Georgia Institute of Technology College of Sciences

Atlanta Combinatorics Colloquium - Jinyoung Park

October 30th from 4:30pm – 5:30pm
@ Skiles 006
Reception at 4pm

Title: The Convexity Conjecture, the Kahn-Kalai Conjecture, and introduction to k-thresholds

Abstract: The "Convexity Conjecture" by Talagrand asks (very roughly) whether one can "create convexity" in constant steps regardless of the dimension of the ambient space. Talagrand also suggested a discrete version of the Convexity Conjecture and called it "my lifetime favorite problem," offering $1,000 prize for its solution. We introduce a reformulation of the discrete Convexity Conjecture using the new notion of "k-thresholds," which is an extension of the traditional notion of thresholds, introduced by Talagrand. Some ongoing work on understanding k-thresholds, along with a (vague) connection between the Kahn-Kalai Conjecture and the discrete Convexity Conjecture, will also be discussed. Joint work with Michel Talagrand.

 

About the Speaker

Jinyoung Park is an Assistant Professor of Mathematics at the Courant Institute, NYU. Her research interests include extremal and probabilistic combinatorics. She earned her Ph.D. in mathematics from Rutgers University in 2020, under the supervision of Jeff Kahn. Following her Ph.D., she was a postdoctoral researcher at the Institute for Advanced Study (IAS) and Stanford University before joining the faculty of the Courant Institute at NYU in 2023. Park received her bachelor's degree in mathematics education from Seoul National University in 2005 and worked as a secondary school mathematics teacher until 2011. She has received several honors and awards, including the AMS Conant Prize, the Dénes König Prize, and the Maryam Mirzakhani New Frontiers Prize.

Photo by Rod Searcey

School of Mathematics Professor John Etnyre has been selected as a section lecturer for the 2026 International Congress of Mathematicians (ICM 2026). Featuring world-leading mathematicians at the forefront of their fields, ICM 2026 will be held in Philadelphia, Pennsylvania in July — coinciding with the 250th anniversary of the signing of the Declaration of Independence and marking the first time in 40 years that the conference will be held in the United States. 

“Speaking as a section lecturer at ICM is a rare distinction and prestigious honor,” says School of Mathematics Chair and Professor Mike Wolf. "Fewer than two dozen researchers in the world are asked to speak on geometry and topology at this event, which happens just once every four years. We are thrilled, but not surprised, that John has been selected. His top-notch research and teaching are truly world class.” 

Known for his expertise in the area, Etnyre will speak on the topic of Geometry. “I was surprised and excited to receive the invitation to speak,” says Etnyre. “It is a great honor to represent my branch of mathematics and the School of Mathematics at ICM. The School of Mathematics has had several ICM speakers in the past, and I am very happy to continue that legacy.”

H. Milton Stewart School of Industrial and Systems Engineering Professor Katya Scheinberg has also been selected as a section lecturer and will speak on Control Theory and Optimization.

About John Etnyre

Etnyre is known for his expertise in topology, including knot theory, which is crucial to understanding three- and four-dimensional spaces, with applications ranging from string theory to DNA recombination and understanding the shape of the universe. He also studies contact and symplectic geometry and three- and four-dimensional manifolds.

“A large part of my work over the years has been to demonstrate that special subspaces of contact and symplectic manifolds are the keys to unlocking their subtle nature,” he explains. “This goes back, at least, to Bennequin in the early 1980s and then Eliashberg in the late 1980s and 1990s. My talk at ICM will survey this research, starting with Bennequin’s work and ending with current trends in the field.”

Etnyre’s previous distinctions include being in the Inaugural Class of Fellows of the American Mathematical Society. He has also received a National Science Foundation CAREER grant award and was a Simons Fellow in Mathematics.

Senior Academic Professional Chris Jankowski is being recognized for his excellence in teaching and for winning the institute level Dean George C. Griffin Award for faculty member of the year.

Chris Jankowski Wins the Dean George C. Griffin Award for Faculty Member of the Year

At the "Up with the White & Gold (UWWG)" award ceremony, the Student Government Association -- Undergraduate House of Representative -- awarded Chris Jankowski the Dean George C. Griffin Award for faculty member of the year.

Chris Jankowski is a key contributor to the success of the SoM’s teaching mission, and I’m happy to see him win this award.

- Matt Baker

This unique honor to a single faculty member at the Institute represents the opinion of the students as which faculty member has had an especially positive impact on them while at Tech. The citation reads (in part),

The House voted you the winner for your outstanding teaching and bringing clarity to a very difficult course for many students, Linear Algebra. There was very high praise from many students regarding your passion to teach and meet students where they are at.

Congratulations to Chris for this well-deserved important distinction! We in the school greatly appreciate your efforts and are privileged to work beside you as you constantly help to advance the teaching mission of the school. Well done!

Chris Jankowski Wins Fulmer Prize 2025

The SoM is pleased to announce that Chris Jankowski is this year's recipient of the Fulmer award.

