March 11, 2022 By Mitchell Diggs

Numbers are Micah Milinovich’s passion. A respected mathematician, the University of Mississippi professor of mathematics has spent years studying the properties of integers and recently was awarded a National Science Foundation grant to fund his research.

Milinovich’s award for “The Distribution of Zeros of L-Functions and Related Questions” supports his research in number theory, a branch of mathematics that studies properties of integers.

Within all integers is a special set of numbers called primes: numbers whose only factors are 1 and themselves, such as 2, 5, or 17. Numbers such as 4, 12, and 26 are not prime.

“Since the time of the ancient Greeks, number theorists have tried to find patterns within the integers,” Milinovich said. “It turns out that every positive integer can be written as a product of primes, much in the same way that molecules can be written in terms of atoms.

“So, we can think of primes as the ‘building blocks’ or ‘atoms’ of the integers.”

Despite thousands of years of investigation, many basic questions about primes, such as how far apart primes can be or how often primes can be close together, remain unsolved and subject to conjecture.

“Unlike atoms, which can be listed in the periodic table, the ancient Greeks knew that the list of primes goes on forever,” Milinovich said. “No matter how long the list, there will be always be a bigger prime not on it.”

Some pairs of primes can be close together. For instance, the primes 5 and 7, 11 and 13, or 41 and 43 are all two integers two apart.

“We call these ‘twin primes,’” he said. “Mathematicians believe the list of these ‘twin primes’ should go on forever. This is known as the twin prime conjecture.”

Despite tremendous recent progress, the answer to this question remains elusive.

In Milinovich’s research, he has studied the complementary problem of how far apart primes can be.

Micah Milinovich

“Given a prime, how far forward in the list of integers do we have to go until we are guaranteed to find the next one?” he said. “My co-authors, Emanuel Carneiro, of the International Centre for Theoretical Physics, and Kannan Soundararajan, of Stanford University, and I studied this problem in a recent paper.”

The paper, “Fourier optimization and prime gaps,” which was published in the journal Commetarii Mathematici Helvetici, uses a mathematical theory known as Fourier analysis and techniques originally developed to study signal processing and the spreading of waves.

Similar ideas in Fourier analysis recently were used to study sphere packing, finding the most efficient way to stack objects such as oranges and cannonballs.

“My joint work with Carneiro and Soundararajan studies properties of the Riemann zeta-function,” Milinovich said. “In modern mathematics, the Riemann zeta-function, and its generalizations known as L-functions, have played a pivotal role in our understanding of the primes and many problems in number theory can be rewritten in terms of questions about L-functions.”

Understanding this relationship is the focus of the research proposed in Milinovich’s grant, which is funded for $208,000. Besides providing for his research, this grant also includes money to train new graduate students and to support travel to confer with research collaborators.

“Though it is considered pure mathematics, number theory has many important applications outside of math to fields such as theoretical computer science and cryptography,” Milinovich said. “These applications are not the focus of my research, but I find that it is extremely important to keep connections to other fields in mind.”

For instance, his work has some interesting connections to mathematical physics, which can be used to predict how L-functions behave statistically.

“Having these predictive tools available is incredibly important,” he said. “As a mathematician, my job is much easier if I have a solid intuition for what the answer to a question should be ahead of time.”

Talmage James Reid, chair and professor of mathematics, said Milinovich’s award is the latest achievement in his stellar career.

“Dr. Milinovich’s cutting-edge research in number theory has resulted in numerous publications in the best mathematical journals in the world,” Reid said. “His work has profoundly impacted both undergraduate and graduate education at the university.

“For example, an undergraduate research student of Dr. Milinovich is now a doctoral student in statistics at the University of Michigan, while several doctoral students have gone on to prestigious research postdoctoral positions and are rising stars in the mathematical research community.”

For more information, read the project’s abstract at https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101912.

Duo will use funding to develop their pioneering research in geology and mathematics

Samuel Lisi, UM assistant professor of mathematics, has been awarded a National Science Foundation fellowship to advance his innovative research into a branch of mathematics known as topology. Photo by Shea Stewart/University of Mississippi

JANUARY 6, 2020 BY SHEA STEWART

For the second time in three years, a pair of University of Mississippi professors have been awarded National Science Foundation fellowships to advance their innovative research.

Samuel Lisi, assistant professor of mathematics, and Brian Platt, assistant professor of geology and geological engineering, were awarded funding through a foundation program that provides opportunities for young university investigators to boost their research abilities via extended collaborative visits to research centers across the country.

