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Blair D. Sullivan, B.S. MATH/CS 2003, will be one of four plenary speakers at the Spring Southeast Sectional Meeting of the American Mathematical Society, scheduled for March 18-19 at Georgia Tech. Sullivan is now an associate professor in the School of Computing at the University of Utah, with an adjunct appointment at North Carolina State University. Her research interests include parameterized algorithms, structural graph theory, applied discrete mathematics, random graphs, and combinatorial scientific computing. Sullivan was also a research scientist in the Computer Science and Mathematics Division at Oak Ride National Laboratory. Several School of Mathematics faculty members are organizing panels and other discussions during the Spring Southeast Sectional Meeting.
Georgia Tech’s School of Mathematics is dedicated to exploring the frontiers of computational and experimental research in its discipline, so much so that one of the leading math research centers in the country now has a School of Mathematics professor serving as the chair of its board of trustees.
Rachel Kuske’s new role with the Institute for Computational and Experimental Research in Mathematics (ICERM) is just one Georgia Tech connection to the Center, based at Brown University in Providence, Rhode Island.
More than 20 School of Mathematics faculty members and graduate students have participated in recent ICERM programs, including a series of seminars during Fall 2022’s semester on harmonic analysis and convexity, mathematical processes that help researchers navigate large collections of data.
“The participation of School of Mathematics members at different levels in ICERM, one of several leading math research institutes in the U.S., is representative of the School’s leadership in the broader research community,” said Kuske, a former School of Mathematics chair. “Georgia Tech’s multi-faceted involvement benefits our research groups as well as research advances and development of talent in the wider research community.”
John Etnyre, a professor specializing in low-dimensional topology, is also involved in various ICERM activities, and helped co-organize a semester-long program on braid theory in Spring 2022. “ICERM is an excellent research center,” Etnyre said. “They provide a great environment to collaborate with others, as well as great conference facilities. They are certainly one of the best mathematical research centers in the world, and they are unique in their focus on bringing computation and experimentation into mathematics.”
Mathematics labs and more
ICERM’s mission is to expand the use of computational and experimental methods in mathematics, support theoretical advances related to computation, and address problems posed by the existence and use of the computer through mathematical tools, research and innovation.
ICERM pursues these goals by supporting what Kuske calls “mathematics labs,” which are typically human resource-intensive, and highly collaborative nationally and globally.
“ICERM’s goals include catalyzing new directions in research and collaborations, as well as exploiting and expanding the interface between mathematics, computations, and experiments, computational and otherwise,” she said. These intersections have historically been represented in ICERM’s scientific board, with Kuske citing Dana Randall, ADVANCE Professor of Computing in the School of Computer Science, and an adjunct professor in the School of Mathematics, as an example.
Kuske views chairing ICERM’s Board of Trustees as a way to “provide an opportunity to contribute in several directions, including making sure that present and future resources, policies, and procedures support ICERM’s mission.” These include increasing diverse and inclusive participation in mathematical sciences and relevant areas, raising public awareness of the impact of mathematics, and continued service and leadership in the research community.
Etnyre said an important computational aspect to topology — the study of surfaces that can be twisted, bent, or otherwise deformed but never broken — has been around for a while, “but its importance has been increasing over the years,” he said. For example, software called SnapPea/SnapPy “is a program where you can input a three-dimensional space and it will compute a myriad of data about the space. It also has a list of thousands of spaces and data about them. When trying to determine if something you are interested in is true or not, it is always helpful to be able to check its validity on such a large sample of spaces.”
More recently, Etnyre says several teams of people have been using machine learning algorithms to explore relations involving knot theory, the study of closed curves in three dimensional spaces. “There are many other ways in which computation and experimentation is important in topology, and it was great that the ICERM program was able to expose these techniques to a large number of researchers during our program.”
Knot theory and an associated subdiscipline, braid theory, help bring structure to large, complex data problems. Possible applications include finding out more about DNA recombination, Etynre said. “There are also connections with physics through string theory and gauge theory. There are connections between braids and many areas in mathematics. That was really the focus of the program at ICERM last spring,” referring to the Spring 2022 program he helped co-organize at the center.
