Georgia Institute of Technology College of Sciences

On Sept. 13, starting at 11 A.M., mathematicians, musicians, and dancers will breathe life into the classic problem known as the Seven Bridges of Königsberg (7BK). The interactive exposition and performance celebrates this problem’s journey from 18th-century Prussia, in the small town of Königsberg, to 21st-century Atlanta.

On the Georgia Tech campus is a representation of Königsberg and the seven bridges that connect its four land masses, which are divided by a river. The rendition – along the Atlantic Drive Promenade, on a site called the Seven Bridges Plaza – affirms that art, science, and mathematics are but different ways to grasp the world.

The Sept. 13 music and dance performance – The Seven Bridges of Königsberg – especially hopes to demystify and humanize mathematics, says Evans Harrell, an emeritus professor in the School of Mathematics.

The event begins with an interactive exposition by members of the Georgia Tech student organization Club Math. On each of the four land masses representing Königsberg, Club Math members will be at stations to discuss the 7BK problem; the life and times of Leonhard Euler, whose solution to the 7BK problem gave birth to graph theory; the role of graph theory in the modern world, and a special aspect of graph theory called the four-color theorem.

“Our project is also an experiment about how scientific stories can be told and about how the sciences can inspire original art.”

Euler, the Seven Bridges, and Graph Theory

The 7BK problem asks: Can one walk around Königsberg, crossing each of the seven bridges exactly once?

The Swiss mathematician Leonhard Euler proved that it is impossible to cross each of the seven bridges of Königsberg only once. The proof considered not only the case of Königsberg, but all possible ways a city could be connected by bridges and when it is possible to cross each bridge only once. In developing the proof, Euler invented a new field of mathematics, now called graph theory.

Euler’s insight was to simplify the problem, says Georgia Tech mathematics professor Dan Margalit. Graph theory reduces the problem to one about points – called vertices – and the lines – called edges – connecting them. The vertices correspond to the land masses and the edges are the bridges. The 7BK problem thus rendered, it is not hard to see  the answer.

Margalit explains: Any attempted solution has two variations. In the first case, the journey starts and ends at the same land mass. Here, the number of bridges – or edges – associated with each land mass – or vertex – is even. Because every arrival at a land mass comes with a departure, every vertex has an even number of edges. Therefore any continuous path uses an even number of edges at each vertex. This is impossible in Königsberg because each land mass has an odd number of bridges.

In the second case, the journey ends in another place from where it started. Again the number of edges at each vertex must be even except at the start and end. Because to leave the start requires only one bridge, as does arriving at the end. Therefore, at the start and end vertices, the total number of edges will be an odd number. But at all other vertices, the number of edges would be even, as before. 

Again this is impossible in Königsberg, because all the land masses have an odd number of bridges.

Graph theory permeates the modern world. “Facebook is a graph: vertices are people and edges are friendships," Margalit says. "Graph theory has many other applications all over science and mathematics."

“Facebook is a graph: vertices are people and edges are friendships.”

Math in Motion

For the Sept. 13 performance, Harrell partnered with the Georgia Tech School of Music’s Chaowen Ting. She will conduct the Georgia Tech Symphony Orchestra in performing original music by composer Marshall Coats. The music will accompany dances choreographed by artistic director Kristel Tedesco.

In a behind-the-scenes video, Tedesco says she recognized the similar struggles of mathematicians and artists working in imaginary worlds “and trying to find truth within them.” The resulting performance, she adds, aims to spark and stimulate the public’s curiosity about mathematics.

“We wish to engage the public in the wonder of mathematics and science, of music and dance, and the surprising ways that they can work together,” Harrell says. “Our project is also an experiment about how scientific stories can be told and about how the sciences can inspire original art.”

Support for the event came from Science in Vivo and Georgia Tech's College of Design, College of Sciences, and Office of the Arts.

As microorganisms evolve to resist antibiotics, the world risks running out of drugs to treat bacterial infections. One way to slow this trend is to find new modes of using existing drugs, even those now ineffective because of microbial resistance.

