Cheikh Ndiaye
Howard University, DC·Mathematics & Statistics
Many equations from geometry and physics describe states that minimize or balance an energy, such as an optimal shape or a stable vortex in a fluid. For a large class of these partial differential equations, two difficulties appear together.
- Fields
Mathematics & Statistics
- Topics
geometric analysis
partial differential equations
nonlinear PDE
- Grants
- 1 known grant, last ended Sep 2026
What they work on
- Develop a full degree theory and Morse theory to prove existence of solutions to Q-curvature type equations, which describe angle-preserving reshaping of curved spaces to a prescribed curvature.
- Establish existence results in the resonant case, where non-compact flow lines collect at true critical points at infinity, using Abbas Bahri's theory of critical points at infinity.
- Apply Morse-theoretic and algebraic topology arguments to the Mean Field and Toda systems from Chern-Simons theory, whose energies are critical for Moser-Trudinger type inequalities.
More about the research
Solutions can strongly interact, and the energy comes only in fixed, quantized amounts. Ordinary methods for proving that solutions exist then break down. This project develops mathematical tools to show when solutions exist anyway. It has applications to optimal shapes in geometry and to energy-critical states in physics models of fluid vortices and condensates.
Why it matters
This is pure mathematics. It adds general methods for proving that solutions exist for equations with quantization and strong interaction, a feature shared by many models of real phenomena. These results could eventually inform the study of optimal shapes in geometry and of energy-critical states such as vortices in Euler flows and condensates in Chern-Simons-Higgs models.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Daniel Stern
Cornell University, NY·Mathematics & Statistics
Use new constructions of extremal metrics for Laplacian and Steklov eigenvalues, the natural vibration frequencies of a shape, to produce many new minimal surfaces of prescribed topology in low-dimensional spheres and balls.
Xuwen Zhu
Northeastern University, MA·Mathematics & Statistics
Study how gravitational instantons, four-dimensional spaces that solve Einstein's equations, degenerate and how their moduli spaces are structured, using geometric microlocal analysis and PDE techniques.
Artem Zvavitch
Kent State University at Kent, OH·Mathematics & Statistics
Prove Brunn–Minkowski and isoperimetric-type inequalities for principal eigenvalues of real Monge–Ampère operators on convex bodies, using convex geometry and symmetrization.
The school
Howard University: 8 labs on PI Pages, $102M a year on research, 102 research doctorates a year.
Reach out
A start for your email. Fill in the brackets.