The Herman K. Fulmer Faculty Teaching Fund Endowment for the School of Mathematics was established by the late Howard Woodham (Georgia Tech alumnus, Engineering ’48), in memory of Professor Herman Fulmer, his former mathematics professor. Each year the Fulmer award recognizes one of our faculty who exhibit genuine regard for undergraduate students during the first few years of their Engineering studies at Georgia Tech.

Through his teaching of Math 1553, and in his coordinating role, Chris has stood out as a dedicated instructor that delivers masterful lectures, is always available to his students, and helps each student have a strong start in their undergraduate studies.

Congratulations Chris on this well-deserved award!

Imagine a restaurant door swinging open, then closing again. You hear a burst of noise: chattering voices, clattering silverware, and shuffling feet. Now imagine if the sound were to go on for a long time — hours, days, or even years. You’d probably start to hear it as something you could tune out, a drone of noise without the individual “frequencies” that make up the noise.

School of Mathematics Associate Professor Benjamin Jaye has been awarded a prestigious Simons Fellowship in support of his research into these types of related qualities (like time and frequency), and how precisely we can know one without knowing the other.

Called the Fourier uncertainty principle, it’s a centuries-old subject, but progress on this topic is still in its infancy, Jaye says. Using mathematics, he will unravel how much information, or partial information, is needed about a function’s frequencies in order to determine what the original function is.

Advancing fundamental mathematics

While his work centers on theory, there are a number of fields that can benefit from it.

“These problems interest me as basic questions in harmonic analysis,” a fundamental area of mathematics used for research in fields ranging from quantum mechanics to neuroscience, Jaye says, adding that “the specific forms I am interested in arose from work in probability theory — in particular, understanding the probability that a system is reliable over a long period of time.” 

This type of reliability analysis and probability theory plays a crucial role in predicting how safe equipment and processes are, and could lead to advancements in more reliable public transportation to safer planes.

Jaye also notes applications for partial differential equations, in particular “dampening” a wave function to ensure that energy is lost at a certain rate — a critical area of research for controlling and predicting waves, with applications in engineering, physics, and optics.

“I have had many amazing colleagues who have piqued my interest in these questions over the last ten years,” Jaye says. “The Simons Fellowship gives me an important opportunity to develop mathematical theories around them in a systematic way.”

School of Mathematics Professor Anton Leykin has been awarded a prestigious Simons Fellowship for his proposal of applying nonlinear algebra to tackle one of the key mathematical questions of the 21st century.  Leykin is one of two mathematicians in the School awarded the Fellowship, and is joined by Associate Professor Benjamin Jaye.

The work could lead to new discoveries and a deeper understanding of how celestial bodies like planets, moons, and asteroids interact. The fellowship will fund one year of work, during which Leykin also plans to finish writing a book on nonlinear algebra for advanced undergraduate and beginning graduate students.

Leykin explains that the mathematical problem — known as “Smale's sixth problem” for its position as number six on the list of questions for the 21st century compiled by Fields Medalist Stephen Smale — involves understanding the number of ways celestial bodies can be arranged in space so that they stay at relative equilibrium, growing neither further apart nor closer to each other as they orbit.

“People have been trying to solve this problem for more than two hundred years — since Euler and Lagrange — but even proving that the number of relative equilibria in an n-body problem is finite is extremely difficult,” Leykin says. One reason for this? “Even for small cases, (e.g. n=5) the brute-force approach leads to an enormous amount of computation.”  

Each of Leykin’s initial experiments for the case n=6 required a CPU year — the computational power equivalent to a single computer running for an entire year. 

This difficulty is partially what draws Leykin to the problem. “We use supercomputers to help with the computation time, but this isn’t an area that AI and machine learning can advance,” he explains. “For this type of problem, we need human intelligence — and even with our current technology, there are no easy solutions. It’s a challenge, but that is what makes it interesting.”

Stellar pathways

Imagine the Moon and Earth as two celestial bodies on a plane. Both exert gravitational force on each other — we can see the result as tides on Earth and the Moon’s orbit. Now add the Sun and other planetary bodies to the plane: asteroids, satellites, and other planets. These bodies also exert gravitational force — a function of their masses and distances apart — creating a complex system of orbits and trajectories.

Smale’s sixth problem imagines a plane like this, with any number of celestial bodies arranged on it. The problem considers an arrangement of the bodies in a way that the gravitational forces balance, so that even while they are interacting and orbiting, none of the bodies travel further away or closer to each other.

“It has been conjectured, but so far not shown, that the number of such configurations is finite,” Leykin says. “It seems simple. Is it finite or infinite? But progress is minimal at the moment.  Several approaches settle the question for almost all values of n=5 masses with the case n=6 wide open, even for a non-special choice of masses.”

Leykin is taking a different approach than many researchers, leveraging a field of mathematics called tropical geometry, which simplifies the geometry of curved equations as straight lines. 

“We're not trying to compute or describe the original solution manifold but rather replace it with its tropicalization, a combinatorial shadow which captures the finiteness aspect,” he explains. 

Leykin’s method has already found success for the case n=5. “A recent paper proved that for five bodies, if the masses are general enough, there are a finite number of relative equilibria,” Leykin shares. “Using our approach, we were able to reproduce the result for five celestial bodies in a simpler way.”

