Georgia Institute of Technology College of Sciences

While in high school, Bharath Hebbe Madhusudhana wanted to be a mathematician or a physicist. Now, he takes home degrees in the two fields he esteems the most: an M.S. in Mathematics and a Ph.D. in Physics.

The mathematics degree was almost an afterthought. When Bharath began his Ph.D. program in physics, he also started taking one graduate-level class in mathematics per semester. Before long, he needed only a few more, as well as a thesis, to complete the requirements of the master’s degree.

Prior to Tech, Bharath completed his undergraduate degree in physics in the Indian Institute of Technology (IIT) Kanpur, in Uttar Pradesh, India. He knew he would do a Ph.D. “I joined Georgia Tech in the pursuit of a place where cutting-edge research was being done,” he says.

At Tech, Bharath not only studied his major fields but also pushed himself to communicate his science well. In 2016, he participated in Georgia Tech’s Three Minute Thesis Competition. Competitors explained their research to a diverse audience in just three minutes.

At the time, Bharath was a fourth-year Ph.D. student. He had discovered something fundamental about rubidium atoms: When cooled to about 170 nanoKelvins – almost absolute zero – and exposed to a magnet that traces a circle around them, the very-low-energy rubidium atoms can remember something abstract. They can tell the area of an abstract surface – called the Boy’s surface – corresponding to the real traced circle.

For his spirited explanation of how atoms, when cooled to almost immobility, remember abstract geometric phenomena, the judges named Bharath the third-place winner and the audience voted him as one of two winners of the People’s Choice award.

What is the most important thing you learned at Georgia Tech?
Apart from the technical knowledge necessary to conduct scientific research in my area, I learned the art of academic communication and collaboration in research. The papers I wrote and the conferences I attended helped me learn the basics of communicating my research work. While working with multiple faculty members at Georgia Tech, I gained experience in scientific collaboration.  

What is your proudest achievement at Georgia Tech?
One of my research papers was rejected three times in a row by the same journal. However, with a carefully crafted rebuttal, I got it published after the fourth resubmission. The process was challenging, but I was supported extensively by the faculty members at Georgia Tech. 

Which professor(s) or class(es) made a big impact on you?
I gained a lot from the technical guidance of my advisor, Professor Michael Chapman. I owe my experimental skills and my intuitive understanding of atomic physics to him. He also provided valuable advice on crucial career-related decisions that I had to make in the later part of my Ph.D. work. His guidance has been pivotal in my professional development.

Professors Brian Kennedy and Carlos Sa de Melo also made a significant impact on my understanding of physics.

Professor Kennedy was always welcoming and available to talk about the theoretical aspects of our experiment. The discussions he had with me helped steer my research work into a productive direction. He also helped me extensively in writing a theoretical research paper and getting it published. During this process, with Professor Kennedy’s support, I learned how to respond to critical reviews of a research paper.

Being an experimental atomic physicist, I owe almost all my understanding of condensed-matter theory to Professor Sa de Melo. He is very friendly and always enthusiastic to talk about physics. I remember several late-night discussions with him in the laboratory, which resulted in a research paper that he and I wrote.  

I am grateful to two professors from the School of Mathematics, Greg Blekherman and John Etnyre.

As my master’s thesis advisor, Professor Blekherman is responsible for my technical knowledge in the area of convex optimization.  He was kind and accommodating as a thesis supervisor.

Professor Etnyre helped me understand the mathematical basis of my thesis project, which involved the fascinating subject of topology. He was always made himself available for discussions, from which I benefited greatly.   

What is your most vivid memory of Georgia Tech?
I have several.

Professor Sa de Melo would sometimes come to our lab at 9 PM. Along with a freshly brewed pot of tea, we talked about physics. Sometimes, we would lose track of time, only to realize that it is past 1 AM and we should call it a day. These discussions alone have resulted in a couple of research papers.

In the evenings, I would go on long walks, circling the campus area, occasionally stopping at the Campus Recreation Center for a swim or rock climbing or a game of ping-pong.

In what ways did your time at Georgia Tech transform your life?
Professionally, I now have a clear view of what I am going to do. At Georgia Tech, along with the acquiring the necessary technical skills, I developed an understanding of the goals of the specific research field. This understanding helped me decide what I want to do next.  

