Georgia Institute of Technology College of Sciences

At the start of his class on differential equations, Rafael de la Llave invites students to watch a mesmerizing demonstration.

He hangs two one-inch-diameter hex nuts from a clothesline through strings of the same length. With both hex nuts at rest, the School of Mathematics professor taps one slightly.

Given the slight energy input, the nut moves. In a while, the nut at rest also starts to swing. Eventually, a dance commences, the two hex nuts gracefully oscillating as they transfer energy from one to the other.

When more oscillators are involved, beautiful geometric patterns emerge, as this video shows.    

Designers of space missions can harness the dynamics creating these dazzling motions to save fuel. “If we could make the mathematical details very explicit, we can make these work to our advantage,” de la Llave says. “We could move spacecraft with very small amounts of fuel. We could extend the life of satellites – or send robots to the moon – inexpensively.”

NASA recently awarded a $100,000 grant to de la Llave, Marian Gidea of Yeshiva University, and Rodney Anderson of NASA’s Jet Propulsion Laboratory (JPL)to take the first steps to realize the potential of mathematics to lower the fuel cost of space travel. The project – “Accelerating Diffusion to Enable Rapid Tour Design” – has a duration of one year.  

As part of the project goals, this week in the Skiles Building, space mission designers from JPL and mathematicians from Georgia Tech and Yeshiva University are gathering for a four-day workshop. The participants will work together with the mathematical tools of the Arnold diffusion mechanism and trajectory design. The goal is to incorporate what is also known as the “butterfly effect” – which is the ability of minuscule changes to cause gigantic effects in certain systems – into space mission design.

“If we want to go around jumping from moon to moon, applying these new advances in mathematics can help us get there at much, much lower cost, making such a mission so much more doable.”

The Arnold diffusion mechanism is the underlying mathematical concept. Both de la Llave and Gidea are world-renowned experts in this field.

“In a nutshell, the Arnold diffusion mechanism states that small amounts of force, applied at the right moments, can produce large effects over time,” Gidea explained last year. “A familiar example is pushing a playground swing: with a tiny push on the swing each time it comes back to you, the amplitude of the swing will keep increasing.

“In the case of space missions, this small forcing translates into firing the rocket’s engine at the right place and the right moment to accelerate in orbit when the natural dynamics is slow.” Other possible small forcings could be the tugs of gravitational tides induced by stars, planets, moons, and even asteroids.

At other times, “the spacecraft will coast along the space superhighway at zero cost,” Gidea said.

“Celestial bodies are moving all the time,” de la Llave says. “And they generate forces that depend on time. If you can ride the wave of those forces, then you can move and accelerate using just the gravitational forces of astronomical objects.”

The Arnold diffusion mechanism is rooted in the Kolmogorov-Arnold-Moser (KAM) theorem. The theorem provides a general framework for understanding what happens when a simple physical system is modified slightly, according to School of Mathematics Professor Howard “Howie” Weiss. “Rafael and others played a big role in extending the KAM theorem,” Weiss says. “Rafael is extremely modest. He is probably the world’s leader in this business.”

Design of space mission routes historically has been based mostly on patching orbits of conical geometry. Recent mathematical advances in the Arnold diffusion mechanism have uncovered other geometries that reveal new potential pathways leveraging the gravitational dynamics in space. Adding small maneuvers at precise times and locations to the pathways found via the Arnold diffusion mechanism could significantly drop the cost of space missions.

While de la Llave and Gidea work on the mathematics, JPL’s Anderson will focus on applying the mathematical methods to mission concepts. Anderson is an expert on the application of dynamical systems theory to trajectory design problems. He is the coauthor of a 2013 NASA monograph that explores the use of low-energy paths to transfer a spacecraft from Earth to its moon.  

One space endeavor of great interest is to visit the moons of Jupiter systematically, de la Llave says. “If we want to go around jumping from moon to moon, applying these new advances in mathematics can help us get there at much, much lower cost, making such a mission so much more doable.”

Story Collider: Part 1 - Lew Lefton tries to succeed as both a math professor and a math comedian

Lew Lefton is a faculty member in the Georgia Tech School of Mathematics and the Assistant Dean of Information Technology for the Georgia Tech College of Sciences.  He also has the role of Assistant Vice President for Research Cyberinfrastructure at Georgia Tech. Lefton co-founded and is the acting executive director of Decatur Makers, a family-friendly makerspace in downtown Decatur.  He is on the board of the Southeast Makers Alliance and has been involved as a co-producer of Maker Faire Atlanta since 2014. Lefton has a bachelor of science degree in math and computer science from New Mexico Tech, and a Ph.D. in mathematics from the University of Illinois. He moved to Decatur in 1999.  Lefton is also an accomplished and experienced comedian who has done stand up and improv comedy for more than 30 years.

External News: https://www.storycollider.org/stories/2018/1/5/math-problems-stories-about-math

Recently the AMS published a book written by Michael Damron and two co-authors, titled 50 Years of First-Passage Percolation.

First-passage percolation (FPP) is a fundamental model in probability theory that has a wide range of applications to other scientific areas (growth and infection in biology, optimization in computer science, disordered media in physics), as well as other areas of mathematics, including analysis and geometry. FPP was introduced in the 1960s as a random metric space. Although it is simple to define, and despite years of work by leading researchers, many of its central problems remain unsolved.