Collaboration on convexity
Galyna Livshyts, associate professor in the Georgia Tech School of Mathematics; Ben Jaye, assistant professor, and postdoctoral researcher and visiting assistant professor Naga Manasa Vempati recently completed the semester-long program on harmonic analysis and convexity at ICERM. Livshyts said the center is one of several institutions around the world that provide such lengthy research opportunities for various areas in mathematics.
“The harmonic analysis and convexity research program presented us with the opportunity to collaborate with other researchers in the area during this time,” Livshyts said. “Also, ICERM often hosts various interesting and stimulating workshops. ICERM is located in the buzzing town of Providence, and it has excellent facilities to allow people to discuss mathematics, and also provide some great views.”
The following Georgia Tech School of Mathematics faculty and students have participated in various recent ICERM programs:
The Combinatorial Algebraic Geometry virtual workshop in early February 2021 included Trevor Gunn, Arvind Ramaswami, Matthew Baker, Cvetelina Hill, and Alperen Ozdemir.
Matthew Baker, Justin Chen, Tianyi Zhang, Anton Leykin, Josephine Yu, and Josiah Park also took part in Collaborate@ICERM projects in 2021.
School of Mathematics students Yvon Verberne, Sudipta Kolay, Justin (Yi-Chang) Chen and Jiaqi Yang were named ICERM Postdoctoral Fellows for 2021-22.
Professor John Entyre co-organized the Spring 2022 Braids semester program at ICERM, and Professor Dan Margalit participated in Braids in Symplectic and Algebraic Geometry. Postdoctoral students Miriam Kuzbary and Hannah Turner took part in the entire Braids program. Graduate student Sally Collins and undergraduate student Sarah Pritchard participated in the Braids in Low-Dimensional Topology conference in April.
Georgia Tech faculty and students taking part in the Harmonic Analysis and Convexity program in September 2022 at ICERM include Manuel Fernandez, Orli Herscovici, Galyna Livshyts, Naga Manasa Vempati, Shixuan Zhang, Ben Jaye.
Mohit Singh and Swati Gupta are scheduled to participate in an ICERM program, Combinatorics and Optimization, March 27-31, 2023.
Josephine Yu and her coauthors have had a recent work published in the AMS Notices.
https://www.ams.org/journals/notices/202301/rnoti-p34.pdf
From the article:
In this article we introduce some of the basic constructions in tropical geometry, focusing on linear spaces and Grassmannians for their combinatorial significance. We give pointers to some recent research frontiers and discuss applications in matroid theory, phylogenetic trees, and auction theory.
Professor Josephine Yu
Dr. Josephine Yu was recently promoted to a Full Professor in the School of Mathematics, in 2022. Prof. Yu's research lies in the area of Tropical Algebraic Geometry and its applications to combinatorics, matroid theory, and analytic and computational geometry, and since arriving at Tech her work has been continually supported by the NSF. Prof. Yu has organized many conferences since 2011 including the Meeting on Applied Algebraic Geometry in 2019, and the Computational Tropical Geometry Minisymposium at the SIAM Algebraic Geometry meeting in 2017. Prof. Yu was also the Program co-chair of SIAM AG21 and has served on the Advisory Board of MEGA (Effective Methods in Algebraic Geometry) since 2019.
Prof. Yu is also currently an Editor for two journals, Combinatorial Theory and Algebraic Statistics (AStat), is the Associate Editor for the Journal of Software for Algebra and Geometry (JSAG), and was the Editor in Chief for the Journal of Combinatorial Theory, Series A (JCTA) from 2018 until 2020.
In addition to her impressive research and organizational work, Prof. Yu has mentored three graduate students and two masters students including award winning graduate students Cvetelina Hill and Marcel Celaya.
The College of Sciences is pleased to welcome Svetlana Jitomirskaya, Distinguished Professor at the University of California, Irvine, and a prize-winning mathematician, as the inaugural Hubbard Chair Professor in the School of Mathematics at the Georgia Institute of Technology. Jitormirskaya will arrive on campus in January 2023.
