Participants of the MATH REU conference gather gather outside the Amphitheatre.

Participants of the MATH REU conference gather outside the Amphitheatre.
Image Credit: College of Science

CLEMSON – A National Science Foundation grant awarded in 2016 to a team of five mathematical sciences professors continues to pay extensive dividends for the College of Science.

The NSF Research Training Groups (RTG) grant awarded to Jim Brown, Shuhong Gao, Kevin James, Felice Manganiello and Gretchen Matthews has been supporting efforts to improve research training by involving undergraduate students, graduate students, postdoctoral associates and faculty members in structured research groups centered on a common research theme. The RTG assigned to Clemson University focuses on Coding Theory, Cryptography and Number Theory, which are research areas aligned with the University’s ClemsonForward and  ScienceForward plans.

The long-range goal of the RTG program is to strengthen the nation’s scientific competitiveness by increasing the number of well-prepared U.S. citizens, nationals and permanent residents who pursue careers in the mathematical sciences.

Here are some recent program highlights provided by Manganiello:

pREU (May 14-June 15): The five-week summer program called “preliminary Research Experience for Undergraduates” focused on code-based cryptography. These cryptosystems are known for their resiliency to quantum algorithms, making them possible candidates for secure communication when large-scale quantum computers become available. A team of five undergraduates – Austin Allen (Carnegie Mellon University), Keller Blackwell (University of South Florida), Olivia Fiol (Vassar College), Bethany Matsick (Liberty University) and Zoe Nelson (Oglethorpe University) – was led by Matthews and assisted by Ph.D. student Rutuja Kshirsagar. The team studied finite fields, curves and other algebraic structures, and applied this knowledge to define a family of codes based on Hermitian curves that meet the indistinguishability criteria for use in a code-based cryptosystem.

REU (May 21-July 13): This was an eight-week Research Experience for Undergraduates (REU) led by Brown, Gao and Manganiello. Nine students – Jacob Brazeal (Bob Jones University), Alice Chudnovsky (University of Illinois at Urbana-Champaign), Haley Colgate (Colorado College), Beren Gunsolus (University of Minnesota Twin Cities), Timothy Jackman (Northeastern University), Jake Januzelli (Cornell University), Jeremy Lilly (Oregon State University), Rico Vicente (California State University, Long Beach) and Alexandra Walsh (Brown University) – were divided into four groups with the following research topics.

pREU students remain smiling after completing an intense five-week program.

pREU students remain smiling after completing an intense five-week program.
Image Credit: College of Science

Exploring Batch Codes: These codes have applications to communication problems such as database design and optimized data storage for information retrieval on distributed storage systems (e.g., cloud systems). The students, led by Manganiello and supported by Ph.D. student Travis Baumbaugh, performed an original study of the batch properties of a class of special evaluation codes.
Eigenform Product Identities of Degree-Two Siegel Modular Forms: Modular forms are complex analytic functions that play an essential role in modern number theory. They arise in the study of many arithmetic identities, such as counting the number of representations of an integer as the sums of squares. Led by Brown and supported by Ph.D. student Hugh Geller, the participants determined some conditions of when the product of two degree-two Siegel eigenforms is also an eigenform.
Learning with Errors and Fully Homomorphic Encryption: One of the most recent developments in cryptography is on fully homomorphic encryptions and computational problems in lattices. These mathematical techniques are crucial in providing security of massive sensitive data stored in clouds and blockchains. Gao and Ph.D. student Benjamin Case directed their students in the study of efficient Fully Homomorphic Encryption based on a problem called Learning with Errors.
Codes, Lattices, and Modular Forms: This group, under the guidance of Brown and Manganiello, studied the relationship between linear codes (structures used in applications to communication) and lattices, with the end-goal being to look at modular forms.

2018 MATH REU CONFERENCE (June 9): For the first time, the RTG grant supported a conference for regional NSF-funded REUs in mathematics. The department of mathematical sciences hosted more than 30 students and their mentors for a total of more than 40 participants. The conference consisted entirely of student talks, with the goal being that students from different institutions would be able to see each other’s research and give a first conference talk in a friendly environment. Clemson University also hosted the REU in Algebra and Discrete Mathematics at Auburn University and the REU Math-Biology at the University of North Carolina at Greensboro. Research results and experiences were shared extensively.

ECRW (June 11-June 15): This one-week Early Career Research Workshop (ECRW) focused on Coding Theory, Cryptography and Number Theory. Three prominent researchers – Venkatesan Guruswami (Carnegie Mellon University), Jeremy Rouse (Wake Forest University) and Daqing Wan (University of California at Irvine) – led teams consisting of postdocs and junior faculty. This year’s participants were Alison Beemer (Arizona State University), Ryan Coatney (University of Arizona), Rachel Davis (University of Wisconsin-Madison), Duc Huynh (Armstrong State University), Rodney Keaton (East Tennessee State University), Tim Lai (Indiana University Bloomington), Hiram López (Clemson University), Alicia Marino (University of Hartford), Fernando Piñero (University of Puerto Rico at Ponce) and Angela Robinson (Florida Atlantic University). This workshop helped postdocs and junior faculty foster new collaborations and expand their existing research programs.

The principal investigators of the grant would like to thank the department of mathematical sciences and its staff for the support received in order to run these events.

For more information, please visit https://www.math.clemson.edu/ccnt/