## News & Events - Spotlights

### May 17, 2016

• As part of Mathematics Awareness Month, some of the professors in the School of Mathematics participated in interviews that explored their research focus, highlights of their career, and their personal insights. The interviews became a series of Get to Know the Math Professor articles that were featured on the School of Math website last month. All together, sixteen articles were published, and links to all of the articles are listed below.    more...

### May 11, 2016

My research area is topology. In topology, we study properties of shapes that persist even when we stretch or bend the shapes. For example, if you have two metal rings that are linked, then they stay linked even if you bend or stretch the metal. A typical question in topology is the following: Someone hands you two rings made of metal; if you are allowed to bend and stretch the metal, can you pull the rings apart or not?

Most of my research in topology is about surfaces. The surface could be that of a ball or a donut. Surfaces are central in mathematics. They can describe the possible motions of a robot arm or all the possible solutions of a polynomial. My particular research is on the symmetries of surfaces - if we really want to understand an object, we must also understand its symmetries! Some symmetries of surfaces are easy to understand. But when we allow bending and stretching, they more challenging.

Mathematics is important because it describes the world in a beautiful and coherent way. Even the most far-fetched and abstract mathematical ideas can make their way into everyday life. For example, I was very pleased recently to attend a lecture at Georgia Tech by Jesse Johnson, a topologist who is currently working at Google. He described an application of his research on the topology of three-dimensional manifolds to the analysis of large data sets. This was shocking to me and very satisfying.

### April 29, 2016

I work on partial differential equations that arise in biology and fluid dynamics.

Collective behavior is a phenomenon that is commonly observed for many animal species, such as a flock of birds in flight or bacteria aggregating and forming patterns. If we want to track the movement of each individual, the whole system would consist of thousands or millions of equations, which would be tough to analyze or simulate. However, if we treat the system as a cloud of particles and track how the density function evolves in time, then the whole system can be described much more easily using one or two partial differential equations.

Although fluid dynamics seem unrelated to animal swarming, both phenomena can be described by partial differential equations with some long-range interactions between the particles. I analyze these nonlocal equations from the mathematical aspect, such as studying whether the solution converges to some equilibrium pattern as the time goes to infinity.

### April 28, 2016

I work at the interface of discrete mathematics and molecular biology. For instance, most viruses code their genomes in RNA rather than DNA, which is then packaged into a protein capsid. Understanding how this happens is a fundamental biomedical problem with important therapeutic applications. The "branching" of these large RNA molecules (much like a tree in nature) is a critical characteristic that I've studied using techniques from analytic, geometric, and probabilistic combinatorics.

### What has been the most exciting time so far in your research life?

The most recent breakthrough always seems the most exciting. As one of my undergraduate researchers, who is now a PhD student in the School of Electrical and Computer Engineering and works at the Georgia Tech Research Institute, said: "It's hard but rewarding. Half the time it feels like you are banging your head against the wall, but every now and then you get something to work, and it's such a rush."

### April 27, 2016

My research is in a field of mathematics called partial differential equations. These equations describe the evolution with time of various physical systems, ranging from the motion of water in the ocean or of air in the atmosphere, to the strength of an electromagnetic signal, and all the way to the motion of galaxies according to Einstein's equations of general relativity.

Partial differential equations have fundamental importance in everyday applications. For example, they have allowed us over the past 50 years to make more accurate weather predictions, improve how we handle turbulence in the atmosphere in relation to air travel, and invent faster ways to transfer information by electromagnetic signals (like using fiber optics for fast speed internet communication). Partial differential equations also appear in finance, where they are used to model changes in stock prices.

The study of partial differential equations is broad because of the various applications. I work on a particular class called nonlinear dispersive equations. This class includes equations governing ocean and atmospheric sciences, plasma physics, nonlinear optics (including fiber optics), and Einstein's equations of general relativity.

The first question when analyzing such equations is whether a solution exists, or not. The second question is whether we can describe the properties of this solution. In most cases, this solution cannot be written explicitly, so we have little to no way of inspecting its properties beyond analyzing the equation itself. This leads to a very nice and delicate study, in which we try to figure out the properties of the solution without having a formula for it, but only by looking at the equation that it solves.

### April 26, 2016

I work in an area of mathematics called harmonic analysis. This field grew from the fundamental fact that many functions defined over an interval can be decomposed as sums of the simple sine and cosine functions.

I study cases where the above decomposition does not hold - or holds but is not efficient enough - say, because the functions are no longer defined over an interval. The question is whether similar decompositions are possible also in such cases, with the sines and cosines being replaced by other functions with a simple structure.

Usually, the goal is to use functions which mimic the structure of the sines and cosines, in one way or another. By finding good replacements for the trigonometric functions, one obtains a good way to understand the behavior of functions and the interrelationships between them. With this we get an excellent tool to study the mathematical aspect of the way the world around us behaves.

This area is of much interest in natural sciences and engineering, including in sound and image processing, wireless communications and data transmission, methods in quantum mechanics and quantum computing, and the analysis of signals in geophysics and medicine.

