“Liquid likes to lie flat, so what nature does with these particles that are distributed in the liquid is to lump the objects together so the milk can avoid having to crawl up too many things, that minimizes the effort that the liquid has to make,” said Rolf Ryham, Ph.D., associate professor of mathematics at Fordham College at Rose Hill.

Ryham has been working in computational biology for 20 years and he recently received a Fordham Faculty Fellowship to explain these phenomena on a microscopic level. The research will have implications for understanding molecular functions in biology and “the design of novel self-assembling materials,” he said, like the Cheerios in the bowl or, say, oil particles on the surface of water. It can be used when studying proteins in an aqueous cellular environment or in the field of microfluidics, one application of which is separating plasma from other cells in fingerstick blood sampling.

One way to describe the phenomenon of self-assembly in computational biology is through a hybrid approach that Ryham is pioneering. The hybrid employs both a particle-based approach and a continuum approach.

“In the particle-based approach, I can say, ‘I’m going to take my microscope and look deeply into this substance and resolve what these molecules are,’ and in some cases, I don’t necessarily have to resolve the individual molecules, instead I can lump them together and call that a coarse-grained particle.”

A continuum approach is the opposite. In a continuum, mathematicians use differential equations to describe a continuous substance like milk, water, or oil. Liquids have certain physical properties that can be represented by mathematical formulas.

The hybrid that Ryham and his colleagues are working on merges the continuum and the particle-based approaches. To visualize the problem, Ryham returned to his breakfast bowl.

“The particles are the bits of cereal and the continuum is the surface of the milk,” he said. “The interaction is what we’re studying.”

Beyond the Cereal Bowl 

Another example beyond the breakfast table would be two boats that are moored next to each other in choppy waters and inevitably drift toward each other until they collide.

“In between the boats the water is not choppy, because the two boats shield that area from the waves,” said Ryham. “Again, nature says, ‘Well I don’t like having this patch of water that’s not choppy. I’m just gonna get rid of it and push these boats together.” 

All of biology is a delicate balance between these forces, he said, some of which are understood and some of which are less understood. In the 1960s and 1970s, chemists came to identify the phenomenon as hydrophobic forces.

“If you take two hydrophobic substances, like oil on water, you put them together and the oil particles become attracted and come together like the boats in the choppy water,” he said. “The chemists measured this and noted that there was an exponential curve describing the force of attraction. It turns out that the simple representation they came up with could be generalized to objects of arbitrary shapes—like Cheerios. So, we’re trying to calculate that attraction.”

Ryham co-authored a paper with on the subject with Szu-Pei P. Fu, Ph.D., the Peter M. Curran visiting associate professor of mathematics at Fordham. Their article was published this past winter in the journal Multiscale Modeling & Simulation.

A Computational Challenge

In mathematics, these forces of attraction are given voice through “functionals” which are calculated using numerical simulations. Ryham said the computational challenge has been to reduce the time it takes to model a simulation from months to days.

“That’s the thing that keeps up at night, when is this simulation going to finish? How can we make it faster? Then it becomes a question of algorithms and mathematical shortcuts,” he said. “If you’re studying 10 particles it will take one hour to compute, a hundred particles take 10 hours, a thousand particles take a hundred hours.”

One of the mathematical shortcuts the group is using is called an “integral equation method” that allows the simulations to be completed much more quickly.

It’s akin to having many computers simultaneously working on the same problem at the same time through the use of parallel computing or using several graphics processing units. This could potentially turn that one hour into one minute enabling much shorter simulations times, he said.

Merging Science and Math

Ryham said biologists understand the computational challenge but don’t they necessarily want to take it on. Similarly, mathematicians don’t always know what the units in their simulations represent. Ryham’s interests, however, lie somewhere in between.

“You have to be a little bit diplomatic when you go between those two spheres,” he said.

He first became attracted to the sweet spot where math meets science when he worked with a group of physiologists on a problem called membrane fusion.

“These were people who understand the biology and the mathematical language and so we could talk to each other and they gave me a huge appreciation for the complexity of biology,” he said.

The physiologists opened his eyes to modeling challenges in biology that he had trained for as a mathematician.

“It’s the first time that I saw, ‘OK, so I’ve learned all these computational techniques, here is actually a testing ground where we can see how our methods work out,” he recalled.

He noted that computational methods are being developed all the time. But the mathematicians, like many academic communities, can be insularly focused on problems in their discipline.

