Mast has mentored women in STEM student groups. Last fall, her office secured funding to create the ASPIRES Scholarship program, which provides mentorship and monetary support to underrepresented students in STEM—including young women. A few months ago, she became the director of the Clare Boothe Luce Program at Fordham, which provides scholarships for outstanding women undergraduate students and graduate fellows in the sciences. 

“I’m very excited to work with the other deans and with the faculty to strengthen the support that we give to the Clare Boothe Luce scholars and to create a stronger community for women in STEM at Fordham,” Mast said. “We’ve got some amazing scientists and mathematicians here, and I’m really excited about bringing them together so that we can be even stronger.”

Mast will be honored at the AWM Reception and Awards Presentation in Denver on Jan. 16, 2020. 

Hein, a Princeton-educated mathematician who moved to the U.S. from his native Germany more than a decade ago, is the first Kim B. and Stephen E. Bepler Chair in Mathematics at Fordham University. Since his appointment last fall, he’s had more time to pursue his research in differential geometry, or geometry in any dimension. 

On a daily basis, he develops methods and equations that lead to new shapes. These shapes can surpass three-dimensional space. It’s impossible to sketch some of them. But the point of his papers, often with titles as abstract as “A Liouville Theorem for the Complex Monge-Ampere Equation on Product Manifolds,” is to explore uncharted territory in the realm of mathematics and develop new ways of thinking that can describe complex phenomena like black holes, though perhaps only decades or centuries from now. 

“[Mathematicians] try to figure out patterns, describe certain things that they observe, purely within math,” Hein said. “These methods and equations have a life of their own. They exist abstractly, without any specific application. And then 20, 30, 50 years later, it may turn out that this is exactly the right kind of math that you need to describe something that actually exists in the real world—like gravitational waves or black holes.” 

How did your love for math begin? 

There used to be these TV programs in Germany for people who didn’t finish high school or wanted to brush up on high school material before they went to college. I started watching the trigonometry program, just out of curiosity, when I was 9 or 10. I liked the shapes. They were explaining how to graph sin and cosine. I sat down after the lesson and tried to recreate that on paper on my own. And I got a shape that looked like the thing that I saw on TV. 

How do you define mathematics? 

Simple ideas that solve problems, that, in the end, are correct. It doesn’t depend on anybody’s opinion. It’s some pattern or idea that’s going to be correct a thousand years from now, if humanity still exists. 

Your branch of math is differential geometry. What does that mean? 

There’s the more elementary stuff, like basic differential geometry that actually happens in three-dimensional space, that actually exists in the real world, that engineers and physicists use all the time. Then there’s my work—the rarified, cutting-edge stuff in theoretical math.

What does it look like for you to do research? It’s obvious for chemists and biologists. They have microscopes, petri dishes, beakers—things like that. But for you, a mathematician, how does that work? 

I lie on the couch all day. I imagine shapes and connections between shapes and quantities and try to figure out if some quantity is going to be large or small—how different quantities interact with each other. It’s a little bit like art in the sense that you create shapes and patterns. And then if I have the complete picture in my mind, I’m usually able to see the solution.

My wife is a mathematician, too, so at least it’s not weird for her. She knows what’s going on … that I’m actually working. 

When you’re brainstorming, do you map out your thoughts on a blackboard? 

No. It’s just in my head. If I’ve really thought something through, I can just go to my laptop and write like 10 pages of equations and formulas and arguments and reasoning, based on what I have been imagining. Sometimes I have to do some calculations on paper, but that usually comes later.

What’s the most difficult part of your research?

That you’re stuck constantly. You don’t know what you’re doing most of the time. It’s not like you’re applying some method that you learned in grad school, and you’re trying to use that to create something new. I mean, sometimes it’s like that. But more often, you’re working on some problem that nobody’s really thought about before—that certainly no one has ever solved before. What that usually means is that the methods that exist aren’t sufficient enough to solve that problem. 

So you have to come up with new methods? 

Right. Usually it’s a tweak on some method that you learned in grad school or from somebody else’s paper. But, you know, once in a while, you have to create something completely new. 

To a non-mathematician, how do you explain the importance of your work? 

This kind of math that I do is incredibly abstract. Right now, nobody knows if it’s ever going to have an application to anything real. Much of the math is developed completely independently of any applications to physics [for example]. We often create new ideas for their own sake. And then [decades or even centuries later]it turns out to be exactly the right math that’s needed to make sense of things like quantum mechanics. 

You discover these new beasts, specimens. You can see them in your head. Somehow, they’re out there.

This interview has been edited and condensed for clarity.

“Ed, aren’t you a math teacher?” he asked.

Yes, said Burger, president of Southwestern University, a mathematics professor, and the author of a new book called “Making Up Your Own Mind: Thinking Effectively through Creative Puzzle-Solving,” which was the subject of his presentation at Fordham College at Rose Hill on Oct. 3.