"The School of Mathematics is just delighted to welcome Professor Jitomirskaya,” said Michael Wolf, professor and chair of the School of Mathematics. “We had hoped that the Hubbard Chair would be transformational for the School of Mathematics, and the appointment of Svetlana to this position exceeds our wildest ambitions. Known for her penetrating insights into mathematical physics and dynamics, she adds to our already premiere presence in mathematical physics — an additional depth that rivals any other such center in North America.”
Jitomirskaya is one of seven new faculty members starting in Fall 2022 in the School of Mathematics. That number includes Wolf, who was named school chair in December 2021 and officially arrived at Georgia Tech last summer.
The inspiration for the Hubbard Chair
The chair is named for Elaine M. Hubbard (MATH 1972, M.S. MATH 1974, Ph.D. MATH 1980), who died in 2016 after a 28-year career as a mathematics professor at Kennesaw State University. Hubbard was a long-time friend and supporter of the School of Mathematics and a member of the College of Sciences Advisory Board.
Hubbard “was a true innovator — her delight in mathematics served to inspire her students,” said Paul Goldbart, who spoke at Hubbard’s memorial service six years ago and was then College of Sciences Dean. “She piloted the use of graphical calculators and gained national recognition and the Kennesaw Distinguished Teaching Award for her groundbreaking uses of technology. She spoke on mathematics education at conferences and campuses around the nation and received the Kennesaw State Alumni Association Achievement Award in 1994,” Goldbart said. “Elaine co-authored an amazing 13 textbooks on mathematics, important for their incorporation of the scholarship of teaching and learning to promote student success.”
Hubbard included a provision in her estate that established the Elaine M. Hubbard Endowed Chair for the School of Mathematics. Colleagues note that her passion was teaching, and the fund serves to support robust, leading-edge mathematics education and research at Georgia Tech.
“Elaine Hubbard was a gentle champion of mathematics at Georgia Tech, and I believe she would be pleased by how this first result of her generosity and vision has propelled forward the school and all of its missions,” Wolf added.
‘A scientific granddaughter’ of Russia’s greatest mathematician
Jitomirskaya was born in Kharkov, Ukraine, to parents who were both mathematicians. She has described both of them as survivors, having barely escaped as young children from the German invasion of Kiev in 1941.
Mathematics excellence runs in her family. Her mother chaired the famed Department of Analysis at Kharkov State University, and was also the only female professor of mathematics in Ukraine for some twenty years. Her father was a long-time chair of the Department of Mathematics at KhADI, an engineering school.
Svetlana left Kharkov at 16 to study at Moscow State University, and graduated under the supervision of Yaklov G. Sinai, himself a student of A.N. Kolmogorov, whom Leonid Bunimovich, Regents’ Professor in the Georgia Tech School of Mathematics called “the greatest Russian mathematician ever.”
Jitomirskaya “brings to Georgia Tech a brilliant scientific genealogy which is really hard to match,” Bunimovich said. “She is a scientific granddaughter of A.N. Kolmogorov and her advisor Sinai is an Abel Prize winner. The spirit of this School is that mathematics is rigorously proved. In other words, mathematicians should not just prove what scientists and engineers already understood, but uncover why their ideas are right and show them the way further, and even bring in new ideas, which mathematicians must rigorously justify, especially in cases when these new ideas contradict ‘preexisting’ physics intuition.”
International recognition
Jitomirskaya was awarded the Ruth Lyttle Satter Prize from the American Mathematical Association in 2005, and the Dannie Heineman Prize for Mathematical Physics awarded by the American Physics Association and the American Institute of Physics in 2020. She has been a member of the American Academy of Arts and Sciences (AAAS) since 2018, and was elected to the NAS in 2022. Jitomirskaya was invited to deliver a plenary lecture at the International Congress of Mathematicians, held as a virtual event in July 2022.
In July 2022, Jitomirskaya was also announced as the first winner of a new award for mathematical physics: the Olga Alexandrovna Ladyzhenskaya Prize. Wolf, the Georgia Tech School of Mathematics chair, said the prize celebrates the “extraordinary mathematical contributions in the middle part of the previous century of the Russian mathematician Olga Ladyzhenskaya.”
Jitomirskaya has also dedicated a large portion of her career to teaching. She received the University of California, Irvine Chancellor’s Award for Excellence in Fostering Undergraduate Research in 2018. She has advised many graduate students and post-doctoral researchers, who eventually found positions in the academic world.