### April 25, 2016

My research is in a general area called applied and computational mathematics. The primary goal is developing efficient and accurate methodologies, based on modern mathematics, to improve computer simulations for a wide range of problems in science, engineering, and even social sciences. For example: How to find the best path for a robot to travel from one place to another with the least energy cost while avoiding collisions with possibly moving obstacles? How to predict viral news propagation on social media, such as Facebook or Twitter?

### What has been the most exciting time so far in your research life?

My research career has brought many exciting times, including having my first paper accepted, receiving an offer from Caltech as a postdoc, and coming to Georgia Tech as an assistant professor. The happiest times were when I collaborated with engineering colleagues, including Ali Adibi and Magnus Egerstedt of the School of Electrical and Computer Engineering. Together we developed methodologies that are now being applied to optical devices for medical imaging, and to robotics.  It is very rewarding to see my research in action, and it is even more exciting to discover new challenging mathematical problems emerging from practical problems.

### April 22, 2016

I work closely with biologists to combine mathematical modeling with experiments. We study how populations of bacteria grow and evolve, as well as the dynamics of bacterial killing by antibiotics and viruses, especially in physically structured environments such as biofilms.

Physically structured habitats are particularly critical for treatment of surface-associated infections, such as endocarditis, osteomyelitis, and infections of prosthetic heart valves and joints. Our goal is to improve the efficacy of antimicrobial therapy.

I am also the principal investigator of the Boeing-sponsored FlyHealthy™ research study, whose goal is to understand the rates and routes of transmission of infectious diseases in an airplane cabin during flight and to find strategies to mitigate transmission.

### April 21, 2016

I am a probabilist, a mathematician studying probability theory - a specialist of the study of chance and randomness. Because we do not really have a definition of randomness, it might appear, at first, contradictory to try to have a mathematical theory dealing with something that is undefined. But everyone has some intuition about randomness.

When flipping a fair coin, the outcome cannot be predicted with certainty, but a couple of assumptions are reasonable. First, one expects to get on each flip either a head or a tail with equal probability. Second, one expects that when flipping this same coin a very large number of times (say 10,000 times) one would approximately get 5,000 heads and 5,000 tails.   more...

### April 20, 2016

I've been all over the place in terms of research.  In fact, majority of my publications have been outside of mathematics proper - in electrical engineering (network theory), physics (quantum mechanics and fractals), operations research (optimization, games), education (cross-disciplinary learning, MOOCS), etc. My latest paper was with a student on nuclear engineering. What ties these research activities all together is that the problems involve a combination of discrete mathematics, matrices, and computers. I find much of mathematics all by itself to be sterile and too inward-looking.

### What has been the most exciting time so far in your academic life?

Working with really good students even before they get to Georgia Tech. Through the Distance Calculus Program I started teaching in 2005, high school students in the Georgia Public School System have been able to take college credits for courses in calculus. Using audio or video links, they take the courses at the time I teach them in Tech, and they take the same quizzes and tests that I give to on-campus students.   more...

### April 19, 2016

The world intrinsically contains multiple scales, and one theme of my research is to develop theories and algorithms that help us understand how different scales interact.

One classic example is the following astronomical problem. It is well known that planets rotate around their host star due to mutual gravitational attraction. For instance, Earth finishes one period of rotation around the Sun in exactly one year. Pairs of planets also experience mutual gravitational attractions, but such interactions are much weaker and the immediate effects are not obvious, especially if one considers only hundreds of years in the stellar system.   more...

### April 18, 2016

I am fascinated by analogies. Much of my work involves so-called "p-adic" numbers, which are analogous to real numbers like 2 or π, but with important differences. For example, in p-adic geometry, every triangle is isosceles! This world might sound exotic and useless, but p-adic numbers play an important role in modern life, including cryptography, which is the making and breaking of secret codes.

A lot of things in mathematics appear to have no applications, but in fact, down the road, they turn out to be incredibly useful.   more...

### April 15, 2016

My research involves using mathematics to understand human behavior and has largely focused on criminal behavior. I try to use math to help predict, solve, and defend against crimes. This research has led to tangible societal benefits, including a software program now in use by several police departments, including Atlanta's, that helps police better predict where crimes may occur today

### What has been the most exciting time so far in your research life?

The most exciting time was probably when I was still a young graduate student and everything was so new. I still remember when my first paper was accepted for publication: It gave me a huge sense of validation and served as a tangible symbol of my entrance into academia as a researcher.   more...

### April 11, 2016

I work on theory and algorithms of graphs. A graph consists of nodes and links joining nodes. Many real-world situations, including social networks and communication networks, can be modeled by graphs.

My current research has two components: basic mathematics research in graph theory and application of graph theory to other areas of mathematics and engineering.

Examples of basic research in graph theory are problems related to the Four Color Theorem, which states: Given a map of countries, one can always color the countries with at most four colors such that countries sharing borders always have different colors. Techniques we developed may be used to solve other problems in graph theory, as well as related problems in theoretical computer science and engineering.

An example of applications of graph theory is a project I'm working on with engineering colleagues about radio-frequency, or spectrum, allocations for wireless communications. We use graph theory techniques to find good solutions to resource allocation problems formulated by engineering colleagues to address the technological challenges in spectrum trading.   more...