The boy asked him for help. Burger scanned the math problems. Try the question about donuts, he suggested. But Seamus was stumped.

“He was, basically, intellectually constipated,” Burger remembered. “He was pushing and pushing, and not one idea would come out.”

So they tried something different.

“When I say go, I want you to give me an answer that you’re confident is wrong,” said Burger.

“16!” Seamus immediately said.

The boy was wrong, but he was close. And that was the point. Instead of just sitting there, trying to solve a problem he couldn’t, he took action. He ignored the pressure of getting the right answer, got a wrong answer, and learned from his mistakes. And once he did, the right answer wasn’t far behind.

“Effective failure is how you respond to failure. So the failing itself—not that interesting or important,” he said. “It’s what you do next.”

Solving Real-World Problems through Effective Thinking

This concept—intentionally failing, and then gaining a different perspective that allows you to grow and react in a meaningful way—was the foundation of Burger’s presentation, “Making Up Your Own Mind: Thinking Effectively through Creative Puzzle-Solving.”

“Now, to be clear, I’m not suggesting that you do this on your final exam,” he told the students at the lecture. “But it’s an intermediate step.”

The lecture, co-sponsored by the Rose Hill dean’s office, the math department, and the Graduate School of Education, was about how to think more effectively and creatively in our daily lives. Burger’s advice, he said, could apply to all aspects of life: work, academic, personal.

Failing effectively was just one part of a three-part formula that Burger developed and described to the audience gathered in Keating Hall.  

Understand simple things deeply. Don’t try to understanding something that, at first glance, seems very complicated, Burger said. Focus on something simple that’s related to that complicated problem. Examine it at a “subatomic particle level,” until you’ve noticed some feature that didn’t register before. “You’ll begin to understand that simple thing deeper, and then the more complex thing that you were first struggling with becomes easier to think about.”

Add the adjective. When you approach a math question, your first instinct is to solve it, said Burger. That can be a bad approach if you’re not fluent in the material. “Back up. Force yourself to describe it. Add as many descriptors as you can to describe the thing that you’re looking at. And every word that you add is going to reveal something you otherwise wouldn’t have seen,” he said.

Fail intentionally. Failure is good for you, he emphasized—as long as you gain some new insight from that failure, and then respond in a thoughtful way.

The audience was riveted. “How did you come up with this stuff?” asked one man.

It was an idea that was more than 20 years in the making, said Burger. He integrated these three steps into a class he developed at Southwestern University, where he gave students puzzles on which to practice this mindset, and eventually encouraged them to apply it to their own everyday lives.

After explaining the three-part approach, Burger quizzed his audience with a problem from his new book.

On a projector screen, he presented pictures of three black-and-white chess boards. The first one was normal. The other two were truncated: the second board was missing its northwestern and northeastern corners; the third one was missing its northwestern and southeastern corners.

“Imagine that you have a whole bunch of dominos, and each domino will cover exactly two squares,” he said. “And so the question is, can you cover each chess board with dominos so that every domino covers two squares, every square is covered, and no square is covered by two dominos?”

For two minutes, he said, apply those three practices to a puzzle. Brainstorm with the strangers next to you. And don’t actually solve the puzzles. Practice this way of thinking, and see where it leads you.

Each chess board had an even number of squares. The first board had 64.

“Sixty-four is a big number. What if I thought about a smaller chess board?” Burger said. “So understand simple things deeply.” Using a black ballpoint pen, he sketched the simplest version of the normal board—a two-by-two version.

And then, after drawing mini versions of the other two boards, an audience member “added the adjective.” The chess boards, said the audience member, were “white and black.” In a regular chessboard, a Domino needs to cover a white square and a black square, Burger reasoned. Then the audience noticed something different: the colors of the tiles that were removed. In the second board, a black and white tile were removed. In the third board, two black tiles were removed. Thus, the third board would not work.

“Now you realize that the missing squares in the second case were a different color, and the missing squares in the last case were actually the same color,” Burger said. “Just within a matter of minutes, you’ve looked at something that you’ve seen your whole life in a different way.”

“Just using those three things, you now see a chess board differently. And in some sense, I think that’s a metaphor for what we have within ourselves—the power we have through our own thoughts, when harnessed effectively, to see everything differently,” he continued. “And more magically.”

Peter M. Curran, Ph.D., FCRH ’47, a professor of mathematics at Fordham for nearly five decades, died on April 18. He was 92.

Curran, a native of Queens, began his studies at Fordham in 1943 and graduated summa cum laude four years later; he was first out of a class of 320 graduates. He began teaching immediately after graduation and went on to earn a master’s and Ph.D. at Columbia University.