“There is no better choice than Svetlana Jitomirskaya to occupy this inaugural Chair,” added Jean Bellissard, professor emeritus in the School of Mathematics at Georgia Tech, “as she is both a worldwide recognized expert in analysis, and widely appreciated among her students and her university for her dedication to teaching mathematics at the highest level of excellence.”
“Mathematics progresses through a sustained conversation among a community of scholars, and Svetlana will deepen and broaden that dialogue for our scholars while exciting and inspiring our students,” Wolf said.
More new faces at the School of Mathematics
Michael (Mike) Wolf joined the School of Mathematics as chair and professor in the fall of 2022. Wolf comes to Georgia Tech from Rice University, where he served most recently as Milton B. Porter Professor. During his three-decade tenure at Rice, Wolf has held many positions, including two periods as chair of the Department of Mathematics, head of a residential college, and co-founder and co-director of the Rice Emerging Scholars Program.
“Georgia Tech’s Mathematics faculty is world-renowned for its strength and scope, and it is an honor to participate in its leadership,” Wolf said in the announcement of his new role. “Mathematics is an engine for modern science and technology — from codes for cybersecurity, to differential equations that explain black holes and the interfaces of materials, to machine learning and mathematical neuroscience, and through beautiful advances whose applications will only be revealed to our grandchildren. Mathematics is everywhere, and Georgia Tech’s mathematicians are at the frontier.”
Three assistant professors joined Wolf in the Fall 2022 semester as new School of Math faculty: Gong Chen, Vesselin Dimitrov, and Tom Kelly; along with Academic Professionals Hunter Lehmann and Kalila Lehmann.
The School of Mathematics is also welcoming 15 new visiting assistant professors and postdoctoral scholars for the 2022-2023 academic year.
This years Tech Topology Conference will be from December 9 to 11. We have a great line up this year:
• Wade Bloomquist (Georgia Institute of Technology)
• Ruth Charney (Brandeis University)
• Dave Gabai (Princeton University)
• Maggie Miller (Stanford University)
• Abdoul Karim Sane (Georgia Institute of Technology)
• Hannah Schwartz (Princeton University)
• Jonathan Simone (Georgia Institute of Technology)
• Hannah Turner (Georgia Institute of Technology)
• Yvon Verberne (University of Toronto)
You can find out more at
ttc.gatech.edu
Please register if you plan to attend any of the talks so that we can add you to the participants page and put you on the e-mail list for the conference. If you would like to give a lightning talk, you can also apply for that on the web page too.
About the Conference
The 2022 conference features several sessions of five-minute pre-recorded and live lightning talks. If you are interested in giving such a talk please see the "Registration and Support" page.
The deadline for submitting proposals for Lightning Talks is October 14.
Organizers: W. Bloomquist A. Christian, J. Etnyre, J. Hom, M. Kuzbary, D. Margalit, N. Saglam, J. Simone, H. Turner
Supported by the NSF and the Georgia Institute of Technology
Jen Hom, an associate professor in the School of Mathematics, has been named to the 2023 Class of Fellows of the American Mathematical Society for her “contributions to low-dimensional topology, Heegaard Floer homology, and service to the mathematical community.”
The Fellows of the American Mathematical Society program recognizes members who have made outstanding contributions to the creation, exposition, advancement, communication, and utilization of mathematics. “It is an honor to welcome a new class of AMS Fellows and to congratulate them for their notable contributions to mathematics research and service to the profession,” said AMS President Ruth Charney.
Hom joined Georgia Tech in 2015 and is a previous recipient of the College of Sciences Cullen-Peck Fellowship Award. “It’s a great honor to be named an AMS Fellow, and to join this esteemed list of mathematicians that includes many of my mentors,” Hom said.
Hom’s research focuses on “knots, surfaces, and their higher dimensional analogs,” referring to certain mathematical structures embedded in three-dimensional space. Rather than the kinds of knots used in ropes and shoelaces, the ends of strings in mathematical knots are joined together. Surfaces, meanwhile, refer to the outsides of malleable geometric shapes.