In this book, the authors describe the main results of FPP, with two purposes in mind. First, they give self-contained proofs of seminal results obtained until the 1990s on limit shapes and geodesics. Second, they discuss recent perspectives and directions including (1) tools from metric geometry, (2) applications of concentration of measure, and (3) related growth and competition models. The authors also provide a collection of old and new open questions. This book is intended as a textbook for a graduate course or as a learning tool for researchers.

See the CoS full story here:

http://www.cos.gatech.edu/hg/item/602241

Michael Damron is an Associate Professor in the SoM at Georgia Tech, who has been highly active in the fields of Continuous and Discrete Probability, has been awarded several teaching awards including the LexisNexis Dean’s award from Georgia Tech in 2016, and is mentoring several post-docs and graduate students here at Tech. This is Michael's second book.

Larry Rolen and several collegues have won the PROSE Award in Mathematics, for their work in Harmonic Maas Forms and Mock Modular Forms.

The PROSE Award by Category:

American Mathematical Society

Harmonic Maass Forms and Mock Modular Forms: Theory and Applications

By Kathrin Bringmann, Amanda Folsom, Ken Ono, and Larry Rolen

The PROSE Awards annually recognize the very best in professional and scholarly publishing by bringing attention to distinguished books, journals, and electronic content in 58 categories including Mathematics, Biomedicine & Neuroscience, Chemistry & Physics, as well as Computing & Information Sciences, Psychology and many others.

Larry Rolen is a current Ussher Assistant Professor in
Number Theory & Cryptography at Georgia Tech SoM.

It's a good month for SoM in the Notices.

Dana Randall is on the cover for her JMM lecture and Mohammad Ghomi has an article there about his work on unfolding polyhedra (See the full story). 

http://www.ams.org/journals/notices/201801/201801FullIssue.pdf

Also Christine Heitsch was featured in a recent issue:

http://www.ams.org/publications/journals/notices/201708/rnoti-p816.pdf

Additionally, Matt Baker also has an article in the current Bulletin:

http://www.ams.org/journals/bull/2018-55-01/S0273-0979-2017-01599-6/

A prominent article highlighting the research of our own Sung Ha Kang, Seong Jun Kim (postdoc), and Haomin Zhou was featured in a SIAM News Bulliten: Streamlined Security: Optimizing Sensor Placement with Mathematics.

https://sinews.siam.org/Details-Page/streamlined-security-optimizing-sensor-placement-with-mathematics

Vinayak Agarwal and Lutz Warnke are among 126 outstanding U.S. and Canadian researchers selected to receive the 2018 Sloan Research Fellowships. Awarded yearly by the Alfred P. Sloan Foundation, the fellowships honor early-career faculty whose achievements mark them as among the very best scientific minds working today.

Also named 2018 Research Fellow is Bilal Haider, assistant professor in the Wallace H. Coulter Department of Biomedical Engineering at the Georgia Institute of Technology and Emory University.

Vinayak Agarwal is an assistant professor in School of Chemistry and Biochemistry. His research seeks to understand how natural products are produced, what biological forces drive their synthesis in nature, and how the knowledge can advance the well-being of humans. In particular, he is interested in the genesis of polybrominated compounds in ocean systems.

Agarwal seeks to understand how biosynthetic pathways manipulate halogens, especially bromine, to create polyhalogenated compounds, some of which are of great interest as anticancer agents. He has been at the forefront of efforts to decipher the routes for the natural production of polyhalogenated aromatic compounds by marine bacteria.  

Just like synthetic chlorofluorocarbons, which are now banned because they destroy Earth’s protective ozone layer in the stratosphere, polybrominated compounds can generate halogen species that can deplete stratospheric ozone. Agarwal aims to establish the biogenic basis for oceanic bromine radical production from polybrominated compounds.

To gain a deep understanding of biosynthetic pathways, he combines chemical and biochemical approaches, including molecular biology, phylogenetic and bioinformatics analysis, microbial genetics and metagenomics, structural and biochemical enzymology, protein biochemistry, synthetic chemistry, and analytical chemistry.

“The brightest minds tackling the hardest problems, and succeeding brilliantly – Fellows are quite literally the future of 21st-century science.”

Lutz Warnke is an assistant professor in the School of Mathematics. His research area lies at the interface of discrete mathematics and probability theory. In particular, he has extensively studied various random graphs (or networks), focusing on phase transition phenomena and other fascinating properties. 

In 2016, Warnke received the biennial Dénes König Prize from the Society of Industrial and Applied Mathematics for his contribution to the study of random graph processes. Because of Warnke’s work, theorems are now available to help mathematicians understand the behavior of a large class of discrete random graph processes close to phase transition.  

Earlier, Warnke received the biennial 2014 Richard-Rado-Prize, for his outstanding dissertation in discrete mathematics, titled “Random Graph Processes with Dependencies.”

In announcing the prize, Jiri Matousek of Charles University Prague, said: “[Warnke’s] work has all the attributes one may expect from excellent mathematics: very interesting topics appealing to [a] broad audience, in this case to a community including many theoretical physicists; improved methods and new ideas that both solve old problems and new horizons; considerable depth, breadth, and technical difficulty; and last but not least, careful and accessible presentation.   

“The Sloan Research Fellows represent the very best science has to offer,” said Sloan Foundation President Adam Falk. “The brightest minds tackling the hardest problems, and succeeding brilliantly – Fellows are quite literally the future of 21st-century science.”

This story was originally posted on Feb 15th, 2018 by the College of Science.