One strategy is based on the phenomenon of collateral sensitivity: When some microbes develop resistance to one antibiotic, they become hypersensitive to another. For example, when an Escherichia coli strain became indifferent to chloramphenicol, it also became highly vulnerable to polymyxin B. For this strain, chloramphenicol and polymyxin B form a collaterally sensitive pair.

Sometimes the drug pair exhibits mutual collateral sensitivity (MCS) for a pathogen: The pathogen’s evolution of resistance to drug A increases its sensitivity to drug B and vice versa.

Researchers have identified several MCS pairs for pathogens like E. coli and Pseudomonas aeruginosa. Some have proposed exploiting the phenomenon to treat infections by cycling through the drugs, A-B-A-B.

“This sounds very clever,” says Georgia Tech biomathematician Howard “Howie” Weiss. “Bbut what could prevent this scheme from working is the rapid emergence and ascent of a population of cells that are resistant to both antibiotics.”

The prospect is exciting, but no experiments have yet been performed to test efficacy.

 “This was a real team effort between a microbiologist and a biomathematician.”

With Stockholm University microbiologist Klas Udekwu, Weiss has tested the plausibility of such schemes, using a mathematical model that considers factors affecting efficacy. Applying treatment protocols consisting of pairs MCS antibiotics, they examined how fast multiply-resistant mutants would emerge. They reported results in Drug Design, Development and Therapy.

They found some treatments that did not produce multiply-resistant mutants for several weeks, for several months, and even indefinitely. That means some combinations of an MCS pair prevented further development of the bacteria’s resistance to either drug.

 “This was a real team effort between a microbiologist and a biomathematician,” Weiss says. “My job was to construct the model using a system of differential equations and very carefully simulate their solution using a computer.” 

The first experiment used low to moderate concentrations of antibiotics and daily cycling: drug A on day 1, drug B on day 2, drug A on day 3. At these drug levels, treatment failed. Resistant mutants rapidly developed and dominated.

Simulation results improved with higher drug concentrations. “We found that one-day cycling of certain antibiotics kept the double-resistant mutants in check for over two weeks, which would be sufficient to cure many infections,” Weiss says.

The simulations also showed that three-day cycling of antibiotics that only inhibit bacterial growth – not kill – would never result in double-resistant mutants. “This was striking,” Udekwu says, “but in line with ecological theory.”

Udekwu is now conducting in-vitro cycling experiments. The next step would likely be experiments in mice. “It is far too early for clinicians to think about this strategy,” he says, “other than to keep an ear out for it, perhaps in a Cochrane report someday. 

The College of Sciences has named Jennifer Hom, Takamitsu Ito, and Scott Moffat as the 2018 recipients of the Cullen-Peck Faculty Fellowship Awards in the College of Sciences. The awards recognize innovative research by faculty at the associate professor or advanced assistant professor level. The goal is to help recipients take their research programs in new directions.

The fellowships are made possible by a generous gift to the College of Sciences from alumni Frank H. Cullen (B.S. in Mathematics with Honors 1973, M.S. in Operations Research 1975, Ph.D. Engineering 1984) and Libby Peck (B.S. in Applied Mathematics 1975, M.S. in Industrial Engineering 1976). The alumni couple wish to recognize and support faculty development in the College of Sciences

“We continue to be grateful for the generosity of alumni who encourage our faculty to take intellectual risks in their research,” says College of Sciences Dean and Sutherland Chair Paul M. Goldbart. “The Cullen-Peck fellowships help ensure that our research is pushing the frontiers of knowledge. Congratulations to the latest Cullen-Peck fellows.”

Knot Theory

Jennifer C. Hom is an associate professor in the School of Mathematics. The award recognizes her outstanding research in knot theory, which has led to fundamental contributions to the study of knots and development of powerful tools in topology.

Knots can be conceived as loops of strings with ends glued together. Their study is a beautiful subject, central to understanding low-dimensional space, as well as some modern trends in physics.  Hom’s work centers on knots in three-dimensional space. She has enriched the field by introducing deep new ideas.

A much-studied question asks whether a knot can bound a disk in four-dimensional space in certain nice ways. Such knots were previously known. But Hom was able to find a huge new family of such knots, inspiring a flurry of activity in the use of Heegaard-Floer theory to study such objects.