What unique learning activities did you undertake?
In 2016, Professor Chapman encouraged me to participate in the Three Minute Thesis (3MT) Competition at Tech. The challenge was to communicate my thesis work in three minutes to a nonexpert audience.

While preparing for 3MT, I learned the art of oral communication, and it changed the way I presented my work at conferences thereafter. I was fortunate to win prize money, which I used to attend a conference. Professor Chapman had the foresight to know that participating in 3MT would be a good step in my professional development.

What advice would you give to incoming graduate students at Georgia Tech? Georgia Tech has vast intellectual wealth, held by the numerous knowledgeable faculties in various disciplines. I would advise incoming graduate students to make use of this resource, as well as the facilities available on campus, to maximize their intellectual development during their time here.

Where are you headed after graduation?
I am starting a postdoctoral position at the Max Planck Institute of Quantum Optics, in Garching, Germany.

Professors Chapman, Kennedy, and Sa de Melo helped me develop the skills and confidence to continue in academia.They prepared me for an academic career, particularly for this postdoctoral position.

Volume X Contents

• Welcome from the Chair
• Benefits Add Up for Undergrads in SoM REU Programs
• Seven REUs Planned for Summer 2018
• Georgia Tech Hosts Annual High School Competition
• Libby Taylor Feature: Georgia Tech Undergrad Takes Home AWM Math Prize
• Annual TA Student Award Winners
• Recent Graduates Give Advice to Incoming Freshmen
• Geometric Group Theory Gets an Informal Take from Tech Professor
• Donor Awards
• PhD Program
• Events
• Awards
• Featured Article: Researchers Determine Routes of Respiratory Infection Disease Transmission on Aircraft
• Members of SoM at the Helm of National Research Programs
• Discrete Math/Combinatorics Moves Up to No. 2 in US News Graduate School Rankings
• Faculty Profiles
• Teaser: New Frontiers Beckon Math and Biology in Multi-million Dollar NSF-Simons Project
• ProofReader Article Picked Up by Notices of American Math Society
• SoM Professor Called to Give Expert Testimony in Jury Selection Case

Please see the Proofreader page on our website or click here to view a .pdf of the new ProofReader. 

Please send comments to Sal Barone at comm@math.gatech.edu, with subject line "ProofReader".

Mysteries of Floating

-By John McCuan

We are used to seeing a light object, like a beach ball, float on the surface of water while a heavy one, like a solid silver ball, sinks to the bottom (Fig.1-Fig.2). Over two-thousand years ago, based on similar observations, Archimedes proposed a simple and beautiful rule to determine which objects float, which objects sink, and how much liquid will be displaced by a floating object. He asserted that everything should be determined by relative densities.

Archimedes might be surprised to see this green plastic ball (Fig. 3-Fig. 5) which sinks to the bottom if pushed below the surface but also floats on the surface of the water if it is gently released there. The framework needed to understand the behavior of a “heavy” floating ball like this one was introduced by the mathematician Carl Friedrich Gauss in 1830. He applied his ideas about minimizing energy to the geometrical and analytical concepts of surface tension and contact angle introduced by Thomas Young and Pierre Simone Laplace in 1805 and 1806.

Nevertheless, theoretical verification of the possibility of a heavy floating object like the green ball was first obtained by Rajat Bhatnagar and Robert Finn of Stanford University in 2006. To obtain their result various simplifications were made. One of those simplifications was to assume the liquid bath was infinite in extent with the walls of the container infinitely far away. John McCuan of the School of Mathematics has been interested in floating objects in laterally bounded containers since about the same time. In 2013 he was able, along with Ray Treinen of Texas State University, to analyze the energy landscape for problems that include the green ball floating in a finite cylindrical container as in the photo above. They showed, in particular, that if such a ball, floating on the surface of the water is pushed downward, the energy of the system will increase at first, eventually reaching a single maximum, at which point, as the ball moves lower, the energy of the system decreases and eventually the ball slips below the surface and sinks.

While relaxing the assumption of an infinite sea on which the ball floats, McCuan and Treinen introduced an additional symmetry assumption, effectively requiring the ball to be constrained to a frictionless vertical wire through its center keeping the ball in the middle of a circular cylindrical container. The characterization of parameters (density, surface tension versus gravity, the size of the ball relative to that of the container, and adhesion properties) for which a floating ball will remain in the center without the guide-wire is still a major open problem.