“That’s not a critical statement if you want to be an expert in something you really have to focus—and maybe that’s the right approach—but I like to see other things as well.”

Titled, “Comets, Craters, and Calendars,” the lecture celebrated the 480th anniversary of the birth of the great mathematician and astronomer Christopher Clavius, S.J. Father Mueller is the vice director of the Vatican Observatory and superior to its Jesuit communities in Rome and Arizona. Following the talk, a panel of Fordham mathematicians and scientists shared how Clavius influenced their lives and work.

Champion of Mathematics

For the lecture, Father Mueller took listeners on a tour through the history of a man who, among his many accomplishments, is perhaps most noted for championing and helping design the Gregorian calendar that is still in use today. But the calendar, famous as it may be, was not the primary focus of Father Mueller’s talk. Instead he equated the scene from 2001 to Clavius’ championing of mathematics.

“In my view, Clavius played a historical role somewhat akin to that of Kubrik’s monolith: Through his teaching and textbooks and curricular reform, Clavius was the midwife of one of humanity’s great leaps forward: the scientific revolution,” he said.

Father Mueller was careful to point out that Clavius, who was born in Bamberg, Germany, may not have played an “active first-person role in the scientific revolution of the first half of the 17th century,” as he died in 1612. But through his many years of teaching at the Jesuit Roman College and through the wide use of his textbooks on mathematics and astronomy, Clavius greatly influenced the innovators and discoverers. He also “amplified” the importance of mathematics at universities, Father Mueller said, which theretofore had been taught at the 16th-century equivalence of high-school level.

“In the 16th-century curriculum, mathematics did not have the high status of a science,” he said. “A science was understood to be a discipline capable of knowledge of true causes. Mathematics was held to deal not in true causes but in mere abstractions—in mere possibility rather than reality. Mathematics teachers and faculty members were seen as being of lower status—they didn’t have much of a say.”

Forming a Pedagogy

Having joined the Jesuits in 1555, Clavius was received into the order by St. Ignatius himself, placing him amidst the “explosive growth” of the young order and its ever-expanding roster of Jesuit schools. Near the time of Clavius’ death there were 372 Jesuit schools that would have been the equivalent of junior colleges today. Even outside of the Jesuit institutions, from the 1580s to the 1640s “just about everyone got their introduction to math and astronomy from textbooks written by Clavius,” said Father Mueller.

Father Mueller noted a few examples of Clavius’ fame, which he cemented through his role as the official public explainer and defender of the calendar.  As a young professor, Galileo relied on lecture notes obtained from Clavius’ math and astronomy class, which led him to conclude Clavius “worthy of immortal fame.” During Clavius’ own lifetime he was known as “the Euclid of our Time.” And at St. Peter’s Basilica in Rome, there are only two Jesuits who are portrayed: St. Ignatius of Loyola and Clavius, who can be found on the tomb of Pope Gregory XIII, presenting the new calendar.

In Part VI of his Constitutions for the Society of Jesus, St. Ignatius called for a separate document to provide guidance for Jesuits schools. That “separate treatise” eventually became the famous Ratio studiorum of 1599, and Clavius held significant sway in its treatment of mathematics, said Father Mueller.

“An earlier 1591 draft of the Ratio includes an admonition to university administrators to make sure that philosophy teachers do not disparage the dignity of mathematics,” he said. “In a document written as a contribution to preparing the Ratio, Clavius wrote: ‘Since … the mathematical disciplines in fact require, delight in, and honor truth… there can be no doubt that they must be conceded the first place among all the other sciences.’”

Post-Lecture Q&A 

Father Mueller said that both the new Gregorian calendar and Clavius’ proposals for an augmented role for mathematics in the curriculum met with significant resistance, and in both cases Clavius took on a “prominent role as explainer and defender”—of the “validity of the calendar and of the dignity and worth of mathematics as a path to objective truth.” It’s a role that resonates to this day, with science and truth under attack at the highest levels of government and religion. In a conversation after the lecture, Father Mueller drew parallels between Clavius’ role and that of contemporary Jesuit institutions.

In your talk, you noted that at the time of his death Clavius had not finished publishing his theory of astronomy. It sounds like he was very cautious with the truth.

At the end of his life, he mentions in a letter to one of his colleagues that he’s in the middle of writing a theory of astronomy, but he also mentions explicitly that he’s waiting for more data from [Danish astronomer] Tycho Brahe, to help him know what he wants to do. He never ended up publishing it in his lifetime. I personally think what happened was he saw that astronomy was changing so rapidly, the time was not right to write a theory.