Knots and surfaces are found in topology, the study of surfaces and shapes that can be bent, twisted and otherwise deformed but never broken or torn. Heegaard Floer homology helps mathematicians make sense of these shapes, Hom said. “Heegaard Floer homology is a powerful tool for studying these objects that helps translate questions about shapes into questions about algebra.”
Before coming to Georgia Tech, Hom was Ritt Assistant Professor at Columbia University, and was a member of the Institute for Advanced Study in Princeton, N.J. She received her B.S. in Applied Physics from Columbia, and her Ph.D. in Mathematics from the University of Pennsylvania.
NOTE: This news release first appeared on the website of the American Mathematical Society.
Diana M. Thomas (Ph.D. Mathematics 1996) will receive the 2023 Mary P. Dolciani Prize for Excellence in Research from the American Mathematical Society (AMS).
Thomas is currently a professor of mathematics at the United States Military Academy, an adjunct professor at the Pennington Biomedical Research Center, and a research associate at the New York Obesity Research Center at Columbia University. She was awarded the prize for her outstanding research at the interface of mathematics with nutrition and obesity; her work in number theory, combinatorics, and dynamical systems; and her impressive work with undergraduates.
Thomas has an extensive publication record with more than 150 articles, book chapters, and conference proceedings. Much of her research is interdisciplinary and has been published in a diverse set of journals, including those specializing in nutrition, obesity, behavioral science, biology, and pure mathematics.
Thomas’ work on obesity and metabolism has been particularly impactful. Her nominators write that “she has published a remarkable series of highly original and imaginative papers that display creativity and quantitative rigor, and more recently, on the dynamics of energy exchange and weight gain in pregnancy. Each of these areas suffered substantial quantitative assessment gaps. The reports by Thomas provide not only important new biological insights, but also important clinical advances and assessment tools. She is rapidly filling the gap between classical mathematics and biological processes. In so doing, she adds a new dimension to the study of human obesity that is so pervasive across adults and children.”
Work by Thomas has led to the design of innovative software that assists users with weight-related health issues, and which has been covered by several media outlets. The work of Thomas and her colleagues has been funded by numerous grants, including six projects funded by the National Institutes of Health. Thomas received an American Heart Association Most Impactful Publications Award in 2014 and the Obesity Society George A. Bray Founders Award in 2017.
Thomas has advised undergraduate research in both pure and applied mathematics, and has coauthored more than 50 publications with undergraduates, including first-generation college students. She inspires undergraduates with informal discussions inside and outside of the classroom, and masterfully draws them into research projects that are appropriate for their background and interests. Undergraduates she has mentored have pursued many different professions, including enrollment into doctoral programs and careers in education and medicine.
Passionate about transforming mathematics education, Thomas has served in several important leadership roles in this regard. She directed the Mathematical Association of America’s undergraduate research poster session competition, and while serving as Director of the Center for Quantitative Obesity Research at Montclair State University, grouped together STEM students engaged in quantitative research, medicine, and nutrition to develop and integrate their knowledge across disciplines.
In addition, Thomas teaches an annual short course on Mathematical Science in Obesity Research, and she recently served as a Remote Teaching Dean’s Fellow at the United States Military Academy. As her nominators note, “Her leadership, collegiality, and results-oriented focus are three strengths that drive any program that she takes on to use science to answer hard questions. She has inspired, educated, and mentored generations of mathematics and nutrition researchers to choose fact and science to make policy decisions.”
From Diana M. Thomas:
"To be nominated for this award by my colleagues is the ultimate recognition and reflects the level of support that I experience daily. My continued intellectual and personal development have been made possible by my relationships with the nominating team, which includes Colonels Hartley, Scioletti, Lindquist, and Gist; Lieutenant Colonels Bluman and Wallen; and Doctors Misiurewicz, Calkin, Heymsfield, and Allison. What we, as professors, live for is the opportunity to play a role in the lives of our students and our mentees. The former students and early career faculty who have reached out because of this award have warmed my heart and remind me of the impact we make. Finally, I would like to thank my mother, Mary Thomas. No career is without obstacles. Every time I hit the big ones, she’s the person I turned to. As the tears and the heartache flooded, she would hold my hands and tell me to be patient and continue to work hard. She was confident that as long as I stuck to this work ethic, I would be successful. It is my hope she will be at the JMM awards ceremony this year to know that her words are why I persevered."