He served as assistant chairman of Fordham’s math department from 1967 to 1972 and chair from 1979 to 1984. He was a member of the University Faculty Senate’s tenure review and appeals committee and served on numerous committees concerned with the University statutes, faculty memorials, and faculty hearings.

Curran was awarded a National Science Foundation Science Faculty Fellowship for 1961–1962 and was appointed a member of the Institute for Advanced Study at Princeton for the fall term of 1968. His research specialty was in infinite group theory and cohomology.

In 1987, the University honored him for 40 years of service with a Bene Merenti medal; he was cited for his unwavering loyalty to the University and its students.

“As a teacher, mathematician, humanist, friend, Pete Curran in 40 years here has earned the respect and affection of students and colleagues,” the citation read.

He retired in 1992 after 49 years at Fordham. In 2006, the department established the Peter M. Curran Visiting Assistant Professorship, a two-year appointment offered to recent Ph.D. graduates who demonstrate strong research potential and success in undergraduate teaching.

Janusz Golec, Ph.D., current chair of the department, said Curran, whom he pegged a “human being of high caliber,” and “a true gentleman,” was an clear choice for the honor at the time.

“It was just a quick decision; we didn’t have to deliberate on it. It was obvious to us,” he said.

Armand Brumer, Ph.D., a professor emeritus of mathematics who worked for nearly 20 years with Curran, said he’d miss most the spirited yet respectful political debates he had with Curran, who, he noted, had an uncanny knack for remembering students he’d taught decades before.

“I had trouble remembering the names of everyone in my class currently, but he would remember people from his classes from 20, 30 years ago. It almost felt as if he had a photographic memory,” he said.

“He certainly cared. It was awesome to hear him say ‘Oh, this is a student from 1955.’”

Curran’s dedication to the University was undeniable; he visited the campus as recently as 2015 to hear one of his former students, Frank Connolly, Ph.D., FCRH ’61, professor emeritus at Notre Dame, speak. Connolly called him a linchpin of the department whose reputation for helping undergraduates who were serious about math came with a “quiet and wicked sense of humor.”

“His political outlook was a bit to the starboard side, but it was far more gently enunciated than is the current fashion,” he said.

“In the late 50s he spent a lot of time in informal seminars with mathematics majors.

He was very generous with his time that way. I was one of the students in those seminars, and I am greatly indebted to him.”

Curran’s niece Patricia Chimento said he had a heart of gold.

“He was the primary caregiver for both his sickly mother and brother during the last years of their lives. He went out of his way to help family, friends, former students, and members of the church and community,” she said.

“He was always available to tutor, give advice, and provide service to those in need. He will be greatly missed.

Curran is survived by Chimento, her husband Jim; a nephew, Brian Hummel; and six grand nieces and nephews.

Experience First, Vocabulary Second

Mathematical jargon is a huge turnoff to students, especially in New York City where English is a second language for many of them. If I tell you what a “slope” is, and I make you memorize its definition, that’s not engaging. But say I give you a task, and in trying to solve the task you need to interact with the concept of “slope?” If you’re standing in front of the class and you’re stumbling trying to describe this concept, that’s when I give you the vocabulary behind the concept. It’s a real turnoff to come into a classroom and get a bunch of vocabulary that you don’t understand, and can’t even conceive the purpose of.

Put Students in the Center of Learning

Students already come to the classroom with a lot of mathematical knowledge and understanding of numbers. By putting them at the center of learning, and trying to elicit information and ideas from them, we get a lot more engagement. You can do this is by engaging in rich tasks that have applications in the real world, and having them generate questions about the situation.

If you’re working with younger kids, use something called a three-act task: You show an intriguing video that begs some questioning and ask students to generate a list of questions to investigate. It could be as simple as a video of a football player running with a football down the field. ‘How fast are they running?’ ‘How many yards did they cover?’ One of my favorite activities to do with little kids is to watch a video of Cookie Monster eating cookies from a box, and getting them to answer, ‘How many cookies did he eat?’ You want them to mathematize the world, instead of imposing mathematics onto their world.

Play Games

Most good games have a lot of mathematics inherent in them. If you’re talking to students about probability and expected value, you could look critically at the game Risk, and talk about different situations in the game and when it makes sense to attack or not. If you’re working on multi-step problems, and you’re trying to get them to think ahead, play a game like Nim or even chess, in which thinking steps ahead is a strategy to help you win. Appeal to their competitive nature. Give them a challenge. Let their desire to win be the motivation behind their mathematics.

Work in Groups

In math class, anxiety is very real. By allowing students to work in groups, we give them resources in each other. We give them models for people who already understand it, and we also give them a little bit of camaraderie because they’re inevitably going to work with people who don’t understand what’s going on. We know that people solve problems better in groups, and we know from research that they solve problems better on their own from having had a group-solving experience.

In the real world, people do math in teams. In the real world, programmers and scientists work in teams. The only time you really do mathematics by yourself is in these contrived educational settings.