Last spring’s Clavius lecture dealt with artificial intelligence. Things are moving so quickly in that sphere that peer-reviewed articles are almost an impossibility. What would Clavius make of that?

Be careful about imposing the whole notion of peer review back on him. His time is where the whole idea that empirical observation of the heavens can affect astronomical theory. That itself is a new idea. You knew your astronomical theories from ancient authorities. That’s part of what got Galileo in trouble, but I think Clavius was doing it more cautiously. The whole thing with Galileo blew up after Clavius died, so he had no notion that there was theological danger. But, I think, as a teacher, he didn’t want to go in print with a textbook with stuff that’s not right. In the late 1500s there was an extraordinary series of comets and novas in the heavens, which really wowed people because physics at the time said that the heavens should be immutable and unchanging. Clavius participated in proving this stuff shouldn’t be changing. So, I think that’s part of what slowed him down. He says at one point, ‘Maybe we need to rethink at a fundamental level, our astronomy.’ Today, we all agree what the standards are. Clavius was in a time when the very notion of standards themselves were changing.

Clavius had the ability to speak to the public and the men in power to convince them of truth and the importance of empirical standards. What lesson we can draw from that today?

Clavius had to explain and defend. I hope I don’t sound too despairing here, but he lived in a time when everyone knew and believed that there was such a thing as an objective truth. They might have disagreed about how to get there, the standards were changing, but no one disagreed that there is objective truth. We are now in a very strange time, a new time, where for many reasons, people’s confidence in the very notion of objective truth has been shaken. And, I find that to be a really dangerous and a sad step backwards. … I’m very discouraged by the fact that, as a culture, we seem to be backing away from the notion that we’re capable of coming to agreement on objective truth. It’s hard, you’ve got to work at it, you’ve got to be willing to fight, but there’s a truth out there.

Yet so much of the recent challenges to science come from people of faith. How do you, as a Jesuit, deepen your faith in light of scientific truths?  

My colleague at the observatory, Guy Consolmagno [S.J.], is always clear in saying that when he does science, for him it’s an act of worship. If God is the way, the truth, and the life, then when you look for the truth via science, you are also looking for God. I believe that the very search for truth means that I trust that my reason participates in God’s reason. I’m made in God’s image and I really think that matters. We really need to work together to come to some kind of agreement on these things, because it’s a great, lovely mystery that should bring us together.

“He didn’t pay much attention to the dark side of life, there was always a fun story to be told instead,” said Mila Martynovsky, an adjunct professor who shared an office with Zigelbaum for several years.

“I will always remember him as a very caring personality,” said Janusz Golec, Ph.D., chair of the Department of Mathematics, who said Zigelbaum had worked at Fordham for 13 years..

Martynovsky said that Zigelbaum always put his “students’ personal situation in the mix” when assessing their abilities, taking extra care if he knew they were going through tough times. It’s something he did with colleagues as well.

“He was always supportive whenever you wanted to share a problem. He never behaved, like, ‘Well that’s too bad,’” she said. “He listened and offered suggestions.”

Zigelbaum’s daughter, Elaine Itkin, said that his love of math extended well beyond the gates of the Rose Hill campus. In addition to teaching college students, he also did after-school tutoring of grade school students in New Jersey, the Bronx, Brooklyn, Manhattan, and in Queens. He developed his own teaching techniques that he willingly shared with parents.

“He encouraged all of the parents to come into the lesson to observe how they could help their children at home,” she said. “He wanted them to understand.”

Itkin said that her father, who was Russian, had started teaching math in his homeland after college in the 1980s. She said he wanted to attend Moscow State University, but was hampered by his Jewish last name.

“Back then there was a quota for how many Jews got to attend the university,” she said, and Jews faced discrimination in testing. Even tough her father scored very high—the equivalent of 100—in the mathematics portion of the exam, he was given a low score on an essay section.

“They would specifically fail you in the essay to make an excuse why they wouldn’t take you,” she said.

When he married, Zigelbaum took his wife’s Ukrainian last name so that the family would not have to face similar bigotries. But when he entered the United States, fleeing religious persecution, “he got his name back,” she said.

Though he held a master’s in mathematics from another Russian university, in the United States he earned a second master’s in education. With the two degrees, he devoted the rest of his life to teaching math.

Itkin said that in October her father began to tutor his youngest student in math ever: his two-year-old granddaughter, Sophia. The two counted birds, compared big numbers to small numbers, and before leaving he gave her homework.