Biographical Sketch of Diana M. Thomas
Diana M. Thomas received her Ph.D. from the Georgia Institute of Technology in 1996. Dr. Thomas has been an active research mathematician for more than 25 years with a focus on nutrition and obesity related modeling. She co-invented the remote weight loss program SmartLoss™, which has been clinically applied worldwide to guide and improve individual patient weight loss adherence through smartphone technology. Dr. Thomas has published more than 150 peer-reviewed articles and book chapters, including more than 50 articles with undergraduates. Her work has been covered by The New York Times, The Wall Street Journal, Fitness magazine, Good Housekeeping, CBS News, and ABC News. Dr. Thomas holds the 2012 Mathematical Association of America of New Jersey Distinguished Teaching Award and the 2017 Obesity Society George A. Bray Founders Award.
About the Dolciani Prize
The AMS Mary P. Dolciani Prize for Excellence in Research recognizes a mathematician from a department that does not grant a Ph.D., who has an active research program in mathematics and a distinguished record of scholarship. The primary criterion for the prize is an active research program, as evidenced by a strong record of peer-reviewed publications. This prize is funded by a grant from the Mary P. Dolciani Halloran Foundation. Mary P. Dolciani Halloran (1923-1985) was a gifted mathematician, educator, and author.
The 2023 Dolciani Prize will be recognized during the Joint Mathematics Meetings in January 2023 in Boston.
Contact: AMS Communications
The American Mathematical Society is dedicated to advancing research and connecting the diverse global mathematical community through our publications, meetings and conferences, MathSciNet, professional services, advocacy, and awareness programs.
A team led by Georgia Tech mathematician Anton Leykin has developed a powerful new technique for solving problems related to 3D reconstruction. The research team’s open access paper, "Learning to Solve Hard Minimal Problems", has also won the prestigious best paper award at CVPR 2022, the Computer Vision and Pattern Recognition Conference (CVPR) — selected from a pool of over 8,000 papers submitted this year.
The team’s research idea revolved around developing a new way to solve a family of problems known as hard minimal problems, which are essential for 3D reconstruction. “A minimal problem is a smallest geometric problem one can consider in the 3D reconstruction context,” Leykin explained. “For example, recovering a 3D scene consisting of 5 points from 2 views (2-dimensional images of 5 points in the plane) without knowing the relative position and orientation of the second camera with respect to the first.”
In other words, the problem focuses on “solving” how to see in three dimensions by analyzing multiple two-dimensional perspectives — this is how humans and self-driving cars see in 3D. One way to understand this is by imagining our eyes as cameras. Both eyes capture two-dimensional images, each from a slightly different perspective. By considering the perspective of the image sent by each eye, our brains create a 3D rendering of these two-dimensional images. While our brains might do this with seeming ease in the case of our vision, solving these problems mathematically can be more difficult.
Petr Hruby, currently a Ph.D. student at the ETH Zurich Department of Computer Science, with a recent Master’s degree from Czech Technical University, serves as the paper’s lead author. He is joined by co-authors Leykin, a professor in the School of Mathematics at Georgia Tech; Timothy Duff, NSF Postdoctoral Fellow at the University of Washington (Georgia Tech Ph.D. in Algorithms, Combinatorics, and Optimization, 2021); and Tomas Pajdla, professor at the Czech Technical University in Prague Czech Institute of Informatics, Robotics and Cybernetics. The core of the team started working together during the Institute for Computational and Experimental Research in Mathematics (ICERM) semester on Nonlinear Algebra in 2018, of which Leykin was the primary organizer.
After their first project won the best student paper award at the 2019 International Conference on Computer Vision (ICCV), the team decided to pursue research in hard minimal problems.
Since the technique the researchers developed is general, Leykin said it can be applied to many other situations with similar mathematical problems. In addition, the software pieces derived from the researchers’ findings are in the public domain, and can be used by a broad computer vision community.