“Our goal now is to collect more evidence by solving the problem for six bodies,” he adds. “If this helps lead us to a general solution to the problem as a whole — that would be great.”

Space ‘storage spots’

While the project is theoretical, it could lead to a greater understanding of celestial mechanics.

Leykin is collaborating on a separate but related project with aerospace departments around the country. “We're working to understand the trajectories of a massless spacecraft, assuming it is primarily affected by gravitation of the Moon and the Earth,” he shares.

The 18th century mathematics developed for this type of problem, a restricted three-body problem, could help teams use the gravitational pull of the Earth and Moon to place a small spacecraft near a Lagrangian point — a space “storage spot” where it would remain stationary relative to the Earth.

“You can place something at a Lagrangian point, and it will stay stationary relative to the system,” Leykin explains. “It's a way to place things so they don't move.” For example, in the Sun-Earth system, the James Webb Space Telescope was placed at one of these points, where it conveniently stays in Earth’s shadow — avoiding the bright light and heat of the Sun, Earth, and Moon.

“Smale’s sixth problem is about acquiring more theoretical knowledge,” Leykin adds. “If we discover something on the theoretical front, it can be of practical importance for applied scientists and designing missions for exploratory spacecraft going far into the solar system.”

What’s the shape of the universe? Mathematicians use topology to study the shape of the world and everything in it

written by John Etnyre
originally appeared in The Conversation nonprofit news source

When you look at your surrounding environment, it might seem like you’re living on a flat plane. After all, this is why you can navigate a new city using a map: a flat piece of paper that represents all the places around you. This is likely why some people in the past believed the earth to be flat. But most people now know that is far from the truth.

You live on the surface of a giant sphere, like a beach ball the size of the Earth with a few bumps added. The surface of the sphere and the plane are two possible 2D spaces, meaning you can walk in two directions: north and south or east and west.

What other possible spaces might you be living on? That is, what other spaces around you are 2D? For example, the surface of a giant doughnut is another 2D space.

Through a field called geometric topology, mathematicians like me study all possible spaces in all dimensions. Whether trying to design secure sensor networks, mine data or use origami to deploy satellites, the underlying language and ideas are likely to be that of topology.

The shape of the universe

When you look around the universe you live in, it looks like a 3D space, just like the surface of the Earth looks like a 2D space. However, just like the Earth, if you were to look at the universe as a whole, it could be a more complicated space, like a giant 3D version of the 2D beach ball surface or something even more exotic than that.

While you don’t need topology to determine that you are living on something like a giant beach ball, knowing all the possible 2D spaces can be useful. Over a century ago, mathematicians figured out all the possible 2D spaces and many of their properties.

In the past several decades, mathematicians have learned a lot about all of the possible 3D spaces. While we do not have a complete understanding like we do for 2D spaces, we do know a lot. With this knowledge, physicists and astronomers can try to determine what 3D space people actually live in.

While the answer is not completely known, there are many intriguing and surprising possibilities. The options become even more complicated if you consider time as a dimension.

To see how this might work, note that to describe the location of something in space – say a comet – you need four numbers: three to describe its position and one to describe the time it is in that position. These four numbers are what make up a 4D space.

Now, you can consider what 4D spaces are possible and in which of those spaces do you live.

Topology in higher dimensions

At this point, it may seem like there is no reason to consider spaces that have dimensions larger than four, since that is the highest imaginable dimension that might describe our universe. But a branch of physics called string theory suggests that the universe has many more dimensions than four.

There are also practical applications of thinking about higher dimensional spaces, such as robot motion planning. Suppose you are trying to understand the motion of three robots moving around a factory floor in a warehouse. You can put a grid on the floor and describe the position of each robot by their x and y coordinates on the grid. Since each of the three robots requires two coordinates, you will need six numbers to describe all of the possible positions of the robots. You can interpret the possible positions of the robots as a 6D space.

As the number of robots increases, the dimension of the space increases. Factoring in other useful information, such as the locations of obstacles, makes the space even more complicated. In order to study this problem, you need to study high-dimensional spaces.

There are countless other scientific problems where high-dimensional spaces appear, from modeling the motion of planets and spacecraft to trying to understand the “shape” of large datasets.

Tied up in knots

Another type of problem topologists study is how one space can sit inside another.

For example, if you hold a knotted loop of string, then we have a 1D space (the loop of string) inside a 3D space (your room). Such loops are called mathematical knots.

The study of knots first grew out of physics but has become a central area of topology. They are essential to how scientists understand 3D and 4D spaces and have a delightful and subtle structure that researchers are still trying to understand.

In addition, knots have many applications, ranging from string theory in physics to DNA recombination in biology to chirality in chemistry.

What shape do you live on?

Geometric topology is a beautiful and complex subject, and there are still countless exciting questions to answer about spaces.

For example, the smooth 4D Poincaré conjecture asks what the “simplest” closed 4D space is, and the slice-ribbon conjecture aims to understand how knots in 3D spaces relate to surfaces in 4D spaces.

Topology is currently useful in science and engineering. Unraveling more mysteries of spaces in all dimensions will be invaluable to understanding the world in which we live and solving real-world problems.