Demetrios Christodoulou
A summary of his career can be found on his autobiography for the Shaw prize. In short, his works to date has focused on the use of geometric methods to resolve problems in nonlinear PDEs, and much of his work is in the field of mathematical general relativity. Some of the highlights:
He proved the nonlinear stability of Minkowski space with Sergiu Klainerman, which established the use of a fully intrinsic-geometrical method of applying the vector field method which allows proving weighted energy estimates that respect the intrinsic null structure of the Lorentzian metric induced by the quasilinear coefficients. The same method is crucial in his later work on formation of trapped surfaces in general relativity and also formation of shocks in three dimensional fluids, and is now a (somewhat) standard method in the geometric analysis of quasilinear hyperbolic equations.
In a series of papers from late 1980s to 1999, he examined the spherically symmetric collapse of self-gravitating scalar field. This is a model problem for gravitation waves (one cannot study pure gravitation waves in spherical symmetry due to Birkhoff's theorem). He was able to prove for this model both the existence of naked singularity solutions and their instability, thereby showing a version of the weak cosmic censorship conjecture for generic solutions and showing that genericity in the statement is required.
His work on shock formation in fluids is essentially the study of blow-up behaviour of small data solutions to a quasilinear wave equation where the null condition fails. Alinhac has already previously shown that such solutions will blow-up generically in finite time, found the asymptotic lifespan as the size of data goes to zero, and described the local geometry at the first blow-up point as that of shock formation (that the main blow-up mechanism is essentially that of Burger's equation, the crossing of characteristic surfaces). The main improvements are (a) a new proof using only finitely many derivatives; Alinhac required $C^\infty$ data (b) full tracking of geometric quantities along the evolution and (c) description of the singular boundary of the globally hyperbolic domain [or physically, description of the shock front]. This improvements are done with a view toward perhaps being able to extend the notion of "entropy weak solutions" of 1+1 dimensional conservation laws to fluids in higher spatial dimensions.
His work on formation of trapped surfaces gave the first rigorous proof that a black hole can form dynamically from the focussing of gravitational waves. In this work he introduced the short pulse method (which grew out of his work on self gravitating scalar fields), which, in one guise, can be thought of as a way of getting large data, long time existence for quasilinear wave equations when some version of null condition is available. (Basically, one way of reading the null condition, which is so successful in small data theory, is that wave packets can only weakly self interact. In particular, this prevents the self-reinforcing ODE type blow-up to occur, at least to leading order. If one carefully keeps track of the full hierarchy of sizes from the nonlinear effects [roughly speaking the weak self interaction of the wave packets generate other wave packets in other directions, and we need to keep track of all of them], especially the "anomalous" (compared to linear theory) terms, one can get long-time existence while also extracting information unavailable from naive linear analysis.)
Remark: the short pulse method is in some sense dual to the peeling estimates implied by the nonlinear stability of Minkowski space. These peeling estimates show, in particular, that even in the small data regime, the asymptotic behaviour of gravitational waves cannot be captured purely as the behaviour of a linear spin-2 field. There will necessarily be some non-linear effects which manifest at top order. This "Christodoulou memory effect" in principle may affect the efforts of experimentally observing gravitation radiation.