Solve-and-Pick vs. Pick-and-Solve
Solving minimal problems can be difficult, because they often have many spurious solutions (solutions that might solve the equation, but are ultimately unhelpful or unexpected).
Previously, the state-of-the-art technique for solving minimal problems used a “solve-and-pick” approach. Solve-and-pick involves first determining all of the possible solutions to a problem, and then picking the optimal solutions — this is done by removing non-real solutions, using inequalities, and evaluating how well they support the solution. But, when there are many spurious solutions, this type of optimization can be costly and time-consuming.
Instead of using this traditional solve-and-pick approach, the researchers investigated the opposite: a “pick-and-solve” technique that learns, for a given data sample, how to first pick a promising starting point and then continue it to a meaningful solution. This approach is unique in that it avoids computing large numbers of spurious solutions.
By selecting a suitable starting point and solving from that point (instead of solving from all points), the method can quickly find and track a path to the solution more quickly, learning how to find that target solution more efficiently.
“Instead of finding all possible solutions and then deciding which one is relevant, we aimed at ‘guessing’ which path leads to one physically meaningful solution — as long as the guess is correct with high probability, this becomes practically useful,” said Leykin. “For a ‘hard’ minimal problem, this is like finding a needle in a haystack — we need to guess one correct path out of several hundreds.”
To do so, the research combined concepts spanning several fields of mathematics: algebra, geometry, numerical analysis, and statistics. Computer science and engineering components also played a vital role: “We had to use neural networks for one particular task and, of course, implement the algorithms efficiently,” Leykin said. Since the minimal problem solvers are executed as subroutines millions, billions, or trillions of times, efficiency was essential.
Solving the hard problems
To test their method, the researchers developed a solver using their pick-and-solve technique for a well-known problem in the field. They benchmarked and studied their engineering choices with another familiar problem.
Finally, they applied their technique to a harder problem – reconstructing a 3D view using four 2D points in three views. The researchers’ implementation of their method solves this problem in about 70 microseconds on average – ten times faster than any other method.
The team hopes that their solution could change how these problems are approached and solved in the future. “Previously, ‘hard’ minimal problems were avoided in practical applications, since there were no fast reliable solvers for them,” Leykin said. “We hope that, over time, our work will convince the industry to reconsider – the ‘hard’ problems are not that hard after all!”
Leykin will soon deliver a colloquium on the work with the School of Mathematics. Learn more.
Citation:
Hruby, Petr & Duff, Timothy & Leykin, Anton & Pajdla, Tomas. (2021). Learning to Solve Hard Minimal Problems.
A team led by Georgia Tech mathematician Anton Leykin has developed a powerful new technique for solving problems related to 3D reconstruction. The research team’s open access paper, "Learning to Solve Hard Minimal Problems", has also won the prestigious best paper award at CVPR 2022, the Computer Vision and Pattern Recognition Conference (CVPR) — selected from a pool of over 8,000 papers submitted this year.
The team’s research idea revolved around developing a new way to solve a family of problems known as hard minimal problems, which are essential for 3D reconstruction. “A minimal problem is a smallest geometric problem one can consider in the 3D reconstruction context,” Leykin explained. “For example, recovering a 3D scene consisting of 5 points from 2 views (2-dimensional images of 5 points in the plane) without knowing the relative position and orientation of the second camera with respect to the first.”
In other words, the problem focuses on “solving” how to see in three dimensions by analyzing multiple two-dimensional perspectives — this is how humans and self-driving cars see in 3D. One way to understand this is by imagining our eyes as cameras. Both eyes capture two-dimensional images, each from a slightly different perspective. By considering the perspective of the image sent by each eye, our brains create a 3D rendering of these two-dimensional images. While our brains might do this with seeming ease in the case of our vision, solving these problems mathematically can be more difficult.
Petr Hruby, currently a Ph.D. student at the ETH Zurich Department of Computer Science, with a recent Master’s degree from Czech Technical University, serves as the paper’s lead author. He is joined by co-authors Leykin, a professor in the School of Mathematics at Georgia Tech; Timothy Duff, NSF Postdoctoral Fellow at the University of Washington (Georgia Tech Ph.D. in Algorithms, Combinatorics, and Optimization, 2021); and Tomas Pajdla, professor at the Czech Technical University in Prague Czech Institute of Informatics, Robotics and Cybernetics. The core of the team started working together during the Institute for Computational and Experimental Research in Mathematics (ICERM) semester on Nonlinear Algebra in 2018, of which Leykin was the primary organizer.