Buoyed up by some success, McCuan and Treinen attempted to characterize the equilibrium configurations (maxima and minima of the Gauss energy) for balls like the beach ball with density lower than that of the liquid. They were able to obtain a number of results, but they were also in for a big surprise. The natural expectation would be that for the light ball there is a unique equilibrium (energy minimum) with the energy increasing monotonically as the ball is pushed downward (and constrained to the center) in a cylindrical container. This is true for a beach ball in, say, a swimming pool. Sometimes, however, for certain collections of parameters, the energy will, in fact, increase but then decrease to another local minimum before increasing as the ball is submerged. (See chart, first image)

Note: For purposes of illustration the figure is neither to scale nor accurately proportioned.

There are several consequences of this 2018 discovery. One is that a ball floating in a cylinder need not have a unique floating height; the ball may rest at equilibrium in two different positions. If, for example, the ball is positioned as on the left, it will remain there, but if the ball is manually moved to the position on the right, it will also float in position there. Such a ball in a cylinder might be used as a two position switch. Furthermore, the phenomenon first encountered with the heavy green ball is not isolated to the heavy floating ball. Even with a light floating ball, the observed floating configuration can depend on where one positions the ball initially. The only known instances of this behavior for a light ball occur when the ball fits within the cylinder leaving only a small gap (several one hundredths of a millimeter) between the ball and the wall, so the phenomenon would likely never have been discovered without considering the case of laterally bounded containers.

Other “fun” facts:

• 1. It was about 200 years between the time a mathematical framework describing floating objects (including capillarity and adhesion energies) was proposed and the time it was actually used with any success to describe floating objects.

Part of the groundwork for this kind of application of the theory was laid in McCuan’s 2007 paper which adapts the framework of Gauss to situations which allow floating. Previous to this, force phenomena such as buoyancy were viewed as separate from capillary equilibrium theory. McCuan showed all conditions for equilibrium (including various generalized force equations) follow from the basic approach of Gauss.

• 2. An essential difference between the analysis of floating objects (say balls) based on Archimedes’ principle and that based on capillarity is that in the former the liquid surface is assumed to be a flat plane, while in the latter the geometric shape of the liquid surface can be curved and plays a central role. Sometimes the liquid surface surrounding a floating ball can be so far from a plane that it bends back over itself as suggested by the exaggerated figure below.[PP] Several results in the paper of McCuan and Treinen (2013) give conditions under which this cannot happen. They show, for example, that if the ball is too heavy (dense) or the ball is too small, then such “folding over” is not possible. Also, if the ball is too light and the adhesion of the liquid with the ball is too small (resulting in an angle between the liquid and the ball measured within the liquid which is too big), then, again, folding over is not possible.
• 3. Another factor in the recent progress on problems like this (in spite of interest in them from antiquity) is the new capability to numerically analyze the model equations.
• 4. One approach (and perhaps the only approach) to understanding when a floating ball will remain centered in the container (rather than move to the side) requires an extension of McCuan’s 2007 first variation formula to the second variation of energy. In some instances (experimentally) when the outer edge of the liquid interface is higher than the edge on the ball, and the ball is heavy, the ball will stay in the center. Similarly, when the outer edge is lower than the inner edge, then a heavy ball will tend to the side. These observations can be reversed for a light ball. These experimentally observed conditions are (first of all) far from a mathematical analysis; it is very unlikely that they capture the entire range of possibilities.
• 5. Most of the known results are for a system which is simplified in dimension. Mathematically, we are really considering (in the drawings above for example) a two dimensional problem which can be viewed as treating an infinite log (extending directly out of the paper) floating in a trough. It seems likely that all equilibria for this simplified problem can be identified/classified within the next decade. A similar time frame applies to the spherical ball in a cylindrical container as indicated in the photographs. Some fundamental advance, like obtaining a second variational formula for energy as mentioned in the previous point will be necessary for understanding/classifying the conditions characterizing central floating versus moving to the side.