The Heegard-Floer theory is a much-studied technique that revolutionized low-dimensional topology. Yet, Hom found new subtle features, which she formalized as the epsilon invariant. The epsilon invariant is a number associated to each knot. By using the properties of these numbers, Hom proved that an infinite number of knots could bound certain disks in four-dimensional space and not others.

​Her work inspired leaders in the field, including the developers of Heegaard-Floer theory themselves, to pursue ​new avenues of research​. Among other things, this work gives a new proof that in a sense there is more than one way to do calculus in four dimensions.

The epsilon invariant is now part of the Heegard-Floer theory; it is taught in graduate courses around the world; it is considered one of the top five spectacular advances in the past decade. A mark of top-notch mathematics is that it inspires other people and takes a life of its own. Hom’s epsilon invariant belongs to this category. 

“It's a great honor to receive this award,” Hom says. “I look forward to using this fellowship to help develop new techniques for studying knots and low-dimensional spaces.”

Biogeochemical Cycling and Ocean Deoxygenation

Takamitsu “Taka” Ito is an associate professor in the School of Earth and Atmospheric Sciences. The Cullen-Peck award recognizes his outstanding research in biogeochemical cycling and ocean deoxygenation.

Ito uses models to better understand the interactions of physical, chemical, and biological processes that regulate the cycling of chemical elements in the ocean. He develops theories of the partitioning of dissolved gases between the ocean and the atmosphere. He is renowned for recent work on the distribution of dissolved oxygen in the subsurface ocean.

In the 2017 paper “The Upper Ocean Oxygen Trend: 1958-2015,” Ito analyzed historical, global datasets of dissolved oxygen. He found that the amount of dissolved oxygen in the water – an important measure of ocean health – has been declining for more than 20 years.

This paper garnered media attention for the implications of declining oxygen in the ocean: It could affect the habitat of marine organisms worldwide. It could lead to more frequent “hypoxic events,” which kill or displace populations of fish, crabs, and other organisms.

Furthermore, the analysis showed that ocean oxygen is falling more rapidly than anticipated from the rise in water temperature due to climate change.

Ito has also been exploring the previously under-appreciated role of polluted aerosols in altering ocean biogeochemistry. In a 2016 paper in Nature Geoscience, he and his collaborators showed that air pollution can deliver additional iron and reactive nitrogen to the ocean and affect oxygen levels.

The transport of highly insoluble iron to the ocean and its availability for biological productivity are not well understood. Ito’s modelling approach will help translate into new insights the oceanic iron data from the large observational program GEOTRACES. His research could reveal how iron cycling affects ocean productivity, carbon uptake, and oxygen concentrations over various time scales.

Cognitive Neuroscience of Aging

Scott Moffat is an associate professor in the School of Psychology. His selection as Cullen-Peck fellow is based on his outstanding research in the cognitive neuroscience of aging.

With aging comes cognitive decline, which affect mental faculties including memory and the ability to navigate. Moffat has embarked on research addressing metabolism and aging. In particular, he studies the role of diabetes in cognitive aging.

Peripheral insulin crosses the blood–brain barrier to modulate memory processes. Insulin resistance in the periphery goes with insulin resistance in the brain and memory impairment. The hope is to associate variations in peripheral insulin secretion and insulin sensitivity to cognitive and neural endpoints.

Meanwhile, type 2 diabetes is a public health crisis in the U.S. and many developed countries. The disease is a risk factor for other serious health conditions, such as brain and cognitive dysfunction, as well as Alzheimer’s disease. Using functional magnetic resonance imaging, Moffat is examining the association of glucose and insulin metabolism with cognitive and brain function.

The research is still in its early days, but already Moffat and his colleagues are realizing remarkable results. For example, they’ve found that individuals with higher fasting glucose levels or insulin insensitivity – even within the non-diabetes range – have poorer performance in episodic and working memories. They also have thinner gray matter in key prefrontal cortical areas. 

The implications for prediabetes are profound. Prediabetes is prevalent among adults; the National Center for Chronic Disease Prevention reports that majority of all adults older than 65 have prediabetes. Discovering the impact of prediabetes on cognition and cognitive decline could bring about interventions, pharmaceutical or otherwise.