After their first project won the best student paper award at the 2019 International Conference on Computer Vision (ICCV), the team decided to pursue research in hard minimal problems.
Since the technique the researchers developed is general, Leykin said it can be applied to many other situations with similar mathematical problems. In addition, the software pieces derived from the researchers’ findings are in the public domain, and can be used by a broad computer vision community.
Solve-and-Pick vs. Pick-and-Solve
Solving minimal problems can be difficult, because they often have many spurious solutions (solutions that might solve the equation, but are ultimately unhelpful or unexpected).
Previously, the state-of-the-art technique for solving minimal problems used a “solve-and-pick” approach. Solve-and-pick involves first determining all of the possible solutions to a problem, and then picking the optimal solutions — this is done by removing non-real solutions, using inequalities, and evaluating how well they support the solution. But, when there are many spurious solutions, this type of optimization can be costly and time-consuming.
Instead of using this traditional solve-and-pick approach, the researchers investigated the opposite: a “pick-and-solve” technique that learns, for a given data sample, how to first pick a promising starting point and then continue it to a meaningful solution. This approach is unique in that it avoids computing large numbers of spurious solutions.
By selecting a suitable starting point and solving from that point (instead of solving from all points), the method can quickly find and track a path to the solution more quickly, learning how to find that target solution more efficiently.
“Instead of finding all possible solutions and then deciding which one is relevant, we aimed at ‘guessing’ which path leads to one physically meaningful solution — as long as the guess is correct with high probability, this becomes practically useful,” said Leykin. “For a ‘hard’ minimal problem, this is like finding a needle in a haystack — we need to guess one correct path out of several hundreds.”
To do so, the research combined concepts spanning several fields of mathematics: algebra, geometry, numerical analysis, and statistics. Computer science and engineering components also played a vital role: “We had to use neural networks for one particular task and, of course, implement the algorithms efficiently,” Leykin said. Since the minimal problem solvers are executed as subroutines millions, billions, or trillions of times, efficiency was essential.
Solving the hard problems
To test their method, the researchers developed a solver using their pick-and-solve technique for a well-known problem in the field. They benchmarked and studied their engineering choices with another familiar problem.
Finally, they applied their technique to a harder problem – reconstructing a 3D view using four 2D points in three views. The researchers’ implementation of their method solves this problem in about 70 microseconds on average – ten times faster than any other method.
The team hopes that their solution could change how these problems are approached and solved in the future. “Previously, ‘hard’ minimal problems were avoided in practical applications, since there were no fast reliable solvers for them,” Leykin said. “We hope that, over time, our work will convince the industry to reconsider – the ‘hard’ problems are not that hard after all!”
Leykin will soon deliver a colloquium on the work with the School of Mathematics. Learn more.
Citation:
Hruby, Petr & Duff, Timothy & Leykin, Anton & Pajdla, Tomas. (2021). Learning to Solve Hard Minimal Problems.
Jinyoung Park, who will take up her position as Assistant Professor of Mathematics in the School in January, has been awarded a 2023 Maryam Mirzakhani New Frontiers Prize. The prize is awarded to researchers "For contributions to the resolution of several major conjectures on thresholds and selector processes."
This year, three Maryam Mirzakhani New Frontiers Prizes, of $50,000 each, were awarded to women mathematicians who have recently completed their PhDs and produced important results.
Jinyoung Park
Jinyoung Park is a Szegö Assistant Professor at Stanford University, working with her mentor Jacob Fox. Previously a postdoctoral member of Institute for Advanced Study (CSDM program, led by Avi Wigderson), Dr. Park will be joining SoM as an incoming faculty member in 2023.
Dr. Park's research interests include
extremal and probabilistic combinatorics,
asymptotic enumeration, and
graph theory.
See below for more information on the Mizakhani prize story.
https://breakthroughprize.org/News/73
For another story featuring Incoming SoM Faculty Jinyoung Park see this story.
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