Jonathan Hauenstein
University of Notre Dame, IN·Mathematics & Statistics
Many practical problems come down to solving systems of polynomial equations, equations built from powers and products of unknowns. Examples include designing a robot's mechanical linkage to perform given tasks, analyzing chemical reaction networks, and finding bifurcations, the tipping points where a biological process changes behavior.
- Fields
Mathematics & Statistics
Computer Science
- Topics
polynomial systems
real algebraic geometry
computational algebraic geometry
- Grants
- 1 known grant, through Dec 2027
What they work on
- Develop homotopy continuation algorithms that compute at least one, or all, real solutions of polynomial systems with finitely many solutions, without computing all complex solutions.
- Design an algorithm that samples smooth points in every connected component of the real solution set of polynomial systems with infinitely many solutions.
- Use real sample points to determine the dimension and connectedness of real solution sets, and test the new methods on science and engineering problems such as linkage design and reaction networks.
More about the research
In these problems only the real-number solutions usually matter. Yet standard methods compute every complex solution, and complex solutions can outnumber real ones by many orders of magnitude, which wastes effort.
Why it matters
Faster, rigorous solvers for real solutions could help engineers design robotic mechanisms. They could also help chemists and biologists analyze reaction networks and tipping points in biological systems, where large polynomial systems now take a long time to solve. Avoiding the many unneeded complex solutions could make larger real-world problems practical.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
John Lesieutre
Pennsylvania State University, PA·Mathematics & Statistics·Early-career
Study lower bounds on how fast the degrees of repeated rational maps grow, using methods from the Sarkisov program, a way of decomposing geometric transformations into simpler steps.
Hugo Woerdeman
Drexel University, PA·Mathematics & Statistics
Resolve open problems on when noncommutative rational expressions in bounded operator variables become singular or vanish, combining complex analysis, representation theory and operator theory.
Emily Witt
University of Kansas, KS·Mathematics & Statistics·Early-career
Use the Frobenius map, which raises ring elements to a prime power, to study hypersurfaces in prime characteristic, sharpening the understanding of test ideals and Frobenius jumping exponents that quantify how singular a shape is.
The school
University of Notre Dame: 19 labs on PI Pages, $346M a year on research, 277 research doctorates a year.
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