The 2012 speaker is Dr. Emmanuel Candès from Stanford University. He holds the Simons Chair in Mathematics and Statistics. His research areas include: compressive sensing, mathematical signal processing, computational harmonic analysis, multiscale analysis, scientific computing, stastistical estimation and detection, high-dimensional statistics. Applications to the imaging sciences and inverse problems. Other topics of recent interest include theoretical computer science, mathematical optimization, and information theory.

There will be two lectures. One (for a general audience) will be on September 10, at 4:25 pm, in Clough Commons, Room 144. Another one will be at 11:05 am on September 11 in Skiles 006.

Lecture 1: General Audience

Robust principal component analysis? Some theory and some applications

This talk is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. In the second part of the talk, we present applications in computer vision. In video surveillance, for example, our methodology allows for the detection of objects in a cluttered background. We show how the methodology can be adapted to simultaneously align a batch of images and correct serious defects/corruptions in each image, opening new perspectives.

Lecture 2: Mathematics Lecture

PhaseLift: Exact Phase Retrieval via Convex Programming

This talks introduces a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that any complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we present some novel theory showing that our entire approach may be provably surprisingly effective.

Professor Howie Weiss is the Georgia Power Professor of Excellence from Science this year. He was featured at the October 20, 2012 Georgia Tech vs. Boston College football game. The Athletic Association continues the "Professor of the Excellence" program at each home football game this season. The corporate sponsorship this year is Georgia Power.

At each game, one professor will be highlighted, a picture on the scoreboard, information about their research is read to the crowd, etc. A professor from each college will be selected for each of the next six home games.

Weiss' research interests include mathematical biology, analysis, dynamical systems as well as geometry and topology. He is currently working on a $1.3M study with Emory University and Delta Airlines to analyze the transmission of infectious diseases on aircraft. The study is being funded by Boeing.

The benefits to the faculty member are:

• Their research is featured at the game and through the website
• They receive tickets to attend the game where they are recognized
• The faculty member is honored on the field during the game between quarters
• Georgia Power will make a donation of $1,000 in their name to their college
• Each Professor of Excellence receives a football autographed by Coach Johnson

Twelve faculty from Georgia Tech's School of Mathematics were named today as Fellows of the American Mathematical Society (AMS). The listing represents the society's inaugural class and includes 1,119 fellows from more than 600 institutions.

The faculty from Tech include math professors: Matt Baker, Jean Bellissard, John Etnyre, Wilfrid Gangbo, Michael Lacey, Michael Loss, Doron Lubinsky, Prasad Tetali, Robin Thomas and associate professor Brett Wick. Adjunct math professors who were recognized also include Bill Cook, from the School of Industrial and Systems Engineering, and Dana Randall, from the School of Computer Science.

“I am delighted that such a large number of Georgia Tech faculty members have been named as Fellows of the AMS,” said Doug Ulmer, chair of the School of Mathematics. “It is an indication of the quality of work being done here and its impact in the wider world.”

The Fellows of the AMS designation recognizes members who have made outstanding contributions to the creation, exposition, advancement, communication and utilization of mathematics. Among the goals of the program are to create an enlarged class of mathematicians recognized by their peers as distinguished for their contributions to the profession and to honor excellence.

“The AMS is the world's largest and most influential society dedicated to mathematical research, scholarship and education,” said AMS President Eric M. Friedlander. “Recent advances in mathematics include solutions to age-old problems and key applications useful for society. The new AMS Fellows Program recognizes some of the most accomplished mathematicians - AMS members who have contributed to our understanding of deep and important mathematical questions, to applications throughout the scientific world and to educational excellence.”

To see the names of individuals who are in this year's class, their institutions, and a description of the fellows program, visit www.ams.org/profession/ams-fellows

For the second time in as many years, Ms. Klara Grodzinsky has received the Class of 1934 Course Survey Teaching Award from Center for the Enhancement of Teaching and Learning (CETL) in recognition of high Course/Instructor Opinion Survey (CIOS) ratings and high response rates. The award includes a monetary bonus and recognition at Celebrating Teaching Day in March. This most recent award was for spring and fall semesters 2012. The previous award was for spring and fall 2011. Out of the 52 recipients this year, 12 also received the award last year.