Lisi, who joined the UM faculty in 2014, will continue his study of a branch of mathematics known as topology with collaborators at Ohio State University. Platt will conduct his research into geological drill core samples from Mississippi at the Continental Scientific Drilling Coordination Office at the University of Minnesota.

“I am very honored and excited to receive this fellowship,” Lisi said. “This is a great opportunity for me to expand my areas of interest and branch out into a new – though related – field of research.

“My colleague, Dr. (Sasa) Kocic, received this award two years ago, and it was a very transformative experience for his work. I have high hopes that this will have a similar impact on my career and work.”

In 2017, Kocic, associate professor of mathematics, and Ryan Garrick, UM associate professor of biology, were awarded fellowships through the same program. Garrick also conducted his fellowship at Ohio State, while Kocic visited the University of California at Irvine.

For Platt, who joined the UM faculty in 2013, the fellowship will help him meet a personal objective.

“When I began working at the University of Mississippi, one of my goals was to establish a research program that used Mississippi geology to address broader-scale climatic and environmental issues,” he said. “To me, receiving this fellowship means that I am well on my way to meeting that goal.”

The National Science Foundation program is funded through the Established Program to Stimulate Competitive Research, or EPSCoR, program. The Research Infrastructure Improvement Track 4: EPSCoR Research Fellows program allows fellows to “learn new techniques, develop new collaborations or advance existing partnerships, benefit from access to unique equipment and facilities, and/or shift their research toward potentially transformative new directions.”

While enhancing their research trajectories, Lisi and Platt also are expected to use their new knowledge to benefit the university’s research capacity.

Brian Platt, UM assistant professor of geology and geological engineering, is using a National Science Foundation fellowship to research geological drill core samples from Mississippi at the Continental Scientific Drilling Coordination Office in Minnesota. Photo by Shea Stewart/University of Mississippi

“UM is proud of these new awards for Drs. Lisi and Platt,” said Josh Gladden, the university’s vice chancellor for research and sponsored programs. “A unique aspect of these awards is that they help fund collaborative work between faculty across institutions.

“Such collaborations are crucial for all robust research environments and often lead to new ideas and breakthroughs.”

Lisi’s research at Ohio State will focus on topology, which studies the shape of space, and specifically applied symplectic topology, which Lisi describes as “the study of the geometric consequences of conservation of energy.”

“My interest in the field comes, first and foremost, from my interest in the parts of mathematics that are most closely connected to physics,” he said. “The great mathematician Vladimir Arnold said that ‘mathematics is the part of physics where experiments are cheap.’

“I’ve also always loved geometry and geometric problems. Symplectic topology combines these two interests of mine in a beautiful way.”

The award creates a new collaboration between UM and an interdepartmental group at Ohio State called the TGDA group, whose interests span topics in the intersection of topology, geometry, probability, statistics and data analysis. Lisi will work with Ohio State professors Tamal Krishna Dey and Matthew Kahle in opening up a new field of study: probabilistic symplectic topology.

“This collaboration will … build ties between our two institutions, which is beneficial for our graduate and undergraduate students,” said Lisi, as the collaboration will broaden his ability to work with Ole Miss colleagues in engineering and the sciences, and also strengthen UM’s research into topology and dynamics by bringing a new area of expertise to the university and state.

A rock collection purchased by his parents at a garage sale introduced Platt to geology as a boy, and he vividly recalls smashing the pieces of granite with a hammer to try to separate and identify the individual minerals that composed the rocks.

His research is focused on employing geological drill core samples from deep below the Earth that can be used to reconstruct long-term changes in ancient environments and climates.

Platt’s fellowship will allow him and a graduate assistant to make two extended visits to the Continental Scientific Drilling Coordination Office, where they will examine existing core from Mississippi drilled by the UM Mississippi Mineral Resources Institute.

“I plan to learn 14 new techniques for analyzing core; this will produce multiple sets of data spanning up to 360 feet of core,” he said. “The (Mississippi) core contains rocks and sediments from (an epoch) known as the Paleocene, meaning that they are over 56 million years old.

“The time interval sampled by the core preceded several episodes of rapid global warming, which are thought to be the best ancient analogs for projected future climate change. But to properly understand the effects of the ancient warming episodes, we need to know about background climate and environmental conditions before the changes occurred. A result of this project will allow me and my colleagues to interpret what Mississippi was like during the Paleocene.”

Funded by National Science Foundation award No. 1929176 for $190,024, Lisi’s project is titled “Applied Symplectic Topology.” Funded by foundation award No. 1929145 for $174,430, Platt’s project is titled “Establishing Baseline Critical Zone Conditions in Mississippi Prior to the Onset of Paleocene-Eocene Hyperthermals.”

Both projects are projected to run through Nov. 30, 2021.