John Lesieutre
Pennsylvania State University, PA·Mathematics & Statistics·Early-career
Systems of polynomial equations in many variables show up throughout mathematics, but their solution sets are usually very hard to describe unless the equations have some special structure. One powerful kind of structure is symmetry: a map that sends solutions to other solutions.
- Fields
Mathematics & Statistics
- Topics
algebraic geometry
birational geometry
algebraic dynamics
- Grants
- 1 known grant, through Apr 2027
What they work on
- 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.
- Extend the open-source SageMath system so it can compute dynamical degrees, the long-run growth rates of degrees under iteration, supporting both computation and theory.
- Investigate the growth rates of spaces of sections of tensor powers of line bundles, which measure how many functions of a given type live on a geometric space.
- Formulate and study higher-dimensional versions of the bounded negativity conjecture, which concerns limits on how negative self-intersections of curves can be.
More about the research
Algebraic geometry is the study of these solution sets. Dynamical systems is the study of what happens when a function is applied over and over to a set of points. This research sits where the two fields meet. It aims to use the tools of each field to answer questions in the other.
Why it matters
This is basic research in pure mathematics. It aims to deepen understanding of how symmetries of polynomial equation systems behave when applied repeatedly, connecting algebraic geometry and dynamical systems. Improved tools in the open-source SageMath system for computing dynamical degrees could help other mathematicians explore examples. New teaching materials could introduce algebraic dynamics to graduate students and give younger students a view of mathematics beyond calculus.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Laura DeMarco
Harvard University, MA·Mathematics & Statistics
Characterize the special subvarieties — geometric pieces preserved by the map — that arise in families of dynamical systems built from endomorphisms of algebraic varieties, aiming at a general classification within a unifying conjectural framework.
Claudiu Raicu
University of Notre Dame, IN·Mathematics & Statistics
Compute the cohomology of line bundles on flag varieties in positive characteristic, where the Borel–Weil–Bott theorem no longer gives the answer, and describe how the result depends on the prime p.
Jonathan Hauenstein
University of Notre Dame, IN·Mathematics & Statistics
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.
The school
Pennsylvania State University: 60 labs on PI Pages, $1.3B a year on research, 743 research doctorates a year.
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