Math prof uses optimal control, game theory to attack cancer cells, move pedestrians, plan paths for robots

Alex Vladimirsky is the type of mathematician who draws inspiration from real-world problems. This was one of his main reasons for joining Cornell, where interdisciplinary collaborations are both valued and encouraged.

“I appreciate the opportunity to interact with colleagues in engineering, in medicine, in sociology, in economics, in physics, in computer science,” he said. “Both in my teaching and research, I’m frequently intrigued by something happening in other areas. Aside from suggesting new problems for existing tools, applications can be also an important stimulus for new mathematical ideas.”

Alex Vladimirsky
Vladimirsky, director of Cornell’s Center for Applied Mathematics, has tackled mathematical questions both theoretical and applied across a number of areas ranging from control theory to numerical analysis and dynamical systems. But lately he finds himself veering toward mostly applied projects.

His recent paper, "Anisotropic Challenges in Pedestrian Flow Modeling," published in October 2018 in Communications in Mathematical Sciences, focuses on the mathematics of crowd motion. How do the individual choices of pedestrians change the overall crowd density? How much will two groups of people slow down each other as they rush to their respective destinations through a crowded environment? The mathematical models studied by Vladimirsky jointly with PhD student Elliot Cartee are based on Mean Field Games, a framework suitable for modeling interactions of a large number of independent “selfish” agents. 

“These are not people trying to delay others, but they are just trying to get out of a burning building or rushing to meet a train,” he said. “Similar models are also useful in economics, traffic engineering and biology.”

In another recent project, Vladimirsky collaborated with Cornell PhD student Mark Gluzman and Cleveland Clinic oncologist Jason Scott to improve a drug therapy. The key observation exploited in their approach is that different sub-types of cancer cells are “competing” to grow inside a tumor. The proposed methods could help doctors use less drugs to cure a patient by exploiting this competition and carefully timing the drug doses to maximize their effect.

“I could see that there were many different therapy schedules leading to patient recovery under this model. As an optimizer, I could determine which of them would use the smallest amount of drugs,” Vladimirsky said. “It’s close to the optimal control of  evolutionary games – a topic I have been working on with Gluzman, motivated by applications in economics and sociology. When Scott visited Cornell and explained that similar equations come up in cancer modeling, it was natural for us to bring our tools into this new domain.

“Surprisingly many real-world problems can be written down using similar equations,” he said. “The problems can be as different as preventing stampedes during a building evacuation, or guiding a robot to minimize exposure to enemy observers, or finding the best place to drill for oil, or improving drug therapies. But if you find a good way to solve one of these, your methods might have an impact in other areas as well.”

Vladimirsky’s ongoing projects also focus on optimization under uncertainty, with conflicting objectives, random changes in the global environment and adversarial interactions of many agents. Such features pose additional computational challenges but are often needed to make mathematical models realistic.

Vladimirsky discovered his passion for applied mathematics as a college student at the University of California Berkeley and quickly became involved with an international Mathematical Contest in Modeling (MCM). As part of that competition, teams of undergraduates use mathematical tools to solve real-world problems. The contest attracts diverse students from more than 900 institutions around the world.

As a young professor in 2002, Vladimirsky brought math modeling activities to Cornell undergraduates. Since then, Cornell teams have done well in the international MCM, finishing in the top 1 percent in 2009 and 2012 and with at least one team receiving a meritorious ranking (in the top 10 percent worldwide) every year.   

To help feed students’ interest in modeling, Vladimirsky and colleagues also organize a university-wide annual contest and teach a popular bi-annual class, where students are encouraged to work on problems of local interest:

  • How is global warming affecting different types of grapes in Upstate New York and how can winery owners prepare?
  • On which routes should our local transit authority (TCAT) use available hybrid-electric buses to maximize fuel savings?
  • How could Cornell officials reconfigure the Arts Quad to maximize the contiguous lawn space but allow for the most efficient walking paths so that students could get to class without trampling the grass?
  • Where should Ithaca officials place emergency response vehicles to minimize waiting times for patients?

“I definitely believe that for students it’s much more interesting to be presented with open-ended problems first and then figure out what kind of mathematics they need to learn to solve them,” he said. These continuing efforts to highlight real-world problems in mathematical curriculum are also partially supported by an Engaged Curriculum Grant in support of Engaged Cornell. The grant is led by Assistant Professor Alex Townsend, with an interdisciplinary faculty team including Vladimirsky and others in math, applied math and operations research. 

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 Crowds rushing through a station, photo by Nicolai Berntsen on Unsplash