Have a Problem of the Week

Problem solving is central to the study of mathematics, and it’s a big focus of Common Core. In order to keep kids engaged and challenged, try posting a non-routine problem somewhere in the classroom at the start of each week. If students are ever feeling unengaged or just want a challenge, they can get this problem and work on it. The problem of the week can help at the end of an activity when some students are finished and others are not. It gives students who are excited by mathematics an avenue in which to express that excitement. That creates a classroom culture in which students see others engaging in public extracurricular mathematics—which is a very powerful thing.

Six years later, however, McCauley has embraced a different calling. The Houston native is graduating with a major in mathematics and a minor in economics. This year, he realized he had a knack for computer coding—and the response he received at a recent interview at Google has convinced him to pursue it as a career.

He discovered an interest in coding after spending his junior year studying abroad at the London School of Economics. Even though he was able to line up several internship interviews for positions in management consulting, nothing came of them. He found himself with no plans for the summer.

So he turned to his thesis adviser, David Swinarski, Ph.D., assistant professor of mathematics, to see if he could assist in any research. With a summer research grant, McCauley created an analysis suite using raw data from motion capture technology that Columbia Presbyterian Hospital doctors are using to study breathing.

“Breathing looks very different in a person with emphysema compared with a normal person,” he said. His resulting research shows how those differences can be expressed mathematically.

“When I started this project, I realized I could take it as far as I wanted to,” he said. “It ended up being a much more valuable experience, from the standpoint of developing skills and having some experience, than a management consultancy would have been.”

As his interests have evolved from theater to finance to coding, McCauley  also feels like he’s learned how to relax more. A 4.0 student in his first semester, he joked that he felt like he learned more during some of those subsequent “3.75” semesters. His four years at Fordham also afforded him the chance to form long term relationships outside his immediate family. This year, he shared an apartment in Harlem with two fellow honors students who he first met his freshman year, and spent a great deal of his free time at a poetry collective that one of them started.

The college’s small size and liberal arts focus made it the perfect fit, he said.

“If you put yourself out there, you have direct access to everybody—from the professors all the way up to [FCLC Dean] Father Grimes,” he said.

“People really express an interest in the individual student’s well-being.”

In the beginning, he focused on the mathematical theory behind experiments that Fredric Cohen, Ph.D. and Robert Eisenberg, Ph.D., physiologists at the Rush University Medical Center in Chicago, had been working on— until they advised that he move from theory to “reality.”

“They said, ‘this is what the experiments say. This is what actually happens in the experiments when we change conditions,’” said Ryham of his collaborators. “’Your mathematical description must account for these effects.’”

According to Ryham, if researchers know precisely where and how a particular virus and its machinery expend energy, it may be possible to develop treatments that interrupt the delivery of genetic material by that virus and prevent infection.

Last year, he partnered with Cohen and other researchers from the National Institutes of Health to publish results of a funded study about the properties of influenza virus haemagglutinin. The study, which was published in Nature Microbiology, explores how a deeper understanding of the structural details of membrane fusion machinery can help to effectively combat influenza.

While there have been several mathematical models that estimate activation energies, Ryham helped to identify a transition pathway that previous physicists had missed in their explorations of viral fusion: a new structure called ‘lipidic junction.’ The researchers made further headway in the study by quantifying the activation energy for this new pathway of membrane fusion, which had never been done in this capacity, he said.

Ryham calculated the minimum-energy paths necessary for membrane fusion. He said an approach to calculation called the “string method” allowed him to provide a mathematical explanation for what the biologists were observing under the microscope.

“The thing that they were seeing, which was totally unexpected, was that when the virus attaches itself to the cell or fuses with the cell, it doesn’t follow the conjectured pattern,” he said. “It seems to break open and re-attach in an abrupt process. You wouldn’t suspect that nature would permit something so abrupt.”

The experiments also revealed that the activation energy for membrane fusion was significantly lower than what researchers previously predicted, which has biological significance. Another unexpected outcome was that the amount of cholesterol in the cell governed whether the event was an abrupt or smoothly transitioning process.

“It just reinforces the idea that cholesterol is a control parameter for things occurring inside the body,” he said. “In this case, it was for influenza.”

Related work on the subject deals with gaining entry into a cell—but without the use of viruses.

“The application of this study is more toward pharmacology,” he said. “The idea is that you want to load up minute spherical vesicles with a drug, launch them in the bloodstream, and then release the drug in a controlled fashion. Having an equation describing how and when the vesicle is going to burst is part of what makes these treatments possible.”

Ryham said mathematical models like the ones he develops, whether big or small, could contribute to breakthroughs in medicine in the future.

“Mathematicians are often motivated by questions in basic science and research something because it’s interesting,” he said. “But it may end up having an application down the road, and you never know what that application is going to be.”