Currently working on boundary field equalities that generalize the idea of conservation laws, new methods for guiding stress in structures, bounds that provide limitations to cloaking, novel methods for obtaining bounds on the spectrum of operators, novel metamaterials, novel minimization variational principles for acoustic and electromagnetic scattering, characterizing the possible elasticity tensors of 3d printed … Continued
I study the complexity of mathematics. In mathematics there are constructions, proofs, and objects that are clearly more complicated than others. In computability theory we look for tools to measure this complexity and we measure it.
My work is in algebraic geometry. I have been studying invariants that measure the singularities of algebraic varieties, such as minimal log discrepancies, log canonical thresholds, multiplier ideals, Bernstein-Sato polynomials, and F-thresholds. Various points of view and techniques come in the picture when studying these invariants: resolutions of singularities, jet schemes, D-modules, and positive characteristic … Continued
I am a mathematician who works on analysis and geometry. My research investigates the extent to which abstract geometries with an intrinsic notion of distance (metric spaces) can be faithfully represented as points in better-understood geometries, such as Euclidean space. Such abstract geometries are ubiquitous in mathematics, computer science, and statistics. My work on understanding … Continued
Most of my work is concerned with questions related to the long-term dynamics of solutions of partial differential equations. These equations describe physical phenomena, such as water waves, plasma evolutions, and gravitation, and their relevance is often verified numerically. Ionescu studies the solutions of these equations rigorously, and recovers quantitative and qualitative information about their … Continued
I conduct laboratory experiments on fundamental problems in fluid mechanics.
A major focus of my work has been the development of “fast algorithms” for particle simulations, wave propagation, and heat flow. These algorithms (such as the Fast Mutipole Method) are now used in electromagnetic and acoustic design, quantum chemistry and a variety of other application areas. For the last few decades, I have been working … Continued
My research is concerned with representation theory, quantum field theory, integrable systems, and the interrelations between them.
I work on number theory and novel connections and applications from number theory to algebraic geometry, Euclidean lattices, coding theory, and other areas of mathematics. One major area of interest is K3 surfaces of large Picard number and their applications to various questions mostly in Diophantine and algebraic geometry. Applications include explicit formulas for Shimura … Continued
My recent research deals with billiards in rational polygons. The elementary questions posed in this area are connected to many branches of modern mathematics. More broadly, we can motivate the problem as follows: Some natural phenomena are “chaotic” (i.e. unpredictable). These are often studied by statistical methods. Others are “integrable” (i.e. predictable and regular). Other … Continued