References:

250 B.C. Archimedes, On floating bodies

1805 Thomas Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. Lond. 95[PP]

1806 Pierre Simone Laplace, Mécanique céleste

2006 Raj Bhatnagar and Robert Finn, Equilibrium configurations of an infinite cylinder in an unbounded fluid. Phys. Fluids 18 no. 4

2007 John McCuan, A variational formula for floating bodies, Pac. J. Math. 231 no. 1

2009 John McCuan, Archimedes’ principle revisited, Milan J. Math. 77

2013 John McCuan and Ray Treinen, Capillarity and Archimedes’ principle of flotation, Pacific J. Math. 265 no 1

2018 John McCuan and Ray Treinen, On floating equilibria in a laterally finite container, SIAM J. Appl. Math. 78 no. 1

Congratulations go to Dan Margalit and Chongchun Zeng, who have been named American Mathematical Society (AMS) Fellows.

A special issue of the journal Discrete and Continuous Dynamical Systems-A has been dedicated to Prof. Rafael de la Llave.

The issue 38-12 of the journal Discrete and Continuos Dynamical Systems-A contains the proceedings of the international conference LLAVEFEST, which was celebrated June, 2017 in Barcelona. The conference was devoted to the interface of dynamics and partial differential equations and applications.

The goal of the conference was to present new advances in different aspects on Dynamical Systems and Partial Differential Equations.

There were 151 participants in attendance.

Topics covered included:
-  Dynamical systems and ergodic theory
-  Global dynamics in Hamiltonian systems
-  KAM theory
-  Arnol'd diffusion
-  PDEs and their applications
-  Lattice systems
-  Action-minimizing orbits and measures
-  Invariant manifold theory
-  Hyperbolic systems
-  Renormalization group methods

The main goal of the conference was bringing together many researchers from different disciplines, who presented high level talks. The conference also served as a celebration of Prof. de la Llave 60th birthday.

Several of the presentations in the conference have been written up, refereed for correctness and relevance, and gathered in a special volume of Discrete and Continuous Dynamical Systems-A. 

Prof. de la Llave and Prof. C. Zeng have been editors of the journal for several years.

Conference website:

http://www.crm.cat/en/Activities/Curs_2016-2017/Pages/C_FIDDS.aspx

Preface of Llavefest:

http://aimsciences.org//article/doi/10.3934/dcds.201812i#FullText

Discrete and Continuous Dynamical Systems-A Issue 38-12:

http://aimsciences.org/journal/1078-0947/2018/38/12

Episode 9 of ScienceMatters' Season 1 stars Dan Margalit.  Listen to the podcast or read the transcript here. 

Dan Margalit is a professor in the School of Mathematics. 

Margalit's research area is topology. He studies the properties of shapes that persist even when the shapes are stretched or bent. 

For example, two metal rings that are linked 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 Margalit's 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.

Margalit's particular research is on the symmetries of surfaces. Some symmetries of surfaces are easy to understand. But when bending and stretching are allowed, the symmetries are more challenging.

For Margalit, "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."

In Episode 9, Margalit talks about the beauty of mathematics and offers advice to overcome "math phobia."

Take a listen at sciencematters.gatech.edu.

Enter to win a prize by answering the question for Episode 9: 

According to Episode 9, what group of people can’t tell the difference between a coffee cup and a donut?

Submit your entry by 11 AM on Monday, Oct. 22, at sciencematters.gatech.edu. 

Elizabeth Ann “Libby” Peck earned two degrees from Georgia Tech: a B.S. in applied mathematics in 1975, and -- from the H. Milton Stewart School of Industrial and Systems Engineering -- an M.S. in industrial engineering in 1976. For over 40 years she applied the knowledge she learned from Tech in building mathematical models to answer myriad questions of the Coca-Cola Company—from supply chains to strategic infrastructure to delivery routes.

Libby was the first woman to use mathematical models for supply chain analysis at the Coca-Cola Company. Often, she was the only woman among male colleagues working on global problems. By standing up to defend her work vigorously and completing projects with assiduous diligence, she proved herself equal to the best of the men around her.

Read a Q&A with Peck in which she talks about her roles at Coca-Cola, how her ISyE degree prepared her for her work there, and her memories of being a student at Georgia Tech: https://b.gatech.edu/2RaZaiT.