Artem Zvavitch
Kent State University at Kent, OH·Mathematics & Statistics
How does the shape of an object control what can happen inside it or along its edge? This project studies that question for the Monge–Ampère equations, an important family of nonlinear equations that arise in geometry, analysis and mathematical physics.
- Fields
Mathematics & Statistics
- Topics
Monge-Ampère equation
Monge-Ampère operator
principal eigenvalue
- Grants
- 1 known grant, through Aug 2028
What they work on
- Prove Brunn–Minkowski and isoperimetric-type inequalities for principal eigenvalues of real Monge–Ampère operators on convex bodies, using convex geometry and symmetrization.
- Establish Rogers–Shephard and reverse Brunn–Minkowski-type inequalities for Monge–Ampère eigenvalues, drawing on shadow systems and the asymptotic theory of normed spaces.
- Study duality and affine-invariant products of Monge–Ampère eigenvalues for a convex body and its dual (polar) body.
- Address existence, uniqueness, regularity and variational formulations for complex, quaternionic and octonionic Monge–Ampère operators, and extend geometric inequalities to these settings.
More about the research
The investigators, collaborating with mathematicians in Israel, seek general principles for how solutions change when shapes are enlarged, combined, symmetrized (made more symmetric) or replaced by related shapes.
Why it matters
This is pure mathematics. Its main contribution is a deeper, more unified understanding of how geometric shape controls the behavior of Monge–Ampère equations, connecting convex geometry, nonlinear partial differential equations and functional analysis. These results could eventually inform other areas of geometry, analysis and mathematical physics where these equations appear.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Galyna Livshyts
Georgia Institute of Technology, GA·Mathematics & Statistics
Approach the log-Brunn-Minkowski, (B)-, and Dimensional Brunn-Minkowski conjectures, which predict stronger volume inequalities for symmetric convex bodies, using local versions of isoperimetric-type inequalities.
Cheikh Ndiaye
Howard University, DC·Mathematics & Statistics
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.
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.
The school
Kent State University at Kent: 6 labs on PI Pages, $61M a year on research, 162 research doctorates a year.
Reach out
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